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A $3 \times 3 \times 3$ cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes makes a $45^{\circ}$ angle with the horizontal plane. | 60 | 0.125 |
There is a magical tree with 46 fruits. On the first day, 1 fruit will fall from the tree. Every day, the number of fallen fruits increases by 1 compared to the previous day. However, if on any day the number of fruits on the tree is less than the number of fruits that should fall that day, then the process restarts by dropping 1 fruit, following the original pattern. Continue in this manner. On which day will all the fruits have fallen from the tree? | 10 | 0.375 |
The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence to be number 0, determine after what time interval (in minutes) they will coincide the 19th time. If necessary, round the answer to two decimal places following the rounding rules. | 1243.64 | 0.875 |
When is the distance between the real image produced by a convex lens and the object the smallest? | 4f | 0.25 |
Four people (A, B, C, D) are practicing passing a ball. The ball is initially passed by A, and each person who receives the ball has an equal probability of passing it to one of the other three people. Let \( p_{n} \) represent the probability that the ball returns to A after \( n \) passes. What is \( p_{6} \)? | \frac{61}{243} | 0.75 |
The function \( f(x)=a x^{2}+b x+c \), where \( a \), \( b \), and \( c \) are integers, has two distinct roots in the interval \((0,1)\). Find the smallest positive integer \( a \) for which the given condition holds. | 5 | 0.75 |
Losyash is walking to Sovunya's house along the river at a speed of 4 km/h. Every half-hour, he launches paper boats that travel to Sovunya at a speed of 10 km/h. What is the time interval at which the boats arrive at Sovunya's house? | 18 \text{ минут} | 0.375 |
If 10 points are placed at equal intervals on a line, they occupy a segment of length \( s \). If 100 points are placed, they occupy a segment of length \( S \). By what factor is \( S \) greater than \( s \)? | 11 | 0.875 |
Can 1965 points be arranged inside a square with side 15 so that any rectangle of unit area placed inside the square with sides parallel to its sides must contain at least one of the points? | Yes | 0.625 |
What is the maximum possible area of a triangle with sides \(a\), \(b\), and \(c\) that fall within the following ranges:
\[ 0 \leq a \leq 1, \quad 1 \leq b \leq 2, \quad 2 \leq c \leq 3? \] | 1 | 0.625 |
Given a regular $n$-sided prism, where the area of the base is $S$. Two planes intersect all the lateral edges of the prism in such a way that the volume of the part of the prism between the planes is $V$. Find the sum of the lengths of the segments of the lateral edges of the prism that are enclosed between the planes, given that the planes do not have any common points inside the prism. | \frac{nV}{S} | 0.5 |
The positive integers \( a \) and \( b \) are relatively prime. The greatest common divisor of the numbers \( A = 8a + 3b \) and \( B = 3a + 2b \) is not 1. What is the greatest common divisor of \( A \) and \( B \)? | 7 | 0.875 |
Among 2017 natural numbers, there is at least one two-digit number, and for any two of these numbers, at least one is a three-digit number. Determine the number of three-digit numbers among these 2017 numbers. | 2016 | 0.875 |
Clara takes 2 hours to ride her bicycle from Appsley to Bancroft. The reverse trip takes her 2 hours and 15 minutes. If she travels downhill at 24 km/h, on level road at 16 km/h, and uphill at 12 km/h, what is the distance, in kilometres, between the two towns? | 34 \text{ km} | 0.5 |
Given \( w = \sqrt{2p - q} + \sqrt{3q - 2p} + \sqrt{6 - 2q} \), where \( p \) and \( q \) are real numbers that make \( w \) meaningful, determine the maximum value of \( w \). | 3\sqrt{2} | 0.875 |
In a regular triangular prism \(P-ABC\) where all edge lengths are \(1\), points \(L\), \(M\), and \(N\) are the midpoints of edges \(PA\), \(PB\), and \(PC\) respectively. Determine the area of the cross-section of the circumscribed sphere intersected by the plane \(LMN\). | \frac{\pi}{3} | 0.5 |
A \(15 \times 15\) square is divided into \(1 \times 1\) small squares. From these small squares, several were chosen, and in each chosen square, one or two diagonals were drawn. It turned out that no two drawn diagonals have a common endpoint. What is the maximum number of diagonals that can be drawn? (In the solution, provide the answer, the method of drawing the diagonals, and proof that this number of diagonals is indeed the maximum possible.) | 128 | 0.5 |
How many integers at minimum must be selected from the set $\{1, 2, \ldots, 20\}$ to ensure that this selection includes two integers \(a\) and \(b\) such that \(a - b = 2\)? | 11 | 0.625 |
A fast-food chain offers summer jobs to students with a salary of 25,000 rubles per month. Those who perform well receive an additional monthly bonus of 5,000 rubles.
How much will a well-performing student working at the fast-food chain earn per month (take-home pay) after income tax deduction?
Provide only the number without units in the answer! | 26100 | 0.75 |
Distribute 16 identical books among 4 students so that each student gets at least one book, and each student gets a different number of books. How many distinct ways can this be done? (Answer with a number.) | 216 | 0.625 |
Four points form a convex quadrilateral. If the smallest distance between any two of them is \( d \), show that there are two whose distance apart is at least \( d\sqrt{2} \). | d\sqrt{2} | 0.5 |
What square is equal to the product of four consecutive odd numbers? | 9 | 0.75 |
In Figure 1, \(ABCD\) is a square, \(M\) is the midpoint of \(AD\), and \(N\) is the midpoint of \(MD\). If \(\angle CBN : \angle MBA = P : 1\), find the value of \(P\). | 2 | 0.5 |
Given any $n$, let
$$
a_{n}=\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\ldots+\frac{1}{n(n+1)(n+2)}
$$
Is the sequence $a_{n}$ convergent, and if it is, what is its limit? | \frac{1}{4} | 0.875 |
A group of 9 boys and 9 girls is randomly paired up. Find the probability that at least one pair consists of two girls. Round your answer to two decimal places. | 0.99 | 0.875 |
How many 8-digit numbers begin with 1, end with 3, and have the property that each successive digit is either one more or two more than the previous digit, considering 0 to be one more than 9? | 21 | 0.25 |
On December 24, 1800, First Consul Bonaparte was heading to the Opera on Saint-Nicaise Street. A bomb exploded on his route with a delay of a few seconds. There were many killed and injured. Bonaparte blamed the republicans for the conspiracy; he deported 98 of them to the Seychelles and Guyana. Several people were executed.
Suppose the number of injured is equal to twice the number of killed (in the explosion) plus four thirds of the number of executed, and the sum of the number of killed or injured and the number of executed is slightly less than the number deported. Also, if you subtract 4 from the number of killed, you get exactly twice the number of executed.
Can you, without consulting history books, determine how many people Bonaparte executed after the assassination attempt on Saint-Nicaise Street? | 9 | 0.875 |
$$\frac{4 \sin ^{2}(\alpha - 5\pi) - \sin ^{2}(2\alpha + \pi)}{\cos ^{2}\left(2\alpha - \frac{3}{2}\pi\right) - 4 + 4 \sin ^{2} \alpha}.$$ | -\tan^4 \alpha | 0.75 |
Find the smallest natural number \( n \) for which \( (n+1)(n+2)(n+3)(n+4) \) is divisible by 1000. | 121 | 0.625 |
In a two-story house that is inhabited in both floors as well as on the ground floor, 35 people live above someone and 45 people live below someone. One third of all the people living in the house live on the first floor.
How many people live in the house in total? | 60 | 0.625 |
There are 2016 cards, each with a unique number from 1 to 2016. A certain number \( k \) of these cards are selected. What is the smallest \( k \) such that among these selected cards, there exist two cards with numbers \( a \) and \( b \) satisfying the condition \( |\sqrt[3]{a} - \sqrt[3]{b}| < 1 \)? | 13 | 0.625 |
Find the first few terms of the series expansion in powers of $z$ for the function $f(z)=\operatorname{tg} z$ and determine the radius of convergence of the series. | R = \frac{\pi}{2} | 0.625 |
On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \( k \) cards were taken. What is the smallest \( k \) such that among them there will be two cards with numbers whose square root difference is less than 1? | 45 | 0.875 |
An isosceles triangle has a side of length \(20 \text{ cm}\). Of the remaining two side-lengths, one is equal to two-fifths of the other. What is the length of the perimeter of this triangle? | 48 \text{ cm} | 0.875 |
When 15 is added to a number \( x \), it becomes a square number. When 74 is subtracted from \( x \), the result is again a square number. Find the number \( x \). | 2010 | 0.875 |
Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the marked 100 points and the vertices of the square, and for each triangle in the division, each marked point either lies outside this triangle or is a vertex of it (such divisions are called triangulations). Find the number of triangles in the division. | 202 | 0.625 |
Given positive integers \( a, b, \c \), define the function \( f(x, y, z) = a x + b y + c z \), where \( x, y, z \in \mathbb{Z} \). Find the minimum positive integer value of \( f(x, y, z) \). | \gcd(a, b, c) | 0.875 |
Find the equation of a line \( L \) such that the graph of the function
\[ y = x^4 + 4x^3 - 26x^2 \]
lies entirely on one side of this line, with two points of intersection. | y = 60x - 225 | 0.625 |
Given that \( 169(157 - 77x)^2 + 100(201 - 100x)^2 = 26(77x - 157)(1000x - 2010) \), find the value of \( x \). | 31 | 0.375 |
On a table, there is a certain quantity \( N \) of candies. Aline and Bruna agree that, alternately, each must eat at least one but no more than half of the existing quantity. The winner of the game is the one who eats the last candy. Aline always starts the game.
a) For \( N = 5 \), which of the two has the winning position? (A winning position is one in which the player can ensure a win regardless of the opponent's sequence of moves.)
b) For \( N = 20 \), which of the two has the winning position?
c) What are the values of \( N \), where \( 100 < N < 200 \), that give Bruna the winning position? | 191 | 0.125 |
Show that the value of the number $A$ is an integer:
$$
\begin{aligned}
A & = \frac{8795689 \cdot 8795688 \cdot 8795687 \cdot 8795686}{8795688^2 + 8795686^2 + 8795684^2 + 8795682^2} \\
& - \frac{8795684 \cdot 8795683 \cdot 8795682 \cdot 8795681}{8795688^2 + 8795686^2 + 8795684^2 + 8795682^2}.
\end{aligned}
$$ | 43978425 | 0.375 |
Find the Lagrange interpolation polynomial that takes the values $y_{0}=-5$, $y_{1}=-11$, $y_{2}=10$ at the points $x_{0}=-3$, $x_{1}=-1$, $x_{2}=2$, respectively. | 2x^2 + 5x - 8 | 0.25 |
A $7 \times 7$ table is filled with zeroes. In one operation, the minimum number in the table is found (if there are several such numbers, any one of them is chosen), and one is added to it as well as to all the numbers in the cells adjacent to it by side or corner. What is the largest number that can appear in one of the cells of the table after 90 operations?
Answer: 40. | 40 | 0.5 |
Bryce has 7 blue socks and 7 red socks mixed in a drawer. He plays a game with Sean. Blindfolded, Bryce takes two socks from the drawer. Sean looks at the socks, and if they have the same color, Sean gives Bryce 1 point. Bryce keeps drawing socks until the drawer is empty, at which time the game ends. The probability that Bryce's score is at most 2 can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers. Find \( p+q \). | 613 | 0.5 |
Through the point \((1,2)\), two tangent lines to the circle \(x^2 + y^2 = 1\) are drawn. What is the area of the quadrilateral formed by these two tangent lines and the \(x\)-axis and \(y\)-axis? | \frac{13}{8} | 0.5 |
There are 6 positive integers \( a, b, c, d, e, f \) arranged in order, forming a sequence, where \( a=1 \). If a positive integer is greater than 1, then the number that is one less than it must appear to its left. For example, if \( d > 1 \), then one of \( a, b, \) or \( c \) must have the value \( d-1 \). For instance, the sequence \( 1,1,2,1,3,2 \) satisfies the requirement; \( 1,2,3,1,4,1 \) satisfies the requirement; \( 1,2,2,4,3,2 \) does not satisfy the requirement. How many different permutations satisfy the requirement? | 203 | 0.375 |
A food factory has made 4 different exquisite cards. Each bag of food produced by the factory randomly contains one card. If all 4 different cards are collected, a prize can be won. Xiaoming buys 6 bags of this food at once. What is the probability that Xiaoming will win the prize? | \frac{195}{512} | 0.75 |
Find the largest real number \( m \) such that for all positive numbers \( a, b, \) and \( c \) satisfying \( a + b + c = 1 \),
$$
10\left(a^{3}+b^{3}+c^{3}\right)-m\left(a^{5}+b^{5}+c^{5}\right) \geqslant 1.
$$ | 9 | 0.75 |
In how many ways can an \( n \times n \) grid be filled with zeros and ones such that each row and each column contains an even number of ones? Each cell of the grid must contain either a zero or a one. | 2^{(n-1)^2} | 0.875 |
Find all three-digit numbers that are 11 times the sum of their digits. | 198 | 0.625 |
Given that \( 990 \times 991 \times 992 \times 993 = \overline{966428 A 91 B 40} \), find the values of \( \overline{A B} \). | 50 | 0.25 |
Find the largest real number \(\lambda\) such that for the real coefficient polynomial \(f(x) = x^3 + ax^2 + bx + c\) with all non-negative real roots, it holds that \(f(x) \geqslant \lambda(x - a)^3\) for all \(x \geqslant 0\). Additionally, determine when the equality in the expression is achieved. | -\frac{1}{27} | 0.875 |
Let \( n \) be a natural number less than 50. Find the sum of all possible values of \( n \) such that \( 4n + 5 \) and \( 7n + 6 \) have a common divisor greater than 1. | 94 | 0.75 |
Given the quadratic equations in \( x \),
\[ x^{2} + kx - 12 = 0 \]
and
\[ 3x^{2} - 8x - 3k = 0, \]
having a common root, find all possible values of the real number \( k \). | 1 | 0.625 |
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y = f(x) \) and \( y = g(x) \) are parallel lines, and not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^2 + 8 f(x) \), if the minimum value of the function \( (f(x))^2 + 8 g(x) \) is -29. | -3 | 0.375 |
In a group at a pioneer camp, there are children aged 10, 11, 12, and 13 years old. There are 23 people in total, and together they are 253 years old. How many 12-year-olds are in the group if it is known that there are one and a half times more 12-year-olds than 13-year-olds? | 3 | 0.125 |
In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, what is the measure of the dihedral angle $A-BD_{1}-A_{1}$? | 60^\circ | 0.75 |
A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes? | 74 \text{ m} | 0.375 |
Given a positive integer $n$, how many quadruples of ordered integers $(a, b, c, d)$ satisfy the condition
$$
0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n ?
$$ | \binom{n+4}{4} | 0.875 |
Calculate the area of the figure bounded by the graphs of the functions:
$$
x = 4 - y^2, \quad x = y^2 - 2y
$$ | 9 | 0.75 |
Is it possible to replace stars with pluses or minuses in the following expression
\[ 1 \star 2 \star 3 \star 4 \star 5 \star 6 \star 7 \star 8 \star 9 \star 10 = 0 \]
so that to obtain a true equality? | \text{No} | 0.75 |
What is the minimum number of points that can be chosen on a circle of length 1956 so that for each chosen point there is exactly one chosen point at a distance of 1 and exactly one chosen point at a distance of 2 (distances are measured along the circle)? | 1304 | 0.125 |
Given a regular triangular pyramid \( S-ABC \) with height \( SO=3 \) and base side length of 6. A perpendicular line is drawn from point \( A \) to the corresponding lateral face \( SBC \), with the foot of the perpendicular being \( D \). A point \( P \) is taken on \( AD \), such that \( \frac{AP}{PD} = 8 \). Find the area of the cross-section that passes through point \( P \) and is parallel to the base. | \sqrt{3} | 0.375 |
Given the parabola \( C: x^{2} = 2py \) with \( p > 0 \), two tangents \( RA \) and \( RB \) are drawn from the point \( R(1, -1) \) to the parabola \( C \). The points of tangency are \( A \) and \( B \). Find the minimum area of the triangle \( \triangle RAB \) as \( p \) varies. | 3 \sqrt{3} | 0.75 |
Points \( K \) and \( M \) are marked on the sides \( BC \) and \( AD \) of the inscribed quadrilateral \( ABCD \), respectively, such that \( BK : KC = AM : MD \). A point \( L \) is chosen on the segment \( KM \) such that \( KL : LM = BC : AD \). Find the ratio of the areas of triangles \( ACL \) and \( BDL \), given that \( AC = p \) and \( BD = q \). | \frac{p}{q} | 0.875 |
Given that \( n \) is a positive integer and \( S = 1 + 2 + 3 + \cdots + n \). Determine the units digits that \( S \) cannot have and find the sum of these forbidden digits. | 22 | 0.5 |
In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the basketball shooting competition. How many students participate in all three events? | 22 | 0.25 |
The diagonals \(AC\) and \(BD\) of a convex quadrilateral \(ABCD\), equal to 3 and 4 respectively, intersect at an angle of \(75^\circ\). What is the sum of the squares of the lengths of the line segments connecting the midpoints of opposite sides of the quadrilateral? | 12.5 | 0.5 |
The function \( f(x) \) defined on the set of real numbers \( \mathbf{R} \) satisfies \( f(x+1) = \frac{1+f(x+3)}{1-f(x+3)} \). Determine the value of \( f(1) \cdot f(2) \cdots f(2008) + 2009 \). | 2010 | 0.75 |
Let \(M\) and \(N\) be the midpoints of the sides \(CD\) and \(DE\) of a regular hexagon \(ABCDEF\). Find the angle between the lines \(AM\) and \(BN\). | 60^{\circ} | 0.875 |
The points $E$ on the side $BC$ and $F$ on the side $AD$ of a convex quadrilateral $ABCD$ are such that $BE=2EC$ and $AF=2FD$. There is a circle with radius $r$ centered on segment $AE$, which is tangent to the sides $AB$, $BC$, and $CD$. There is another circle with the same radius $r$ centered on segment $BF$, which is tangent to the sides $AB$, $AD$, and $CD$. Find the area of the quadrilateral $ABCD$, given that the mentioned circles touch each other externally. | 8r^2 | 0.125 |
What is the value of \(a + b + c + d\) if
$$
\begin{gathered}
6a + 2b = 3848 \\
6c + 3d = 4410 \\
a + 3b + 2d = 3080
\end{gathered}
$$ | 1986 | 0.625 |
Inflation over two years will be:
$$
\left((1+0,025)^{\wedge 2-1}\right)^{*} 100 \%=5,0625 \%
$$
The real interest rate of a bank deposit with reinvestment for the second year will be $(1.06 * 1.06 /(1+0,050625)-1) * 100=6,95 \%$ | 6.95\% | 0.625 |
The calculation result of the expression \(\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}\) is | 6500 | 0.125 |
Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$
Find all possible values of $a_{2021}$. | 2021 | 0.25 |
\( \binom{n}{0} \binom{n}{\left\lfloor \frac{n}{2} \right\rfloor} + 2 \binom{n}{1} \binom{n-1}{\left\lfloor \frac{n-1}{2} \right\rfloor} + 2^2 \binom{n}{2} \binom{n-2}{\left\lfloor \frac{n-2}{2} \right\rfloor} + \ldots + 2^{n-1} \binom{n}{n-1} \binom{1}{\left\lfloor \frac{1}{2} \right\rfloor} + 2^n = \binom{2n+1}{n} \)
Show that the above equation holds true. | \binom{2n+1}{n} | 0.25 |
Two people, A and B, agree to meet at a certain location within 10 days. They agree that the person who arrives first will wait for the other person, but only for 3 days before leaving. Assuming that their arrivals at the destination are equally likely to occur within the given period, what is the probability that the two will meet? | \frac{51}{100} | 0.25 |
The year 2009 has the following property: by rearranging the digits of the number 2009, it is not possible to obtain a smaller four-digit number (numbers do not start with zero). In which year will this property occur again for the first time? | 2022 | 0.375 |
In how many ways can an \( n \times n \) grid be filled with zeros and ones such that each row and each column contains an even number of ones? Each cell of the grid must contain either a zero or a one. | 2^{(n-1)^2} | 0.875 |
Let \(ABCD\) be a right trapezoid with bases \(AB\) and \(CD\), featuring right angles at \(A\) and \(D\). Given that the shorter diagonal \(BD\) is perpendicular to the side \(BC\), determine the minimum possible value for the ratio \(\frac{CD}{AD}\). | 2 | 0.875 |
In triangle \(ABC\), it is known that \(AB = 3\), \(AC = 3\sqrt{7}\), and \(\angle ABC = 60^\circ\). The bisector of angle \(ABC\) is extended to intersect at point \(D\) with the circle circumscribed around the triangle. Find \(BD\). | 4\sqrt{3} | 0.375 |
In the figure, the rays \( O A, O B, O C, O D, O E, O F \) are such that:
- \( О B \) is the bisector of \(\angle A O C\)
- \( O E \) is the bisector of \(\angle D O F \)
- \(\angle A O F = 146^\circ\)
- \(\angle C O D = 42^\circ\)
How many degrees is the angle \( B O E \)? | 94^{\circ} | 0.75 |
Given \(\alpha \in \left[0, \frac{\pi}{2}\right]\), \(\beta \in \left[0, \frac{\pi}{2}\right]\), find the minimum value of \(\cos^{2} \alpha \sin \beta + \frac{1}{\sin \beta}\). | 1 | 0.875 |
Find the sum of all three-digit numbers that are divisible by 7. | 70336 | 0.75 |
There are nine cards, each with the numbers $2, 3, 4, 5, 6, 7, 8, 9, 10$. Four people, A, B, C, and D, each draw two of these cards.
Person A says: "The two numbers I drew are relatively prime because they are consecutive."
Person B says: "The two numbers I drew are not relatively prime and are not multiples of each other."
Person C says: "The two numbers I drew are both composite numbers and they are relatively prime."
Person D says: "The two numbers I drew are in a multiple relationship and they are not relatively prime."
Assuming all four people are telling the truth, what is the number on the remaining card? $\quad \quad$ | 7 | 0.375 |
There are three water pipes, \(A\), \(B\), and \(C\), which can be used to fill a water truck. If pipes \(A\) and \(C\) are used simultaneously, the truck gets filled when pipe \(A\) has injected 4 tons of water. If pipes \(B\) and \(C\) are used simultaneously, the truck gets filled when pipe \(B\) has injected 6 tons of water. It is known that the water injection rate of pipe \(B\) is twice that of pipe \(A\). How many tons of water can the water truck hold at most? | 12 | 0.875 |
If the eight digits \(1, 2, 3, 4, 5, 6, 7, 8\) are randomly permuted, what is the probability that the resulting eight-digit number is divisible by 8? | \frac{1}{8} | 0.375 |
\(ABCD\) is a square-based pyramid with base \(ABCD\) and apex \(E\). Point \(E\) is directly above point \(A\), with \(AE = 1024\) units and \(AB = 640\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \(ABCD\), at a height \(h\) above the base. The portion of the pyramid above the plane forms a new smaller pyramid. For how many integer values of \(h\) does the volume of this new pyramid become an integer? | 85 | 0.375 |
Find the greatest common divisor of all nine-digit numbers in which each of the digits 1, 2, 3, ..., 9 appears exactly once. | 9 | 0.5 |
The diagonal \( AC \) of a convex quadrilateral \( ABCD \) serves as the diameter of the circumscribed circle around it. Find the ratio of the areas of triangles \( ABC \) and \( ACD \), given that the diagonal \( BD \) divides \( AC \) in the ratio 2:5 (counting from point \( A \)), and \( \angle BAC = 45^\circ \). | \frac{29}{20} | 0.5 |
Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1. | 2010 | 0.125 |
A unit square is called "colorful" if three of its four sides are colored in three different colors. Consider the $1 \times 3$ grid, which contains 10 unit length line segments. These 10 line segments are to be colored using red, yellow, or blue, such that all three unit squares are colorful. How many ways can this coloring be done? (Express the answer as a numerical value.) | 5184 | 0.125 |
Two right triangles \( \triangle AXY \) and \( \triangle BXY \) have a common hypotenuse \( XY \) and side lengths (in units) \( AX=5 \), \( AY=10 \), and \( BY=2 \). Sides \( AY \) and \( BX \) intersect at \( P \). Determine the area (in square units) of \( \triangle PXY \). | \frac{25}{3} | 0.375 |
In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number) | -21600 | 0.125 |
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012? | 2012 | 0.875 |
There are 15 players participating in a Go tournament. Each pair of players needs to play one match. A win earns 2 points, a draw earns 1 point for each player, and a loss earns 0 points. If a player scores at least 20 points, they will receive a prize. What is the maximum number of players that can receive a prize? | 9 | 0.625 |
Calculate the area of the regions bounded by the curves given in polar coordinates.
$$
r=3 \sin \phi, \quad r=5 \sin \phi
$$ | 4 \pi | 0.5 |
Let \( p(x) \) be the product of the digits of the decimal number \( x \). Find all positive numbers \( x \) for which \( p(x) = x^2 - 10x - 22 \). | 12 | 0.125 |
Mark on the line with red all points of the form $81x + 100y$, where $x, y$ are natural numbers, and paint the remaining integer points blue. Find a point on the line such that any symmetrical integer points relative to it are painted in different colors. | 4140.5 | 0.125 |
For the set $\{1,2,\cdots,n\}$ and each of its non-empty subsets, define a unique "alternating sum" as follows: Arrange the numbers in each subset in descending order, then start from the largest number and alternately subtract and add subsequent numbers to obtain the alternating sum (for example, the alternating sum of the set $\{1, 3, 8\}$ is $8 - 3 + 1 = 6$). For $n=8$, find the total sum of the alternating sums of all subsets. | 1024 | 0.25 |
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