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a) Each vertex of a cube has a number, either 1 or 0. On each face of the cube, the sum of the four numbers at its vertices is written. Can it be that all the numbers written on the faces are different?
b) The same question, but with the numbers 1 or -1 written at the vertices.
|
No
| 0.75 |
Eliane wants to choose her schedule for swimming. She wants to attend two classes per week, one in the morning and one in the afternoon, not on the same day nor on consecutive days. In the morning, there are swimming classes from Monday to Saturday at 9 AM, 10 AM, and 11 AM, and in the afternoon, from Monday to Friday at 5 PM and 6 PM. How many different ways can Eliane choose her schedule?
|
96
| 0.25 |
Simplify the following expression:
$$
\frac{\cos \left(2 x+\frac{\pi}{2}\right) \sin \left(\frac{3 \pi}{2}-3 x\right)-\cos (2 x-5 \pi) \cos \left(3 x+\frac{3 \pi}{2}\right)}{\sin \left(\frac{5 \pi}{2}-x\right) \cos 4 x+\sin x \cos \left(\frac{5 \pi}{2}+4 x\right)}
$$
|
\tan(5x)
| 0.5 |
A quality controller checks products for standard compliance. It is known that the probability of a product meeting the standard is 0.95. Find the probability that:
a) Both of the two checked products will meet the standard, assuming the events of the products meeting the standard are independent.
b) Out of the two checked products, only one will meet the standard.
|
0.095
| 0.875 |
Candies are in the shape of $1 \times 1 \times 1$ cubes. The teacher arranged them into a $3 \times 4 \times 5$ rectangular prism and asked the children to pick candies. In the first minute, Petya took one of the corner candies. Each subsequent minute, the children took all candies that had a neighboring face with already missing candies (for example, 3 candies were taken in the second minute). How many minutes did it take for the children to take all the candies?
|
10 \text{ minutes}
| 0.625 |
Find all pairs of integers \((x, y)\) that are solutions to the equation
$$
7xy - 13x + 15y - 37 = 0.
$$
Indicate the sum of all found values of \(x\).
|
4
| 0.625 |
As shown in the figure to the right, three circles intersect to form seven regions. Assign the integers $0 \sim 6$ to these seven regions such that the sum of the four numbers within each circle is equal. What is the maximum possible sum? $\qquad$
|
15
| 0.25 |
There are $n$ people, and it is known that any two of them communicate at most once. The number of communications among any $n-2$ of them is equal and is $3^k$ (where $k$ is a positive integer). Find all possible values of $n$.
|
5
| 0.375 |
Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight 2015 pounds?
|
13
| 0.25 |
Given \( f(x)=a \sin x+\sqrt[2019]{x}+1 \) where \( a \in \mathbf{R} \), and \( f\left(\lg \log _{2} 10\right)=3 \), find \( f(\lg \lg 2) \).
|
-1
| 0.875 |
In how many ways can 25 forints be paid using 1, 2, 5, 10, and 20 forint coins?
|
68
| 0.125 |
Anna and Berta are playing a game where they take turns removing marbles from the table. Anna makes the first move. If at the beginning of a turn there are \( n \geq 1 \) marbles on the table, then the player whose turn it is takes \( k \) marbles, where \( k \geq 1 \) is either an even number with \( k \leq \frac{n}{2} \) or an odd number with \( \frac{n}{2} \leq k \leq n \). A player wins the game if she takes the last marble from the table.
Determine the smallest number \( N \geq 100000 \) such that Berta can force a win if there are initially exactly \( N \) marbles on the table.
|
131070
| 0.25 |
Through point \( A \) located on a circle, a diameter \( AB \) and a chord \( AC \) are drawn, where \( AC = 8 \) and \( \angle BAC = 30^\circ \).
Find the length of the chord \( CM \), which is perpendicular to \( AB \).
|
8
| 0.875 |
Let \( n = 1990 \), then evaluate the following expression:
$$
\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{4}{ }^{n}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right)
$$
|
-\frac{1}{2}
| 0.75 |
Three sides \(OAB, OAC\) and \(OBC\) of a tetrahedron \(OABC\) are right-angled triangles, i.e. \(\angle AOB = \angle AOC = \angle BOC = 90^\circ\). Given that \(OA = 7\), \(OB = 2\), and \(OC = 6\), find the value of
\[
(\text{Area of }\triangle OAB)^2 + (\text{Area of }\triangle OAC)^2 + (\text{Area of }\triangle OBC)^2 + (\text{Area of }\triangle ABC)^2.
\]
|
1052
| 0.25 |
Calculate the area of the figure bounded by the curves given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=8(t-\sin t) \\
y=8(1-\cos t)
\end{array}\right. \\
& y=12(0<x<16 \pi, y \geq 12)
\end{aligned}
$$
|
48\sqrt{3}
| 0.875 |
Positive real numbers \( a, b, c \) are in a geometric progression \((q \neq 1)\), and \( \log _{a} b, \log _{b} c, \log _{c} a \) are in an arithmetic progression. Find the common difference \( d \).
|
-\frac{3}{2}
| 0.875 |
Point \( M \) is located on the lateral side \( AB \) of trapezoid \( ABCD \), such that \( AM : BM = 2 : 1 \). A line passing through point \( M \) parallel to the bases \( AD \) and \( BC \) intersects the lateral side \( CD \) at point \( N \). Find \( MN \), if \( AD = 18 \) and \( BC = 6 \).
|
10
| 0.75 |
Let \( n \) be an integer with \( n \geqslant 2 \). On a slope of a mountain, \( n^{2} \) checkpoints are marked, numbered from 1 to \( n^{2} \) from the bottom to the top. Each of two cable car companies, \( A \) and \( B \), operates \( k \) cable cars numbered from 1 to \( k \); each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any \( i \) and \( j \) with \( 1 \leqslant i < j \leqslant k \), the starting point of car \( j \) is higher than the starting point of car \( i \); similarly, the finishing point of car \( j \) is higher than the finishing point of car \( i \). Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed).
Determine the smallest \( k \) for which one can guarantee that there are two checkpoints that are linked by each of the two companies.
|
n^2 - n + 1
| 0.125 |
Among the natural numbers from 1 to 1000 (inclusive), how many are divisible by 2 or 3 or 5, but not by 6?
|
568
| 0.125 |
Let \( n \) be a positive integer. Given \(\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2n} x^{2n} \), find the value of \( a_{0}+a_{3}+a_{6}+a_{9}+\cdots \).
|
3^{n-1}
| 0.75 |
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a surjective function and \( g: \mathbb{N} \rightarrow \mathbb{N} \) be an injective function such that for all \( n \in \mathbb{N} \), \( f(n) \geq g(n) \). Show that \( f = g \).
|
f = g
| 0.875 |
A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments of lengths $6 \mathrm{~cm}$ and $7 \mathrm{~cm}$. Calculate the area of the triangle.
|
42 \text{ cm}^2
| 0.875 |
Can the difference of the squares of two prime numbers be equal to 4048?
|
No
| 0.625 |
A chord \( AB \) of fixed length slides its ends along a circle of radius \( R \). A point \( C \) on this chord, which is at distances \( a \) and \( b \) from the ends \( A \) and \( B \) of the chord respectively, traces another circle as the chord makes a full rotation. Compute the area of the annulus enclosed between the given circle and the circle traced by point \( C \).
|
\pi ab
| 0.875 |
Calculate the areas of figures bounded by lines given in polar coordinates.
$$
r = 1 + \sqrt{2} \sin \phi
$$
|
2\pi
| 0.625 |
If \( 2^{n+2} \cdot 3^n + 5n - a \) can be divided by 25, what is the smallest positive value of \( a \)?
|
4
| 0.25 |
Find the value of the expression \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\), given that \(a, b, c\) are three distinct real numbers satisfying the conditions:
\[ a^{3} - 2022a + 1011 = 0, \]
\[ b^{3} - 2022b + 1011 = 0, \]
\[ c^{3} - 2022c + 1011 = 0. \]
|
2
| 0.75 |
Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=a, a_{2}=b, 2a_{n+2}=a_{n+1}+a_{n}$. If $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)=4$, find the values of $a$ and $b$.
|
a = 6, b = -3
| 0.875 |
Given \( y z \neq 0 \) and the set \( \{2x, 3z, xy\} \) can also be represented as \( \{y, 2x^2, 3xz\} \), find \( x \).
|
x=1
| 0.625 |
In a regular truncated quadrilateral pyramid, the height is 2, and the sides of the bases are 3 and 5. Find the diagonal of the truncated pyramid.
|
6
| 0.375 |
Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3} + y^{2} = 1 \). A line \( l \) with positive slope passes through \( F \) and intersects the ellipse \( E \) at points \( A \) and \( B \). From points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn respectively, where \( AM \perp l \) and \( BN \perp l \), intersecting the x-axis at points \( M \) and \( N \). Find the minimum value of \( |MN| \).
|
\sqrt{6}
| 0.625 |
The number of moles of potassium hydroxide and nitric acid in the solutions:
$$
\begin{gathered}
v(\mathrm{KOH})=\omega \rho V / M=0,062 \cdot 1,055 \text { g/ml } \cdot 22.7 \text { ml / } 56 \text { g/ mol }=0.0265 \text { mol. } \\
v\left(\mathrm{HNO}_{3}\right)=\text { C }_{\mathrm{M}} \cdot \mathrm{V}=2.00 \text { mol/l } \cdot 0.0463 \text { l }=0.0926 \text { mol. }
\end{gathered}
$$
Since $v(\mathrm{KOH})<v(\mathrm{HNO} 3)$, and the coefficients for these substances in the reaction equation are equal to one, potassium hydroxide is in deficit, and its quantity will determine the thermal effect of the reaction, which will be:
$$
\text { Q = 55.6 kJ/mol } \cdot \text { 0.0265 mol = 1.47 kJ. }
$$
|
1.47 \, \text{kJ}
| 0.125 |
Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction, the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?
|
55 \text{ meters}
| 0.125 |
Let the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) be defined as follows:
\[ a_{1} = 3, \quad b_{1} = 1 \]
and for any \( n \in \mathbb{Z}_{+} \), we have
\[
\begin{cases}
a_{n+1} = a_{n} + b_{n} + \sqrt{a_{n}^{2} - a_{n} b_{n} + b_{n}^{2}}, \\
b_{n+1} = a_{n} + b_{n} - \sqrt{a_{n}^{2} - a_{n} b_{n} + b_{n}^{2}}.
\end{cases}
\]
(1) Find the general terms of the sequences \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\).
(2) Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). Define \(S_{n} = \sum_{i=1}^{n} \left[a_{i}\right]\) and \(T_{n} = \sum_{i=1}^{n} \left[b_{i}\right]\). Find the smallest \(n \in \mathbb{Z}_{+}\) such that
\[
\sum_{k=1}^{n} \left( S_{k} + T_{k} \right) > 2017.
\]
|
9
| 0.75 |
The numbers \( x \), \( y \), and \( z \) satisfy the equations
\[
xy + yz + zx = xyz, \quad x + y + z = 1
\]
What values can the sum \( x^3 + y^3 + z^3 \) take?
|
1
| 0.75 |
How many pairs of natural numbers \(a\) and \(b\) exist such that \(a \geq b\) and the equation
\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6} \]
is satisfied?
|
5
| 0.75 |
Find the number of pairs of positive integers $(x, y)$ which satisfy the equation $2x + 3y = 2007$.
|
334
| 0.625 |
Let \( a_{n} = \frac{1}{(n+1) \sqrt{n} + n \sqrt{n+1}}, \) for \( n=1,2,3, \ldots \). Find the value of \( a_{1} + a_{2} + \cdots + a_{99} \).
|
\frac{9}{10}
| 0.75 |
What is the minimum number of weights necessary to be able to measure any number of grams from 1 to 100 on a balance scale, given that weights can be placed on both sides of the scale?
|
5
| 0.625 |
On side \( BC \) of the rhombus \( ABCD \), a point \( M \) is chosen. Lines drawn through \( M \) perpendicular to the diagonals \( BD \) and \( AC \) intersect line \( AD \) at points \( P \) and \( Q \) respectively. It turned out that lines \( PB, QC \), and \( AM \) intersect at one point. What can the ratio \( BM : MC \) be?
|
1:2
| 0.375 |
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[3]{n}-n}
$$
|
5
| 0.75 |
Given the sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{n+1} \leq \frac{a_{n+2} + a_{n}}{2}\), with \(a_{1}=1\) and \(a_{404}=2016\), find the maximum value of \(a_{6}\).
|
26
| 0.875 |
On a line, there are blue and red points, with at least 5 red points. It is known that on any interval with endpoints at red points that contains a red point inside, there are at least 4 blue points. Additionally, on any interval with endpoints at blue points that contains 3 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on an interval with endpoints at red points, not containing other red points inside?
|
4
| 0.375 |
As shown in the figure, the abacus has three sections, each with 10 beads. Divide the beads in each section into an upper and lower part to form two three-digit numbers. The upper part must form a three-digit number with distinct digits, and it must be a multiple of the three-digit number formed by the lower part. What is the three-digit number in the upper part? $\qquad$
|
925
| 0.875 |
Suppose that \( n \) is a positive integer and that the set \( S \) contains exactly \( n \) distinct positive integers. If the mean of the elements of \( S \) is equal to \( \frac{2}{5} \) of the largest element of \( S \) and is also equal to \( \frac{7}{4} \) of the smallest element of \( S \), determine the minimum possible value of \( n \).
|
5
| 0.25 |
Let \( S = \{1, 2, 3, 4\} \) and let the sequence \( a_{1}, a_{2}, \cdots, a_{n} \) have the following property: For any non-empty subset \( B \) of \( S \) (with the number of elements in set \( B \) denoted as \( |B| \)), there exist \( |B| \) consecutive terms in the sequence that exactly form the set \( B \). Find the smallest possible value of \( n \).
|
8
| 0.125 |
Two circles of radius \( r \) touch each other. Additionally, each of them is externally tangent to a third circle of radius \( R \) at points \( A \) and \( B \) respectively.
Find the radius \( r \), given that \( AB = 12 \) and \( R = 8 \).
|
24
| 0.875 |
ABCD is a rectangle. AEB is isosceles with E on the opposite side of AB to C and D and lies on the circle through A, B, C, D. This circle has radius 1. For what values of |AD| do the rectangle and triangle have the same area?
|
\frac{2}{5}
| 0.5 |
As shown in Figure 2, in triangle \( \triangle ABC \), angle \( \angle B \) is an acute angle. Circles \( \odot O \) and \( \odot I \) are the circumcircle and excircle corresponding to angle \( \angle B \), respectively. Let \( R \) and \( r \) be the radii of \( \odot O \) and \( \odot I \), respectively. Then:
$$
OI^{2} > R^{2} - 2 R r
$$
|
OI^2 > R^2 - 2Rr
| 0.125 |
In the set of positive integers less than 10,000, how many integers \( x \) are there such that \( 2^x - x^2 \) is divisible by 7?
|
2857
| 0.125 |
The six-digit number begins with the digit 2. If this digit is moved from the first place to the last, keeping the order of the other five digits, the resulting number will be three times the original number. Find the original number.
|
285714
| 0.875 |
There is a five-digit odd positive integer \( x \). By changing all 2s in \( x \) to 5s and all 5s in \( x \) to 2s, while keeping all other digits unchanged, we obtain a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x+1) \), then \( x \) is ________.
|
29995
| 0.75 |
A perfect power is an integer \( n \) that can be represented as \( a^{k} \) for some positive integers \( a \geq 1 \) and \( k \geq 2 \). Find the sum of all prime numbers \(0 < p < 50\) such that \( p \) is 1 less than a perfect power.
|
41
| 0.625 |
How many triangles exist that have a perimeter of 15 units and sides measuring integer numbers?
|
7
| 0.375 |
There are two positive integers, \(A\) and \(B\). The sum of the digits of \(A\) is \(19\), the sum of the digits of \(B\) is \(20\), and their addition results in carrying over twice. What is the sum of the digits of \((\mathbf{A} + B)\)?
|
21
| 0.875 |
Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A \), \( B \), and \( C \) such that \( AB = BC \). From points \( A \), \( B \), and \( C \), perpendiculars \( AD \), \( BE \), and \( CF \) are drawn to line \( m \) with feet \( D \), \( E \), and \( F \) respectively. Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \).
|
\sqrt{6}
| 0.125 |
Anya, Vanya, Dania, Manya, Sanya, and Tanya were picking apples. It turned out that each of them picked a whole number of percentages of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?
|
25
| 0.25 |
There are two types of containers: 27 kg and 65 kg. How many containers of the first and second types were there in total, if the load in the containers of the first type exceeds the load of the container of the second type by 34 kg, and the number of 65 kg containers does not exceed 44 units?
|
66
| 0.25 |
In the quadrilateral \(ABCD\), \(AB = 1\), \(BC = 2\), \(CD = \sqrt{3}\), \(\angle ABC = 120^\circ\), and \(\angle BCD = 90^\circ\). What is the exact length of side \(AD\)?
|
\sqrt{7}
| 0.25 |
The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles.
|
90^\circ
| 0.5 |
The first \( n \) terms of the expansion of \((2 - 1)^{-n}\) are \(2^{-n} \left( 1 + \frac{n}{1!} \left(\frac{1}{2}\right) + \frac{n(n + 1)}{2!} \left(\frac{1}{2}\right)^2 + \ldots + \frac{n(n + 1) \cdots (2n - 2)}{(n - 1)!} \left(\frac{1}{2}\right)^{n-1} \right)\). Show that they sum to \(\frac{1}{2}\).
|
\frac{1}{2}
| 0.375 |
Given the point \( M_{1}(7, -8) \) and the normal vector of the line \( \vec{n} = (-2, 3) \), find the equation of the line.
|
2x - 3y - 38 = 0
| 0.875 |
During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if it is known that Vasya did not play with either Petya or Kolya?
|
30
| 0.5 |
Find the real roots of the following equations:
(1) $\sqrt[3]{x+1}+\sqrt[3]{2 x+3}+3 x+4=0$;
(2) $126 x^{3}+225 x^{2}+141 x+30=0$.
|
x = -\frac{1}{2}
| 0.75 |
Find all values of \( x \) for each of which one of the three given numbers \( \log _{x^{2}}\left(x^{2}-10 x+21\right) \), \( \log _{x^{2}} \frac{x^{2}}{x-7} \), and \( \log _{x^{2}} \frac{x^{2}}{x-3} \) is equal to the sum of the other two.
|
x = 8
| 0.75 |
For each positive integer \( n \), define the point \( P_{n} \) to have coordinates \(\left((n-1)^{2}, n(n-1)\right)\) and the point \( Q_{n} \) to have coordinates \(\left((n-1)^{2}, 0\right)\). For how many integers \( n \) with \( 2 \leq n \leq 99 \) is the area of trapezoid \( Q_{n} P_{n} P_{n+1} Q_{n+1} \) a perfect square?
|
6
| 0.75 |
In $\triangle ABC$, if $\overrightarrow{AB} \cdot \overrightarrow{AC} = 7$ and $\left|\overrightarrow{AB} - \overrightarrow{AC}\right| = 6$, then the maximum area of $\triangle ABC$ is
|
12
| 0.875 |
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (with masses of 1 kg, 2 kg, 3 kg, ..., up to 20 kg), balancing a watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on paper the mass of the weights he used. What is the minimum number of different numbers that could appear in his records, given that the mass of each weight is an integer in kilograms?
|
6
| 0.375 |
Given two regular triangular pyramids \( P-ABC \) and \( Q-ABC \) with the same base \( ABC \) such that both are inscribed in the same sphere. If the angle between the side face and the base of the regular triangular pyramid \( P-ABC \) is \( 45^{\circ} \), find the tangent of the angle between the side face and the base of the regular triangular pyramid \( Q-ABC \).
|
4
| 0.125 |
One evening, 21 people communicate by phone $n$ times. It is known that among them, there are $m$ ($m$ is an odd number) people $a_{1}, a_{2}, \cdots, a_{m}$ such that $a_{i}$ communicates with $a_{i+1}$ $\left(i=1,2, \cdots, m; a_{m+1}=a_{1}\right)$. If none of these 21 people had a three-way conversation, find the maximum value of $n$.
|
101
| 0.125 |
Let \( S \) be a set of \( n \) points in the plane such that the greatest distance between two points of \( S \) is 1. Show that at most \( n \) pairs of points of \( S \) are a distance 1 apart.
|
n
| 0.375 |
Paris is a city with 2,300,000 inhabitants. There are at most 500,000 hairs on the head of a Parisian. Show that there are at least 5 Parisians who have the same number of hairs.
|
5
| 0.875 |
Color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly 2 black squares. How many different ways are there to color the chessboard?
|
90
| 0.25 |
Ten points are given in the plane, and no three points are collinear. Four distinct segments connecting pairs of these points are chosen at random, all with the same probability. What is the probability that three of the chosen segments will form a triangle?
|
\frac{16}{473}
| 0.625 |
Through vertex $C$ of square $ABCD$, a line passes, intersecting diagonal $BD$ at point $K$ and the perpendicular bisector of side $AB$ at point $M$ ($M$ lies between $C$ and $K$). Find the angle $\angle DCK$ if $\angle AKB = \angle AMB$.
|
15^\circ
| 0.125 |
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 |
The area of triangle \( ABC \) is \( S \). Find the area of the triangle whose sides are equal to the medians of triangle \( ABC \).
|
\frac{3}{4} S
| 0.125 |
Point $B$ is the midpoint of segment $AC$. The square $ABDE$ and the equilateral triangle $BCF$ are located on the same side of the line $AC$. Find (in degrees) the measure of the acute angle between lines $CD$ and $AF$.
|
75^\circ
| 0.75 |
There are two docks, A and B, on a river. Dock A is upstream and dock B is downstream. Two people, person 1 and person 2, start rowing from A and B respectively at the same time, rowing towards each other, and meet after 4 hours. If person 1 and person 2 start rowing from A and B respectively at the same time, rowing in the same direction, person 2 catches up with person 1 after 16 hours. Given that person 1's rowing speed in still water is 6 km per hour, determine the speed of person 2's rowing in still water in km per hour.
|
10
| 0.875 |
Given \( k, m, n \in \mathbf{N}^{\cdot}, 1 \leqslant k \leqslant m \leqslant n \), find the value of \( \sum_{i=0}^{n}(-1)^{n+i} \frac{1}{n+k+i} \cdot \frac{(m+n+i)!}{i!(m-i)!(m+i)!} \).
|
0
| 0.5 |
A divisor of a natural number is called proper if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is a prime number, and the other is a perfect square, and their sum is also a perfect square. How many interesting numbers are there that do not exceed 1000?
|
70
| 0.375 |
A circle passing through the vertices $B, C,$ and $D$ of parallelogram $ABCD$ touches line $AD$ and intersects line $AB$ at points $B$ and $E$. Find the length of segment $AE$, if $AD = 4$ and $CE = 5$.
|
\frac{16}{5}
| 0.375 |
A team of athletes, one-third of which are snowboarders, descended from the mountain. Some of them took a cable car that holds no more than 10 people, while the rest descended on their own. The number of people who descended on their own was more than 45% but less than 50% of the total number. Determine the number of snowboarders (if the total number of snowboarders is ambiguous based on the problem's conditions, provide the sum of all possible values).
|
5
| 0.25 |
Find the sum of the first 10 elements that appear both in the arithmetic progression $\{4, 7, 10, 13, \ldots\}$ and the geometric progression $\{10, 20, 40, 80, \ldots\}$. (10 points)
|
3495250
| 0.125 |
Point \( M \) divides the side \( BC \) of parallelogram \( ABCD \) in the ratio \( BM: MC = 1: 3 \). Line \( AM \) intersects diagonal \( BD \) at point \( K \). Find the area of quadrilateral \( CMKD \) if the area of parallelogram \( ABCD \) is 1.
|
\frac{19}{40}
| 0.625 |
Given real numbers \( x, y, z \) satisfying
\[
\frac{y}{x-y}=\frac{x}{y+z}, \quad z^{2}=x(y+z)-y(x-y)
\]
find the value of
\[
\frac{y^{2}+z^{2}-x^{2}}{2 y z}.
\]
|
\frac{1}{2}
| 0.375 |
How many ordered pairs \((b, g)\) of positive integers with \(4 \leq b \leq g \leq 2007\) are there such that when \(b\) black balls and \(g\) gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is \(\frac{1}{2}\)?
|
59
| 0.875 |
Given the function \( f(x) = \frac{2+x}{1+x} \), let \( f(1) + f(2) + \cdots + f(1000) = m \) and \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \cdots + f\left(\frac{1}{1000}\right) = n \). What is the value of \( m + n \)?
|
2998.5
| 0.375 |
In triangle \( ABC \), \( AC = 2BC \), \(\angle C = 90^\circ\), and \( D \) is the foot of the altitude from \( C \) onto \( AB \). A circle with diameter \( AD \) intersects the segment \( AC \) at \( E \). Find the ratio \( AE: EC \).
|
4
| 0.875 |
Let \(a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of all positive integers that are relatively prime to 75, where \(a_{1}<a_{2}<a_{3}<\cdots\). (The first five terms of the sequence are: \(a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8\).) Find the value of \(a_{2008}\).
|
3764
| 0.375 |
Given the sets \( A = \{2, 4, a^3 - 2a^2 - a + 7\} \) and \( B = \{-4, a + 3, a^2 - 2a + 2, a^3 + a^2 + 3a + 7\} \) where \( a \in \mathbb{R} \), if \( A \cap B = \{2, 5\} \), find the real number \( a \).
|
2
| 0.75 |
The base of a pyramid is a square with side length \( a = \sqrt{21} \). The height of the pyramid passes through the midpoint of one of the edges of the base and is equal to \( \frac{a \sqrt{3}}{2} \). Find the radius of the sphere circumscribed around the pyramid.
|
3.5
| 0.125 |
A circular coin \(A\) is rolled, without sliding, along the circumference of another stationary circular coin \(B\) with radius twice the radius of coin \(A\). Let \(x\) be the number of degrees that the coin \(A\) makes around its center until it first returns to its initial position. Find the value of \(x\).
|
1080
| 0.75 |
In a plane, two vectors \(\overrightarrow{O A}\) and \(\overrightarrow{O B}\) satisfy \(|\overrightarrow{O A}|=a\) and \(|\overrightarrow{O B}|=b\), with \(a^2 + b^2 = 4\), and \(\overrightarrow{O A} \cdot \overrightarrow{O B} = 0\). If the vector \(\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}\) \((\lambda, \mu \in \mathbb{R})\) satisfies
\[
\left(\lambda-\frac{1}{2}\right)^{2} a^{2}+\left(\mu-\frac{1}{2}\right)^{2} b^{2}=1,
\]
then the maximum value of \(|\overrightarrow{O C}|\) is __________.
|
2
| 0.625 |
Plot the set of points on the plane \((x, y)\) that satisfy the equation \( |3x| + |4y| + |48 - 3x - 4y| = 48 \), and find the area of the resulting figure.
|
96
| 0.875 |
\[
\left(\cos 70^{\circ}+\cos 50^{\circ}\right)\left(\cos 310^{\circ}+\cos 290^{\circ}\right)+\left(\cos 40^{\circ}+\cos 160^{\circ}\right)\left(\cos 320^{\circ}-\cos 380^{\circ}\right)=1
\]
|
1
| 0.5 |
There are 21 nonzero numbers. For each pair of these numbers, their sum and product are calculated. It turns out that half of all the sums are positive and half are negative. What is the maximum possible number of positive products?
|
120
| 0.75 |
Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \), which satisfy \( x \leq y \leq z \), \( x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine the relationship between \( x + y \) and \( z \). Fill in the blank with ">", "<" or "=".
|
=
| 0.5 |
It is known that the number of birch trees in a certain mixed forest plot ranges from $13\%$ to $14\%$ of the total number of trees. Find the minimum possible total number of trees in this plot.
|
15
| 0.5 |
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