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 |
|---|---|---|---|---|
math | 13.198. A certain substance absorbs moisture, thereby increasing its mass. To absorb 1400 kg of moisture, 300 kg more of the uncrushed substance is required than of the crushed substance. What percentage of the mass of the substance is the mass of the absorbed moisture in the case of crushed substance and in the case o... | 280175 | 98 | 6 |
math | 6. The smallest natural number $n$ that satisfies $n \sin 1 > 1 + 5 \cos 1$ is $\qquad$ . | 5 | 34 | 1 |
math | One. (This question is worth 40 points) Let real numbers $a_{1}, a_{2}, \cdots, a_{2016}$ satisfy $9 a_{i}>11 a_{i+1}^{2}(i=1,2, \cdots, 2015)$. Find the maximum value of $\left(a_{1}-a_{2}^{2}\right) \cdot\left(a_{2}-a_{3}^{2}\right) \cdots \cdot\left(a_{2015}-a_{2016}^{2}\right) \cdot\left(a_{2016}-a_{1}^{2}\right)$. | \frac{1}{4^{2016}} | 154 | 12 |
math | Example 4 (2002 National Girls' Mathematical Olympiad) Find all positive integers $k$ such that for any positive numbers $a, b, c$ satisfying the inequality $k(ab + bc + ca) > 5(a^2 + b^2 + c^2)$, there must exist a triangle with side lengths $a, b, c$.
Find all positive integers $k$ such that for any positive numbers... | 6 | 140 | 1 |
math | II. (40 points) Let $p$ be a prime number, and the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0, a_{1}=1$, and for any non-negative integer $n$, $a_{n+2}=2 a_{n+1}-p a_{n}$. If -1 is a term in the sequence $\left\{a_{n}\right\}$, find all possible values of $p$.
保留了原文的换行和格式。 | 5 | 121 | 1 |
math | 3. The equation of the line passing through the intersection points of the parabolas
$$
y=2 x^{2}-2 x-1 \text { and } y=-5 x^{2}+2 x+3
$$
is $\qquad$ . | 6 x+7 y-1=0 | 57 | 9 |
math | Example 4 Let $\lambda>0$, find the largest constant $c=c(\lambda)$, such that for all non-negative real numbers $x, y$, we have
$$
x^{2}+y^{2}+\lambda x y \geqslant c(x+y)^{2} .
$$ | c(\lambda)=\left\{\begin{array}{ll}
1, & \lambda \geqslant 2, \\
\frac{2+\lambda}{4}, & 0<\lambda<2 .
\end{array}\right.} | 65 | 54 |
math | 9.065. $\frac{15}{4+3 x-x^{2}}>1$.
9.065. $\frac{15}{4+3 x-x^{2}}>1$. | x\in(-1;4) | 45 | 8 |
math | Example 2 Find
$$
I=\sin \frac{\pi}{n} \cdot \sin \frac{2 \pi}{n} \cdots \cdots \sin \frac{(n-1) \pi}{n}
$$
the value of ( $n$ is a natural number greater than 1). | \frac{n}{2^{n-1}} | 68 | 10 |
math | 7.1. In a hotel, rooms are arranged in a row in order from 1 to 10000. Masha and Alina checked into the hotel in two different rooms. The sum of the room numbers they are staying in is 2022, and the sum of the room numbers of all the rooms between them is 3033. In which room is Masha staying, if her room has a lower nu... | 1009 | 99 | 4 |
math | 3. [5] Let $D E F$ be a triangle and $H$ the foot of the altitude from $D$ to $E F$. If $D E=60, D F=35$, and $D H=21$, what is the difference between the minimum and the maximum possible values for the area of $D E F$ ? | 588 | 76 | 3 |
math | Example. Compute the limit
$$
\lim _{n \rightarrow \infty} \frac{(2 n+1)^{2}-(n+1)^{2}}{n^{2}+n+1}
$$ | 3 | 49 | 1 |
math | Task 1 - 331211 Determine all natural numbers $n$ for which the following conditions are satisfied:
The number $n$ is ten-digit. For the digits of its decimal representation, denoted from left to right by $a_{0}, a_{1}$, $\ldots, a_{9}$, it holds that: $a_{0}$ matches the number of zeros, $a_{1}$ matches the number of... | 6210001000 | 115 | 10 |
math | 10. For any $x \in \mathbf{R}$, the function $f(x)$ satisfies the relation $f(x+2008)=f(x+2007)+f(x+2009)$, and $f(1)=\lg \frac{3}{2}, f(2)=\lg 15$, then the value of $f(2007)$ is $\qquad$ . | 1 | 94 | 1 |
math | 3. In space, there are $n(n \geqslant 3)$ planes, among which any three planes do not have a common perpendicular plane. There are the following four conclusions:
(1) No two planes are parallel to each other;
(2) No three planes intersect in a single line;
(3) Any two lines of intersection between planes are not parall... | 4 | 110 | 1 |
math | 10. A. Given positive real numbers $a$, $b$, $c$ satisfy $9a + 4b = abc$. Then the minimum value of $a + b + c$ is $\qquad$. | 10 | 46 | 2 |
math | ## Task B-3.3.
Determine all real numbers $x$ for which $3^{\frac{1}{\log x}}-2 \cdot 3^{\frac{\log 10 x^{2}}{\log x^{2}}}=27$. | \sqrt[4]{10} | 58 | 8 |
math | ## 1. a) Show that the number $\log _{2015} 2016$ is irrational;
## b) Compare the numbers $\log _{5} 6$ and $\log _{6} 7$;
c) Calculate $E=\lg ^{3} 5+\lg ^{3} 20+\lg 8 \cdot \lg (0.25)$. | \log_{6} | 91 | 5 |
math | 3. Given a triangle $A B C$, where $\angle B A C=60^{\circ}$. Point $S$ is the midpoint of the angle bisector $A D$. It is known that $\angle S B A=30^{\circ}$. Find DC/BS. | 2 | 62 | 1 |
math | 8-8. The numbers $a, b$, and $c$ (not necessarily integers) are such that
$$
a+b+c=0 \quad \text { and } \quad \frac{a}{b}+\frac{b}{c}+\frac{c}{a}=100
$$
What is $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ ? | -103 | 91 | 4 |
math | Z1) Find all integer values that the expression
$$
\frac{p q+p^{p}+q^{q}}{p+q}
$$
where $p$ and $q$ are prime numbers.
Answer: The only integer value is 3 . | 3 | 56 | 1 |
math | Let $a_1, a_2, a_3, \ldots$ be an infinite sequence of positive integers such that $a_1=4$, $a_2=12$, and for all positive integers $n$, \[a_{n+2}=\gcd\left(a_{n+1}^2-4,a_n^2+3a_n \right).\] Find, with proof, a formula for $a_n$ in terms of $n$. | a_n = 4 \cdot (2^n - 1) | 102 | 15 |
math | 2. Solve the equation $\left(\frac{x}{400}\right)^{\log _{5}\left(\frac{x}{8}\right)}=\frac{1024}{x^{3}}$. | \frac{8}{5},16 | 45 | 9 |
math | 6.063. $\frac{\sqrt{x}+\sqrt[3]{x}}{\sqrt{x}-\sqrt[3]{x}}=3$. | 64 | 33 | 2 |
math | 11. B. Given the parabola $y=x^{2}$ and the moving line $y=(2 t-1) x-c$ have common points $\left(x_{1}, y_{1}\right) 、\left(x_{2}, y_{2}\right)$, and $x_{1}^{2}+x_{2}^{2}=t^{2}+2 t-3$.
(1) Find the range of real number $t$;
(2) For what value of $t$ does $c$ attain its minimum value, and find the minimum value of $c$. | \frac{11-6 \sqrt{2}}{4} | 130 | 15 |
math | Example 3 Given $a^{2}+a+1=0$. Then, $a^{1992}+$ $a^{322}+a^{2}=$ $\qquad$
(Harbin 15th Junior High School Mathematics Competition) | 0 | 57 | 1 |
math | Example 10 Let $p(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants, and $p(1)=1993$, $p(2)=3986$, $p(3)=5979$. Try to calculate $\frac{1}{4}[p(11)+p(-7)]$. (1993 Macau Olympiad Question) | 5233 | 105 | 4 |
math | For 8 $\star \star$ positive integers $r, n$ satisfying $1 \leqslant r \leqslant n$, find the arithmetic mean $f(r, n)$ of the smallest numbers in all $r$-element subsets of $\{1,2, \cdots, n\}$. | \frac{n+1}{r+1} | 68 | 10 |
math | 4. [4] Let $a, b$ be constants such that $\lim _{x \rightarrow 1} \frac{(\ln (2-x))^{2}}{x^{2}+a x+b}=1$. Determine the pair $(a, b)$. | (-2,1) | 58 | 5 |
math | 7.20 A single-player card game has the following rules: Place 6 pairs of different cards into a backpack. The player draws cards randomly from the backpack and returns them, but when a pair is drawn, it is set aside. If the player always draws three cards at a time, and if the three cards drawn are all different (i.e.,... | 394 | 128 | 3 |
math | 1. Simplify the expression
$$
1-a+a^{2}-a^{3}+\ldots+a^{1980}-\frac{a^{1981}}{1+a}
$$
and find its value for $a=-\frac{4}{5}$. | 5 | 61 | 1 |
math | 2. Find all triples of real numbers $(x, y, z)$ satisfying the system of equations
$$
\begin{aligned}
x^{3}+y^{3} & =9 z^{3} \\
x^{2} y+y^{2} x & =6 z^{3} .
\end{aligned}
$$ | (,2t,)(2t,,)forevery\neq0,(,-,0)forevery | 69 | 23 |
math | 2. The sequence $\left(a_{n}\right)_{n=1}^{\infty}$ is defined by $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}-a_{n}^{2}$. Is $a_{999}<\frac{1}{1000}$? | a_{999}<\frac{1}{1000} | 74 | 16 |
math | We know that the following equation has two pairs of roots whose sum is 0. Solve the equation.
$$
x^{6}-2 x^{5}-9 x^{4}+14 x^{3}+24 x^{2}-20 x-20=0
$$ | \sqrt{2},-\sqrt{2},\sqrt{5},-\sqrt{5},1+\sqrt{3},1-\sqrt{3} | 61 | 32 |
math | For every positive integer $n$, determine the greatest possible value of the quotient
$$\frac{1-x^{n}-(1-x)^{n}}{x(1-x)^n+(1-x)x^n}$$
where $0 < x < 1$. | 2^n - 2 | 56 | 7 |
math | If $\log _{3 n} 675 \sqrt{3}=\log _{n} 75$, determine the value of $n^{5}$. | 5625 | 37 | 4 |
math | 1. In the field of real numbers, solve the equation
$$
\sqrt{2}(\sin t+\cos t)=\tan^{3} t+\cot^{3} t .
$$ | \frac{1}{4}\pi+2k\pi | 41 | 13 |
math | 3. (50 points) Find all positive integer solutions $(x, y, n)$ to the equation $1+2^{x}+2^{2 x+1}=y^{n}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (x,y,n)=(4,23,2) | 67 | 11 |
math |
1. Find all real solutions of the system
$$
\begin{aligned}
& \sqrt{x^{2}-y}=z-1, \\
& \sqrt{y^{2}-z}=x-1, \\
& \sqrt{z^{2}-x}=y-1 .
\end{aligned}
$$
| (x,y,z)=(1,1,1) | 68 | 10 |
math | 6. Given $f(x)=x^{3}$ on $[1, b]$ satisfies
$$
\frac{f(b)-f(1)}{b-1}=f^{\prime}(t)(1<t<b) \text {. }
$$
then $\lim _{b \rightarrow 1} \frac{t-1}{b-1}=$ | \frac{1}{2} | 77 | 7 |
math | 【Question 2】
The ratio of the length, width, and height of a rectangular prism is $4: 3: 2$, and the total length of all edges is 72 centimeters. Try to find the volume of this rectangular prism. | 192 | 54 | 3 |
math | 5. Find all values of the parameter $a$ for which the equation
$$
3|x+3 a|+\left|x+a^{2}\right|+2 x=a
$$
has no solution. | (-\infty;0)\cup(10;+\infty) | 44 | 16 |
math | 11. (20 points) Determine all complex numbers $\alpha$ such that for any complex numbers $z_{1} , z_{2}\left(\left|z_{1}\right| , \left|z_{2}\right|<1, z_{1} \neq z_{2}\right)$, we have
$$
\left(z_{1}+\alpha\right)^{2}+\alpha \overline{z_{1}} \neq\left(z_{2}+\alpha\right)^{2}+\alpha \overline{z_{2}} .
$$ | \{\alpha|\alpha \in \mathbf{C},| \alpha \mid \geqslant 2\} | 126 | 27 |
math | $O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$, and $C$. What is the magnitude of $\angle B AC$ in degrees? | 60^\circ | 70 | 4 |
math | 33rd Putnam 1972 Problem B2 A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity v a distance d from the start. What is the maximum time it could have taken to travel the distance d? Solution | 2d/v | 58 | 3 |
math | Example 3 Find a primitive root modulo 43. | 3^{42} \equiv 1 + 2 \cdot 43 \pmod{43^2} | 12 | 26 |
math | 7. (10 points) Calculate: $(100+15+17) \times(21+36+11)+(100-21-36-11) \times(15+17)=$ | 10000 | 55 | 5 |
math | 4. In $\pm 1 \pm 2 \pm 3 \pm 5 \pm 20$, by appropriately choosing + or -, different algebraic sums can be obtained $\qquad$.
| 24 | 43 | 2 |
math | For what values of the parameter $a$ does the equation $\frac{\log _{a} x}{\log _{a} 2}+\frac{\log _{x}(2 a-x)}{\log _{x} 2}=\frac{1}{\log _{\left(a^{2}-1\right)} 2}$ have:
(1) solutions?
(2) exactly one solution? | 2 | 87 | 1 |
math | 10. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy
$$
\begin{array}{l}
a_{1}=-1, b_{1}=2, \\
a_{n+1}=-b_{n}, b_{n+1}=2 a_{n}-3 b_{n}\left(n \in \mathbf{Z}_{+}\right) .
\end{array}
$$
Then $b_{2015}+b_{2016}=$ | -3 \times 2^{2015} | 121 | 12 |
math | 2. (2 points) The numbers $p$ and $q$ are distinct non-zero roots of the quadratic equation $x^{2} - a x + b = 0$, and the numbers $a$ and $b$ are distinct non-zero roots of the quadratic equation $x^{2} - p x - q = 0$. What can these numbers be equal to? | =1,b=-2,p=-1,q=2 | 79 | 11 |
math | G1.2 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 | 95 | 4 |
math | 6. The function $f(x)$ defined on $\mathbf{R}$ satisfies: when $x \in[0,1)$, $f(x)=2^{x}-x$, and for any real number $x$, $f(x)+f(x+1)=1$. Let $a=\log _{2} 3$, then the value of the expression $f(a)+f(2 a)+f(3 a)$ is $\qquad$ . | \frac{17}{16} | 96 | 9 |
math | 17. (1993 3rd Macau Mathematical Olympiad) $x_{1}, x_{2}, \cdots, x_{1993}$ satisfy
$$
\begin{array}{l}
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993, \\
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}(k=1,2, \cdots, 1993) .
\end{array}
$$
Then what is the maximum possible value... | 1992 | 205 | 4 |
math | 1. (8 points) The calculation result of the expression $2016 \times\left(\frac{8}{7 \times 9}-\frac{1}{8}\right)$ is | 4 | 42 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi}\left(\operatorname{ctg}\left(\frac{x}{4}\right)\right)^{1 / \cos \left(\frac{x}{2}\right)}$ | e | 56 | 1 |
math |
Problem 9.1. a) Draw all points in the plane with coordinates $(x ; y)$ such that
$$
(|3 x-y|-3)(|3 x+y|-3)=0
$$
b) Find all $x$ and $y$ for which
$$
\begin{array}{ccc}
(|3 x-y|-3)(|3 x+y|-3) & = & 0 \\
y-\{4 x\} & = & 0 \\
-1 \leq x \leq 1 & &
\end{array}
$$
(For a real number $x$ we denote the unique number in t... | \frac{5}{7},\frac{6}{7};\frac{6}{7},\frac{3}{7};1,0;-1,0 | 162 | 35 |
math | Shapovalov A.V.
A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. No... | 98 | 97 | 2 |
math | A sequence of real numbers $a_{0}, a_{1}, \ldots$ is said to be good if the following three conditions hold.
(i) The value of $a_{0}$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1}=2 a_{i}+1$ or $a_{i+1}=\frac{a_{i}}{a_{i}+2}$.
(iii) There exists a positive integer $k$ such that $a_{k... | 60 | 172 | 2 |
math | ## Task A-4.1.
Determine all prime numbers $p$ and $q$ such that $p^{q}+1$ is also prime. | p=2,q=2 | 34 | 6 |
math | 14. Let $f:[0,1] \rightarrow[0,1]$ be a continuous function such that $f(f(x))=1$ for all $x \in[0,1]$. Determine the set of possible values of $\int_{0}^{1} f(x) d x$. | (\frac{3}{4},1] | 65 | 9 |
math |
Problem 8.1. Find all natural numbers $x$ and $y$ such that:
a) $\frac{1}{x}-\frac{1}{y}=\frac{1}{3}$
b) $\frac{1}{x}+\frac{1}{y}=\frac{1}{3}+\frac{1}{x y}$.
| 4,9;5,6;6,5;9,4 | 77 | 15 |
math | 9. If $5 \pi$ is a period of the function $f(x)=\cos n x \cdot \sin \frac{80}{n^{2}} x$, then all possible values of the positive integer $n$ are $\qquad$ | 2,10 | 54 | 4 |
math | 2.48. In an oblique triangular prism, the distances between the lateral edges are equal to $a, b$ and $c$. The lateral edge is equal to $l$, and the height of the prism is $h$. Determine the total surface area of the prism. | \frac{2}{}(\sqrt{p(p-)(p-b)(p-)}+p) | 58 | 22 |
math | 19. Find the last two digits of the sum
$$
1^{2}+2^{2}+\ldots+50^{2}-51^{2}-\ldots-100^{2}+101^{2}+\ldots 150^{2}-151^{2}-\ldots 200^{2}+\ldots-2000^{2}+2001^{2}+\ldots+2017^{2}
$$
(i.e., 50 numbers with a plus sign, 50 with a minus sign, and so on.) | 85 | 136 | 2 |
math | Problem 10.1. Consider the equations
$$
3^{2 x+3}-2^{x+2}=2^{x+5}-9^{x+1}
$$
and
$$
a .5^{2 x}+|a-1| 5^{x}=1
$$
where $a$ is a real number.
a) Solve the equation (1).
b) Find the values of $a$ such that the equations (1) and (2) are equivalent.
Kerope Chakarian | \in[0,1]\cup{-1} | 112 | 11 |
math | Find the largest number $n$ having the following properties:
(a) No two digits of $n$ are equal.
(b) The number formed by reversing the digits of $n$ is divisible by 8 .
Remark. $n$ cannot start with 0 , but it can end with 0 . | 8697543210 | 62 | 10 |
math | Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated text:
Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^... | 8 | 130 | 1 |
math | 2. Find all right-angled triangles with integer side lengths, in which the hypotenuse is one unit longer than one of the legs. | all\triangles\with\legs\2k+1\\2k(k+1)\\hypotenuse\2k^2+2k+1,\where\k\is\any\natural\ | 29 | 45 |
math | 8. (10 points) In the expression $(x+y+z)^{2028}+(x-y-z)^{2028}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1030225 | 69 | 7 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{4 n^{2}-\sqrt[4]{n^{3}}}{\sqrt[3]{n^{6}+n^{3}+1}-5 n}$ | 4 | 61 | 1 |
math | Example 1. Solve the system of equations
$$
\frac{d y}{d x}=z, \frac{d z}{d x}=-y
$$ | C_{1}\cosx+C_{2}\sinx;-C_{1}\sinx+C_{2}\cosx | 36 | 25 |
math | 5. Why are the parameters $a$ and $b$ of a uniformly distributed random variable $X$ estimated by the formulas
$$
a^{*}=\bar{x}_{\mathrm{B}}-\sqrt{3} \sigma_{\mathrm{B}}, \quad b^{*}=\bar{x}_{\mathrm{B}}+\sqrt{3} \sigma_{\mathrm{B}} ?
$$ | ^{*}=\bar{x}_{\mathrm{B}}-\sqrt{3}\sigma_{\mathrm{B}},\quadb^{*}=\bar{x}_{\mathrm{B}}+\sqrt{3}\sigma_{\mathrm{B}} | 85 | 51 |
math | Find all positive integer $n$ such that for all $i=1,2,\cdots,n$, $\frac{n!}{i!(n-i+1)!}$ is an integer.
[i]Proposed by ckliao914[/i] | n = p-1 | 53 | 6 |
math | One TV station has $n$ ad breaks in a day, during which a total of $m$ ads were broadcast. In the first ad break, one ad and $\frac{1}{8}$ of the remaining $(m-1)$ ads were broadcast. In the second ad break, 2 ads and $\frac{1}{8}$ of the remaining ads were broadcast. This pattern continued for each subsequent ad break... | 49 | 130 | 2 |
math | Given a triangle with sides $A B=2, B C=3, A C=4$. A circle is inscribed in it, and the point $M$ where the circle touches side $B C$ is connected to point $A$. Circles are inscribed in triangles $A M B$ and $A M C$. Find the distance between the points where these circles touch the line $A M$. | 0 | 85 | 1 |
math | 1. A printing house determines the cost of printing a book as follows: it adds the cost of the cover to the cost of each page, and then rounds the result up to the nearest whole number of rubles (for example, if the result is 202 rubles and 1 kopeck, it is rounded up to 203 rubles). It is known that the cost of a book ... | 77 | 156 | 2 |
math | Task 5. (20 points)
Find $x_{0}-y_{0}$, if $x_{0}$ and $y_{0}$ are the solutions to the system of equations:
$$
\left\{\begin{array}{l}
x^{3}-2023 x=y^{3}-2023 y+2020 \\
x^{2}+x y+y^{2}=2022
\end{array}\right.
$$ | -2020 | 101 | 5 |
math | There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
[i]i.)... | 5000 | 176 | 4 |
math | Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$, such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$. | 3454 | 54 | 4 |
math | Claudine has $p$ packages containing 19 candies each.
If Claudine divides all of her candies equally among 7 friends, there are 4 candies left over. If Claudine divides all of her candies equally among 11 friends, there is 1 candy left over. What is the minimum possible value of $p$ ? | 40 | 71 | 2 |
math | We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more ele... | 436 | 115 | 3 |
math | ## Task B-4.3.
Three different real numbers $a$, 2016, and $b$ are three consecutive terms of a geometric sequence. If the numbers $a+2016, b+2016$, and $a+b$ are three consecutive terms of an arithmetic sequence, determine the numbers $a$ and $b$. | -1008,-4032 | 76 | 10 |
math | 9. [55] Let $N$ be the smallest positive integer for which
$$
x^{2}+x+1 \quad \text { divides } \quad 166-\sum_{d \mid N, d>0} x^{d} \text {. }
$$
Find the remainder when $N$ is divided by 1000 . | 672 | 79 | 3 |
math | 61 (1161). When the polynomial $2 x^{3}-5 x^{2}+7 x-8$ is multiplied by the polynomial $a x^{2}+b x+11$, the resulting polynomial does not contain either $x^{4}$ or $x^{3}$. Find the coefficients $a$ and $b$ and determine what polynomial results from the multiplication. | =4,b=10;8x^{5}-17x^{2}-3x-88 | 84 | 23 |
math |
1. Find all real roots of the equation
$$
4 x^{4}-12 x^{3}-7 x^{2}+22 x+14=0,
$$
if it is known that it has four distinct real roots, two of which sum up to 1.
| \frac{1}{2}+\sqrt{2},\frac{1}{2}-\sqrt{2},1+\sqrt{3},1-\sqrt{3} | 62 | 36 |
math | Let $x$ be a number such that $x +\frac{1}{x}=-1$. Determine the value of $x^{1994} +\frac{1}{x^{1994}}$. | -1 | 49 | 2 |
math | 6. Given point $A(0,1)$, curve $C: y=\log _{a} x$ always passes through point $B$. If $P$ is a moving point on curve $C$, and $\overrightarrow{A B} \cdot \overrightarrow{A P}$ has a minimum value of 2, then the real number $a=$ $\qquad$. | e | 81 | 1 |
math | Example 3. Find $\int \arcsin x d x$. | x\arcsinx+\sqrt{1-x^{2}}+C | 15 | 15 |
math | 3. Find the values of the following expressions:
(1) $\sin 10^{\circ} \cdot \sin 30^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ}$;
(2) $\sin ^{2} 20^{\circ}+\cos ^{2} 80^{\circ}+\sqrt{3} \sin 20^{\circ} \cdot \cos 80^{\circ}$;
(3) $\cos ^{2} A+\cos ^{2}\left(60^{\circ}-A\right)+\cos ^{2}\left(60^{\circ}+A\right)... | \frac{1}{16},\frac{1}{4},\frac{3}{2},\frac{1}{128} | 257 | 31 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{3}-7}+\sqrt[3]{n^{2}+4}}{\sqrt[4]{n^{5}+5}+\sqrt{n}}$ | 0 | 65 | 1 |
math | Solve the following system of equations:
$$
\begin{aligned}
x+y+z & =2 \\
x^{3}+y^{3}+z^{3} & =20 \\
x^{7}+y^{7}+z^{7} & =2060
\end{aligned}
$$ | 3,1,-2 | 68 | 5 |
math | 6. $\underbrace{2 \times 2 \times \ldots \times 2}_{20 \uparrow 2}-1$
The unit digit of the result is $\qquad$ | 5 | 42 | 1 |
math | 7.1. Solve the equation
$$
3 \cos \frac{4 \pi x}{5}+\cos \frac{12 \pi x}{5}=2 \cos \frac{4 \pi x}{5}\left(3+\operatorname{tg}^{2} \frac{\pi x}{5}-2 \operatorname{tg} \frac{\pi x}{5}\right)
$$
In the answer, write the sum of its roots on the interval $[-11 ; 19]$. | 112.5 | 111 | 5 |
math | 6. 66 $a, b, c$ are positive real numbers, $\alpha$ is a real number, assume
$$
\begin{array}{c}
f(\alpha)=a b c\left(a^{\alpha}+b^{\alpha}+c^{\alpha}\right), \\
g(\alpha)=a^{\alpha+2}(b+c-a)+b^{\alpha+2}(a-b+c)+c^{\alpha+2}(a+b-c),
\end{array}
$$
Determine the relationship in size between $f(\alpha)$ and $g(\alpha)$. | f(\alpha)\geqslant(\alpha),\alpha\inR | 128 | 16 |
math | There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.
(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.) | 944 | 106 | 3 |
math | Let's consider a four-digit natural number with the following property: if we swap its first two-digit number with the second, we get a four-digit number that is 99 less.
How many such numbers are there in total, and how many of them are divisible by 9? (K. Pazourek) | 89 | 65 | 2 |
math | Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that for all $x, y \in \mathbf{R}$, we have
$$
f(x f(y))=(1-y) f(x y)+x^{2} y^{2} f(y) .
$$ | f(x) \equiv 0 \text{ or } f(x)=x-x^{2} | 68 | 20 |
math | 3. Given $O$ is the circumcenter of $\triangle A B C$, $|A B|=2,|A C|=1, \angle B A C=\frac{2}{3} \pi$, let $\overrightarrow{A B}=\boldsymbol{a}, \overrightarrow{A C}=\boldsymbol{b}$, if $\overrightarrow{A O}=\lambda_{1} a+\lambda_{2} \boldsymbol{b}$, then $\lambda_{1}+\lambda_{2}=$ $\qquad$ | \frac{13}{6} | 115 | 8 |
math | 2. Find the smallest constant $C$, such that for all real numbers $x, y, z$ satisfying $x+y+z=-1$, we have
$$
\left|x^{3}+y^{3}+z^{3}+1\right| \leqslant C\left|x^{5}+y^{5}+z^{5}+1\right| \text {. }
$$ | \frac{9}{10} | 88 | 8 |
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