task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
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math | ## Task 20/73
Given a circle with radius $r$ and a regular $n$-sided polygon inscribed in the circle. The task is to find the sum of the squares of the distances from any point $A$ on the circumference of the circle to the vertices of the $n$-sided polygon. | 2nr^{2} | 71 | 5 |
math | Exercise 2. Let $x, y$ and $z$ be three real numbers such that $0 \leqslant x \leqslant y \leqslant z$ and $x+y+z=1$. Find the maximum value that the expression
$$
(x-y z)^{2}+(y-z x)^{2}+(z-x y)^{2}
$$
can take. | 1 | 87 | 1 |
math | Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$. | (2, 3, 7) | 53 | 10 |
math | Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties:
i) $P(0) = 1$
ii) for any real numbers $x$ and $y,$
\[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] | P(x) = x^2 + 1 | 84 | 11 |
math | Example 13 Let $a, b$ be non-zero complex numbers, and $\frac{a}{b}$ is not a real number. Define:
$$
\begin{array}{l}
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\}, \\
R_{a, b}=\left\{z \mid z \text { is a non-zero complex number, and } L_{a, b}=L_{z a, z}\right\} .
\end{array}
$$
Try to find the maximum number of el... | 6 | 144 | 1 |
math | 1. (8 points) The calculation result of the expression $5 \times 13 \times(1+2+4+8+16)$ is | 2015 | 34 | 4 |
math | 14. Given $A\left(x_{1}, y_{1}\right) 、 B\left(x_{2}, y_{2}\right)$ are two moving points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=$ $1(a>b>0)$, $O$ is the origin, and $O A \perp$ $O B$. Find the minimum length of segment $A B$. | \frac{2 a b \sqrt{a^{2}+b^{2}}}{a^{2}+b^{2}} | 101 | 28 |
math | ## Task 1 - 210931
Four pairs of statements are made about a natural number $x$:
Pair A: (1) $x$ is a two-digit number.
(2) $x$ is less than 1000.
Pair B: (1) The second digit of the number $x$ is 0.
(2) The sum of the digits of the number $x$ is 11.
Pair C: (1) $x$ is written with exactly three digits, and all t... | 703,740 | 233 | 7 |
math | A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings. | 576 | 82 | 3 |
math | 4. The largest integer not exceeding $(\sqrt{5}+\sqrt{3})^{6}$ is
$\qquad$ | 3903 | 27 | 4 |
math | Example 2.10. $I=\int_{0}^{\pi / 2} \sin ^{3} x \sin 2 x d x$.
| \frac{2}{5} | 37 | 7 |
math | 3. The number of real roots of the equation $x^{2}|x|-5 x|x|+2 x=0$ is $\qquad$.
The equation $x^{2}|x|-5 x|x|+2 x=0$ has $\qquad$ real roots. | 4 | 60 | 1 |
math | The squares of the natural numbers from 1 to 99 were written one after another, forming the number 14916253649... What is the digit that occupies the 100th position? (The positions are counted from left to right, so the $1^{\underline{a}}$ position is the 1, the $2^{\underline{a}}$ is the 4, and so on.) | 9 | 95 | 1 |
math | 3. Find the smallest distance from the point with coordinates $(10 ; 5 ; 10)$ to a point whose coordinates are positive and satisfy the inequality $(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \geq 9 \sqrt{1-(2 x+y)^{2}}$. In your answer, write the square of the found distance. | 115.2 | 92 | 5 |
math | 8. Given that $A B C D$ is a square with side length 4, $E, F$ are the midpoints of $A B, A D$ respectively, $G C \perp$ plane $A B C D$, and $G C=2$. Find the distance from point $B$ to plane $E F G$. | \frac{2\sqrt{11}}{11} | 74 | 14 |
math | 10. A positive number, if its fractional part, integer part, and the number itself form a geometric sequence, then the number is | \frac{1+\sqrt{5}}{2} | 28 | 12 |
math | 7.275.
$$
\left\{\begin{array}{l}
(x+y) \cdot 3^{y-x}=\frac{5}{27} \\
3 \log _{5}(x+y)=x-y
\end{array}\right.
$$ | (4;1) | 59 | 5 |
math | he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit. | L = 1 | 106 | 5 |
math | Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers? | 88 | 72 | 2 |
math | Problem 4. Mira had a rectangular cardboard with sides $55 \mathrm{~cm}$ and $20 \mathrm{~cm}$. She divided it into squares with a perimeter of $20 \mathrm{~cm}$. How many squares did Mira get? | 44 | 59 | 2 |
math | II. (16 points) Find all natural numbers $n$ such that $2^{8}+2^{11}+2^{n}$ is a perfect square. | 12 | 37 | 2 |
math | Find all monic polynomials $P, Q$ in $\mathbb{Q}[X]$ such that:
$$
P(x)^{3}+Q(x)^{3}=x^{12}+1
$$ | P=1,Q=x^4 | 47 | 7 |
math | The equation with integer coefficients $x^{4}+a x^{3}+b x^{2}+c x+d=0$ has four positive roots, counting multiplicities.
Find the smallest possible value of the coefficient $b$ under these conditions. | 6 | 53 | 1 |
math | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$,
for all real $x$. | f(x) = \frac{1}{2} | 55 | 11 |
math | 3. (3 points) Solve the system of equations: $\left\{\begin{array}{l}x y=5(x+y) \\ x z=4(x+z) \\ y z=2(y+z)\end{array}\right.$ | -40,\frac{40}{9},\frac{40}{11} | 51 | 20 |
math | Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$
(a) Prove that $\hat B=\hat C=80^\circ$.
(b) Find the measure of the angle $D \hat{A} C$. | 5^\circ | 98 | 3 |
math | 2. Find the sum of all four-digit numbers that can be formed using the digits 1, 2, 3, and 4, where the digits do not repeat. | 66660 | 37 | 5 |
math | 19. Different positive 3-digit integers are formed from the five digits $1,2,3,5,7$, and repetitions of the digits are allowed. As an example, such positive 3-digit integers include 352, 577, 111, etc. Find the sum of all the distinct positive 3-digit integers formed in this way. | 49950 | 79 | 5 |
math |
4. Determine all integers $n \geq 2$ for which there exist integers $x_{1}, x_{2}, \ldots, x_{n-1}$ satisfying the condition that if $0<i<n, 0<j<n, i \neq j$ and $n$ divides $2 i+j$, then $x_{i}<x_{j}$.
Proposed by Merlijn Staps, NLD
The answer is that $n=2^{k}$ with $k \geq 1$ or $n=3 \cdot 2^{k}$ where $k \geq 0$.... | n=2^{k}withk\geq1orn=3\cdot2^{k}wherek\geq0 | 134 | 27 |
math | ## Task A-3.2.
Determine all pairs of real numbers $(x, y)$ that satisfy the equation
$$
\left(4^{x}+1\right)\left(9^{y}+1\right)+70=10\left(2^{x}+1\right)\left(3^{y}+1\right)
$$ | (1,1)(\log_{2}3,\log_{3}2) | 79 | 18 |
math | The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$. | 6 | 95 | 1 |
math | Find the closed form (as a function of $k$ and $n$) of the following series sum:
$$
\binom{k}{1} \sum_{i=1}^{n} i+\binom{k}{2} \sum_{i=1}^{n} i^{2}+\ldots+\binom{k}{k-1} \sum_{i=1}^{n} i^{k-1}
$$ | (n+1)^{k}-(n+1) | 93 | 12 |
math | 1. Tyler has an infinite geometric series with sum 10 . He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15. Compute the common ratio of Tyler's series.
Proposed by: Isabella Quan | \frac{1}{5} | 68 | 7 |
math | 1. Given $f(x)=\frac{10}{x+1}-\frac{\sqrt{x}}{3}$. Then the number of elements in the set $M=\left\{n \in \mathbf{Z} \mid f\left(n^{2}-1\right) \geqslant 0\right\}$ is $\qquad$. | 6 | 79 | 1 |
math | 40. Find the length of a segment of a line parallel to the bases of a trapezoid and passing through the point of intersection of the diagonals, if the bases of the trapezoid are equal to $a$ and $b$. | \frac{2}{+b} | 53 | 8 |
math | 6. 115 Let $r_{1}, r_{2}, \cdots, r_{m}$ be $m$ given positive rational numbers, such that $\sum_{k=1}^{m} r_{k}=1$. For every positive integer $n$, define the function $f$ as
$$f(n)=n-\sum_{k=1}^{m}\left[r_{k} n\right]$$
Find the minimum and maximum values of $f(n)$. | 0 \text{ and } m-1 | 104 | 9 |
math | 3.1. (12 points) An eraser, 3 pens, and 2 markers cost 240 rubles. Two erasers, 4 markers, and 5 pens cost 440 rubles. What is the total cost (in rubles) of 3 erasers, 4 pens, and 6 markers? | 520 | 74 | 3 |
math | In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. | 120 | 105 | 3 |
math | 5. A rectangle has perimeter 10 and diagonal $\sqrt{15}$. What is its area? | 5 | 23 | 1 |
math | Given the number sets $A=\left\{a+2,(a+1)^{2}, a^{2}+3 a+3\right\}, B=\{a+b, 1, a-$ $b+5\}$. If $A=B$, find the values of the real numbers $a, b$. | =0,b=2or=0,b=3 | 69 | 11 |
math | 1. Find all real solutions of the system of equations
$$
\frac{1}{x+y}+z=1, \quad \frac{1}{y+z}+x=1, \quad \frac{1}{z+x}+y=1
$$ | (-1,-1,\frac{3}{2}),(\frac{3}{2},-1,-1),(-1,\frac{3}{2},-1) | 58 | 35 |
math | Let's write down the numbers $x_{k}=[k \sqrt{2}](k=1,2, \ldots)$ in a row, and below them, the integers $0<y_{1}<y_{2}<\ldots$ that do not appear among the $x_{k}$ numbers. Determine the difference $y_{k}-x_{k}$ as a function of $k$. | 2r | 85 | 2 |
math | 2. Usually, Nikita leaves home at 8:00 AM, gets into Uncle Vanya's car, who drives him to school by a certain time. But on Friday, Nikita left home at 7:10 and ran in the opposite direction. Uncle Vanya waited for him and at $8: 10$ drove after him, caught up with Nikita, turned around, and delivered him to school 20 m... | 13 | 113 | 2 |
math | The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*}
Calculate $f(14,52)$. | 364 | 93 | 3 |
math | 3. Given $a, b, c \in \mathbf{R}_{+}$, and $abc=4$. Then the minimum value of the algebraic expression $a^{a+b} b^{3b} c^{c+b}$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 64 | 81 | 2 |
math | Example 4 Several 1s and 2s are arranged in a row
$$
1,2,1,2,2,1,2,2,2,1,2, \cdots
$$
The rule is: the 1st number is 1, the 2nd number is 2, the 3rd number is 1, ... Generally, first write a row of 1s, then insert $k$ 2s between the $k$th 1 and the $(k+1)$th 1 ($k=1$, $2, \cdots$). Try to answer:
(1) Is the 2005th num... | 2 | 151 | 1 |
math | 11. Given the line $y=x$ intersects the ellipse $C: \frac{x^{2}}{16}+\frac{y^{2}}{11}=1$ at points $A, B$, and a line $l$ passing through the right focus $F$ of the ellipse $C$ with an inclination angle of $\alpha$ intersects the chord $A B$ at point $P$, and intersects the ellipse $C$ at points $M, N$.
(1) Express the... | -\frac{1}{2}x+\frac{\sqrt{5}}{2} | 151 | 18 |
math | ## Task 4 - 260824
a) Determine all two-digit natural numbers $z$ that satisfy the following condition:
If you place a third digit in front of the two digits of $z$, a three-digit number is formed that is 29 times as large as $z$.
b) State how further natural numbers $z^{\prime}$ can be formed that satisfy the follo... | 25,5,75 | 207 | 7 |
math | 6. Let $f_{1}(x)=-\frac{2 x+7}{x+3}, f_{n+1}(x)=f_{1}\left(f_{n}(x)\right), x \neq-2, x \neq-3$, then $f_{2022}(2021)=$ | 2021 | 73 | 4 |
math | Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\t... | \frac{13}{29} | 116 | 9 |
math | Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive? (4 points)
... | \text{positive} | 54 | 5 |
math | Example 5 Color each vertex of a 2003-gon with one of three colors: red, blue, or green, such that adjacent vertices have different colors. How many such colorings are there? ${ }^{[3]}$
(2002-2003, Hungarian Mathematical Olympiad) | 2^{2003}-2 | 67 | 8 |
math | [Similar auxiliary triangles]
The base of the triangle is $a$, and the height dropped to the base is $h$. A square is inscribed in the triangle, one of its sides lying on the base of the triangle, and two vertices on the lateral sides. Find the ratio of the area of the square to the area of the triangle. | \frac{2ah}{(+)^2} | 70 | 10 |
math | ## 22. We Need Boys
The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Therefore, he decreed that there should not be more than one girl in any family. As a result, in this country, among the children of each woman, the last and only the last was a ... | \frac{1}{2} | 108 | 7 |
math | ## Task 20/61
For the Klingenberg power plant in Berlin-Rummelsburg, two new chimneys were built. Each of them consists of a concrete casing in the form of a hollow frustum of a cone with the following dimensions:
Lower inner diameter $d_{u}=10.00 \, \text{m}$, upper inner diameter $d_{o}=7.50 \, \text{m}$, lower out... | 6(n-1) | 1,157 | 5 |
math | $$
\begin{array}{l}
\text { Four, (20 points) Let } a_{1}=1, \\
a_{n+1}=2 a_{n}+n^{2}\left(1+3^{n}\right)(n=1,2, \cdots) \text {. }
\end{array}
$$
Find the general formula for the term $a_{n}$. | -23 \times 2^{n-1} + 3^{n}(n^2 - 6n + 15) - n^2 - 2n - 3 | 88 | 41 |
math | 69 If $11 z^{10}+10 i z^{9}+10 i z-11=0$, then $|z|=$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
However, the translation is already provided in the response, so there's no need for further action. The text is a mathemati... | 1 | 125 | 1 |
math | 2. (2004 National College Entrance Examination, Fujian Province) Given $f(x)=\frac{2 x-a}{x^{2}+2}(x \in \mathbf{R})$ is an increasing function on the interval $[-1,1]$.
(I) Find the set $A$ of real numbers $a$;
(II) Suppose the equation $f(x)=\frac{1}{x}$ has two roots $x_{1} 、 x_{2}$. Is there a real number $m$ such ... | \leqslant-2or\geqslant2 | 184 | 14 |
math | For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$. | 7 | 68 | 1 |
math | 8.4. Nikita and Egor are running on a circular road, having started from the same place in opposite directions. It is known that Nikita runs a lap 12 seconds faster than Egor, but still takes more than 30 seconds to do so. It turned out that for the seventh time after the start, they met at the same place where they be... | Nikita:36 | 91 | 6 |
math | (Fixed points and limits) Find all functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}$ such that for all $x, y>0, f(x f(y))=y f(x)$ and $\lim _{x \rightarrow+\infty} f(x)=0$.
(IMO 1983) | f(x)=\frac{1}{x} | 83 | 10 |
math | 5. Let the sequence of positive numbers $\left\{a_{n}\right\}$ satisfy
$$
\begin{array}{l}
a_{1}=\sqrt{2}-1, \\
a_{n+1}=\frac{2 n+1}{S_{n}+S_{n+1}+2}(n=1,2, \cdots),
\end{array}
$$
where $S_{n}$ is the sum of the first $n$ terms of $a_{n}$. Then the general term formula of the sequence is $\qquad$ | a_{n}=\sqrt{n^{2}+1}-\sqrt{(n-1)^{2}+1} | 123 | 26 |
math | Find all positive real numbers $\lambda$ such that every sequence $a_{1}, a_{2}, \ldots$ of positive real numbers satisfying
$$
a_{n+1}=\lambda \cdot \frac{a_{1}+a_{2}+\ldots+a_{n}}{n}
$$
for all $n \geq 2024^{2024}$ is bounded.
Remark: A sequence $a_{1}, a_{2}, \ldots$ of positive real numbers is bounded if there ex... | \lambda \leq 1 | 156 | 7 |
math | 271. Find the differentials of the functions:
1) $y=x^{3}-3^{x}$;
2) $F(\varphi)=\cos \frac{\varphi}{3}+\sin \frac{3}{\varphi}$
3) $z=\ln \left(1+e^{10 x}\right)+\operatorname{arcctg} e^{5 x} ;$ calculate $\left.d z\right|_{x=0 ; d x=0,1}$ | 0.25 | 109 | 4 |
math | Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. | 63 | 50 | 2 |
math | Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length $16,$ find the radius of the large circle. | 8 | 33 | 1 |
math | Task B-2.5. What is the last digit of the number $2012^{3}+3^{2012}$? | 9 | 32 | 1 |
math | ## SUBJECT II
a) Dividing the natural number a by the natural number b results in a quotient of 14 and a remainder of 18. If the difference between the numbers a and a-3b is equal to 135, show that the number 2a is a perfect square.
Gazeta matematica
b) How many three-digit numbers, when divided by a natural number ... | 2a=2^2\cdot3^4=(2^2)^2\cdot(3^2)^2= | 107 | 26 |
math | We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle? | 3\sqrt{3} - \pi | 35 | 9 |
math | Problem 11.4. In a sports school, 55 people are training, each of whom is either a tennis player or a chess player. It is known that there are no four chess players who would have the same number of friends among the tennis players. What is the maximum number of chess players that can train in this school? | 42 | 70 | 2 |
math | 9.5. Find all integer solutions of the equation: $x^{3}+y^{3}=2^{30}$. Answer. $\left(0,2^{10}\right),\left(2^{10}, 0\right)$. | (0,2^{10}),(2^{10},0) | 56 | 16 |
math | 7. [30] Five points are chosen on a sphere of radius 1 . What is the maximum possible volume of their convex hull? | \frac{\sqrt{3}}{2} | 29 | 10 |
math | Find the value of the expression $\sqrt{1+2011^{2}+\left(\frac{2011}{2012}\right)^{2}}+\frac{2011}{2012}$. | 2012 | 52 | 4 |
math | 13.041. A 30% hydrochloric acid solution was mixed with a 10% solution to obtain 600 g of a 15% solution. How many grams of each solution were taken? | 150 | 50 | 3 |
math | 1. Calculate: $25 \times 13 \times 2 + 15 \times 13 \times 7=$ | 2015 | 30 | 4 |
math | If $x, y$ are real numbers, and $1 \leqslant x^{2}+4 y^{2} \leqslant 2$. Find the range of $x^{2}-2 x y+4 y^{2}$. | \left[\frac{1}{2}, 3\right] | 55 | 14 |
math | 5. (5 points) Dividing 1722 by a two-digit number, Xiao Ming mistakenly reversed the tens and units digits of this number, resulting in the incorrect answer of 42. What should the correct result be? $\qquad$ . | 123 | 54 | 3 |
math | Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$. | 2\sqrt{2} - 1 | 61 | 11 |
math | Example 6 Let the sequence of positive numbers $a_{0}, a_{1}, a_{2}, \cdots$ satisfy $a_{0}=a_{1}=1$, and $\sqrt{a_{n} \cdot a_{n-2}}-\sqrt{a_{n-1} \cdot a_{n-2}}=2 a_{n-1}, n=2,3, \cdots$ Find the general term formula of this sequence. | a_{n}=\prod_{k=1}^{n}(2^{k}-1)^{2} | 98 | 23 |
math |
C4. Let $p_{1}, p_{2}, \ldots, p_{2005}$ be different prime numbers. Let $\mathrm{S}$ be a set of natural numbers which elements have the property that their simple divisors are some of the numbers $p_{1}, p_{2}, \ldots, p_{2005}$ and product of any two elements from $\mathrm{S}$ is not perfect square.
What is the ma... | 2^{2005} | 108 | 7 |
math | Solve the equation
$$
x^{3}-5 x^{2}-9 x+45=0
$$
if we know that two of the roots differ only in sign. | x_{1}=3,x_{2}=-3,x_{3}=5 | 39 | 16 |
math | 4. Let $[x]$ denote the greatest integer not exceeding $x$. Then the set $\{[x]+[2 x]+[3 x] \mid x \in R\} \mid\{1,2, \ldots, 100\}$ has
$\qquad$ elements. | 67 | 66 | 2 |
math | $\therefore$ Let $x, y, z \geqslant 0$, and $x+y+z=1$, find the maximum and minimum value of $\iota y+y z+z x-3 x y z$. | \frac{1}{4} | 48 | 7 |
math | 13th VMO 1975 Problem B1 Find all terms of the arithmetic progression -1, 18, 37, 56, ... whose only digit is 5. | 5..5with18k+5digitsfork=0,1,2,3, | 43 | 20 |
math | 4. There is an unlimited supply of square glasses in 10 colors. In how many ways can 4 glasses be inserted into a $2 \times 2$ window frame so that some color appears in both the upper and lower halves of the window. | 3430 | 53 | 4 |
math | Given a parallelogram $ABCD$, where $AB=5$, $AD=2\sqrt{3}+2$, and $\angle BAD=30^{\circ}$. On side $AB$, a point $K$ is taken such that $AK:KB=4:1$. A line parallel to $AD$ is drawn through point $K$. On this line, inside the parallelogram, a point $L$ is chosen, and on side $AD$, a point $M$ is chosen such that $AM=KL... | 75 | 137 | 2 |
math | 7.130. $\left\{\begin{array}{l}\lg \left(x^{2}+y^{2}\right)=2-\lg 5, \\ \lg (x+y)+\lg (x-y)=\lg 1.2+1 .\end{array}\right.$ | (4;2)(4;-2) | 65 | 9 |
math | 14. (10 points) Team A and Team B need to transport a batch of relief supplies to an earthquake-stricken area. Team A can transport 64.4 tons per day, which is 75% more than Team B; if Team A and Team B transport simultaneously, when Team A has transported half of all the relief supplies, it has transported 138 tons mo... | 644 | 97 | 3 |
math | 32. If mom gives each of her children 13 notebooks, she will have 8 notebooks left; if she gives them 15 notebooks each, all the notebooks will be distributed. How many notebooks did mom have? | 60 | 47 | 2 |
math | 1. Given the equation $\left|x^{2}-2 a x+b\right|=8$ has exactly three real roots, and they are the side lengths of a right triangle. Find the value of $a+b$.
(Bulgaria) | 264 | 50 | 3 |
math | 11. If $a, b \in \mathbf{R}$ and $a^{2}+b^{2}=10$, then the range of values for $a-b$ is $\qquad$ | [-2 \sqrt{5}, 2 \sqrt{5}] | 45 | 14 |
math | Problem 5. Consider four consecutive numbers $n, n+1, n+2, n+3$. For which $n$ is the LCM of the first three numbers greater than the LCM of the last three? | Any\odd\\not\\than\5 | 47 | 9 |
math | Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit. | p \equiv 1 \pmod{4} | 84 | 12 |
math | Task B-1.6. Solve the equation
$$
\frac{x+3}{12(x+1)}:\left(\frac{2 x-3}{3 x-3}-\frac{3 x-1}{4 x+4}+\frac{x^{2}-7 x+14}{12 x^{2}-12}\right)=2015
$$ | 2012 | 82 | 4 |
math | ## 22. Math Puzzle $3 / 67$
Peter wants to use a balance scale, whose beam lengths $a$ and $b$ are no longer exactly equal. He first places a $5 \mathrm{~kg}$-"weight" on the left pan and weighs, and then the $5 \mathrm{~kg}$ piece on the right pan and weighs the rest of the apples.
Is the weighed amount heavier or l... | G>10 | 102 | 4 |
math | Let $n\geq 2$ be a positive integer. Find the minimum value of positive integer $m$ for which there exist positive integers $a_1,\ a_2,\ \cdots, a_n$ such that :
$\bullet\ a_1<a_2<\cdots <a_n=m$
$\bullet \ \frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2}$ are all square numbers. | 2n^2 - 1 | 131 | 7 |
math | 9. (24th American Mathematics Competition) There are 1990 piles of stones, with the number of stones in each pile being $1, 2, \cdots, 1990$. Perform the following operation: each time, you can choose any number of piles and remove the same number of stones from each chosen pile. How many operations are required at min... | 11 | 86 | 2 |
math | Example 3.14 Solve the recurrence relation
$$
\left\{\begin{array}{l}
a_{n}=3 a_{n-1}-3 a_{n-2}+a_{n-3}+24 n-6 \quad(n \geqslant 3), \\
a_{0}=-4, \quad a_{1}=-2, \quad a_{2}=2 .
\end{array}\right.
$$ | a_{n}=-4+17n-21n^{2}+5n^{3}+n^{4}\quad(n\geqslant0) | 98 | 37 |
math | 373. Find the condition under which the numbers $a, b, c$ represent the $k$-th, $n$-th, and $p$-th terms of the same geometric progression. | (\frac{}{b})^{k-p}=(\frac{}{})^{k-n} | 45 | 19 |
math | The integer 203 has an odd ones digit, an even tens digit, and an even hundreds digit. How many integers between 100 and 999 have an odd ones digit, an even tens digit, and an even hundreds digit? | 100 | 53 | 3 |
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