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 | 4. If a die is rolled five times in succession, the probability of the event “the numbers appearing in the five rolls are neither all the same nor all different, and from the second roll onwards, each number is not less than the previous one” is | \frac{5}{162} | 52 | 9 |
math | $$
\left(\frac{1+x}{1-x}-\frac{1-x}{1+x}\right)\left(\frac{3}{4 x}+\frac{x}{4}-x\right)=?
$$
| 3 | 46 | 1 |
math | 5. The integer part $[x]$ of a number $x$ is defined as the greatest integer $n$ such that $n \leqslant x$, for example, $[10]=10,[9.93]=9,\left[\frac{1}{9}\right]=0,[-1.7]=-2$. Find all solutions to the equation $\left[\frac{x-1}{2}\right]^{2}+2 x+2=0$. | -3 | 101 | 2 |
math | 6. For what values of the parameter a does the equation $\left(x^{2}-a\right)^{2}+2\left(x^{2}-a\right)+(x-a)+2=0$ have exactly one solution? Specify the solution for the found values of the parameter a. (20 points) | =0.75,\quadx_{1}=-0.5 | 66 | 15 |
math | 3. [6 points] On the plane $O x y$, there is a point $A$, the coordinates $(x ; y)$ of which satisfy the equation $5 a^{2}-6 a x-4 a y+2 x^{2}+2 x y+y^{2}=0$, and a circle with center at point $B$, given by the equation $a^{2} x^{2}+$ $a^{2} y^{2}-6 a^{2} x-2 a^{3} y+4 a y+a^{4}+4=0$. Find all values of the parameter $... | (-1;0)\cup(1;2) | 168 | 11 |
math | Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA =
17, PB = 15,$ and $PC = 6.$
What is the length of $PD$? | 10 | 49 | 2 |
math | ## Zadatak A-4.6.
Izračunaj
$$
\sum_{k=1}^{1011}\binom{2022}{2 k-1} 2^{2 k-1}
$$
| \frac{3^{2022}-1}{2} | 55 | 14 |
math | 37th IMO 1996 shortlist Problem 22 Find all positive integers m and n such that [m 2 /n] + [n 2 /m] = [m/n + n/m] + mn. Solution | (,n)=(k,k^2+1)or(k^2+1,k)foranypositiveintegerk | 51 | 24 |
math | 7. (7 points) Given a sequence of numbers with a certain pattern: $1, \frac{2}{3}, \frac{5}{8}, \frac{13}{21}, \frac{34}{55} \cdots$. Then, in this sequence, the 10th number from left to right is . $\qquad$ | \frac{4181}{6765} | 77 | 13 |
math | 3. Given $x \in \mathbf{R}$, then the minimum value of $\frac{4 \sin x \cos x+3}{\cos ^{2} x}$ is $\qquad$ | \frac{5}{3} | 45 | 7 |
math | Problem 7. The dragon has 40 piles of gold coins, and the number of coins in any two of them differs. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the smallest number of different piles of coins that could result... | 14 | 76 | 2 |
math | A Tim number is a five-digit positive integer with the property that it is a multiple of 15 , its hundreds digit is 3, and its tens digit is equal to the sum of its first (leftmost) three digits. How many Tim numbers are there? | 16 | 55 | 2 |
math | 3. Let $a, b \in \mathbf{R}$, and $a+b=1$. Then
$$
f(a, b)=3 \sqrt{1+2 a^{2}}+2 \sqrt{40+9 b^{2}}
$$
the minimum value of $f(a, b)$ is $\qquad$ | 5 \sqrt{11} | 73 | 7 |
math | 15.1. $[9.3$ (15 points)] In trapezoid $ABCD$, diagonal $AC$ is equal to 1 and is also its height. Perpendiculars $AE$ and $CF$ are drawn from points $A$ and $C$ to sides $CD$ and $AB$ respectively. Find $AD$, if $AD=CF$ and $BC=CE$. | \sqrt{\sqrt{2}-1} | 88 | 9 |
math | What is the maximum value that the area of the projection of a regular tetrahedron with an edge of 1 can take?
# | 0.50 | 28 | 4 |
math | Let $a<b<c$ be the solutions of the equation $2016 x^{3}-4 x+\frac{3}{\sqrt{2016}}=0$. Determine the value of $-1 /\left(a b^{2} c\right)$. | 1354752 | 58 | 7 |
math | 1. The range of the function $f(x)=\sqrt{x-2}+\sqrt{3-x}$ is
$\qquad$ | [1, \sqrt{2}] | 29 | 8 |
math | Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$. Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible. Justify your answer.
# | 10 | 70 | 2 |
math | 4. Determine all pairs of integers $(a, b)$ that solve the equation $a^{3}+b^{3}+3 a b=1$. | (,b)=(1,0),(0,1),(-1,-1),(+1,-)foranyinteger | 33 | 24 |
math | 13. The sequence $\left\{a_{n} \mid\right.$ is defined as: $a_{0}=0, a_{1}=a_{2}=1$ $a_{n+1}=a_{n}+a_{n-1}(n \in \mathbf{N})$, find the greatest common divisor of $a_{2002}$ and $a_{1998}$. | 1 | 91 | 1 |
math | [ Trilinear coordinates]
Find the trilinear coordinates of the Brocard points.
# | (\frac{b}{}:\frac{}{}:\frac{}{b})(\frac{}{b}:\frac{}{}:\frac{b}{}) | 17 | 32 |
math | (19) Let $P$ be a moving point on the major axis of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$. A line passing through $P$ with slope $k$ intersects the ellipse at points $A$ and $B$. If $|P A|^{2}+|P B|^{2}$ depends only on $k$ and not on $P$, find the value of $k$.
Let $P$ be a moving point on the major axis ... | \\frac{4}{5} | 201 | 7 |
math | Which is the six-digit number (abcdef) in the decimal system, whose 2, 3, 4, 5, 6 times multiples are also six-digit and their digits are formed by cyclic permutations of the digits of the above number and start with $c, b, e, f, d$ respectively? | 142857 | 66 | 6 |
math | Example 10 Find the minimum value of the function $f(x, y)=\frac{3}{2 \cos ^{2} x \sin ^{2} x \cos ^{2} y}+$ $\frac{5}{4 \sin ^{2} y}\left(x, y \neq \frac{k \pi}{2}, k \in Z\right)$. (Example 3 in [4]) | \frac{29+4 \sqrt{30}}{4} | 91 | 16 |
math | 6. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$ there are always two monochromatic triangles that share exactly one vertex.
---
The translation maintains the original text's format and line breaks as requested. | 9 | 52 | 1 |
math | ## Task 15/90
It is to be investigated whether the equation $3^{n}=10^{4} m+1$ with $m ; n \in N, m ; n>0$ has solutions! | 3^{n}-1=10^{4} | 49 | 11 |
math | 12. (3 points) Xiao Hua and Xiao Jun both have some glass balls. If Xiao Hua gives Xiao Jun 4 balls, the number of Xiao Hua's glass balls will be 2 times that of Xiao Jun; if Xiao Jun gives Xiao Hua 2 balls, then the number of Xiao Hua's glass balls will be 11 times that of Xiao Jun. Xiao Hua originally had $\qquad$ gl... | 20,4 | 105 | 4 |
math | 21.22 * Write an $n$-digit number using the digits 1 and 2, where no two adjacent digits are both 1, and let the number of such $n$-digit numbers be $f(n)$. Find $f(10)$. | 144 | 59 | 3 |
math | 5. Let the even function $f(x)$ satisfy: $f(1)=2$, and when $x y \neq 0$, $f\left(\sqrt{x^{2}+y^{2}}\right)=\frac{f(x) f(y)}{f(x)+f(y)}$. Then $f(5)=$ $\qquad$. | \frac{2}{25} | 76 | 8 |
math | Problem 1. The perimeter of an isosceles triangle is $3 dm$. Calculate its leg $b$, if its base is a side of an equilateral triangle with a perimeter of $18 \mathrm{~cm}$. | 12\mathrm{~} | 49 | 7 |
math | 4.2 For Eeyore's Birthday, Winnie-the-Pooh, Owl, and Piglet decided to give balloons. Winnie-the-Pooh prepared three times as many balloons as Piglet, and Owl prepared four times as many balloons as Piglet. When Piglet was carrying his balloons, he was in a great hurry, stumbled, and some of the balloons burst. Eeyore ... | 4 | 104 | 1 |
math | 3. When $a$ equals $\qquad$, the equations $x^{2}+a x-6$ $=0$ and $x^{2}-6 x+a=0$ have at least one common root. | -6 \text{ or } 5 | 47 | 9 |
math | 44. There are $2 k+1$ cards, numbered with consecutive natural numbers from 1 to $2 k+1$. What is the maximum number of cards that can be selected so that no one of the selected numbers is equal to the sum of two other selected numbers? | k+1 | 58 | 3 |
math | Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the... | 4n^3 | 135 | 4 |
math | Frankin B.P.
Natural numbers $a<b<c$ are such that $b+a$ is divisible by $b-a$, and $c+b$ is divisible by $c-b$. The number $a$ is written with 2011 digits, and the number $b-2012$ is written with 2012 digits. How many digits does the number $c$ have? | 2012 | 85 | 4 |
math | 110. Find the general solution of the equation $y^{\prime \prime}=4 x$. | \frac{2}{3}x^{3}+C_{1}x+C_{2} | 22 | 21 |
math | 46th Putnam 1985 Problem A3 x is a real. Define a i 0 = x/2 i , a i j+1 = a i j 2 + 2 a i j . What is lim n→∞ a n n ? Solution | e^x-1 | 58 | 5 |
math | 16. (YUG 2) Determine all the triples $(a, b, c)$ of positive real numbers such that the system
$$
\begin{aligned}
a x+b y-c z & =0, \\
a \sqrt{1-x^{2}}+b \sqrt{1-y^{2}}-c \sqrt{1-z^{2}} & =0,
\end{aligned}
$$
is compatible in the set of real numbers, and then find all its real solutions. | \begin{pmatrix}\cos,&\cos(+\alpha),\quad\cos(+\beta),&\in[0,\pi-\alpha]\quad\text{or}\\\cos,&\cos(-\alpha),\quad\cos(-\beta),&\in[\alpha,\pi]0\end{pmatrix} | 104 | 70 |
math | Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer
multiple of $1001$.
| 99 | 54 | 2 |
math | A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes. | 4 | 84 | 1 |
math | 17. Find the sum of the reciprocals of all positive divisors of 360. | \frac{13}{4} | 22 | 8 |
math | ## Problem 1
Calculate: $\int \frac{3 x^{4}+2 x^{3}+x^{2}-2015}{\left(x^{4}+x^{3}+x^{2}+2015\right)^{2}} d x, x \in \mathbb{R}$. | \frac{-x}{x^{4}+x^{3}+x^{2}+2015}+C | 73 | 27 |
math | The Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$. Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$. | 256 | 94 | 3 |
math | 1. Two-headed and seven-headed dragons came to a meeting. At the very beginning of the meeting, one of the heads of one of the seven-headed dragons counted all the other heads. There were 25 of them. How many dragons in total came to the meeting? | 8 | 56 | 1 |
math | Find all integers n greater than or equal to $3$ such that $\sqrt{\frac{n^2 - 5}{n + 1}}$ is a rational number. | 3 | 36 | 1 |
math | 1. Determine all integers $m, n$ for which
$$
m^{3}+n^{3}=(m+n)^{2}
$$ | {(,-)\mid\in\mathbb{Z}}\cup{(0,1),(1,0),(1,2),(2,1),(2,2)} | 32 | 35 |
math | 203. Find the derivatives of the following functions:
1) $y=x^{x}$;
2) $r=(\cos \alpha)^{\sin 2 \alpha}$;
3) $s=\frac{2 t}{\sqrt{1-t^{2}}}$;
4) $R=(x-1) \sqrt[3]{(x+1)^{2}(x-2)}$. | \begin{aligned}1)&\quady'=x^x(1+\lnx)\\2)&\quadr'=2(\cos2\alpha\ln\cos\alpha-\sin^2\alpha)(\cos\alpha)^{\sin2\alpha}\\3)&\quad'=\frac{2}{\sqrt{(1-^2)^3}}\\4)&\quad | 86 | 82 |
math | Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions.
(i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$
(ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$
(iii) For all $i=1,2,\cdots, k, a_i\le a_{... | 1997 | 182 | 4 |
math | I2.1 一個等邊三角形及一個正六邊形的周長比率為 $1: 1$ 。若三角形與六邊形的面積比率為 $2: a$, 求 $a$ 的值。
Let the ratio of perimeter of an equilateral triangle to the perimeter of a regular hexagon be $1: 1$. If the ratio of the area of the triangle to the area of the hexagon is $2: a$, determine the value of $a$. | 3 | 108 | 1 |
math | 127. A truncated cone is described around a sphere. The total surface area of this cone is $S$. A second sphere touches the lateral surface of the cone along the circumference of the cone's base. Find the volume of the truncated cone, given that the part of the surface of the second sphere that is inside the first has ... | \frac{1}{3}S\sqrt{\frac{Q}{\pi}} | 74 | 18 |
math | 367. The duration of trouble-free operation of an element has an exponential distribution $F(t)=1$ - $\mathrm{e}^{-0.01 t}(t>0)$. Find the probability that during a time period of $t=50$ hours: a) the element will fail; b) the element will not fail. | 0.3940.606 | 73 | 10 |
math | 1. Given complex numbers $z$ and $\omega$ satisfy the following two conditions:
$$
\text { (1) } z+\omega+3=0 \text {; }
$$
(2) $|z|, 2, |\omega|$ form an arithmetic sequence.
Is there a maximum value for $\cos (\arg z-\arg \omega)$? If so, find it. | \frac{1}{8} | 83 | 7 |
math | 1. If $z$ is a complex number, and $(z+1)(\bar{z}+1)=2004$, then the range of $|z|$ is $\qquad$ . | [\sqrt{2004}-1, \sqrt{2004}+1] | 44 | 21 |
math | Example 3 Given $a=\frac{1}{2} \sqrt{\sqrt{2}+\frac{1}{8}}-\frac{\sqrt{2}}{8}$. Try to find the value of $a^{2}+\sqrt{a^{4}+a+1}$. | \sqrt{2} | 62 | 5 |
math | 445. Find all pairs of natural numbers whose sum is 288, and the greatest common divisor is 36. | 252,36;180,108 | 28 | 14 |
math | 6.209. $\left\{\begin{array}{l}x^{3}+y^{3}=19 \\ x^{2} y+x y^{2}=-6\end{array}\right.$ | (-2,3),(3,-2) | 47 | 9 |
math | 1. Definition:
$$
\begin{array}{l}
f(n)=\sqrt{n(n+1)(n+2)(n+3)+1}, \\
g(n)=(n+1)^{2} .
\end{array}
$$
Given the following conclusions:
(1) For any rational number $n, f(n)$ is a rational number;
(2) $f(1)-g(1)=1$;
(3) If $f(n)-g(n)=2012$, then $n=2012$;
(4) For any rational number $n$, $f(n)>g(n)$.
The correct conclu... | (1)(2)(3) | 153 | 7 |
math | 1. A student wrote a program to recolor a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 19, green - 23, blue - 53, blue - 55, purple - 83, black - 128. If ... | 55 | 221 | 2 |
math | 13.387 A motorboat departed from point A upstream, and simultaneously a raft set off downstream from point B. They met after a hours and continued moving without stopping. Upon reaching point B, the boat turned back and caught up with the raft at point A. The boat's own speed remained constant throughout. How long were... | (1+\sqrt{2}) | 77 | 7 |
math | 19. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. How many different addition equations are there? (Consider $a+b$ and $b+a$ as the same equation)
$\square \square \square \square \square+$ $\square$ $=99999$ | 1536 | 105 | 4 |
math | 2. If $\frac{1}{a}+\frac{1}{b}=\frac{5}{a+b}$, then $\frac{b^{2}}{a^{2}}+\frac{a^{2}}{b^{2}}=$ | 7 | 52 | 1 |
math | 1.001. $\frac{(7-6.35): 6.5+9.9}{\left(1.2: 36+1.2: 0.25-1 \frac{5}{16}\right): \frac{169}{24}}$. | 20 | 68 | 2 |
math | One. (20 points) Let $m, n$ be positive integers, and $m \neq 2$. The quadratic function $y=x^{2}+(3-m t) x-3 m t$ intersects the $x$-axis at two points with a distance of $d_{1}$ between them. The quadratic function $y=-\dot{x}^{2}+(2 t-n) x+2 n t$ intersects the $x$-axis at two points with a distance of $d_{2}$ betwe... | m=3, n=2 \text{ or } m=6, n=1 | 148 | 19 |
math | Three. (25 points) Given the equation
$$
\left(m^{2}-1\right) x^{2}-3(3 m-1) x+18=0
$$
has two positive integer roots, where $m$ is an integer.
(1) Find the value of $m$;
(2) The sides of $\triangle A B C$, $a$, $b$, and $c$, satisfy $c=2 \sqrt{3}$, $m^{2}+a^{2} m-8 a=0$, $m^{2}+b^{2} m-8 b=0$, find the area of $\tria... | 1 \text{ or } \sqrt{9+12 \sqrt{2}} | 145 | 18 |
math | 【Question 11】There is a natural number that when divided by 7 leaves a remainder of 3, and when divided by 9 leaves a remainder of 4. Please write down the first two natural numbers that satisfy the conditions in ascending order here $\qquad$ _. | 31,94 | 58 | 5 |
math | 1. (10 points) $84 \frac{4}{19} \times 1.375 + 105 \frac{5}{19} \times 0.9$. | 210\frac{10}{19} | 46 | 12 |
math | 21.3.5 ** In a triangle with integer side lengths, given one side length as $n$, and the other two side lengths do not exceed $n$, congruent triangles are counted as one. Find the number of such triangles. | [\frac{(n+1)^{2}}{4}] | 50 | 13 |
math | ## Task 11/66
The five numbers of a lottery draw (1 to 90) are sought, about which the following is stated:
a) All digits from 1 to 9 appear exactly once.
b) Only the three middle numbers are even.
c) The smallest number has a common divisor (different from itself) with the largest number.
d) The cross sum of one ... | 9,12,34,68,75 | 114 | 13 |
math | 5. Find the smallest three-digit number with the property that if a number, which is 1 greater, is appended to it on the right, then the result (a six-digit number) will be a perfect square. Answer: 183 | 183 | 51 | 3 |
math | 6.2. Masha and the Bear ate a basket of raspberries and 40 pies, starting and finishing at the same time. At first, Masha was eating raspberries, and the Bear was eating pies, then (at some point) they switched. The Bear ate both raspberries and pies 3 times faster than Masha. How many pies did Masha eat, if they ate t... | 4 | 87 | 1 |
math | In a circumference with center $ O$ we draw two equal chord $ AB\equal{}CD$ and if $ AB \cap CD \equal{}L$ then $ AL>BL$ and $ DL>CL$
We consider $ M \in AL$ and $ N \in DL$ such that $ \widehat {ALC} \equal{}2 \widehat {MON}$
Prove that the chord determined by extending $ MN$ has the same as length as both $ AB$ an... | XY = AB | 105 | 4 |
math | 9. The three-digit number $\overline{a b c}$ consists of three non-zero digits. The sum of the other five three-digit numbers formed by rearranging $a, b, c$ is 2017. Find $\overline{a b c}$.
三位數 $\overline{a b c}$ 由三個非零數字組成。若把 $a 、 b 、 c$ 重新排列, 則其餘五個可組成的三位數之和是 2017 。求 $\overline{a b c}$ 。 | 425 | 125 | 3 |
math | 14. Let $a_{1}=2006$, and for $n \geq 2$,
$$
a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n} .
$$
What is the value of $2005 a_{2005}$ ? | 2 | 72 | 1 |
math | ## Task 1 - 221221
Determine all triples $(x, y, z)$ of real numbers that satisfy the system of equations
$$
\begin{aligned}
& x \cdot(y+z)=5 \\
& y \cdot(x+z)=8 \\
& z \cdot(x+y)=9
\end{aligned}
$$ | (1,2,3),(-1,-2,-3) | 74 | 14 |
math | \section*{Problem 2 - 301032}
As is well known, any sequence of \(n\) numbers of the form
\[
a_{1}=z ; \quad a_{2}=z+d ; \quad a_{3}=z+2 d ; \quad a_{n}=z+(n-1) d
\]
( \(n \geq 1\) natural number; \(z, d\) real numbers) is called a (finite) arithmetic sequence.
Determine the number of all such arithmetic sequences ... | 40 | 186 | 2 |
math | 12.B. Given that $a$ and $b$ are positive integers, the quadratic equation $x^{2}-2 a x+b=0$ has two real roots $x_{1}$ and $x_{2}$, and the quadratic equation $y^{2}+2 a y+b=0$ has two real roots $y_{1}$ and $y_{2}$. It is also given that $x_{1} y_{1}-x_{2} y_{2}=2008$. Find the minimum value of $b$. | 62997 | 116 | 5 |
math | 6. The set $X \backslash Y=\{a \mid a \in X, a \notin Y\}$ is called the difference set of set $X$ and set $Y$. Define the symmetric difference of sets $A$ and $B$ as $A \Delta B=(A \backslash B) \cup(B \backslash A)$.
If two non-empty finite sets $S$ and $T$ satisfy $|S \Delta T|=1$, then the minimum value of $k=|S|+|... | 3 | 119 | 1 |
math | Problem 7.3. Krosh and Yozhik decided to check who would run faster along a straight road from Kopyatych's house to Losyash's house. When Krosh had run 20 meters, Yozhik had run only 16 meters. And when Krosh had 30 meters left, Yozhik had 60 meters left. How many meters is the length of the road from Kopyatych's house... | 180 | 129 | 3 |
math | ## Task B-3.2.
A right circular cone is inscribed in a cylinder such that its base coincides with one base of the cylinder. The vertex of the cone is located at the center of the other base of the cylinder. If the measure of the central angle of the cone's lateral surface is $120^{\circ}$, determine the ratio of the s... | \frac{-2+4\sqrt{2}}{7} | 86 | 14 |
math | Find all real numbers $a, b$ such that $(X-1)^{2}$ divides $a X^{4}+b X^{2}+1$. | (,b)=(1,-2) | 35 | 8 |
math | Example 6 If the three roots of $x^{3}+a x^{2}+b x+c=0$ are $a, b, c$, and $a, b, c$ are rational numbers not all zero. Find $a, b, c$.
(2005, Shanghai Jiao Tong University Independent Recruitment Examination) | (1,-2,0),(1,-1,-1) | 73 | 13 |
math |
N2. Find all triples $(p, q, r)$ of prime numbers such that all of the following numbers are integers
$$
\frac{p^{2}+2 q}{q+r}, \quad \frac{q^{2}+9 r}{r+p}, \quad \frac{r^{2}+3 p}{p+q}
$$
| (2,3,7) | 78 | 7 |
math | Three, (25 points) Try to determine, for any $n$ positive integers, the smallest positive integer $n$ such that at least 2 of these numbers have a sum or difference that is divisible by 21.
| 12 | 48 | 2 |
math | 19. The square $A B C D$ has sides of length 105 . The point $M$ is the midpoint of side $B C$. The point $N$ is the midpoint of $B M$. The lines $B D$ and $A M$ meet at the point $P$. The lines $B D$ and $A N$ meet at the point $Q$.
What is the area of triangle $A P Q$ ? | 735 | 96 | 3 |
math | 3. A non-empty finite set whose sum of the squares of all elements is odd is called a trivial set. If the set $A=\{1,2,3, \cdots, 2016,2017\}$, then the number of trivial sets among all proper subsets of $A$ is $\qquad$ (powers of numbers are allowed in the answer). | 2^{2016}-1 | 82 | 8 |
math | 5. Given a parallelogram $A B C D$ with angle $\angle B$ equal to $60^{\circ}$. Point $O$ is the center of the circumscribed circle of triangle $A B C$. Line $B O$ intersects the bisector of the exterior angle $\angle D$ at point $E$. Find the ratio $\frac{B O}{O E}$. | \frac{1}{2} | 84 | 7 |
math | 15.24. In how many ways can the number $n$ be represented as a sum of several integer terms $a_{i} \geqslant 2$? (Representations differing in the order of the terms are considered different.) | F_{n-1} | 53 | 6 |
math | A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number. | 11 | 119 | 2 |
math | [ Extremal Properties (other) ] [ Examples and Counterexamples. Constructions ]
The number of edges of a convex polyhedron is 99. What is the maximum number of edges that a plane, not passing through its vertices, can intersect? | 66 | 53 | 2 |
math | 3. Petya thought of five numbers. On the board, he wrote down their pairwise sums: $7, 9, 12, 16, 17, 19, 20, 21, 22$, 29. What numbers did Petya think of? | 2,5,7,14,15 | 68 | 11 |
math | Find the smallest natural number such that its half is divisible by three, its third is divisible by four, its quarter is divisible by eleven, and its half gives a remainder of five when divided by seven.
(E. Patáková) | 528 | 48 | 3 |
math | 2.154. What is the value of $\sqrt{25-x^{2}}+\sqrt{15-x^{2}}$, given that the difference $\sqrt{25-x^{2}}-\sqrt{15-x^{2}}=2$ (the value of $x$ does not need to be found)? | 5 | 69 | 1 |
math | A given radius sphere is to be enclosed by a frustum of a cone, whose volume is twice that of the sphere. The radii of the base and top of the frustum, as well as the radius of the circle along which the mantle of the frustum touches the sphere, are to be determined. | \zeta=\frac{2R\sqrt{5}}{5} | 63 | 16 |
math | ## 185. Math Puzzle $10 / 80$
In Moscow, the speed of elevators in high-rise buildings is twice as fast as in ordinary buildings.
Therefore, the travel time to the 20th floor, which is at a height of $81 \mathrm{~m}$, is only five seconds longer than to the eighth floor of an ordinary building, which is at a height o... | 1.5\mathrm{~}/\mathrm{} | 114 | 11 |
math | 83. Each quadratic equation can be reduced, by dividing it by the coefficient of the leading term, to the form
$$
x^{2}+c x+d=0
$$
The total number of different reduced quadratic equations in $p$-arithmetic is $p^{2}$. Calculate how many of them have no roots, how many have one root, and how many have two distinct ro... | \frac{p(p-} | 147 | 7 |
math | Example 1 The function $f(x)$ defined on $\mathrm{R}^{+}$ satisfies the relation $f(x)=f\left(\frac{1}{x}\right) \lg x + 1$, find $f(x)$. | \frac{1+\lgx}{1+\lg^{2}x} | 51 | 16 |
math | Three, (50 points) Find the smallest positive integer $n$ such that there exists an $(n+1)$-term sequence $a_{0}, a_{1}, \cdots, a_{n}$, satisfying $a_{0}=0, a_{n}=2008$, and
$$
\left|a_{i}-a_{i-1}\right|=i^{2}(i=1,2, \cdots, n) .
$$ | 19 | 100 | 2 |
math | The nineteenth question: Find the smallest real number $\lambda$ such that $\sum_{i=1}^{100}\left(a_{i}-a_{i+1}\right)^{2} \leq \lambda\left(100-\left(\sum_{i=1}^{100} a_{i}\right)\right)$ holds for any real numbers $a_{1} 、 a_{2} 、 \ldots 、 a_{100}$ satisfying $\sum_{i=1}^{100} a_{i}^{2}=100$. Here $a_{101}=a_{1}$. | 8 | 141 | 1 |
math | 602. Find the greatest and the least values of the function $y=$ $=x^{5}-5 x^{4}+5 x^{3}+3$ on the interval $[-1,2]$. | y(1)=4,y(-1)=-8 | 47 | 11 |
math | 6. Given the three sides of $\triangle A B C$ are $a, b, c$. If $a+b+c=16$, then
$$
b^{2} \cos ^{2} \frac{C}{2}+c^{2} \cos ^{2} \frac{B}{2}+2 b c \cos \frac{B}{2} \cdot \cos \frac{C}{2} \cdot \sin \frac{A}{2}
$$
$=$ | 64 | 108 | 2 |
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