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 | A boy tried to align 480 cans in the shape of a triangle with one can in the $1^{\text{st}}$ row, 2 cans in the $2^{\text{nd}}$ row, and so on. In the end, 15 cans were left over. How many rows does this triangle have? | 30 | 72 | 2 |
math | 8-2. On a distant planet in the mangrove forests, there lives a population of frogs. The number of frogs born each year is one more than the difference between the number of frogs born in the previous two years (the larger number minus the smaller number).
For example, if 5 frogs were born last year and 2 in the year ... | 2033 | 150 | 4 |
math | (2) If two acute angles $\alpha, \beta$ satisfy: $(\sin \alpha+\cos \alpha)(\sin \beta+\cos \beta)=2$, then $(\sin 2 \alpha+\cos 3 \beta)^{2}+(\sin 2 \beta+\cos 3 \alpha)^{2}=$ $\qquad$ . | 3-2\sqrt{2} | 76 | 8 |
math | Determine all primes $p$ such that there exists an integer $x$ satisfying $x^{2010}+x^{2009}+$ $\cdots+1 \equiv p^{2010}\left(\bmod p^{2011}\right)$. | p\equiv1(\bmod2011) | 62 | 12 |
math | ## Task B-2.3.
Let $f(x)$ be a function for which $f\left(\frac{x+2}{x-3}\right)=x^{2}+x+1$ and let the function $g(x)$ be defined by $g(x)=\sqrt{f(x)-1}$. Determine the natural domain of the function $g(x)$. | \langle-\infty,-\frac{2}{3}]\cup[-\frac{1}{4},1\rangle\cup\langle1,\infty\rangle | 79 | 36 |
math | 15th Irish 2002 Problem B2 n = p · q · r · s, where p, q, r, s are distinct primes such that s = p + r, p(p + q + r + s) = r(s - q) and qs = 1 + qr + s. Find n. | 2002 | 69 | 4 |
math | Numbers $a, b, c$ satisfy $a^{2}+b^{2}+c^{2}=4$, then the maximum value of $(a-b)^{2}+(b-c)^{2}+(c-a)^{2}$ is | 12 | 53 | 2 |
math | [ Divisibility rules for 3 and 9 ] $[\quad$ Case enumeration $]$
Find all two-digit numbers whose digit sum remains unchanged when the number is multiplied by 2, 3, 4, 5, 6, 7, 8, and 9.
# | 18,45,90,99 | 63 | 11 |
math | Let $a,b,c$ be non-zero real numbers. The lines $ax + by = c$ and $bx + cy = a$ are perpendicular and intersect at a point $P$ such that $P$ also lies on the line $y=2x$. Compute the coordinates of point $P$.
[i]2016 CCA Math Bonanza Individual #6[/i] | \left( - \frac{3}{5}, - \frac{6}{5} \right) | 81 | 22 |
math | 7. (3 points) The number of four-digit numbers that satisfy the following two conditions is $\qquad$.
(1) The sum of any two adjacent digits is no more than 2;
(2) The sum of any three adjacent digits is no less than 3. | 1 | 58 | 1 |
math | For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$. | 222 | 78 | 3 |
math | 4. (7 points) Master Li produced a batch of parts one day, and he divided them into two piles, A and B. If 15 parts are moved from pile A to pile B, then the number of parts in both piles will be equal; if 15 parts are moved from pile B to pile A, then the number of parts in pile A will be three times the number of par... | 75,120 | 119 | 6 |
math | ## Task Condition
Find the derivative.
$y=\sqrt{(3-x)(2+x)}+5 \arcsin \sqrt{\frac{x+2}{5}}$ | \sqrt{\frac{3-x}{2+x}} | 36 | 11 |
math | Evin’s calculator is broken and can only perform $3$ operations:
Operation $1$: Given a number $x$, output $2x$.
Operation $2$: Given a number $x$, output $4x +1$.
Operation $3$: Given a number $x$, output $8x +3$.
After initially given the number $0$, how many numbers at most $128$ can he make?
| 82 | 89 | 2 |
math | $$
\begin{array}{l}
\text { 6. Let } a_{n}=1+2+\cdots+n\left(n \in \mathbf{Z}_{+}\right) , \\
S_{m}=a_{1}+a_{2}+\cdots+a_{m}(m=1,2, \cdots) \text {. }
\end{array}
$$
Then among $S_{1}, S_{2}, \cdots, S_{2017}$, the numbers that are divisible by 2 but not by 4 are $\qquad$ in number.
$$ | 252 | 131 | 3 |
math | A function $f$ defined on integers such that
$f (n) =n + 3$ if $n$ is odd
$f (n) = \frac{n}{2}$ if $n$ is even
If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$. | k = 1 | 72 | 5 |
math | 4A. Find all real numbers $a$ and $b, b>0$, such that the roots of the given equations $x^{2}+a x+a=b$ and $x^{2}+a x+a=-b$ are four consecutive integers. | =-1,b=1or=5,b=1 | 55 | 11 |
math | In a triangle two altitudes are not smaller than the sides on to which they are dropped. Find the angles of the triangle. | 45^\circ, 45^\circ, 90^\circ | 26 | 18 |
math | 160. Hens and Eggs. Here is a new version of an old problem. Although it looks very complicated and confusing, it is extremely easy to solve with the right approach.
If one and a half hens lay one and a half eggs in one and a half days, then how many hens plus half a hen, laying eggs one and a half times faster, will ... | 1 | 91 | 1 |
math | $4 \cdot 30$ represents $E(n)$ as the largest integer $k$ such that $5^k$ is a divisor of the product $1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdots \cdots n^n$, then $E(150)$ equals what?
(Shanghai, China High School Mathematics Competition, 1988) | 2975 | 91 | 4 |
math | 1. (5 points) Find the value of $n$ for which the following equality holds:
$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2011
$$ | 4048143 | 88 | 7 |
math | 栵 1 Find the monotonic intervals and range of the function $y=3^{-x^{2}+2 x+3}$.
| (0,81] | 30 | 6 |
math | Four. (50 points) Let $n$ be a positive integer, and let the planar point set be
$$
S=\{(x, y) \mid x, y \in\{0,1, \cdots, n\}, x+y \neq 0\} \text {. }
$$
Question: What is the minimum number of lines in the plane whose union can contain $S$, but not include the point $(0,0)$? | 2n | 99 | 2 |
math | (13) The line $x+y=1$ intersects the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at points $A$ and $B$, and $C$ is the midpoint of segment $AB$. If $|AB|=2 \sqrt{2}$, and the slope of line $OC$ is $k_C=\frac{1}{\sqrt{2}}$, find the equation of the ellipse. | \frac{x^{2}}{3}+\frac{\sqrt{2}y^{2}}{3}=1 | 104 | 24 |
math | 3. Find all values of $c$ for which the inequality $a+\sqrt{b+c}>b+\sqrt{a+c}$ holds for any positive $a, b$ and $a>b$.
(25 points.) | \frac{1}{4} | 48 | 7 |
math | There are a family of $5$ siblings. They have a pile of at least $2$ candies and are trying to split them up
amongst themselves. If the $2$ oldest siblings share the candy equally, they will have $1$ piece of candy left over.
If the $3$ oldest siblings share the candy equally, they will also have $1$ piece of candy lef... | 31 | 120 | 2 |
math | 2.068. $\frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2 c}{a b}\right)(a+b+2 c)}{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{a b}-\frac{4 c^{2}}{a^{2} b^{2}}} ; \quad a=7.4 ; b=\frac{5}{37}$. | 1 | 103 | 1 |
math | 10. $[\mathbf{8}]$ Let $f(n)=\sum_{k=1}^{n} \frac{1}{k}$. Then there exists constants $\gamma, c$, and $d$ such that
$$
f(n)=\ln (n)+\gamma+\frac{c}{n}+\frac{d}{n^{2}}+O\left(\frac{1}{n^{3}}\right),
$$
where the $O\left(\frac{1}{n^{3}}\right)$ means terms of order $\frac{1}{n^{3}}$ or lower. Compute the ordered pair $(... | (\frac{1}{2},-\frac{1}{12}) | 141 | 15 |
math | 8.4. Given a triangle $\mathrm{ABC}$ with angle $\mathrm{BAC}$ equal to $30^{\circ}$. In this triangle, the median $\mathrm{BD}$ was drawn, and it turned out that angle $\mathrm{BDC}$ is $45^{\circ}$. Find angle $\mathrm{ABC}$. | 45 | 73 | 2 |
math | 3. Today is March 24, 2007. The sum of the digits of the year is 9. After how many years will the same sum of digits repeat? How many times will the same sum occur until the year 2107 inclusive? | 9,8 | 57 | 3 |
math | 4. 188 Find all three-digit numbers that satisfy the following condition: the quotient when divided by 11 equals the sum of the squares of its digits. | 550 \text{ and } 803 | 35 | 12 |
math | ## Task 1 - 090831
The age specifications (expressed in full years of life) of a family - father, mother, and their two children - have the following properties:
The product of all four ages is 44950; the father is 2 years older than the mother.
How old are the four family members? | 31,29,10,5 | 76 | 10 |
math | Example 1.21. Form the equation of the plane passing through the line of intersection of the planes $x+y+5z-1=0$, $2x+3y-z+2=0$ and through the point $M(3,2,1)$. | 5x+14y-74z+31=0 | 59 | 15 |
math | 9. Given $\triangle A B C$ satisfies $A B=1, A C=2, \cos B+\sin C=1$, find the length of side $B C$.
---
The translation is provided as requested, maintaining the original format and line breaks. | \frac{2\sqrt{21}+3}{5} | 56 | 15 |
math | 12. Find the product of matrices $A$ and $B$, if
$$
A=\left(\begin{array}{lll}
3 & 1 & 1 \\
2 & 1 & 2 \\
1 & 2 & 3
\end{array}\right), \quad B=\left(\begin{array}{rrr}
1 & 1 & -1 \\
2 & -1 & 1 \\
1 & 0 & 1
\end{array}\right)
$$ | (\begin{pmatrix}6&2&-1\\6&1&1\\8&-1&4\end{pmatrix}) | 106 | 31 |
math | 11 If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\qquad$ . | \sqrt{3} | 45 | 5 |
math | 7 In the subset $S$ of the set $X=\{1,2, \cdots, 50\}$, the sum of the squares of any two elements is not a multiple of 7. Find the maximum value of $|S|$. | 44 | 55 | 2 |
math | Example 4 Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta, \alpha \beta$. Find the sum of the coefficients of this polynomial.
(2005 China Western Olympiad) | 1 | 62 | 1 |
math | 34. In what time will the end of the minute hand of a clock describe an arc equal to the length of this hand? | \frac{30 | 27 | 5 |
math | Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$.
[i]Proposed by pog[/i] | 783 | 95 | 3 |
math | Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form
$ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$,
if it is known that all the roots of them are positive ... | P(X) = X^2 - 8X + 1 | 145 | 15 |
math | Let us have $6050$ points in the plane, no three collinear. Find the maximum number $k$ of non-overlapping triangles without common vertices in this plane. | 2016 | 39 | 4 |
math | 2. Let $2 n$ real numbers $a_{1}, a_{2}, \cdots, a_{2 n}$ satisfy the condition $\sum_{i=1}^{2 n-1}\left(a_{i+1}-a_{i}\right)^{2}=1$, find the maximum value of $\left(a_{n+1}+\right.$ $\left.a_{n+2}+\cdots+a_{2 n}\right)-\left(a_{1}+a_{2}+\cdots+a_{n}\right)$. (2003 Western Mathematical Olympiad) | \sqrt{\frac{n\left(2 n^{2}+1\right)}{3}} | 126 | 21 |
math | 1. How many divisors of the number $30^{2008}$ are not divisors of the number $20^{2007}$? | 2008\cdot2009^{2}+2009 | 35 | 18 |
math | 10,11
A rhombus, with its smaller diagonal equal to its side, which is 1, rotates around a line passing through the end of the larger diagonal and perpendicular to this diagonal. Find the volume of the solid of revolution. | \frac{3\pi}{2} | 52 | 9 |
math | [ Recurrence relations (other)] Fibonacci numbers $\quad]$
a) Find the generating function of the Lucas number sequence (the definition of Lucas numbers see in problem 60585)
b) Using this function, express $L_{n}$ in terms of $\varphi$ and $\hat{\varphi}$ (see problem $\left.\underline{61502}\right)$.
# | L_{n}=\varphi^{n}+\hat{\varphi}^{n} | 84 | 19 |
math | 20. Let $n(n \geqslant 3)$ be a positive integer, and let positive numbers $\alpha$ and $\beta$ satisfy $\frac{\alpha}{\beta}=\frac{n-1}{n-2}$. Also, $x_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, n)$, and $\sum_{i=1}^{n} x_{i}^{\alpha}=1$, find the minimum value of $\sum_{i=1}^{n} \frac{x_{i}^{\beta}}{1-x_{i}^{\alpha}}$. | \frac{n}{n-1}\cdot\sqrt[n-1]{n} | 138 | 17 |
math | 4. A positive integer $a$ consists of four digits, three of which are the same. The square of $a$ consists of seven digits, all of which are different. Number $b$ is obtained by reading number $a$ from back to front. It turns out that number $b$ is greater than $a$. In addition, it turns out that $b^{2}$ is exactly equ... | 1113 | 102 | 4 |
math | The elements of the sequence $\mathrm{Az}\left(x_{n}\right)$ are positive real numbers, and for every positive integer $n$,
$$
2\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{4}=\left(x_{1}^{5}+x_{2}^{5}+\ldots+x_{n}^{5}\right)+\left(x_{1}^{7}+x_{2}^{7}+\ldots+x_{n}^{7}\right)
$$
Determine the elements of the sequence. | x_{n}=n | 125 | 5 |
math | Solve the equation
$$
x^{4}-7 x^{3}+13 x^{2}-7 x+1=0
$$ | \begin{pmatrix}x_{1}=\frac{7+\sqrt{5}+\sqrt{38+14\sqrt{5}}}{4},&x_{2}=\frac{7+\sqrt{5}-\sqrt{38+14\sqrt{5}}}{4},\\x_{3}=\frac{7-\sqrt{5}+\sqrt{38-14\sqrt{5}}}{4},& | 31 | 96 |
math | 10.8. Find all pairs of distinct real numbers \(x\) and \(y\) such that \(x^{100} - y^{100} = 2^{99}(x - y)\) and \(x^{200} - y^{200} = 2^{199}(x - y)\).
(I. Bogdanov) | (x,y)=(2,0)(x,y)=(0,2) | 81 | 14 |
math | 3.357. $\cos \frac{3 \pi}{5} \cos \frac{6 \pi}{5}$. | \frac{1}{4} | 29 | 7 |
math | Problem 11.5. A two-digit number is written on the board. Nезнayka claimed that it is divisible by $3, 4, 5, 9, 10, 15, 18, 30$. Upon hearing this, Znayka saddened Nезнayka by saying that he was wrong exactly 4 times. What number could have been written on the board? List all possible options. | 36,45,72 | 94 | 8 |
math | 29. Jigsaw Puzzle. Everyone is familiar with the puzzle that involves cutting a piece of plywood with a picture drawn on it into small pieces, and then proposing to reassemble the original picture from these mixed pieces. Let the picture be cut into $n$ small pieces. We will call a "move" the joining together of two pi... | n-1 | 106 | 3 |
math | 2. (10 points) From 4 integers, arbitrarily select 3, find their average. Then find the sum of this average and the remaining 1 number, which can yield 4 numbers: $8$, $12$, $10 \frac{2}{3}$, and $9 \frac{1}{3}$. The sum of the originally given 4 integers is $\qquad$. | 20 | 85 | 2 |
math | [ Arithmetic. Mental calculation, etc.]
The Gregorian calendar. An ordinary year contains 365 days, a leap year - 366. The $n$-th year, the number of which is not divisible by 100, is a leap year if and only if $n$ is divisible by 4. The $n$-th year, where $n$ is divisible by 100, is a leap year if and only if $n$ is ... | 365+\frac{97}{407} | 224 | 13 |
math | There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime posit... | 200 | 94 | 3 |
math | 7. Let $a, b, c$ be three, not necessarily distinct, numbers chosen randomly form the set $\{3,4,5,6,7,8\}$. Find the probability that $a b+c$ is even. | 0.5 | 51 | 3 |
math | (11) Given the rectangular cuboid $A B C D-A_{1} B_{1} C_{1} D_{1}$ satisfies $A A_{1}=2, A D=3, A B=$ 251, the plane $A_{1} B D$ intersects $C C_{1}$, $C_{1} B_{1}$, and $C_{1} D_{1}$ at points $L$, $M$, and $N$, respectively. Then the volume of the tetrahedron $C_{1} L M N$ is $\qquad$. | 2008 | 127 | 4 |
math | Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards? | 525 | 57 | 3 |
math | 11. Let the sequence $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $a_{1}=a_{2}=1, a_{3}=2$, and for any positive integer $n$, $a_{n} \cdot a_{n+1} \cdot a_{n+2} \neq 1$. Also, $a_{n} \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+$ $a_{n+3}, a_{1}+a_{2}+\cdots+a_{100}$ has the value $\qquad$ . | 200 | 163 | 3 |
math | 8. Define the length of intervals $(c, d),[c, d],(c, d],[c, d)$ to be $d-c$, where $d>c$. Given real numbers $a > b$, the length sum of intervals formed by $x$ that satisfy $\frac{1}{x-a}+\frac{1}{x-b} \geqslant 1$ is $\qquad$ . | 2 | 87 | 1 |
math | 5. Find the set of values of the expression $\frac{a \cos x - b \sin x + 2c}{\sqrt{a^{2} + b^{2} + c^{2}}}$ where $x, a, b, c$ are arbitrary numbers such that $a^{2} + b^{2} + c^{2} \neq 0$. | [-\sqrt{5};\sqrt{5}] | 81 | 11 |
math | 4. (7 points) When copying a problem, Lin Lin mistakenly wrote a repeating decimal as 0.123456. If the digits are correct but the dots indicating the repeating section were omitted, there are $\qquad$ possible original decimals. | 6 | 54 | 1 |
math | Let $x, y, z$ be integers such that for any triangle $ABC$ we have $16 H N^{2} = x BC^{2} + y CA^{2} + z AB^{2}$, where $H$ is the foot of the altitude from $A$, $M$ is the midpoint of $[BC]$, and $N$ is the midpoint of $[AM]$. Determine $x^{2} + y^{2} + z^{2}$. | 9 | 103 | 1 |
math | 8. (2004 National High School Competition Question) Given the sequence $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ satisfies $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$ and $a_{0}=3$, then $\sum_{i=0}^{n} \frac{1}{a_{i}}$ equals $\qquad$ - | \frac{1}{3}(2^{n+2}-n-3) | 97 | 17 |
math | 9、(16 points) In the sequence $\left\{a_{n}\right\}$, it is known that $a_{1}=1, a_{2}=\frac{1}{4}$, and $a_{n+1}=\frac{(n-1) a_{n}}{n-a_{n}}(n=2,3, \cdots)$
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$
(2) Prove: For all $n \in \mathbb{N}^{*}$, $\sum_{k=1}^{n} a_{k}^{2}<\frac{7}{6}$. | \sum_{k=1}^{n}a_{k}^{2}<\frac{7}{6} | 152 | 24 |
math | 10. Define the "coordinate distance" between two points $P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right)$ as: $d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$. If the "coordinate distance" from $C(x, y)$ to points $A(1,3), B(6,9)$ is equal, where the real numbers $x, y$ satisfy $0 \leqslant x \leqslan... | 5+5\sqrt{2} | 167 | 8 |
math | 5. The function $f(x)$ defined on $\mathbf{R}$, for any real number $x$, satisfies
$$
\begin{array}{l}
f(x+3) \leqslant f(x)+3, \\
f(x+2) \geqslant f(x)+2,
\end{array}
$$
and $f(1)=2$. Let $a_{n}=f(n)\left(n \in \mathbf{Z}_{+}\right)$, then
$$
f(2015)=
$$ | 2016 | 118 | 4 |
math | 4. Given $a, b, c, d \in \mathbf{Z}$,
$$
f(x)=a x^{3}+b x^{2}+c x+d \text {. }
$$
Then among $A(1,1)$, $B(2009,1)$, $C(-6020,2)$, $D(2,-6020)$, at most $\qquad$ of them can be on the curve $y=f(x)$. | 3 | 108 | 1 |
math | Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.
# | \frac{3}{8},\frac{5}{13} | 49 | 15 |
math | [ $[\underline{\text { Games-jokes }}]$
On the board, the numbers 25 and 36 are written. In one move, it is allowed to write down another natural number - the difference between any two numbers already on the board, if it has not appeared before. The player who cannot make a move loses.
# | 34 | 71 | 2 |
math | 4. All natural numbers are divided into "good" and "bad" according to the following rules:
a) From any bad number, you can subtract some natural number not exceeding its half so that the resulting difference becomes "good".
b) From a "good" number, you cannot subtract no more than half of it so that it remains "good"... | 2047 | 98 | 4 |
math | Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1\le n \le 100$, find the maximum value of $\phi(n^2+2n)-\phi(n^2)$.
[i]Proposed by Vincent Huang[/i] | 72 | 73 | 2 |
math | 1. Given $a, b, c$ are all positive integers, and satisfy the conditions
$$
a^{2}-b^{2}-c^{2}=a b c, \quad a^{2}=2(b+c) .
$$
Find $a=$ $\qquad$ ,$b=$ $\qquad$ ,$\dot{c}=$ $\qquad$ | 2,1,1 | 77 | 5 |
math | 2. Calculate $\sin ^{2} 20^{\circ}+\cos ^{2} 50^{\circ}+\sin 20^{\circ} \cos 50^{\circ}=$ | \frac{3}{4} | 48 | 7 |
math | Find a polynomial $P$ of lowest possible degree such that
(a) $P$ has integer coefficients,
(b) all roots of $P$ are integers,
(c) $P(0) = -1$,
(d) $P(3) = 128$. | P(x) = (x - 1)(x + 1)^3 | 58 | 17 |
math | Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$.
[i]Proposed by Michael Ren[/i] | 34 | 85 | 2 |
math | 1. Find the minimum value of the expression $\frac{25 x^{2} \sin ^{2} x+16}{x \sin x}$ for $0<x<\pi$. | 40 | 42 | 2 |
math | 9. (GBR 5) ${ }^{1 \mathrm{M} 03}$ Let $\{f(n)\}$ be a strictly increasing sequence of positive integers: $0<f(1)<f(2)<f(3)<\cdots$. Of the positive integers not belonging to the sequence, the $n$th in order of magnitude is $f(f(n))+1$. Determine $f(240)$. | 388 | 91 | 3 |
math | On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$.
$(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on th... | \pi | 147 | 2 |
math | 5. (10 points) At the end of the term, Teacher XiXi bought the same number of ballpoint pens, fountain pens, and erasers to distribute to the students in the class. After giving each student 2 fountain pens, 3 ballpoint pens, and 4 erasers, she found that there were 48 ballpoint pens left, and the remaining number of f... | 16 | 112 | 2 |
math | 5. In manufacturing electronic digital computers, regardless of what kind of positional notation is used, each digit requires a device. If using $n$-ary notation, each position has $n$ digits, and to represent an $m$-digit $n$-ary number in the machine, $m \times n$ devices are required. Let the maximum number the mach... | Gisminimizedwhenn=3 | 249 | 8 |
math | We have three types of alloys - made of gold, silver, and copper - with the ratios of these metals being $1: 3: 5$, $3: 5: 1$, and $5: 1: 3$. Create an alloy from these alloys weighing $351 \mathrm{~g}$, in which the ratio of gold, silver, and copper is $7: 9: 11$. | 195,78,78 | 91 | 9 |
math | 3.111.
$$
\frac{\cos ^{2}\left(\frac{5}{2} \pi-2 \alpha\right)+4 \cos ^{2}\left(\frac{7}{2} \pi-\alpha\right)-4}{1+\cos (4 \alpha-\pi)-8 \sin ^{2}(5 \pi-\alpha)}
$$ | \frac{1}{2}\operatorname{ctg}^{4}\alpha | 81 | 17 |
math | B2. How many four-digit numbers are there with the following properties:
- the second digit is the average of the first digit and the third digit,
- the third digit is the average of the second digit and the fourth digit? (A number does not start with the digit 0.) | 30 | 59 | 2 |
math | 4. (5 points) Xiao Pang and Xiao Ya have a total of 66 science books and 92 story books. Among them, Xiao Pang's science books are twice as many as Xiao Ya's, and Xiao Ya's story books are three times as many as Xiao Pang's. How many more books does Xiao Ya have than Xiao Pang? $\qquad$ books. | 24 | 78 | 2 |
math | 11. If $a \pm b \mathrm{i}(b \neq 0)$ are the imaginary roots of the equation $x^{3}+q x+r=0$, where $a, b, q$ and $r$ are all real numbers, then $q$ can be expressed in terms of $a, b$ as $\qquad$ | b^{2}-3a^{2} | 76 | 9 |
math | 1. Given $0<x, y<1$. Find the maximum value of
$$
\frac{x y(1-x-y)}{(x+y)(1-x)(1-y)}
$$
(Liu Shixiong) | \frac{1}{8} | 47 | 7 |
math | If $ a$ and $ b$ are nonnegative real numbers with $ a^2\plus{}b^2\equal{}4$, show that: $ \frac{ab}{a\plus{}b\plus{}2} \le \sqrt{2}\minus{}1$ and determine when equality occurs. | \frac{ab}{a+b+2} \le \sqrt{2} - 1 | 65 | 21 |
math | 7. Let $x, y, z$ be real numbers, $3 x, 4 y, 5 z$ form a geometric sequence, and $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ form an arithmetic sequence, then the value of $\frac{x}{z}+\frac{z}{x}$ is . $\qquad$ | \frac{34}{15} | 81 | 9 |
math | 9.26 In a tournament, 16 chess players are participating. Determine the number of different first-round schedules (schedules are considered different if they differ in the participants of at least one game; the color of the pieces and the board number are not taken into account). | 2027025 | 57 | 7 |
math | 12$\cdot$40 Find all solutions of the equation $a^{2}+b^{2}=n!$ where $a, b, n$ are positive integers, and $a \leqslant b, \quad n<14$.
(19th Canadian Mathematics Competition, 1987) | (,b,n)=(1,1,2)\text{}(12,24,6) | 70 | 22 |
math | Task 3. The ratio of the areas of the sides of a cuboid is 2:3:5. Calculate the ratio of the edges of that cuboid. | 10:6:15 | 35 | 7 |
math | 3. Given $a, b, c \in \mathbf{R}$, and
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c} \text {, }
$$
then there exists an integer $k$, such that the following equations hold for
$\qquad$ number of them.
(1) $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^{2 k+1}=\frac{1}{a^{2 k+1}}+\frac{1}{b^{2 k+1}}+\frac{1... | 2 | 344 | 1 |
math | Example 3 A container holds 10 kilograms of water. The first time, $\frac{1}{3}$ is poured out, and then 1 kilogram of pure alcohol is added; the second time, $\frac{1}{3}$ is poured out again, and then $\frac{1}{2}$ kilogram of pure alcohol is added; thereafter, each time $\frac{1}{3}$ is poured out, and the amount of... | \frac{12-9(\frac{3}{4})^{n-1}}{32-9(\frac{3}{4})^{n-1}} | 125 | 36 |
math | 399. Stolen Balm. Three thieves stole a vase containing 24 ounces of balm from a gentleman. Hurriedly making their escape, they met a glassware seller in the forest, from whom they bought three containers. Finding a secluded spot, the thieves decided to divide the loot, but then discovered that the capacities of their ... | 8&8&8&0 | 97 | 7 |
math | 5. Let the line $y=k x-2$ intersect the parabola $y^{2}=8 x$ at points $A$ and $B$. If the x-coordinate of the midpoint of segment $A B$ is 2, then the length of segment $A B$ is $|A B|=$ $\qquad$ | 2\sqrt{15} | 71 | 7 |
math | 3. The product of the areas of the six faces of a rectangular prism is 14641, then the volume of the rectangular prism is | 11 | 31 | 2 |
math | 7. Given a regular tetrahedron $P-ABC$ with the side length of the base being 6 and the side length of the lateral edges being $\sqrt{21}$. Then the radius of the inscribed sphere of the tetrahedron is $\qquad$ | 1 | 59 | 1 |
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