pe-nlp/ORZ-MATH-57k-Filter-difficulty · Datasets at Hugging Face

problem stringlengths 18 4.46k	answer stringlengths 1 942	pass_at_n float64 0.08 0.92
On the board, the natural number \( N \) was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers? (I. Efremov)	6	0.5
As shown in the figure, in quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular. The lengths of the four sides are $AB = 6$, $BC = m$, $CD = 8$, and $DA = n$. Find the value of $m^{2} + n^{2}$.	100	0.875
In the coordinate plane, points with integer values for both coordinates are called lattice points. For a certain lattice point \( P \) and a positive number \( d \), if there are exactly \( k(>0) \) distinct lattice points at a distance \( d \) from \( P \), the range of values for \( k \) is denoted as \( \left\{k_1, k_2, \cdots\right\} \) where \( 0<k_1<k_2<\cdots \). What is \( k_2 \)?	8	0.625
In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \(a\), there are as many numbers smaller than \(a\) among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockwise. A number that is larger than all 30 numbers closest to it is called "huge." What is the smallest possible number of huge numbers?	10	0.5
Quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent \(\ell\) to this circle is drawn. Circle \(\omega\) passes through points \(A\) and \(B\) and touches the line \(\ell\) at point \(P\). Line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is tangent to circle \(\omega\).	1	0.25
Yiyi lives at Huashi and is planning to ride a shared bicycle to the park. There are 3 stations between Huashi station and the park, and each station has shared bicycles of four different colors: yellow, orange, red, and green. If Yiyi does not want to ride a bicycle of the same color at two consecutive stations, how many different riding strategies are there for Yiyi from home to the park?	108	0.75
In a city, there are 9 bus stops and several buses. Any two buses have at most one common stop. Each bus stops at exactly three stops. What is the maximum number of buses that can be in the city?	12	0.375
For a point \(P=\left(a, a^{2}\right)\) in the coordinate plane, let \(\ell(P)\) denote the line passing through \(P\) with slope \(2a\). Consider the set of triangles with vertices of the form \(P_{1}=\left(a_{1}, a_{1}^{2}\right)\), \(P_{2}=\left(a_{2}, a_{2}^{2}\right)\), \(P_{3}=\left(a_{3}, a_{3}^{2}\right)\), such that the intersection of the lines \(\ell\left(P_{1}\right)\), \(\ell\left(P_{2}\right)\), \(\ell\left(P_{3}\right)\) form an equilateral triangle \(\Delta\). Find the locus of the center of \(\Delta\) as \(P_{1} P_{2} P_{3}\) ranges over all such triangles.	y = -\frac{1}{4}	0.375
In triangle \(ABC\) with area 51, points \(D\) and \(E\) trisect \(AB\), and points \(F\) and \(G\) trisect \(BC\). Find the largest possible area of quadrilateral \(DEFG\).	17	0.875
In the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots\). Find \(a_{1999}\).	3331	0.25
In the triangle \( ABC \) the points \( M \) and \( N \) lie on the side \( AB \) such that \( AN = AC \) and \( BM = BC \). We know that \(\angle MCN = 43^\circ\). Find the size in degrees of \(\angle ACB\).	94^\circ	0.375
Given that \( F_1 \) and \( F_2 \) are the left and right foci of the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), and \( P \) is a point on the ellipse \( C \). The incenter of triangle \( \triangle F_{1}PF_{2} \) is \( I \). If there exists a real number \( \lambda \) such that: $$ (1+\lambda) \overrightarrow{PF_{1}} + (1-\lambda) \overrightarrow{PF_{2}} = 3 \overrightarrow{PI}, $$ then the eccentricity of the ellipse \( C \) is $\qquad$ .	\frac{1}{2}	0.75
The work was done together by team $C$ and team $D$. How many days did they take to complete it, given that team $C$ alone would have finished the work $c$ days later, and team $D$ alone would have required $d$ times more time? Apply the result to the cases $c=10, d=2$ and $c=8, d=3$.	16	0.125
How many different integral solutions \((x, y)\) does \(3\|x\| + 5\|y\| = 100\) have?	26	0.375
Small and Big islands have a rectangular shape and are divided into rectangular counties. Each county has a road along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here's how the Small Island, consisting of 6 counties, is arranged (see Figure 1). Draw how the Big Island, which has an odd number of counties, could be arranged. How many counties did you get?	9	0.375
Arsenius has 2018 buckets, each containing 1, 2, 3, ... 2017, 2018 liters of water respectively. Arsenius is allowed to take any two buckets and pour into the second bucket exactly as much water as is already in the second bucket from the first bucket. Can Arsenius collect all the water into one bucket? All buckets are large enough to hold all the water they get.	\text{No}	0.5
Four students, Andrey, Vanya, Dima, and Sasha, participated in a school chess tournament. Each student played two games against each of the other participants. Each win awarded 1 point, a draw awarded 0.5 points, and a loss awarded 0 points. Given the following conditions at the end of the tournament: - Each student scored a different number of points. - Andrey secured first place, Dima secured second, Vanya secured third, and Sasha secured fourth. - Andrey won the same number of games as Sasha. How many points did each student score?	4, 3.5, 2.5, 2	0.125
A plane, perpendicular to an edge of a regular tetrahedron, passes through a point that divides this edge in the ratio 1:4. Find the ratio of the volumes of the resulting parts of the tetrahedron.	\frac{4}{121}	0.375
The height of a right triangle dropped to the hypotenuse is 1, one of the acute angles is $15^{\circ}$. Find the hypotenuse.	4	0.75
A natural number, which does not end in zero, had one of its digits replaced with zero (if it was the leading digit, it was simply erased). As a result, the number became 9 times smaller. How many such numbers exist for which this is possible?	7	0.375
In front of the elevator doors, there are people with masses of 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to transport all the people?	4	0.375
Given that the equation \( \|x\| - \frac{4}{x} = \frac{3\|x\|}{x} \) has \( k \) distinct real root(s), find the value of \( k \).	1	0.875
In a convex quadrilateral \(ABCD\), \(\overrightarrow{BC} = 2 \overrightarrow{AD}\). Point \(P\) is a point in the plane of the quadrilateral such that \(\overrightarrow{PA} + 2020 \overrightarrow{PB} + \overrightarrow{PC} + 2020 \overrightarrow{PD} = \mathbf{0}\). Let \(s\) and \(t\) be the areas of quadrilateral \(ABCD\) and triangle \(PAB\), respectively. Then \(\frac{t}{s} =\) ______.	\frac{337}{2021}	0.25
Find the smallest positive integer \( n \) such that in any two-coloring of the complete graph \( K_{n} \), there exist 3 monochromatic triangles that are pairwise edge-disjoint.	9	0.25
Find the largest positive number \( x \) such that \[ \left(2 x^{3} - x^{2} - x + 1\right)^{1 + \frac{1}{2 x + 1}} = 1. \]	1	0.75
Five runners ran a relay race. If the first runner had run twice as fast, they would have spent 5% less time. If the second runner had run twice as fast, they would have spent 10% less time. If the third runner had run twice as fast, they would have spent 12% less time. If the fourth runner had run twice as fast, they would have spent 15% less time. By what percentage less time would they have spent if the fifth runner had run twice as fast?	8\%	0.75
Find the value of \(a + b + c + d + e\) given the system of equations: \[ \begin{array}{c} 3a + 2b + 4d = 10 \\ 6a + 5b + 4c + 3d + 2e = 8 \\ a + b + 2c + 5e = 3 \\ 2c + 3d + 3e = 4 \\ a + 2b + 3c + d = 7 \end{array} \]	4	0.75
Given a cube \( A B C D A_1 B_1 C_1 D_1 \) with edge length \( a \), find the distance between the lines \( A A_1 \) and \( B D_1 \) and construct their common perpendicular.	\frac{a\sqrt{2}}{2}	0.625
It is known that the sum of the absolute values of the pairwise differences of five nonnegative numbers is equal to one. Find the smallest possible sum of these numbers.	\frac{1}{4}	0.5
In how many different ways can integers \( a, b, c \in [1, 100] \) be chosen such that the points with coordinates \( A(-1, a), B(0, b) \), and \( C(1, c) \) form a right triangle?	974	0.375
Let \( m \) and \( n \) be two integers such that \( n \) divides \( m(n+1) \). Show that \( n \) divides \( m \).	n \mid m	0.125
Determine the largest positive integer \( n \) such that there exist positive integers \( x, y, z \) so that \[ n^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx + 3x + 3y + 3z - 6 \]	8	0.75
In triangle $ABC$, the altitude from vertex $A$ is the harmonic mean of the two segments into which it divides side $BC$. What is $\operatorname{tg} \beta + \operatorname{tg} \gamma$ (where $\beta$ and $\gamma$ are the angles of the triangle)?	2	0.875
Suppose \( a \) and \( b \) are the roots of \( x^{2}+x \sin \alpha+1=0 \) while \( c \) and \( d \) are the roots of the equation \( x^{2}+x \cos \alpha-1=0 \). Find the value of \( \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \).	1	0.5
Given two arbitrary positive integers \( n \) and \( k \), let \( f(n, k) \) denote the number of unit squares that one of the diagonals of an \( n \times k \) grid rectangle passes through. How many such pairs \( (n, k) \) are there where \( n \geq k \) and \( f(n, k) = 2018 \)?	874	0.625
Integrate the equation $$ x \, dy = (x + y) \, dx $$ and find the particular solution that satisfies the initial condition $y = 2$ when $x = -1$.	y = x \ln \|x\| - 2x	0.75
The hedgehogs collected 65 mushrooms and divided them so that each received at least one mushroom, but no two hedgehogs had the same number of mushrooms. What is the maximum number of hedgehogs that could be?	10	0.75
Let \( a = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \) and \( b = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \). Find the value of \( a^4 + b^4 + (a+b)^4 \).	7938	0.375
Dr. Aibolit gave 2006 miraculous pills to four sick animals. The rhinoceros received one more than the crocodile, the hippopotamus received one more than the rhinoceros, and the elephant received one more than the hippopotamus. How many pills did the elephant get?	503	0.875
The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be bought so that there is a probability of 0.99 that at least five of them will be working?	7 \text{ bulbs}	0.25
Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits?	142857	0.375
Point \( P \) is inside triangle \( \triangle ABC \). Line segments \( APD \), \( BPE \), and \( CPF \) are drawn such that \( D \) is on \( BC \), \( E \) is on \( AC \), and \( F \) is on \( AB \). Given that \( AP = 6 \), \( BP = 9 \), \( PD = 6 \), \( PE = 3 \), and \( CF = 20 \), find the area of \( \triangle ABC \).	108	0.125
Let \( f(x) = x^2 + 3x + 2 \). Compute the expression \[ \left(1 - \frac{2}{f(1)}\right) \cdot\left(1 - \frac{2}{f(2)}\right) \cdot\left(1 - \frac{2}{f(3)}\right) \cdots \left(1 - \frac{2}{f(2019)}\right). \]	\frac{337}{1010}	0.875
Given the parabola $y = x^2$ and a point $M(1,1)$ on it as the vertex of right-angled triangles inscribed in the parabola, find the coordinates of the intersection point $E$ of the line segments $AB$ and $CD$.	(-1,2)	0.75
We have one $10 \mathrm{Ft}$ coin and several (at least four) $20 \mathrm{Ft}$ coins in our pocket. We successively draw one coin at a time at random until the sum of the drawn coins becomes a positive integer multiple of $30 \mathrm{Ft}$. On average, how many draws are needed to achieve this?	3	0.75
Given ten 5-element sets \( A_{1}, A_{2}, \cdots, A_{10} \), where the intersection of any two sets contains at least two elements. Let \( A=\bigcup_{i=1}^{10} A_{i}=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\} \). For any \( x_{i} \in A \), let \( k_{i}(i=1,2, \cdots, n) \) be the number of sets among \( A_{1}, A_{2}, \cdots, A_{10} \) that contain the element \( x_{i} \). Define \( m = \max \left\{k_{1}, k_{2}, \cdots, k_{n}\right\} \). Determine the minimum value of \( m \).	5	0.75
The base of the rectangular parallelepiped $\boldsymbol{A}BCD A_{1}B_{1}C_{1}D_{1}$ is a square $ABCD$. Find the maximum possible value of the angle between the line $B D_{1}$ and the plane $BDC_{1}$.	\arcsin \frac{1}{3}	0.25
There are four weights of different masses. Katya weighs the weights in pairs. The results are 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing result in?	2190	0.75
Let \( P(n) \) be the number of permutations \(\left(a_{1}, \ldots, a_{n}\right)\) of the numbers \((1,2, \ldots, n)\) for which \( k a_{k} \) is a perfect square for all \( 1 \leq k \leq n \). Find with proof the smallest \( n \) such that \( P(n) \) is a multiple of 2010.	4489	0.25
A circle inscribed in an angle with vertex \( O \) touches its sides at points \( A \) and \( B \). The ray \( OX \) intersects this circle at points \( C \) and \( D \), with \( OC = CD = 1 \). If \( M \) is the intersection point of the ray \( OX \) and the segment \( AB \), what is the length of the segment \( OM \)?	\frac{4}{3}	0.25
Given distinct real numbers \(a_{1}, a_{2}, a_{3}\) and \(b\). It turns out that the equation \((x-a_{1})(x-a_{2})(x-a_{3}) = b\) has three distinct real roots \(c_{1}, c_{2}, c_{3}\). Find the roots of the equation \((x+c_{1})(x+c_{2})(x+c_{3}) = b\).	-a_1, -a_2, -a_3	0.875
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie. All the students in this class sat at a round table. - Several students said: "There is exactly one troublemaker next to me." - All other students said: "There are exactly two troublemakers next to me." What is the minimum number of troublemakers that can be in the class?	10	0.75
The numerator and denominator of a fraction are natural numbers that add up to 101. It is known that the fraction does not exceed \( \frac{1}{3} \). Determine the greatest possible value of such a fraction.	\frac{25}{76}	0.875
Expand the function $$ f(z)=\frac{z}{z^{2}-2 z-3} $$ into a Taylor series in the vicinity of the point \( z_{0}=0 \), using expansion (12), and find the radius of convergence of the series.	R = 1	0.625
In triangle $ABC$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is given that $AC - AB = 5$ cm. Find the length of the angle bisector of angle $B$.	5 \text{ cm}	0.875
Let \( a, b, c \) be the side lengths of a right triangle, with \( a \leqslant b < c \). Determine the maximum constant \( k \) such that \( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c \) holds for all right triangles, and identify when equality occurs.	2 + 3\sqrt{2}	0.125
There is a ten-digit number. From left to right: - Its first digit indicates the number of zeros in the ten-digit number. - Its second digit indicates the number of ones in the ten-digit number. - Its third digit indicates the number of twos in the ten-digit number. - ... - Its tenth digit indicates the number of nines in the ten-digit number. What is this ten-digit number?	6210001000	0.125
The four-digit number $\overline{abcd}$ and $\overline{cdab}$ have a sum of 3333 and a difference of 693. Find the four-digit number $\overline{abcd}$.	2013	0.75
How many integers from 1 to 2001 have a digit sum that is divisible by 5?	399	0.25
A cube of size \(1000 \times 1000 \times 1000\) is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.	0	0.75
Let \( p \) and \( q \) be distinct prime numbers. How many divisors does the number have: a) \(pq\); b) \(p^2 q\); c) \(p^2 q^2\); d) \(p^m q^n\)?	(m+1)(n+1)	0.875
Given \( f(1) = 1 \), and for any integers \( m \) and \( n \), \( f(m+n) = f(m) + f(n) + 3(4mn - 1) \), find \( f(19) \).	2017	0.5
What is the maximum possible area of a triangle if the sides \(a, b, c\) satisfy the following inequalities: $$ 0 < a \leq 1 \leq b \leq 2 \leq c \leq 3 $$	1	0.625
In triangle \(ABC\), the angles are known: \(\angle A = 45^{\circ}\) and \(\angle B = 15^{\circ}\). On the extension of side \(AC\) beyond point \(C\), point \(M\) is taken such that \(CM = 2AC\). Find \(\angle AMB\).	75^\circ	0.375
Find the integer solutions of the equation $$ x^{4} + y^{4} = 3 x^{3} y $$	(0,0)	0.875
Find the smallest positive integer \( n \) that satisfies the following properties: 1. The units digit of \( n \) is 6. 2. If the units digit 6 is moved to the front of the remaining digits of \( n \), the resulting new number is 4 times \( n \).	153846	0.625
A sandwich and a meal plate cost $\mathrm{R} \$ 5.00$ and $\mathrm{R} \$ 7.00$, respectively. In how many ways can one buy only sandwiches, only meal plates, or some combination of sandwiches and meal plates with $\mathrm{R} \$ 90.00$, without leaving any change?	3	0.75
Determine the coefficients of \(x^{17}\) and \(x^{18}\) after expanding and combining like terms in the expression \[ \left(1 + x^5 + x^7\right)^{20} \]	3420	0.25
Compute the limit of the function: $\lim _{x \rightarrow 0} \frac{2 x \sin x}{1-\cos x}$	4	0.875
Find the smallest prime number $p$ such that $p^{3}+2 p^{2}+p$ has exactly 42 divisors.	23	0.375
Let \(ABC\) be an equilateral triangle. Let \(\overrightarrow{AB}\) be extended to a point \(D\) such that \(B\) is the midpoint of \(\overline{AD}\). A variable point \(E\) is taken on the same plane such that \(DE = AB\). If the distance between \(C\) and \(E\) is as large as possible, what is \(\angle BED\)?	15^\circ	0.375
In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board?	480	0.375
In the sequence $\left\{a_{n}\right\}$, $a_{1}=13, a_{2}=56$, for all positive integers $n$, $a_{n+1}=a_{n}+a_{n+2}$. Find $a_{1934}$.	56	0.875
Show that the expression $$ 3^{2 n+2}-2^{n+1} $$ is divisible by 7 for any positive integer $n$.	7	0.25
In the complex plane, the points corresponding to the complex numbers \( z_1, z_2, z_3 \) are \( Z_1, Z_2, Z_3 \), respectively. Given that \(\left\| z_1 \right\| = \left\| z_2 \right\| = \sqrt{2}\), \(\overrightarrow{OZ_1} \cdot \overrightarrow{OZ_2} = 0\), and \(\left\| z_1 + z_2 - z_3 \right\| = 2\), find the range of the value of \(\left\| z_3 \right\|\).	[0, 4]	0.875
Originally, there were three equally sized grassy fields, and the grass grew at a fixed rate. To eat the grass from all three fields, 50 sheep take exactly 18 days. To eat the grass from two fields, 40 sheep take exactly 12 days. Eating the grass from one field, 70 sheep take $\qquad$ days exactly. How many days will it take?	2	0.625
With what minimum force must you press on a cube with a volume of 10 cm³, floating in water, to submerge it completely? The density of the cube's material is 400 kg/m³, and the density of water is 1000 kg/m³. Provide the answer in SI units. Assume the acceleration due to gravity is 10 m/s².	0.06 \, \text{N}	0.25
Given three composite numbers \( A, B, C \) that are pairwise coprime and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)?	1626	0.375
Given that \( x \) is a real number and \( y = \sqrt{x^2 - 2x + 2} + \sqrt{x^2 - 10x + 34} \). Find the minimum value of \( y \).	4\sqrt{2}	0.625
The difference between the maximum and minimum values of the function \( f(x)=\|\sin x\|+\sin ^{+} 2x+\|\cos x\| \) is equal to ______.	\sqrt{2}	0.75
The number 2 can be represented as \(2 = 2 = 1 + 1\). The number 3 can be represented as \(3 = 3 = 1 + 2 = 2 + 1 = 1 + 1 + 1\). The number 4 can be represented as \(4 = 4 = 1 + 3 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1\), and so on. This kind of representation of positive integers is called ordered partitions. Find the number of ordered partitions of a natural number \(n\).	2^{n-1}	0.875
\[\frac{(1+\tan 2\alpha)^{2}-2 \tan^{2} 2\alpha}{1+\tan^{2} 2\alpha}-\sin 4\alpha-1\]	-2 \sin^2 2 \alpha	0.375
In $\triangle ABC$, $\angle A = 90^\circ$ and $\angle B = \angle C = 45^\circ$. $P$ is a point on $BC$, and $Q, R$ are the circumcenters of $\triangle APB$ and $\triangle APC$ respectively. If $BP = \sqrt{2}$ and $QR = 2$, find $PC$.	\sqrt{6}	0.75
The proportion of top-quality products at this enterprise is $31 \%$. What is the most likely number of top-quality products in a randomly selected batch of 75 products?	23	0.875
Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?	13	0.5
How many boards, each 6 arshins long and 6 vershoks wide, are needed to cover the floor of a square room with a side of 12 arshins? The answer is 64 boards. Determine from this data how many vershoks are in an arshin.	16	0.875
In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be?	1011	0.125
Petrov writes down the odd numbers: $1, 3, 5, \ldots, 2013$, and Vasechkin writes down the even numbers: $2, 4, \ldots, 2012$. Each of them calculates the sum of all digits of all their numbers and tells the result to their classmate Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the final result?	1007	0.375
Find all integers \( x, y \geq 1 \) such that \( x^3 - y^3 = xy + 61 \).	(6,5)	0.375
Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \).	13	0.125
In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers written on the sides of the polygon and obtained the sum \( A \), while Sasha calculated the sum of the numbers written on the diagonals connecting vertices one apart and obtained the sum \( C \). Find the largest possible value of the difference \( C - A \).	1010	0.875
Given an acute triangle \( \triangle ABC \), construct isosceles triangles \( \triangle DAC, \triangle EAB, \triangle FBC \) outside of \( \triangle ABC \) such that \( DA = DC, EA = EB, FB = FC \), and the angles \( \angle ADC = 2 \angle BAC, \angle BEA = 2 \angle ABC, \angle CFB = 2 \angle ACB \). Define \( D' \) as the intersection of line \( DB \) with \( EF \), \( E' \) as the intersection of line \( EC \) with \( DF \), and \( F' \) as the intersection of line \( FA \) with \( DE \). Find the value of \( \frac{DB}{DD'} + \frac{EC}{EE'} + \frac{FA}{FF'} \).	4	0.375
Find the maximum value for \(a, b \geq 1\) of the expression $$ \frac{\|7a + 8b - ab\| + \|2a + 8b - 6ab\|}{a \sqrt{1 + b^2}} $$	9\sqrt{2}	0.875
A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE \).	1:3	0.5
Find the first differential of the function \( y(x) = e^{3x} \ln(1 + x^2) \) and evaluate it at \( x = 0 \) with \( dx = \Delta x = 0.1 \).	dy(0) = 0	0.875
The product of three consecutive even numbers equals $87_{* * * * *}8$. Find these numbers and fill in the blanks in this product.	87526608	0.25
Solve the equation \(\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}\). Write the sum of all obtained solutions as the answer.	2.5	0.375
Color 101 cells blue in an $n \times n$ grid. It is known that there is a unique way to cut the grid along square lines into rectangles such that each rectangle contains exactly one blue cell. Find the smallest possible value of $n$.	101	0.25
Given that \(\alpha\) and \(\beta\) are acute angles, and the following equations hold: $$ \left\{\begin{array}{l} 3 \sin ^{2} \alpha + 2 \sin ^{2} \beta = 1, \\ 3 \sin 2 \alpha - 2 \sin 2 \beta = 0. \end{array}\right. $$ Determine \(\alpha + 2\beta\).	\frac{\pi}{2}	0.75
In the polar coordinate system, the distance from the pole to the center of the circle given by the equation \(\rho=3 \sqrt{2} \cos \left(\theta+\frac{\pi}{4}\right)+7 \sin \theta\) is what?	\frac{5}{2}	0.875

On the board, the natural number $ N $ was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers? (I. Efremov)

6

0.5

As shown in the figure, in quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular. The lengths of the four sides are $AB = 6$, $BC = m$, $CD = 8$, and $DA = n$. Find the value of $m^{2} + n^{2}$.

100

0.875

In the coordinate plane, points with integer values for both coordinates are called lattice points. For a certain lattice point $ P $ and a positive number $ d $, if there are exactly $ k(>0) $ distinct lattice points at a distance $ d $ from $ P $, the range of values for $ k $ is denoted as $ \left\{k_1, k_2, \cdots\right\} $ where $ 0<k_1<k_2<\cdots $. What is $ k_2 $?

8

0.625

In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number $a$, there are as many numbers smaller than $a$ among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockwise. A number that is larger than all 30 numbers closest to it is called "huge." What is the smallest possible number of huge numbers?

10

0.5

Quadrilateral $ABCD$ is inscribed in a circle. At point $C$, a tangent $\ell$ to this circle is drawn. Circle $\omega$ passes through points $A$ and $B$ and touches the line $\ell$ at point $P$. Line $PB$ intersects segment $CD$ at point $Q$. Find the ratio $\frac{BC}{CQ}$, given that $B$ is tangent to circle $\omega$.

1

0.25

Yiyi lives at Huashi and is planning to ride a shared bicycle to the park. There are 3 stations between Huashi station and the park, and each station has shared bicycles of four different colors: yellow, orange, red, and green. If Yiyi does not want to ride a bicycle of the same color at two consecutive stations, how many different riding strategies are there for Yiyi from home to the park?

108

0.75

In a city, there are 9 bus stops and several buses. Any two buses have at most one common stop. Each bus stops at exactly three stops. What is the maximum number of buses that can be in the city?

12

0.375

For a point $P=\left(a, a^{2}\right)$ in the coordinate plane, let $\ell(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_{1}=\left(a_{1}, a_{1}^{2}\right)$, $P_{2}=\left(a_{2}, a_{2}^{2}\right)$, $P_{3}=\left(a_{3}, a_{3}^{2}\right)$, such that the intersection of the lines $\ell\left(P_{1}\right)$, $\ell\left(P_{2}\right)$, $\ell\left(P_{3}\right)$ form an equilateral triangle $\Delta$. Find the locus of the center of $\Delta$ as $P_{1} P_{2} P_{3}$ ranges over all such triangles.

y = -\frac{1}{4}

0.375

In triangle $ABC$ with area 51, points $D$ and $E$ trisect $AB$, and points $F$ and $G$ trisect $BC$. Find the largest possible area of quadrilateral $DEFG$.

17

0.875

In the sequence of positive integers $1, 2, 3, 4, \cdots$, remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: $a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \cdots$. Find $a_{1999}$.

3331

0.25

In the triangle $ ABC $ the points $ M $ and $ N $ lie on the side $ AB $ such that $ AN = AC $ and $ BM = BC $. We know that $\angle MCN = 43^\circ$. Find the size in degrees of $\angle ACB$.

94^\circ

0.375

Given that $ F_1 $ and $ F_2 $ are the left and right foci of the ellipse $ C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $ where $ a > b > 0 $, and $ P $ is a point on the ellipse $ C $. The incenter of triangle $ \triangle F_{1}PF_{2} $ is $ I $. If there exists a real number $ \lambda $ such that: $$ (1+\lambda) \overrightarrow{PF_{1}} + (1-\lambda) \overrightarrow{PF_{2}} = 3 \overrightarrow{PI}, $$ then the eccentricity of the ellipse $ C $ is $\qquad$ .

\frac{1}{2}

0.75

The work was done together by team $C$ and team $D$. How many days did they take to complete it, given that team $C$ alone would have finished the work $c$ days later, and team $D$ alone would have required $d$ times more time? Apply the result to the cases $c=10, d=2$ and $c=8, d=3$.

16

0.125

How many different integral solutions $(x, y)$ does $3|x| + 5|y| = 100$ have?

26

0.375

Small and Big islands have a rectangular shape and are divided into rectangular counties. Each county has a road along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here's how the Small Island, consisting of 6 counties, is arranged (see Figure 1). Draw how the Big Island, which has an odd number of counties, could be arranged. How many counties did you get?

9

0.375

Arsenius has 2018 buckets, each containing 1, 2, 3, ... 2017, 2018 liters of water respectively. Arsenius is allowed to take any two buckets and pour into the second bucket exactly as much water as is already in the second bucket from the first bucket. Can Arsenius collect all the water into one bucket? All buckets are large enough to hold all the water they get.

\text{No}

0.5

Four students, Andrey, Vanya, Dima, and Sasha, participated in a school chess tournament. Each student played two games against each of the other participants. Each win awarded 1 point, a draw awarded 0.5 points, and a loss awarded 0 points. Given the following conditions at the end of the tournament: - Each student scored a different number of points. - Andrey secured first place, Dima secured second, Vanya secured third, and Sasha secured fourth. - Andrey won the same number of games as Sasha. How many points did each student score?

4, 3.5, 2.5, 2

0.125

A plane, perpendicular to an edge of a regular tetrahedron, passes through a point that divides this edge in the ratio 1:4. Find the ratio of the volumes of the resulting parts of the tetrahedron.

\frac{4}{121}

0.375

The height of a right triangle dropped to the hypotenuse is 1, one of the acute angles is $15^{\circ}$. Find the hypotenuse.

4

0.75

A natural number, which does not end in zero, had one of its digits replaced with zero (if it was the leading digit, it was simply erased). As a result, the number became 9 times smaller. How many such numbers exist for which this is possible?

7

0.375

In front of the elevator doors, there are people with masses of 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to transport all the people?

4

0.375

Given that the equation $ |x| - \frac{4}{x} = \frac{3|x|}{x} $ has $ k $ distinct real root(s), find the value of $ k $.

1

0.875

In a convex quadrilateral $ABCD$, $\overrightarrow{BC} = 2 \overrightarrow{AD}$. Point $P$ is a point in the plane of the quadrilateral such that $\overrightarrow{PA} + 2020 \overrightarrow{PB} + \overrightarrow{PC} + 2020 \overrightarrow{PD} = \mathbf{0}$. Let $s$ and $t$ be the areas of quadrilateral $ABCD$ and triangle $PAB$, respectively. Then $\frac{t}{s} =$ ______.

\frac{337}{2021}

0.25

Find the smallest positive integer $ n $ such that in any two-coloring of the complete graph $ K_{n} $, there exist 3 monochromatic triangles that are pairwise edge-disjoint.

9

0.25

Find the largest positive number $ x $ such that \[ \left(2 x^{3} - x^{2} - x + 1\right)^{1 + \frac{1}{2 x + 1}} = 1. \]

1

0.75

Five runners ran a relay race. If the first runner had run twice as fast, they would have spent 5% less time. If the second runner had run twice as fast, they would have spent 10% less time. If the third runner had run twice as fast, they would have spent 12% less time. If the fourth runner had run twice as fast, they would have spent 15% less time. By what percentage less time would they have spent if the fifth runner had run twice as fast?

8\%

0.75

Find the value of $a + b + c + d + e$ given the system of equations: \[ \begin{array}{c} 3a + 2b + 4d = 10 \\ 6a + 5b + 4c + 3d + 2e = 8 \\ a + b + 2c + 5e = 3 \\ 2c + 3d + 3e = 4 \\ a + 2b + 3c + d = 7 \end{array} \]

4

0.75

Given a cube $ A B C D A_1 B_1 C_1 D_1 $ with edge length $ a $, find the distance between the lines $ A A_1 $ and $ B D_1 $ and construct their common perpendicular.

\frac{a\sqrt{2}}{2}

0.625

It is known that the sum of the absolute values of the pairwise differences of five nonnegative numbers is equal to one. Find the smallest possible sum of these numbers.

\frac{1}{4}

0.5

In how many different ways can integers $ a, b, c \in [1, 100] $ be chosen such that the points with coordinates $ A(-1, a), B(0, b) $, and $ C(1, c) $ form a right triangle?

974

0.375

Let $ m $ and $ n $ be two integers such that $ n $ divides $ m(n+1) $. Show that $ n $ divides $ m $.

n \mid m

0.125

Determine the largest positive integer $ n $ such that there exist positive integers $ x, y, z $ so that \[ n^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx + 3x + 3y + 3z - 6 \]

8

0.75

In triangle $ABC$, the altitude from vertex $A$ is the harmonic mean of the two segments into which it divides side $BC$. What is $\operatorname{tg} \beta + \operatorname{tg} \gamma$ (where $\beta$ and $\gamma$ are the angles of the triangle)?

2

0.875

Suppose $ a $ and $ b $ are the roots of $ x^{2}+x \sin \alpha+1=0 $ while $ c $ and $ d $ are the roots of the equation $ x^{2}+x \cos \alpha-1=0 $. Find the value of $ \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} $.

1

0.5

Given two arbitrary positive integers $ n $ and $ k $, let $ f(n, k) $ denote the number of unit squares that one of the diagonals of an $ n \times k $ grid rectangle passes through. How many such pairs $ (n, k) $ are there where $ n \geq k $ and $ f(n, k) = 2018 $?

874

0.625

Integrate the equation $$ x \, dy = (x + y) \, dx $$ and find the particular solution that satisfies the initial condition $y = 2$ when $x = -1$.

y = x \ln |x| - 2x

0.75

The hedgehogs collected 65 mushrooms and divided them so that each received at least one mushroom, but no two hedgehogs had the same number of mushrooms. What is the maximum number of hedgehogs that could be?

10

0.75

Let $ a = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} $ and $ b = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} $. Find the value of $ a^4 + b^4 + (a+b)^4 $.

7938

0.375

Dr. Aibolit gave 2006 miraculous pills to four sick animals. The rhinoceros received one more than the crocodile, the hippopotamus received one more than the rhinoceros, and the elephant received one more than the hippopotamus. How many pills did the elephant get?

503

0.875

The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be bought so that there is a probability of 0.99 that at least five of them will be working?

7 \text{ bulbs}

0.25

Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits?

142857

0.375

Point $ P $ is inside triangle $ \triangle ABC $. Line segments $ APD $, $ BPE $, and $ CPF $ are drawn such that $ D $ is on $ BC $, $ E $ is on $ AC $, and $ F $ is on $ AB $. Given that $ AP = 6 $, $ BP = 9 $, $ PD = 6 $, $ PE = 3 $, and $ CF = 20 $, find the area of $ \triangle ABC $.

108

0.125

Let $ f(x) = x^2 + 3x + 2 $. Compute the expression \[ \left(1 - \frac{2}{f(1)}\right) \cdot\left(1 - \frac{2}{f(2)}\right) \cdot\left(1 - \frac{2}{f(3)}\right) \cdots \left(1 - \frac{2}{f(2019)}\right). \]

\frac{337}{1010}

0.875

Given the parabola $y = x^2$ and a point $M(1,1)$ on it as the vertex of right-angled triangles inscribed in the parabola, find the coordinates of the intersection point $E$ of the line segments $AB$ and $CD$.

(-1,2)

0.75

We have one $10 \mathrm{Ft}$ coin and several (at least four) $20 \mathrm{Ft}$ coins in our pocket. We successively draw one coin at a time at random until the sum of the drawn coins becomes a positive integer multiple of $30 \mathrm{Ft}$. On average, how many draws are needed to achieve this?

3

0.75

Given ten 5-element sets $ A_{1}, A_{2}, \cdots, A_{10} $, where the intersection of any two sets contains at least two elements. Let $ A=\bigcup_{i=1}^{10} A_{i}=\left\{x_{1}, x_{2}, \cdots, x_{n}\right\} $. For any $ x_{i} \in A $, let $ k_{i}(i=1,2, \cdots, n) $ be the number of sets among $ A_{1}, A_{2}, \cdots, A_{10} $ that contain the element $ x_{i} $. Define $ m = \max \left\{k_{1}, k_{2}, \cdots, k_{n}\right\} $. Determine the minimum value of $ m $.

5

0.75

The base of the rectangular parallelepiped $\boldsymbol{A}BCD A_{1}B_{1}C_{1}D_{1}$ is a square $ABCD$. Find the maximum possible value of the angle between the line $B D_{1}$ and the plane $BDC_{1}$.

\arcsin \frac{1}{3}

0.25

There are four weights of different masses. Katya weighs the weights in pairs. The results are 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing result in?

2190

0.75

Let $ P(n) $ be the number of permutations $\left(a_{1}, \ldots, a_{n}\right)$ of the numbers $(1,2, \ldots, n)$ for which $ k a_{k} $ is a perfect square for all $ 1 \leq k \leq n $. Find with proof the smallest $ n $ such that $ P(n) $ is a multiple of 2010.

4489

0.25

A circle inscribed in an angle with vertex $ O $ touches its sides at points $ A $ and $ B $. The ray $ OX $ intersects this circle at points $ C $ and $ D $, with $ OC = CD = 1 $. If $ M $ is the intersection point of the ray $ OX $ and the segment $ AB $, what is the length of the segment $ OM $?

\frac{4}{3}

0.25

Given distinct real numbers $a_{1}, a_{2}, a_{3}$ and $b$. It turns out that the equation $(x-a_{1})(x-a_{2})(x-a_{3}) = b$ has three distinct real roots $c_{1}, c_{2}, c_{3}$. Find the roots of the equation $(x+c_{1})(x+c_{2})(x+c_{3}) = b$.

-a_1, -a_2, -a_3

0.875

There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie. All the students in this class sat at a round table. - Several students said: "There is exactly one troublemaker next to me." - All other students said: "There are exactly two troublemakers next to me." What is the minimum number of troublemakers that can be in the class?

10

0.75

The numerator and denominator of a fraction are natural numbers that add up to 101. It is known that the fraction does not exceed $ \frac{1}{3} $. Determine the greatest possible value of such a fraction.

\frac{25}{76}

0.875

Expand the function $$ f(z)=\frac{z}{z^{2}-2 z-3} $$ into a Taylor series in the vicinity of the point $ z_{0}=0 $, using expansion (12), and find the radius of convergence of the series.

R = 1

0.625

In triangle $ABC$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is given that $AC - AB = 5$ cm. Find the length of the angle bisector of angle $B$.

5 \text{ cm}

0.875

Let $ a, b, c $ be the side lengths of a right triangle, with $ a \leqslant b < c $. Determine the maximum constant $ k $ such that $ a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c $ holds for all right triangles, and identify when equality occurs.

2 + 3\sqrt{2}

0.125

There is a ten-digit number. From left to right: - Its first digit indicates the number of zeros in the ten-digit number. - Its second digit indicates the number of ones in the ten-digit number. - Its third digit indicates the number of twos in the ten-digit number. - ... - Its tenth digit indicates the number of nines in the ten-digit number. What is this ten-digit number?

6210001000

0.125

The four-digit number $\overline{abcd}$ and $\overline{cdab}$ have a sum of 3333 and a difference of 693. Find the four-digit number $\overline{abcd}$.

2013

0.75

How many integers from 1 to 2001 have a digit sum that is divisible by 5?

399

0.25

A cube of size $1000 \times 1000 \times 1000$ is located in space with one vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors is divided by 11.

0

0.75

Let $ p $ and $ q $ be distinct prime numbers. How many divisors does the number have: a) $pq$; b) $p^2 q$; c) $p^2 q^2$; d) $p^m q^n$?

(m+1)(n+1)

0.875

Given $ f(1) = 1 $, and for any integers $ m $ and $ n $, $ f(m+n) = f(m) + f(n) + 3(4mn - 1) $, find $ f(19) $.

2017

0.5

What is the maximum possible area of a triangle if the sides $a, b, c$ satisfy the following inequalities: $$ 0 < a \leq 1 \leq b \leq 2 \leq c \leq 3 $$

1

0.625

In triangle $ABC$, the angles are known: $\angle A = 45^{\circ}$ and $\angle B = 15^{\circ}$. On the extension of side $AC$ beyond point $C$, point $M$ is taken such that $CM = 2AC$. Find $\angle AMB$.

75^\circ

0.375

Find the integer solutions of the equation $$ x^{4} + y^{4} = 3 x^{3} y $$

(0,0)

0.875

Find the smallest positive integer $ n $ that satisfies the following properties: 1. The units digit of $ n $ is 6. 2. If the units digit 6 is moved to the front of the remaining digits of $ n $, the resulting new number is 4 times $ n $.

153846

0.625

A sandwich and a meal plate cost $\mathrm{R} \$ 5.00$ and $\mathrm{R} \$ 7.00$, respectively. In how many ways can one buy only sandwiches, only meal plates, or some combination of sandwiches and meal plates with $\mathrm{R} \$ 90.00$, without leaving any change?

3

0.75

Determine the coefficients of $x^{17}$ and $x^{18}$ after expanding and combining like terms in the expression \[ \left(1 + x^5 + x^7\right)^{20} \]

3420

0.25

Compute the limit of the function: $\lim _{x \rightarrow 0} \frac{2 x \sin x}{1-\cos x}$

4

0.875

Find the smallest prime number $p$ such that $p^{3}+2 p^{2}+p$ has exactly 42 divisors.

23

0.375

Let $ABC$ be an equilateral triangle. Let $\overrightarrow{AB}$ be extended to a point $D$ such that $B$ is the midpoint of $\overline{AD}$. A variable point $E$ is taken on the same plane such that $DE = AB$. If the distance between $C$ and $E$ is as large as possible, what is $\angle BED$?

15^\circ

0.375

In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board?

480

0.375

In the sequence $\left\{a_{n}\right\}$, $a_{1}=13, a_{2}=56$, for all positive integers $n$, $a_{n+1}=a_{n}+a_{n+2}$. Find $a_{1934}$.

56

0.875

Show that the expression $$ 3^{2 n+2}-2^{n+1} $$ is divisible by 7 for any positive integer $n$.

7

0.25

In the complex plane, the points corresponding to the complex numbers $ z_1, z_2, z_3 $ are $ Z_1, Z_2, Z_3 $, respectively. Given that $\left| z_1 \right| = \left| z_2 \right| = \sqrt{2}$, $\overrightarrow{OZ_1} \cdot \overrightarrow{OZ_2} = 0$, and $\left| z_1 + z_2 - z_3 \right| = 2$, find the range of the value of $\left| z_3 \right|$.

[0, 4]

0.875

Originally, there were three equally sized grassy fields, and the grass grew at a fixed rate. To eat the grass from all three fields, 50 sheep take exactly 18 days. To eat the grass from two fields, 40 sheep take exactly 12 days. Eating the grass from one field, 70 sheep take $\qquad$ days exactly. How many days will it take?

2

0.625

With what minimum force must you press on a cube with a volume of 10 cm³, floating in water, to submerge it completely? The density of the cube's material is 400 kg/m³, and the density of water is 1000 kg/m³. Provide the answer in SI units. Assume the acceleration due to gravity is 10 m/s².

0.06 \, \text{N}

0.25

Given three composite numbers $ A, B, C $ that are pairwise coprime and $ A \times B \times C = 11011 \times 28 $. What is the maximum value of $ A + B + C $?

1626

0.375

Given that $ x $ is a real number and $ y = \sqrt{x^2 - 2x + 2} + \sqrt{x^2 - 10x + 34} $. Find the minimum value of $ y $.

4\sqrt{2}

0.625

The difference between the maximum and minimum values of the function $ f(x)=|\sin x|+\sin ^{+} 2x+|\cos x| $ is equal to ______.

\sqrt{2}

0.75

The number 2 can be represented as $2 = 2 = 1 + 1$. The number 3 can be represented as $3 = 3 = 1 + 2 = 2 + 1 = 1 + 1 + 1$. The number 4 can be represented as $4 = 4 = 1 + 3 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1$, and so on. This kind of representation of positive integers is called ordered partitions. Find the number of ordered partitions of a natural number $n$.

2^{n-1}

0.875

\[\frac{(1+\tan 2\alpha)^{2}-2 \tan^{2} 2\alpha}{1+\tan^{2} 2\alpha}-\sin 4\alpha-1\]

-2 \sin^2 2 \alpha

0.375

In $\triangle ABC$, $\angle A = 90^\circ$ and $\angle B = \angle C = 45^\circ$. $P$ is a point on $BC$, and $Q, R$ are the circumcenters of $\triangle APB$ and $\triangle APC$ respectively. If $BP = \sqrt{2}$ and $QR = 2$, find $PC$.

\sqrt{6}

0.75

The proportion of top-quality products at this enterprise is $31 \%$. What is the most likely number of top-quality products in a randomly selected batch of 75 products?

23

0.875

Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?

13

0.5

How many boards, each 6 arshins long and 6 vershoks wide, are needed to cover the floor of a square room with a side of 12 arshins? The answer is 64 boards. Determine from this data how many vershoks are in an arshin.

16

0.875

In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be?

1011

0.125

Petrov writes down the odd numbers: $1, 3, 5, \ldots, 2013$, and Vasechkin writes down the even numbers: $2, 4, \ldots, 2012$. Each of them calculates the sum of all digits of all their numbers and tells the result to their classmate Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the final result?

1007

0.375

Find all integers $ x, y \geq 1 $ such that $ x^3 - y^3 = xy + 61 $.

(6,5)

0.375

Let $ \triangle ABC $ be a triangle such that $ AB = 7 $, and let the angle bisector of $ \angle BAC $ intersect line $ BC $ at $ D $. If there exist points $ E $ and $ F $ on sides $ AC $ and $ BC $, respectively, such that lines $ AD $ and $ EF $ are parallel and divide triangle $ ABC $ into three parts of equal area, determine the number of possible integral values for $ BC $.

13

0.125

In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers written on the sides of the polygon and obtained the sum $ A $, while Sasha calculated the sum of the numbers written on the diagonals connecting vertices one apart and obtained the sum $ C $. Find the largest possible value of the difference $ C - A $.

1010

0.875

Given an acute triangle $ \triangle ABC $, construct isosceles triangles $ \triangle DAC, \triangle EAB, \triangle FBC $ outside of $ \triangle ABC $ such that $ DA = DC, EA = EB, FB = FC $, and the angles $ \angle ADC = 2 \angle BAC, \angle BEA = 2 \angle ABC, \angle CFB = 2 \angle ACB $. Define $ D' $ as the intersection of line $ DB $ with $ EF $, $ E' $ as the intersection of line $ EC $ with $ DF $, and $ F' $ as the intersection of line $ FA $ with $ DE $. Find the value of $ \frac{DB}{DD'} + \frac{EC}{EE'} + \frac{FA}{FF'} $.

4

0.375

Find the maximum value for $a, b \geq 1$ of the expression $$ \frac{|7a + 8b - ab| + |2a + 8b - 6ab|}{a \sqrt{1 + b^2}} $$

9\sqrt{2}

0.875

A point $ D $ is marked on the altitude $ BH $ of triangle $ ABC $. Line $ AD $ intersects side $ BC $ at point $ E $, and line $ CD $ intersects side $ AB $ at point $ F $. It is known that $ BH $ divides segment $ FE $ in the ratio $ 1:3 $, starting from point $ F $. Find the ratio $ FH:HE $.

1:3

0.5

Find the first differential of the function $ y(x) = e^{3x} \ln(1 + x^2) $ and evaluate it at $ x = 0 $ with $ dx = \Delta x = 0.1 $.

dy(0) = 0

0.875

The product of three consecutive even numbers equals $87_{* * * * *}8$. Find these numbers and fill in the blanks in this product.

87526608

0.25

Solve the equation $\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}$. Write the sum of all obtained solutions as the answer.

2.5

0.375

Color 101 cells blue in an $n \times n$ grid. It is known that there is a unique way to cut the grid along square lines into rectangles such that each rectangle contains exactly one blue cell. Find the smallest possible value of $n$.

101

0.25

Given that $\alpha$ and $\beta$ are acute angles, and the following equations hold: $$ \left\{\begin{array}{l} 3 \sin ^{2} \alpha + 2 \sin ^{2} \beta = 1, \\ 3 \sin 2 \alpha - 2 \sin 2 \beta = 0. \end{array}\right. $$ Determine $\alpha + 2\beta$.

\frac{\pi}{2}

0.75

In the polar coordinate system, the distance from the pole to the center of the circle given by the equation $\rho=3 \sqrt{2} \cos \left(\theta+\frac{\pi}{4}\right)+7 \sin \theta$ is what?

\frac{5}{2}

0.875