problem
stringlengths 18
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|---|---|---|
What is the ratio of the volumes of an equilateral cone, an equilateral cylinder, and a sphere, if their surface areas are equal?
|
2 : \sqrt{6} : 3
| 0.75 |
Find all values of the parameters \(a, b, c\) for which the system of equations
\[
\left\{
\begin{array}{l}
a x + b y = c \\
b x + c y = a \\
c x + a y = b
\end{array}
\right\}
\]
has at least one negative solution (where \(x, y < 0\)).
|
a + b + c = 0
| 0.875 |
There are 10 sticks with lengths of 1 cm, 2 cm, $2^2$ cm, ..., $2^9$ cm. Is it possible to form an isosceles triangle using some or all of these sticks?
|
\text{No}
| 0.625 |
Each face and each vertex of a regular tetrahedron is colored red or blue. How many different ways of coloring are there? (Two tetrahedrons are said to have the same coloring if we can rotate them suitably so that corresponding faces and vertices are of the same color.)
|
36
| 0.625 |
Let \( k \) be a circle with center \( O \). Let \( A, B, C \), and \( D \) be four different points on the circle \( k \) in this order, such that \( AB \) is a diameter of \( k \). The circumcircle of the triangle \( COD \) intersects \( AC \) for the second time in \( P \). Show that \( OP \) and \( BD \) are parallel.
|
OP \parallel BD
| 0.875 |
There are more than 5000 toothpicks, which can be divided into small packs of six different specifications. If packed 10 toothpicks per pack, 9 toothpicks will remain in the end. If packed 9 toothpicks per pack, 8 toothpicks will remain. For the third, fourth, fifth, and sixth specifications, if packed 8, 7, 6, and 5 toothpicks per pack, respectively, 7, 6, 5, and 4 toothpicks will remain. How many toothpicks were there originally?
|
5039
| 0.625 |
A point is located at a distance $a$ from the straight line $MN$. A circle with radius $r$ is described such that it passes through point $A$ and is tangent to the straight line $MN$. Find the distance between the point of tangency and the given point $A$.
|
\sqrt{2ar}
| 0.625 |
21. $A$ and $B$ are two different non-zero natural numbers less than 1000. Find the maximum value of $\frac{A-B}{A+B}$.
|
\frac{499}{500}
| 0.75 |
The circle \(C\) has radius 1 and touches the line \(L\) at point \(P\). The point \(X\) lies on the circle \(C\) and \(Y\) is the foot of the perpendicular from \(X\) to the line \(L\). Find the maximum possible value of the area of triangle \(PXY\) as \(X\) varies.
|
\frac{3\sqrt{3}}{8}
| 0.75 |
Given an isosceles triangle with a vertex angle of $20^\circ$, the length of the base is $a$, and the length of the legs is $b$. Find the value of $\frac{a^3 + b^3}{ab^2}$.
|
3
| 0.875 |
As shown in the figure, the squares $\mathrm{ABCD}$ and $\mathbf{EFGH}$ have their sides parallel to each other. Connect $\mathbf{CG}$ and extend it to intersect BD at point $I_{0}$. Given $BD = 10$, the area of $\triangle BFC = 3$, and the area of $\triangle CHD = 5$, find the length of $\mathbf{BI}$.
|
\frac{15}{4}
| 0.25 |
A standard deck of 52 cards has the usual 4 suits and 13 denominations. What is the probability that two cards selected at random, and without replacement, from this deck will have the same denomination or have the same suit?
|
\frac{5}{17}
| 0.75 |
There are 5 integers written on a board. By summing these numbers in pairs, the following set of 10 numbers is obtained: $-1, 2, 6, 7, 8, 11, 13, 14, 16, 20$. Determine which numbers are written on the board. Write their product as the answer.
|
-2970
| 0.5 |
The school plans to arrange 6 leaders to be on duty from May 1st to May 3rd. Each leader must be on duty for 1 day, with 2 leaders assigned each day. If leader A cannot be on duty on the 2nd, and leader B cannot be on duty on the 3rd, how many different methods are there to arrange the duty schedule?
|
42
| 0.5 |
Two individuals, A and B, are undergoing special training on the same elliptical track. They start from the same point simultaneously and run in opposite directions. Each person, upon completing the first lap and reaching the starting point, immediately turns around and accelerates for the second lap. During the first lap, B's speed is $\frac{2}{3}$ of A's speed. During the second lap, A's speed increases by $\frac{1}{3}$ compared to the first lap, and B's speed increases by $\frac{1}{5}$. Given that the second meeting point between A and B is 190 meters from the first meeting point, what is the length of the elliptical track in meters?
|
400
| 0.125 |
An arithmetic sequence consists of two-digit even terms, where the sum of all odd terms is 100. Starting from the first term, each odd term is combined with the following adjacent even term to form a four-digit number without changing the order. Determine the difference between the sum of the new sequence and the sum of the original sequence.
|
9900
| 0.625 |
We have drawn the circumcircle of a right triangle with legs of 3 and 4 units. What is the radius of the circle that is tangent to both legs of the triangle and the circumcircle from the inside?
|
2
| 0.5 |
A set consisting of a finite number of points in the plane has the following property: for any two points $A$ and $B$ in this set, there exists a point $C$ in this set such that the triangle $A B C$ is equilateral. How many points can such a set contain?
|
3
| 0.625 |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\sin (\pi-3 x)} \]
|
-\frac{\sqrt{3}}{3}
| 0.75 |
Let the sequence \( a_1, a_2, \cdots, a_n, \cdots \) be defined such that \( a_1 = a_2 = 1, a_3 = 2 \), and for any positive integer \( n \), \( a_n \cdot a_{n+1} \cdot a_{n+2} \neq 1 \). Also, given that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \), find the value of \( a_1 + a_2 + \cdots + a_{100} \).
|
200
| 0.25 |
Let $k$ be a real number. In the Cartesian coordinate plane $xOy$, there are two sets of points:
\[ A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \]
\[ B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \]
If the intersection of $A$ and $B$ is a single-element set, what is the value of $k$?
|
-2 - \sqrt{3}
| 0.625 |
When the gym teacher whistles, all 10 boys and 7 girls line up in a random order. Find the expected value of the number of girls standing to the left of all the boys.
|
\frac{7}{11}
| 0.75 |
Find the greatest integer less than or equal to \((2+\sqrt{3})^{3}\).
|
51
| 0.875 |
Given a function \( f(x) \) that satisfies for any real numbers \( x \) and \( y \):
\[ f(x+y) = f(x) + f(y) + 6xy \]
and \( f(-1)f(1) \geq 9 \),
find \( f\left(\frac{2}{3}\right) \).
|
\frac{4}{3}
| 0.875 |
Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class. The responses were "Twenty-five" five times, "Twenty-six" six times, and "Twenty-seven" seven times. Can we determine the actual number of students in their class based on their answers?
|
27
| 0.375 |
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle.
|
\frac{1}{9}
| 0.5 |
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 |
Suppose that \(a\) and \(b\) are positive integers with \(2^a \times 3^b = 324\). Evaluate \(2^b \times 3^a\).
|
144
| 0.75 |
For what values of \( k \) does the equation
\[ |x-2007| + |x+2007| = k \]
have \((-\infty, -2007) \cup (2007, +\infty)\) as its solution set?
|
k > 4014
| 0.25 |
A positive integer \( \overline{ABC} \), where \( A, B, C \) are digits, satisfies
\[
\overline{ABC} = B^{C} - A
\]
Find \( \overline{ABC} \).
|
127
| 0.875 |
Given \( n \) points in a plane, where the distance between any two points is \(\geq 1\), the number of pairs of points at a distance of 1 is at most \( 3n \). For \( n \) points in space, where the distance between any two points is \(\geq 1\), the number of pairs of points at a distance of 1 is at most \( 7n \).
|
7n
| 0.375 |
Let \( a, b, c, d \) such that \( a b c d = 1 \). Show that
\[ a^{2} + b^{2} + c^{2} + d^{2} + a b + a c + a d + b c + b d + c d \geq 10 \]
|
10
| 0.5 |
A, B, and C are guessing a two-digit number.
A says: Its number of factors is even, and it is greater than 50.
B says: It is an odd number, and it is greater than 60.
C says: It is an even number, and it is greater than 70.
If each of them only spoke half the truth, what is this number?
|
64
| 0.875 |
Points \(A, A_1, B, B_1, C,\) and \(C_1\) are located on a sphere of radius 11. Lines \(AA_1, BB_1,\) and \(CC_1\) are pairwise perpendicular and intersect at point \(M\), which is at a distance of \(\sqrt{59}\) from the center of the sphere. Find the length of \(AA_1\), given that \(BB_1 = 18\) and point \(M\) divides segment \(CC_1\) in the ratio \((8 + \sqrt{2}) : (8 - \sqrt{2})\).
|
20
| 0.125 |
A sequence of positive integers \(a_{n}\) begins with \(a_{1}=a\) and \(a_{2}=b\) for positive integers \(a\) and \(b\). Subsequent terms in the sequence satisfy the following two rules for all positive integers \(n\):
\[a_{2 n+1}=a_{2 n} a_{2 n-1}, \quad a_{2 n+2}=a_{2 n+1}+4 .\]
Exactly \(m\) of the numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\) are square numbers. What is the maximum possible value of \(m\)? Note that \(m\) depends on \(a\) and \(b\), so the maximum is over all possible choices of \(a\) and \(b\).
|
1012
| 0.125 |
Given the operation defined as \(a \odot b \odot c = a \times b \times c + (a \times b + b \times c + c \times a) - (a + b + c)\), calculate \(1 \odot 43 \odot 47\).
|
4041
| 0.5 |
Solve the inequality \(\left(\sqrt{x^{3}+2 x-58}+5\right)\left|x^{3}-7 x^{2}+13 x-3\right| \leqslant 0\).
|
x = 2 + \sqrt{3}
| 0.375 |
For a real number \( x \), let \( [x] \) be \( x \) rounded to the nearest integer and \( \langle x \rangle \) be \( x \) rounded to the nearest tenth. Real numbers \( a \) and \( b \) satisfy \( \langle a \rangle + [b] = 98.6 \) and \( [a] + \langle b \rangle = 99.3 \). Compute the minimum possible value of \( [10(a+b)] \).
(Here, any number equally between two integers or tenths of integers, respectively, is rounded up. For example, \( [-4.5] = -4 \) and \( \langle 4.35 \rangle = 4.4 \).)
|
988
| 0.5 |
Positive integer \(a\) and integers \(b\) and \(c\), in the three-dimensional coordinate system \(O-xyz\), point \(O(0,0,0)\), \(A(a, b, c)\), and \(B\left(x^{2}, x, 1\right)\) satisfy that the angle between \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) is \(\frac{\pi}{2}\). Given that the real number \(x\) has exactly two distinct solutions \( x_{1} \) and \( x_{2} \) in the interval \((0,1)\), find the minimum value of \(a\).
|
5
| 0.875 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{x \cdot \sin x}
\]
|
\frac{1}{4}
| 0.875 |
The diagram shows a square \(PQRS\) with sides of length 2. The point \(T\) is the midpoint of \(RS\), and \(U\) lies on \(QR\) so that \(\angle SPT = \angle TPU\). What is the length of \(UR\)?
|
\frac{1}{2}
| 0.625 |
This year is 2014. The number 2014 is not a perfect square, but its digits can be rearranged to form a new four-digit number that is a perfect square. For example, \(1024 = 32^2\). Given that using the digits \(2, 0, 1, 4\) exactly once can form another four-digit perfect square, find this new four-digit perfect square.
|
2401
| 0.25 |
An arithmetic sequence \(\{a_{n}\}\) with \(a_1 > 0\) has a sum of the first \(n\) terms denoted by \(S_n\). Given that \(S_9 > 0\) and \(S_{10} < 0\), for which value of \(n\) is \(S_n\) maximized?
|
n = 5
| 0.125 |
Find all natural numbers \(a\) and \(b\) such that for all natural numbers \(n\), the number \((a n + 1)^{6} + b\) is divisible by \(n^{2} + n + 1\).
|
a = 2, b = 27
| 0.375 |
In a warehouse, there are 8 cabinets, each containing 4 boxes, and each box contains 10 mobile phones. The warehouse, each cabinet, and each box are locked with a key. The manager is tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager must take with him?
|
9
| 0.375 |
In the country of Wonderland, an election campaign is being held for the title of the best tea lover, with the candidates being the Mad Hatter, March Hare, and Dormouse. According to a poll, $20\%$ of the residents plan to vote for the Mad Hatter, $25\%$ for the March Hare, and $30\%$ for the Dormouse. The remaining residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter needs to secure to guarantee not losing to both the March Hare and the Dormouse (regardless of how the undecided votes are allocated), given that each undecided voter will vote for one of the candidates. The winner is determined by a simple majority. Justify your answer.
|
70\%
| 0.125 |
A sequence of numbers is arranged in a line, and its pattern is as follows: the first two numbers are both 1. From the third number onward, each number is the sum of the previous two numbers: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55$. How many even numbers are there among the first 100 numbers in this sequence (including the 100th number)?
|
33
| 0.875 |
Find the residue of the function
$$
f(z)=e^{1 / z^{2}} \cos z
$$
at the point $z=0$.
|
0
| 0.75 |
Given positive real numbers \(a\) and \(b\) that satisfy \(ab(a+b) = 4\), find the minimum value of \(2a + b\).
|
2\sqrt{3}
| 0.5 |
Graph the set of points on the $(x; y)$ plane whose coordinates satisfy the system $\left\{\begin{array}{l}(|x|+x)^{2}+(|y|-y)^{2} \leq 16, \\ y-3 x \leq 0\end{array}\right.$, and find the area of the resulting figure.
|
\frac{20}{3} + \pi
| 0.25 |
Given a triangle \(T\),
a) Place the centrally symmetric polygon \(m\) with the largest possible area inside \(T\).
What is the area of \(m\) if the area of \(T\) is 1?
b) Enclose \(T\) in the centrally symmetric convex polygon \(M\) with the smallest possible area.
What is the area of \(M\) if the area of \(T\) is 1?
|
2
| 0.75 |
Let the function \( y = f(x) \) have the domain \( \mathbf{R} \). When \( x < 0 \), \( f(x) > 1 \). Moreover, for any real numbers \( x, y \in \mathbf{R} \), the equation \( f(x+y) = f(x) f(y) \) holds. A sequence \( \left\{a_n\right\} \) is defined such that \( a_1 = f(0) \) and \( f\left(a_{n+1}\right) = \frac{1}{f\left(-2 - a_n\right)} \) for \( n \in \mathbf{N} \).
1. Find the value of \( a_{2003} \).
2. If the inequality \( \left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right) \cdots\left(1+\frac{1}{a_n}\right) \geqslant k \cdot \sqrt{2n+1} \) holds for all \( n \in \mathbf{N} \), determine the maximum value of \( k \).
|
\frac{2\sqrt{3}}{3}
| 0.375 |
The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms?
|
400
| 0.375 |
Teacher Shi distributed cards with the numbers 1, 2, 3, and 4 written on them to four people: Jia, Yi, Bing, and Ding. Then the following conversation occurred:
Jia said to Yi: "The number on your card is 4."
Yi said to Bing: "The number on your card is 3."
Bing said to Ding: "The number on your card is 2."
Ding said to Jia: "The number on your card is 1."
Teacher Shi found that statements between people with cards of the same parity (odd or even) are true, and statements between people with cards of different parity are false. Additionally, the sum of the numbers on Jia's and Ding's cards is less than the sum of the numbers on Yi's and Bing's cards.
What is the four-digit number formed by the numbers on the cards of Jia, Yi, Bing, and Ding, in that order?
|
2341
| 0.5 |
Given one hundred numbers: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}\). Calculate 98 differences: \(a_{1}=1-\frac{1}{3}, a_{2}=\frac{1}{2}-\frac{1}{4}, \ldots, a_{98}=\frac{1}{98}-\frac{1}{100}\). What is the sum of all these differences?
|
\frac{14651}{9900}
| 0.25 |
Let \( ABC \) be any triangle. Let \( D \) and \( E \) be points on \( AB \) and \( BC \) respectively such that \( AD = 7DB \) and \( BE = 10EC \). Assume that \( AE \) and \( CD \) meet at a point \( F \). Determine \( \lfloor k \rfloor \), where \( k \) is the real number such that \( AF = k \times FE \).
|
77
| 0.875 |
In triangle \(ABC\), an incircle \(\omega\) with radius \(r\) is inscribed, touching side \(AB\) at point \(X\). A point \(Y\), diametrically opposite to \(X\), is marked on the circle. Line \(CY\) intersects side \(AB\) at point \(Z\). Find the area of triangle \(ABC\), given that \(CA + AZ = 1\).
|
r
| 0.125 |
Calculate the area of the figure bounded by the graphs of the functions:
$$
y = 4 - x^2, \quad y = x^2 - 2x
$$
|
9
| 0.875 |
Calculate the area of the set of points on the coordinate plane that satisfy the inequality \((y + \sqrt{x})(y - x^2) \sqrt{1 - x} \leq 0\).
|
1
| 0.375 |
Find \( f(1) + f(2) + f(3) + \ldots + f(13) \) if \( f(n) = 4n^3 - 6n^2 + 4n + 13 \).
|
28743
| 0.625 |
The secant passing through the intersection points of two circles, one with center \( O_1 \) and radius \( 4 \text{ cm} \), and the other with center \( O_2 \) and radius \( 6 \text{ cm} \), intersects the segment \( O_1O_2 \) at point \( T \). It is known that the length of \( O_1O_2 \) is not less than \( 6 \text{ cm} \). The larger circle intersects the segment \( O_1O_2 \) at point \( A \), while the smaller circle intersects it at point \( B \), with the ratio \( AT : BT = 1 : 2 \). Calculate the length of the segment \( O_1O_2 \).
|
6 \text{ cm}
| 0.125 |
Let an ellipse have center \(O\) and foci \(A\) and \(B\). For a point \(P\) on the ellipse, let \(d\) be the distance from \(O\) to the tangent at \(P\). Show that \(PA \cdot PB \cdot d^2\) is independent of the position of \(P\).
|
a^2 b^2
| 0.5 |
In the sequence \(\{a_n\}\), \(a_0 = 2\), and \(a_n = (n+2) a_{n-1}\) for \(n \geq 1\), find \(a_n\).
|
(n+2)!
| 0.875 |
Find the millionth digit after the decimal point in the decimal representation of the fraction \( \frac{3}{41} \).
|
7
| 0.875 |
What is the smallest natural number by which 720 must be multiplied to obtain the cube of a natural number?
|
300
| 0.875 |
If \(\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}\) where \(a, b\) are integers, find \(ab\).
|
-3
| 0.875 |
At the first site, higher-class equipment was used, and at the second site, first-class equipment was used. There was less higher-class equipment than first-class equipment. First, 40% of the equipment from the first site was transferred to the second site. Then, 20% of the equipment at the second site was transferred back to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site was 26 units more than at the second site, and the total amount of equipment at the second site increased by more than 5% compared to the original amount. Find the total amount of first-class equipment.
|
60
| 0.625 |
In triangle \( \triangle ABC \), \( AB + AC = 7 \), and the area of the triangle is 4. What is the minimum value of \( \sin A \)?
|
\frac{32}{49}
| 0.625 |
Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence.
|
1039
| 0.125 |
Given a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, a black and a white ant start crawling from point $A$ along the edges. Each time an ant traverses one edge, it is said to have completed one segment. The white ant's path is $A A_{1} \rightarrow A_{1} D_{1} \rightarrow \cdots$, while the black ant's path is $A B \rightarrow B B_{1} \rightarrow \cdots$. Both ants follow the rule that the $(n+2)$-th segment they crawl must be on a line that is skew to the line of the $n$-th segment. Assume that after each of the black and white ants has crawled 2008 segments, each stops at a vertex of the cube. Determine the distance between the black and white ants at this moment.
|
\sqrt{2}
| 0.25 |
Dan is holding one end of a 26-inch long piece of light string that has a heavy bead on it with each hand (so that the string lies along two straight lines). If he starts with his hands together and keeps his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?
|
24 \ \text{inches}
| 0.375 |
For what value of $\lambda$ does the equation
$$
\lambda x^{2}+4 x y+y^{2}-4 x-2 y-3=0
$$
represent a pair of lines?
|
4
| 0.875 |
Given that $\boldsymbol{a}$ and $\boldsymbol{b}$ are unit vectors and $|3 \boldsymbol{a} + 4 \boldsymbol{b}| = |4 \boldsymbol{a} - 3 \boldsymbol{b}|$, and that $|\boldsymbol{c}| = 2$, find the maximum value of $|\boldsymbol{a} + \boldsymbol{b} - \boldsymbol{c}|$.
|
\sqrt{2} + 2
| 0.125 |
Equilateral triangles \(ABC\) and \(A_1B_1C_1\) with side length 10 are inscribed in the same circle such that point \(A_1\) lies on arc \(BC\) and point \(B_1\) lies on arc \(AC\). Find \(AA_1^2 + BC_1^2 + CB_1^2\).
|
200
| 0.625 |
Let $F_{n}$ be the $n$-th term of the Fibonacci sequence (with the usual convention $F_{0}=0$ and $F_{1}=1$), and let $f_{n}$ be the number of ways to tile a $1 \times n$ rectangle with $1 \times 1$ squares and $1 \times 2$ dominos. Show that $F_{n}=f_{n-1}$.
|
F_n = f_{n-1}
| 0.625 |
In triangle \(ABC\), the angles \(\angle B = 30^\circ\) and \(\angle A = 90^\circ\) are known. Point \(K\) is marked on side \(AC\), and points \(L\) and \(M\) are marked on side \(BC\) such that \(KL = KM\) (point \(L\) lies on segment \(BM\)).
Find the length of segment \(LM\), given that \(AK = 4\), \(BL = 31\), and \(MC = 3\).
|
14
| 0.75 |
Given a sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\), where
\[ a_1 = 3, \quad S_n = 2a_n + \frac{3}{2}((-1)^n - 1). \]
If there exist three terms \(a_1, a_p, a_q\) (\(p, q \in \mathbb{Z}_+\), \(1 < p < q\)) that form an arithmetic sequence, find \( q - p \).
|
1
| 0.75 |
Every day from Monday to Friday, an old man went to the blue sea and cast his net. Each day, the number of fish caught in the net was not greater than the number caught the previous day. Over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three days - Monday, Wednesday, and Friday?
|
50
| 0.625 |
In a plane, there are 2020 points, some of which are black and the rest are green.
For every black point, there are exactly two green points that are at a distance of 2020 from this black point.
Determine the minimum possible number of green points.
|
45
| 0.125 |
Let \( a_{1}, a_{2}, a_{3}, \ldots \) be an arithmetic sequence with common difference 1 and \( a_{1} + a_{2} + a_{3} + \ldots + a_{100} = 2012 \). If \( P = a_{2} + a_{4} + a_{6} + \ldots + a_{100} \), find the value of \( P \).
|
1031
| 0.625 |
How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?
A sequence of three numbers \( a, b, c \) is said to form an arithmetic progression if \( a + c = 2b \).
A correct numerical answer without justification will earn 4 points. For full points, a detailed reasoning is expected.
|
45
| 0.375 |
Given that \( x \) and \( y \) are acute angles and \( x + y = 60^\circ \), find the maximum value of \( \sin x + \sin y \).
|
1
| 0.875 |
Sasha chose five numbers from 1, 2, 3, 4, 5, 6, and 7 and informed Anya of their product. Based on this information, Anya realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha inform Anya of?
|
420
| 0.75 |
Each letter in the table represents a different digit, and different letters represent different digits. The leading digit of each number cannot be zero. The three numbers in each row from left to right form an arithmetic sequence, and the three numbers in each column from top to bottom also form an arithmetic sequence. What is the five-digit number $\overline{\mathrm{CDEFG}}$?
\begin{tabular}{|c|c|c|}
\hline
$A$ & $BA$ & $AA$ \\
\hline
$AB$ & $CA$ & $EF$ \\
\hline
$CD$ & $GA$ & $BDC$ \\
\hline
\end{tabular}
|
40637
| 0.5 |
Each of 11 positive numbers is equal to the sum of the squares of the other 10 numbers. Determine the numbers.
|
\frac{1}{10}
| 0.875 |
Given that the hyperbola with asymptotes $y= \pm 2 x$ passes through the intersection point of the lines $x+y-3=0$ and $2 x-y+6=0$, find the length of the hyperbola's real axis.
|
4\sqrt{3}
| 0.625 |
Eighty students stand in a line facing the teacher, and count off from left to right: $1, 2, 3, \cdots, 80$. After the count, the teacher instructs the students whose numbers are multiples of 2 to turn around. Next, the teacher instructs the students whose numbers are multiples of 4 to turn around. Then the multiples of 8 turn around, followed by multiples of 16, 32, and finally 64. How many students are now facing away from the teacher?
|
26
| 0.75 |
An isosceles triangle has a square with a unit area inscribed in it, with one side of the square lying on the base of the triangle. Find the area of the triangle, given that the centers of gravity of the triangle and the square coincide.
|
\frac{9}{4}
| 0.5 |
A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.
|
4
| 0.25 |
A three-digit number is called a "cool number" if there is a digit in one of its places that is half the product of the other two digits. A three-digit number is called a "super cool number" if such digits appear in two or three of its places. How many different "super cool numbers" exist? (Zero is not allowed in the digits of "cool" and "super cool" numbers).
|
25
| 0.25 |
The inhabitants of a village of druids do not get along very well. It is even impossible to place 4 or more druids in a circle such that each druid is willing to shake hands with their two neighbors. Seeing that such a state of affairs tarnishes the reputation of the tribe, the chief tries to undertake a pacifying action to calm the situation. He collects 3 gold coins from each druid, then gives one gold coin to each person in any pair willing to shake hands. Show that the chief can keep at least 3 gold coins in his pocket at the end of the action.
|
3
| 0.75 |
$N$ friends simultaneously learned $N$ news items, with each friend learning one piece of news. They started calling each other and sharing the news.
Each call lasts 1 hour, and during one call, any number of news items can be shared.
What is the minimum number of hours required for everyone to know all the news items? Consider the following cases:
a) $N=64$,
b) $N=55$
c) $N=100$.
|
7 \text{ hours}
| 0.375 |
Solve the following equation where \( n \geq 2 \) is a given natural number and \( x \) is the unknown.
$$
\sum_{i=0}^{n-2} \frac{1}{(x+i)(x+i+1)}=x(x+1)(x+2) \cdot \ldots \cdot (x+n)+\frac{n-1}{x(x+n-1)}
$$
|
x = -n
| 0.25 |
The real numbers \( x, y, z, w \) satisfy
\[
\begin{array}{l}
2x + y + z + w = 1 \\
x + 3y + z + w = 2 \\
x + y + 4z + w = 3 \\
x + y + z + 5w = 25
\end{array}
\]
Find the value of \( w \).
|
\frac{11}{2}
| 0.125 |
795. Calculate the double integral \(\iint_{D} x y \, dx \, dy\), where region \(D\) is:
1) A rectangle bounded by the lines \(x=0, x=a\), \(y=0, y=b\);
2) An ellipse \(4x^2 + y^2 \leq 4\);
3) Bounded by the line \(y=x-4\) and the parabola \(y^2=2x\).
|
90
| 0.375 |
1. Given the equation \( x + y + z = 15 \), find the number of solutions in natural numbers.
2. How many non-negative integer solutions are there for the equation \( 2x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} = 3 \)?
|
174
| 0.75 |
Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$
If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$
then the smallest positive period of \( f(x) \) is ______.
|
\pi
| 0.375 |
Using 9 sheets of $2 \times 1$ rectangular paper to cover a $2 \times 9$ chessboard, there are $\qquad$ different methods.
|
55
| 0.875 |
How many ways can you mark 8 squares of an \(8 \times 8\) chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)
|
21600
| 0.5 |
Let \(ABCD\) be a rectangle with \(AB = 6\) and \(BC = 4\). Let \(E\) be the point on \(BC\) with \(BE = 3\), and let \(F\) be the point on segment \(AE\) such that \(F\) lies halfway between the segments \(AB\) and \(CD\). If \(G\) is the point of intersection of \(DF\) and \(BC\), find \(BG\).
|
1
| 0.875 |
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