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Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2021 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \). | 1011 | 0.125 |
Ten elves are sitting around a circular table, each with a basket of nuts. Each elf is asked, "How many nuts do your two neighbors have together?" and the answers, going around the circle, are 110, 120, 130, 140, 150, 160, 170, 180, 190, and 200. How many nuts does the elf who answered 160 have? | 55 | 0.625 |
The polynomials \( p_n(x) \) are defined by \( p_0(x) = 0 \), \( p_1(x) = x \), \( p_{n+2}(x) = x p_{n+1}(x) + (1 - x) p_n(x) \). Find the real roots of each \( p_n(x) \). | 0 | 0.5 |
As shown in the figure, the distance between two adjacent points in both the horizontal and vertical directions is $m$. If the area of quadrilateral $ABCD$ is 23, what is the area of pentagon $EFGHI$? | 28 | 0.125 |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given:
$$
a_{k}=\sum_{i=k^{2}}^{(k+1)^{2}-1} \frac{1}{i} \quad (k=1,2, \cdots)
$$
Find:
$$
\sum_{k=1}^{n}\left(\left[\frac{1}{a_{k}}\right]+\left[\frac{1}{a_{k}}+\frac{1}{2}\right]\right) .
$$ | \frac{n(n+1)}{2} | 0.5 |
Let \(\mathcal{A}\) denote the set of all polynomials in three variables \(x, y, z\) with integer coefficients. Let \(\mathcal{B}\) denote the subset of \(\mathcal{A}\) formed by all polynomials which can be expressed as
\[ (x + y + z) P(x, y, z) + (xy + yz + zx) Q(x, y, z) + xyz R(x, y, z) \]
with \(P, Q, R \in \mathcal{A}\). Find the smallest non-negative integer \(n\) such that \(x^{i} y^{j} z^{k} \in \mathcal{B}\) for all nonnegative integers \(i, j, k\) satisfying \(i + j + k \geqslant n\). | 4 | 0.375 |
Point \( A \) lies on the line \( y = \frac{5}{12} x - 7 \), and point \( B \) lies on the parabola \( y = x^2 \). What is the minimum length of segment \( AB \)? | \frac{4007}{624} | 0.375 |
The numbers from 1 to 9 are divided into three groups of three numbers, and then the numbers in each group are multiplied. $A$ is the largest of the three products. What is the smallest possible value of $A$? | 72 | 0.5 |
A plane \( P \) slices through a cube of side length 2 with a cross section in the shape of a regular hexagon. The inscribed sphere of the cube intersects \( P \) in a circle. What is the area of the region inside the regular hexagon but outside the circle? | 3 \sqrt{3} - \pi | 0.875 |
In an isosceles triangle \(ABC\) on the base \(AC\), a point \(M\) is taken such that \(AM = a\) and \(MC = b\). Circles are inscribed in triangles \(ABM\) and \(CBM\). Find the distance between the points of tangency of these circles with the segment \(BM\). | \frac{|a-b|}{2} | 0.625 |
In the triangle \( ABC \), \(\angle B = 90^\circ\), \(\angle C = 20^\circ\), \( D \) and \( E \) are points on \( BC \) such that \(\angle ADC =140^\circ\) and \(\angle AEC =150^\circ\). Suppose \( AD=10 \). Find \( BD \cdot CE \). | 50 | 0.625 |
Let \( S = \{1, 2, 3, \ldots, 20\} \) be the set of all positive integers from 1 to 20. Suppose that \( N \) is the smallest positive integer such that exactly eighteen numbers from \( S \) are factors of \( N \), and the only two numbers from \( S \) that are not factors of \( N \) are consecutive integers. Find the sum of the digits of \( N \). | 36 | 0.625 |
Vasily invented a new operation on the set of positive numbers: \( a \star b = a^{\ln b} \). Find the logarithm of the number \(\frac{(a b) \star(a b)}{(a \star a)(b \star b)}\) to the base \( a \star b \). | 2 | 0.75 |
Simplify the expression \(1.6 \frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2 c}{a b}\right)(a+b+2 c)}{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{a b}-\frac{4 c^{2}}{a^{2} b^{2}}}\) given that \(a = 7.4\) and \(b = \frac{5}{37}\). | 1.6 | 0.75 |
If \( x, y \), and \( z \) are real numbers such that \( 2 x^{2} + y^{2} + z^{2} = 2 x - 4 y + 2 xz - 5 \), find the maximum possible value of \( x - y + z \). | 4 | 0.75 |
Given the complex numbers \( Z_1 \) and \( Z_2 \) such that \( \left|Z_1\right| = 2 \) and \( \left|Z_2\right| = 3 \). If the angle between their corresponding vectors is \( 60^\circ \), find the value of \( \left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right| \). | \sqrt{\frac{19}{7}} | 0.625 |
At a worldwide meteorological conference, each participant sequentially announced the average monthly temperature in their hometown. All other participants recorded the product of the temperatures in the announcer's and their own cities at that moment. A total of 48 positive and 42 negative numbers were recorded. What is the least number of times a positive temperature could have been announced? | 3 | 0.875 |
How many natural numbers are divisors of the number 1,000,000 and do not end in 0? | 13 | 0.875 |
Let \(ABCD\) be a convex quadrilateral with all sides and diagonals having integer lengths. Given that \(\angle ABC = \angle ADC = 90^\circ\), \(AB = BD\), and \(CD = 41\), find the length of \(BC\). | 580 | 0.125 |
Calculate \(\cos (\alpha + \beta)\) if \(\cos \alpha - \cos \beta = -\frac{3}{5}\) and \(\sin \alpha + \sin \beta = \frac{7}{4}\). | -\frac{569}{800} | 0.75 |
Among the integers from 1 to 1000, how many are divisible by 2 but not divisible by 3 or 5? | 267 | 0.875 |
How many solutions in integers \( x \) and \( y \) does the inequality \( |y - x| + |3x - 2y| \leq a \) have? a) when \( a = 2 \); b) when \( a = 20 \)? | 841 | 0.625 |
Calculate the sum \(\sum_{n=0}^{502}\left\lfloor\frac{305 n}{503}\right\rfloor\). | 76304 | 0.75 |
Show that in any triangle,
$$
\frac{\operatorname{ctg} \alpha+\operatorname{ctg} \beta}{\operatorname{tg} \alpha+\operatorname{tg} \beta}+\frac{\operatorname{ctg} \beta+\operatorname{ctg} \gamma}{\operatorname{tg} \beta+\operatorname{tg} \gamma}+\frac{\operatorname{ctg} \gamma+\operatorname{ctg} \alpha}{\operatorname{tg} \gamma+\operatorname{tg} \alpha}=1
$$ | 1 | 0.5 |
Let \( \triangle ABC \) have an inscribed circle \( \Gamma \). This inscribed circle is tangent to sides \( AC, BC, \) and \( AB \) at points \( Y, X, \) and \( Z \), respectively. Let \( \gamma_A \) be the A-excircle tangent to sides \( AC, BC, \) and \( AB \) at points \( K_C, T, \) and \( K_B \), respectively. Show that \( BK_A = CX \). | BK_A = CX | 0.75 |
Rectangle \(ABCD\) is congruent to rectangle \(ABEF\), and \(D-AB-E\) forms a dihedral angle. \(M\) is the midpoint of \(AB\). \(FM\) makes an angle \(\theta\) with \(BD\), where \(\sin \theta = \frac{\sqrt{78}}{9}\). Find the value of \(\frac{AB}{BC}\). | \frac{\sqrt{2}}{2} | 0.625 |
Select 5 different numbers from $0,1,2,3,4,5,6,7,8,9$ to form a five-digit number such that this five-digit number is divisible by $3$, $5$, $7$, and $13$. What is the largest such five-digit number? | 94185 | 0.5 |
\(\cos^2 \varphi + \cos^2(\alpha - \varphi) - 2 \cos \alpha \cos \varphi \cos (\alpha - \varphi) = \sin^2 \alpha\) | \sin^2 \alpha | 0.875 |
In a castle, there are 16 identical square rooms arranged in a $4 \times 4$ square. Sixteen people—some knights and some liars—each occupy one room. Liars always lie, and knights always tell the truth. Each of these 16 people said, "At least one of my neighboring rooms houses a liar." Rooms are considered neighbors if they share a common wall. What is the maximum number of knights that could be among these 16 people? | 12 | 0.5 |
Let \( d \) be the greatest common divisor of a natural number \( b \) and 175. If \( 176 \times (b - 11 \times d + 1) = 5 \times d + 1 \), then \( b = \) ________. | 385 | 0.625 |
If the sum of interior angles of \( m \) regular \( n \)-sided polygons is divisible by 27, what is the minimum value of \( m + n \)? | 6 | 0.75 |
Given positive real numbers \( x \) and \( y \) satisfying \( x^{2} + y^{2} + \frac{1}{x} + \frac{1}{y} = \frac{27}{4} \), find the minimum value of \( P = \frac{15}{x} - \frac{3}{4y} \). | 6 | 0.75 |
Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same.
$\square + \square \square = \square \square \square$.
What is the largest possible three-digit number in the equation? | 105 | 0.625 |
A polygon $\mathcal{P}$ is drawn on the 2D coordinate plane. Each side of $\mathcal{P}$ is either parallel to the $x$ axis or the $y$ axis (the vertices of $\mathcal{P}$ do not have to be lattice points). Given that the interior of $\mathcal{P}$ includes the interior of the circle \(x^{2}+y^{2}=2022\), find the minimum possible perimeter of $\mathcal{P}$. | 8 \sqrt{2022} | 0.875 |
According to legend, the poet Li Bai from the Tang Dynasty went to buy wine, singing as he walked down the street. Each time he passed a store, the amount of his wine doubled, and each time he saw a flower, he drank 2 cups. After encountering four stores and flowers in total, he still had 2 cups of wine left at the end. How many cups of wine did he originally have? | 2 | 0.75 |
In a container, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He under pressure $p$. The absolute temperature of the gas is doubled, and all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of the gases are $\mu_{\text{He}} = 4$ g/mol and $\mu_{N_{2}} = 28$ g/mol. Assume the gases are ideal. | \frac{9}{4} p | 0.25 |
There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \( N \) be the number of two-humped camels. How many possible values can \( N \) take within the range from 1 to 99?
| 72 | 0.625 |
The longer base of a trapezoid has a length of 24 cm. Find the length of the shorter base, given that the distance between the midpoints of the diagonals of the trapezoid is 4 cm. | 16 | 0.875 |
Determine the maximal size of a set of positive integers with the following properties:
1. The integers consist of digits from the set {1,2,3,4,5,6}.
2. No digit occurs more than once in the same integer.
3. The digits in each integer are in increasing order.
4. Any two integers have at least one digit in common (possibly at different positions).
5. There is no digit which appears in all the integers. | 32 | 0.125 |
Xiao Hong asked Da Bai: "Please help me calculate the result of $999 \quad 9 \times 999 \quad 9$ and determine how many zeros appear in it."
2019 nines times 2019 nines
Da Bai quickly wrote a program to compute it. Xiao Hong laughed and said: "You don't need to compute the exact result to know how many zeros there are. I'll tell you it's....."
After calculating, Da Bai found Xiao Hong's answer was indeed correct. Xiao Hong's answer is $\qquad$. | 2018 | 0.625 |
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\). | 3:4:5 | 0.375 |
Determine all primes \( p \) such that
\[ 5^p + 4 \cdot p^4 \]
is a perfect square, i.e., the square of an integer. | 5 | 0.5 |
On the sides \(AB, BC, CD,\) and \(DA\) of parallelogram \(ABCD\), points \(M, N, K,\) and \(L\) are taken, respectively, such that \(AM : MB = CK : KD = 1 : 2\) and \(BN : NC = DL : LA = 1 : 3\). Find the area of the quadrilateral whose vertices are the intersections of the segments \(AN, BK, CL,\) and \(DM\), given that the area of parallelogram \(ABCD\) is 1. | \frac{6}{13} | 0.625 |
A function is given by
$$
f(x)=\ln (a x+b)+x^{2} \quad (a \neq 0).
$$
(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.
(2) If $f(x) \leqslant x^{2}+x$ always holds, find the maximum value of $ab$. | \frac{e}{2} | 0.875 |
A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$? | P_{2007} | 0.375 |
Arrange the natural numbers from 1 to 7 in a row such that each number is either greater than all the numbers before it or less than all the numbers before it. For example, the arrangement 4356271 satisfies this condition: starting from the second position, 3 is less than the first number 4; 5 is greater than the first two numbers 4 and 3; 6 is greater than the first three numbers 4, 3, and 5; 2 is less than the first four numbers 4, 3, 5, and 6; 7 is greater than the first five numbers 4, 3, 5, 6, and 2; 1 is less than the first six numbers 4, 3, 5, 6, 2, and 7.
Determine the number of such arrangements where the number 7 does not occupy the fourth position. | 60 | 0.125 |
A natural number \( n \) is such that the number \( 100n^2 \) has exactly 55 different natural divisors. How many natural divisors does the number \( 10n \) have? | 18 | 0.25 |
The numbers \(a, b, c, d\) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 240 | 0.625 |
A society of \(n\) members is selecting one representative from among themselves.
a) In how many ways can an open vote occur if each member votes for one person (possibly themselves)?
b) Solve the same problem if the vote is secret, i.e., only the number of votes each candidate receives is considered, and it does not matter who voted for whom. | \binom{2n-1}{n-1} | 0.25 |
How many ordered pairs of integers \((m, n)\) where \(0 < m < n < 2008\) satisfy the equation \(2008^{2} + m^{2} = 2007^{2} + n^{2}?\) | 3 | 0.625 |
Zhenya had 9 cards with numbers from 1 to 9. He lost the card with the number 7. Is it possible to arrange the remaining 8 cards in a row so that any two adjacent cards form a number divisible by 7? | \text{No} | 0.125 |
Find all prime numbers \( p \) not exceeding 1000 such that \( 2p + 1 \) is a perfect power (i.e., there exist natural numbers \( m \) and \( n \geq 2 \) such that \( 2p + 1 = m^n \)). | 13 | 0.25 |
Through the vertex \(C\) of the base of a regular triangular pyramid \(SABC\), a plane is drawn perpendicular to the lateral edge \(SA\). This plane forms an angle with the base plane, the cosine of which is \( \frac{2}{3} \). Find the cosine of the angle between two lateral faces. | \frac{1}{7} | 0.625 |
Two triangles - an equilateral triangle with side length \(a\) and an isosceles right triangle with legs of length \(b\) - are positioned in space such that their centers of mass coincide. Find the sum of the squares of the distances from all vertices of one triangle to all vertices of the other. | 3a^2 + 4b^2 | 0.375 |
Ria has three counters marked 1, 5, and 11. She wants to place them side-by-side to make a four-digit number. How many different four-digit numbers can she make? | 4 | 0.875 |
Given \( f(x) \) is a function defined on \(\mathbf{R}\), for any \( x, y \in \mathbf{R} \), it always holds that
\[ f(x-f(y)) = f(f(y)) + x f(y) + f(x) - 1 .\]
Find \( f(x) \) and calculate the value of \( f(\sqrt{2014}) \). | -1006 | 0.75 |
A library spends 4500 yuan to buy 300 books of five types: "Zhuangzi," "Kongzi," "Mengzi," "Laozi," and "Sunzi." Their prices per book are 10 yuan, 20 yuan, 15 yuan, 30 yuan, and 12 yuan, respectively. The number of "Zhuangzi" books is the same as "Kongzi" books. The number of "Sunzi" books is 15 more than four times the number of "Laozi" books. How many "Sunzi" books are there in this batch? | 75 | 0.625 |
Find the minimum value of the function \( f(x, y) = 6\left(x^{2} + y^{2}\right)(x + y) - 4\left(x^{2} + xy + y^{2}\right) - 3(x + y) + 5 \) in the region \( D = \{(x, y) \mid x > 0, y > 0\} \). | 2 | 0.875 |
In a box, 10 smaller boxes are placed. Some of the boxes are empty, and some contain another 10 smaller boxes each. Out of all the boxes, exactly 6 contain smaller boxes. How many empty boxes are there? | 55 | 0.75 |
In the triangular prism \( P-ABC \),
\[
\begin{array}{l}
\angle APB = \angle BPC = \angle CPA = 90^{\circ}, \\
PA = 4, \, PB = PC = 3.
\end{array}
\]
Find the minimum sum of the squares of the distances from any point on the base \( ABC \) to the three lateral faces. | \frac{144}{41} | 0.625 |
Calculate the result of the expression:
\[ 2013 \times \frac{5.7 \times 4.2 + \frac{21}{5} \times 4.3}{\frac{14}{73} \times 15 + \frac{5}{73} \times 177 + 656} \] | 126 | 0.875 |
Starting with the numbers \(x_{1}=2\) and \(y_{1}=1\), we define two sequences based on the following rules:
\[ x_{k}=2 x_{k-1}+3 y_{k-1}, \quad y_{k}=x_{k-1}+2 y_{k-1} \]
where \(k=2,3,4, \ldots\). Show that for the terms of the two sequences with the same index, the difference \(x_{k}^{2}-3 y_{k}^{2}\) is constant. | 1 | 0.75 |
Let $ABC$ be a triangle, $M$ the midpoint of segment $[BC]$, and $E$ and $F$ two points on the rays $[AB)$ and $[AC)$ such that $2 \angle AEC = \angle AMC$ and $2 \angle AFB = \angle AMB$. Let $X$ and $Y$ be the intersections of the circumcircle of triangle $AEF$ with the line $(BC)$. Show that $XB = YC$. | X B = Y C | 0.875 |
Find a formula for the sum of the squares of the numbers in the $n$th row of Pascal's triangle (i.e., the numbers $\binom{n}{0}$, $\binom{n}{1}$, ..., $\binom{n}{n}$). | \binom{2n}{n} | 0.75 |
Six people, A, B, C, D, E, and F are each assigned a unique number.
A says: "The six of our numbers exactly form an arithmetic sequence."
B says: "The smallest number in this arithmetic sequence is 2."
C says: "The sum of our six numbers is 42."
D says: "The sum of A, C, and E's numbers is twice the sum of B, D, and F's numbers."
E says: "The sum of B and F's numbers is twice A's number."
What is D's number? ( ) | D = 2 | 0.625 |
Given the function \( f: \mathbf{R} \rightarrow \mathbf{R} \), for any real numbers \( x, y, z \), the inequality \(\frac{1}{3} f(x y) + \frac{1}{3} f(x z) - f(x) f(y z) \geq \frac{1}{9} \) always holds. Find the value of \(\sum_{i=1}^{100} [i f(i)]\), where \([x]\) represents the greatest integer less than or equal to \( x \). | 1650 | 0.75 |
The teacher wrote the number 1818 on the board. Vasya noticed that if you insert a multiplication sign between the hundreds and tens places, the value of the resulting expression is a perfect square (18 × 18 = 324 = 18²). What is the next four-digit number after 1818 that has the same property? | 1832 | 0.375 |
From "The Greek Anthology": Four springs emerge from the ground. The first fills a pool in one day, the second in 2 days, the third in 3 days, and the fourth in 4 days. In how much time will all four springs together fill the pool? | \frac{12}{25} | 0.75 |
A triangular pyramid \( S-ABC \) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \). | \frac{268\pi}{11} | 0.875 |
Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved by raising the ticket price. By what percentage should the ticket price be increased to make the clubs' share 20%? | 12.5\% | 0.375 |
Solve the equation \( 2021 \cdot \sqrt[202]{x^{2020}} - 1 = 2020 x \) for \( x \geq 0 \). | x = 1 | 0.25 |
For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary. | 0.79 | 0.625 |
Let the product of the digits of a positive integer \( n \) be denoted by \( a(n) \). Find the positive integer solution to the equation \( n^2 - 17n + 56 = a(n) \). | 4 | 0.5 |
If the price of a product increased from $R \$ 5.00$ to $R \$ 5.55$, what was the percentage increase? | 11\% | 0.875 |
Determine all pairs \((p, q)\) of positive integers such that \(p\) and \(q\) are prime, and \(p^{q-1} + q^{p-1}\) is the square of an integer. | (2, 2) | 0.875 |
A deck of 54 cards is divided into several piles by a magician. An audience member writes a natural number on each card equal to the number of cards in that pile. The magician then shuffles the cards in a special way and redistributes them into several piles. The audience member again writes a natural number on each card equal to the number of cards in the new pile. This process continues. What is the minimum number of times this process needs to be performed so that the (unordered) array of numbers written on the cards becomes unique for each card? | 3 | 0.375 |
The country Omega grows and consumes only vegetables and fruits. It is known that in 2014, 1200 tons of vegetables and 750 tons of fruits were grown in Omega. In 2015, 900 tons of vegetables and 900 tons of fruits were grown. During the year, the price of one ton of vegetables increased from 90,000 to 100,000 rubles, and the price of one ton of fruits decreased from 75,000 to 70,000 rubles. By what percentage (%) did the real GDP of this country change in 2015, if the base year in Omega is 2014? Round your answer to two decimal places. If the real GDP of the country decreased, put a minus sign in the answer, and if it increased, put a plus sign. | -9.59\% | 0.75 |
Several points were marked on a line. After that, a point was marked between each pair of neighboring points. This "densification" was repeated two more times (a total of 3 times). As a result, there were 113 points marked on the line. How many points were originally marked? | 15 | 0.75 |
In the sequence $5, 8, 15, 18, 25, 28, \cdots, 2008, 2015$, how many numbers have a digit sum that is an even number? (For example, the digit sum of 138 is $1+3+8=12$) | 202 | 0.25 |
In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken? | 123 | 0.625 |
The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\). | \frac{1}{2} | 0.375 |
Determine the integers \( n \in \mathbb{N} \) such that \( 2^{n} + 1 \) is a square or a cube. | n = 3 | 0.875 |
Through the vertex \( A \) of rectangle \( ABCD \), a line \( \ell \) is drawn, as shown in the figure. From points \( B \) and \( D \), perpendiculars \( BX \) and \( DY \) are dropped to the line \( \ell \). Find the length of segment \( XY \), if it is known that \( BX = 4 \), \( DY = 10 \), and \( BC = 2 AB \). | 13 | 0.25 |
Find the number of points on the plane \( xOy \) with natural coordinates \( (x, y) \) that lie on the parabola \( y=-\frac{x^{2}}{4}+9x+19 \). | 18 | 0.625 |
There are three types of plants in the garden: sunflowers, lilies, and peonies.
1. There is only one day in a week when all three types of flowers bloom simultaneously.
2. No type of flower can bloom for three consecutive days.
3. In one week, the number of days when any two types of flowers do not bloom together will not exceed one day.
4. Sunflowers do not bloom on Tuesday, Thursday, and Sunday.
5. Lilies do not bloom on Thursday and Saturday.
6. Peonies do not bloom on Sunday.
On what day of the week do all three types of flowers bloom simultaneously? (Represent the days from Monday to Sunday as numbers 1 to 7.) | 5 | 0.125 |
Let's introduce the notations: $Q$ - center of the first circle, $F, E, B$ - points of tangency of the circles and lines, $G$ - point of tangency of the circles, $AD$ - common tangent (see the figure). Draw the line $OC$ parallel to the line $FA$. Let $FA = a$, $OG = r$, and $AG = a \ (AG = FA$ as tangents drawn from point $A$ to the same circle). Since $QC = 4 - r, OQ = 4 + r, OA = 8 - r, OC = a$, from the right triangles $QOC$ and $AOG$, we obtain the system:
$$
\left\{\begin{array}{c}
(4 - r)^2 + a^2 = (4 + r)^2 \\
r^2 + a^2 = (8 - r)^2
\end{array}\right.
$$
By subtracting the first equation from the second, we get:
$$
4 \cdot(4 - 2r) = 12 \cdot(4 - 2r)
$$
From which it follows that $r = 2$. | 2 | 0.875 |
If $\sin \alpha + \sin \beta = 1$ and $\cos \alpha + \cos \beta = 0$, then what is $\cos 2 \alpha + \cos 2 \beta$? | 1 | 0.875 |
Farmers Ivanov, Petrov, Sidorov, Vasilev and Ermolaev own rectangular plots of land, the areas of which are indicated in the diagram (see the figure). Find the area of the common pasture.
| Ivanov <br> 24 ha | Forest | Ermolaev <br> 30 ha |
| :--- | :--- | :--- |
| Petrov <br> 28 ha | Common <br> pasture | Lake |
| Empty | Sidorov <br> 10 ha | Vasilev <br> 20 ha | | 17.5 | 0.75 |
Calculate the limit of the function:
$$\lim_{x \rightarrow 2\pi}\left(\cos x\right)^{\frac{\cot{2x}}{\sin{3x}}}$$ | e^{-\frac{1}{12}} | 0.5 |
Simplify the expression:
$$
\frac{1-\cos 2 \alpha + \sin 2 \alpha}{1+\cos 2 \alpha + \sin 2 \alpha}
$$ | \tan \alpha | 0.75 |
Let $A B C D$ be a square and $E$ be the point on segment $[B D]$ such that $E B = A B$. Define point $F$ as the intersection of lines $(C E)$ and $(A D)$. Find the value of the angle $\widehat{F E A}$. | 45^\circ | 0.875 |
If \((2x + 4)^{2n} = \sum_{i=0}^{2n} a_i x^i\) (where \(n \in \mathbf{Z}_+\)), what is the remainder when \(\sum_{i=1}^{n} a_{2i}\) is divided by 3? | 1 | 0.625 |
In how many ways can we place two rooks of different colors on a chessboard such that they do not attack each other? | 3136 | 0.75 |
Expanding the expression \((1+\sqrt{5})^{206}\) using the binomial theorem, we get terms of the form \(C_{206}^{k}(\sqrt{5})^{k}\). Find the value of \(k\) at which such a term takes on the maximum value. | 143 | 0.875 |
Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | \frac{\pi}{4} | 0.5 |
The ratio of the sides of a rectangle is \(3:4\), and its diagonal is 9. Find the shorter side of the rectangle. | 5.4 | 0.875 |
In trapezoid \(ABCD\) with bases \(AB\) and \(CD\), it holds that \(|AD| = |CD|\), \(|AB| = 2|CD|\), \(|BC| = 24 \text{ cm}\), and \(|AC| = 10 \text{ cm}\).
Calculate the area of trapezoid \(ABCD\). | 180 \text{ cm}^2 | 0.5 |
A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possible height of the ceiling, such that the following conditions are met: every point on the football field is illuminated by at least one floodlight, and the height of the ceiling must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.). | 27.1 \text{ m} | 0.375 |
Let \( f:[0,1) \rightarrow \mathbb{R} \) be a function that satisfies the following condition: if
\[
x=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{n}}=. a_{1} a_{2} a_{3} \ldots
\]
is the decimal expansion of \( x \) and there does not exist a positive integer \( k \) such that \( a_{n}=9 \) for all \( n \geq k \), then
\[
f(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{2 n}} .
\]
Determine \( f^{\prime}\left(\frac{1}{3}\right) \). | 0 | 0.375 |
In triangle \( \triangle ABC \), \(\angle C = 90^\circ\). The angle bisectors of \(\angle A\) and \(\angle B\) intersect at point \(P\). \(PE \perp AB\) at point \(E\). Given that \(BC = 2\) and \(AC = 3\), find the value of \(AE \cdot EB\). | 3 | 0.625 |
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