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 | Someone invests $24000 \mathrm{~K}$; part of it at $4.5 \%$, the other part at $6 \%$. Their income is the same as if the entire amount earned $5 \%$ interest. How much is invested at $4.5 \%$ and how much at $6 \%$? | 16000koronaat4.5,8000koronaat6 | 71 | 20 |
math | 5. In the Cartesian coordinate system, let $O$ be the origin, point $A(-1,0), B(0, \sqrt{3})$, and the moving point $C$ lies on the circle
$$
(x-3)^{2}+y^{2}=4
$$
Then the maximum value of $|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}|$ is $\qquad$. | \sqrt{7}+2 | 97 | 7 |
math | 218. A discrete random variable $X$ has only two possible values: $x_{1}$ and $x_{2}$, with $x_{2}>x_{1}$. The probability that $X$ will take the value $x_{1}$ is 0.6. Find the distribution law of the variable $X$, if the expected value and variance are known: $M(X)=1.4$; $D(X)=0.24$. | \begin{pmatrix}X&1&2\\p&0.6&0.4\end{pmatrix} | 98 | 27 |
math | 8. A frog starts climbing out of a 12-meter deep well at 8 o'clock. It climbs up 3 meters and then slides down 1 meter due to the slippery well walls. The time it takes to slide down 1 meter is one-third of the time it takes to climb up 3 meters. At 8:17, the frog reaches 3 meters below the well's mouth for the second ... | 22 | 113 | 2 |
math | Are there such cuboids that can be cut in half into two cuboids similar to the original cuboid? | :b:=1:\sqrt[3]{2}:\sqrt[3]{4} | 22 | 17 |
math | 1469. Calculate $\sqrt{1.004}$ with an accuracy of 0.0001. | 1.002 | 27 | 5 |
math | 12.114 Find all prime numbers $x$ and $y$ that satisfy the equation $x^{2}-2 y^{2}=1$.
(Kyiv Mathematical Olympiad, 1966) | 3,2 | 47 | 3 |
math | II. (50 points) There are three types of stocks. The sum of the number of shares of the first two types equals the number of shares of the third type. The total value of the second type of stock is four times that of the first type. The total value of the first and second types of stocks equals the total value of the t... | 12.5\% \leqslant f \leqslant 15\% | 149 | 22 |
math | How many 7-digit numbers divisible by 9 are there, whose second last digit is 5? | 10^5 | 21 | 4 |
math | Triangle $ABC$ is right angled such that $\angle ACB=90^{\circ}$ and $\frac {AC}{BC} = 2$. Let the line parallel to side $AC$ intersects line segments $AB$ and $BC$ in $M$ and $N$ such that $\frac {CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point $K$ such that $OM+OK=KN... | 90^\circ | 145 | 4 |
math | 122. A rectangle is inscribed in a circle of radius $R$, and the midpoints of its sides are sequentially connected. Find the perimeter of the resulting quadrilateral. | 4R | 37 | 2 |
math | Let $p$ be a prime number and $F=\left \{0,1,2,...,p-1 \right \}$. Let $A$ be a proper subset of $F$ that satisfies the following property: if $a,b \in A$, then $ab+1$ (mod $p$) $ \in A$.
How many elements can $A$ have? (Justify your answer.)
| |A| = 1 | 90 | 7 |
math | Anton, Artem, and Vera decided to solve 100 math problems together. Each of them solved 60 problems. We will call a problem difficult if it was solved by only one person, and easy if it was solved by all three. How much does the number of difficult problems differ from the number of easy ones?
# | 20 | 69 | 2 |
math | In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$. | 141 | 54 | 3 |
math | 10. (3 points) 60 adventure team members need to cross a river. There is only one rubber boat on the river that can carry 6 people (a round trip counts as two times), and it takes 3 minutes to cross the river once. The total time required for all team members to cross to the other side of the river is $\qquad$ minutes. | 69 | 79 | 2 |
math | The sequence $a_{1}, a_{2}, a_{3}, \cdots$ satisfies $a_{1}=\frac{1}{2}$, and $a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n}(n \geqslant$ $1)$, determine the value of $a_{n}(n \geqslant 1)$. | a_{n}=\frac{1}{n(n+1)} | 89 | 14 |
math | 1. The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in N\right\}$ is | 2^{90}-1 | 60 | 6 |
math | 1. For every natural number $n$ with 3 decimal digits (so the first digit is not zero), we consider the number $n_{0}$ obtained from $n$ by removing its digits that are equal to zero. For example, if $n=205$ then $n_{0}=25$.
Determine the number of integers $n$ with three digits for which $n_{0}$ is a divisor of $n$ d... | 93 | 99 | 2 |
math |
6. Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function satisfying $f(0) \neq 0, f(1)=0$ and
(i) $f(x y)+f(x) f(y)=f(x)+f(y)$;
(ii) $(f(x-y)-f(0)) f(x) f(y)=0$,
for all $x, y \in \mathbb{Z}$, simultaneously.
(a) Find the set of all possible values of the function $f$.
(b) If $f(10) \neq 0$ and $f(2)=0$, find ... | {5k\midk\in\mathbb{Z}} | 162 | 14 |
math | 3. The maximum value of the function $y=\sin 2x-2(\sin x+\cos x)$ is $\qquad$ . | 1+2 \sqrt{2} | 30 | 8 |
math | 3. From a vessel containing 5 liters of a $6 \%$ acid solution, 1 liter was poured out, after which 2 liters of water were added. Find the concentration of the resulting solution. | 0.04 | 43 | 4 |
math | An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$ | 10^{10} | 90 | 6 |
math | ## Task 12/69
All pairs $(x ; y)$ are to be found that satisfy the system of equations
$$
\begin{aligned}
& |y-x|=|x+1| \\
& \frac{y-3}{4}=\left[\frac{x-1}{5}\right]
\end{aligned}
$$
where $[a]$ is an integer with $a-1<[a] \leq a$. | (1;3)(x_{0};-1)with-4\leqx_{0}<1 | 95 | 22 |
math | 10. Let $A$, $B$, and $C$ be three distinct points on $\odot O$, and $\angle A O B=120^{\circ}$, point $C$ lies on the minor arc $\overparen{A B}$ (point $C$ does not coincide with $A$ or $B$). If $\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}(\lambda, \mu \in \mathbf{R})$, then the range o... | 1<\lambda+\mu\leqslant2 | 122 | 12 |
math | Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers. | 95 | 35 | 4 |
math | 3. Find the sum of the first 10 elements that are found both in the arithmetic progression $\{5,8,11,14, \ldots\}$ and in the geometric progression $\{10,20,40,80, \ldots\} \cdot(10$ points $)$ | 6990500 | 70 | 7 |
math | Let $t$ be TNYWR.
Each of the three lines having equations $x+t y+8=0,5 x-t y+4=0$, and $3 x-k y+1=0$ passes through the same point. What is the value of $k$ ?
## | 5 | 61 | 1 |
math | Let $S_{1}$ and $S_{2}$ be planes that are perpendicular to each other. The line $e$ makes a $30^{\circ}$ angle with both planes. What angle does the line $e$ form with the intersection line of the two planes? | 45 | 58 | 2 |
math | 44*. In how many points do the diagonals of a convex n-gon intersect if no three of them intersect at the same point? | \frac{n(n-1)(n-2)(n-3)}{24} | 29 | 19 |
math | 7. The line $x-2 y-1=0$ intersects the parabola $y^{2}=4 x$ at points $A$ and $B$, and $C$ is a point on the parabola such that $\angle A C B=90^{\circ}$. Then the coordinates of point $C$ are $\qquad$. | (1,-2)or(9,-6) | 76 | 11 |
math | 2. If the equation with respect to $x$
$$
x^{2}+2(m+3) x+m^{2}+3=0
$$
has two real roots $x_{1}$ and $x_{2}$, then the minimum value of $\left|x_{1}-1\right|+\left|x_{2}-1\right|$ is $\qquad$. | 6 | 81 | 1 |
math | 2. $a \in \mathbf{R}$, the complex number $\omega=1+a \mathrm{i}$, and the complex number $z$ satisfies $\bar{\omega} z-\omega=0$. For what value of $a$ does $|z^{2}-z+2|$ have the minimum value, and find this minimum value. | \frac{\sqrt{14}}{4} | 74 | 11 |
math | 6. (10 points) Given $1 ※ 2=1+2=3, 2 ※ 3=2+3+4=9, 5 ※ 4=5+6+7+8=26$, if $a ※ 15=165$, then $a=$ $\qquad$ | 4 | 71 | 1 |
math | Grandfather distributed a bag of candies to Donald, Henry, and John, who got $70 \%$, $\mathbf{25 \%}$, and $5 \%$ respectively. Later Donald gave John 20 candies, and after that Henry and John shared their candies equally. By then Donald had three times as many candies as Henry. The next day Grandfather gave $x$ candi... | 40 | 212 | 2 |
math | 5. $\lim _{x \rightarrow 2} \frac{\ln \left(x^{2}-3\right)}{x-2}=$ | 4 | 32 | 1 |
math | 6. (10 points) A deck of playing cards, excluding the joker, has 4 suits totaling 52 cards, with each suit having 13 cards, numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of these 14 cards Feifei drew is exactly 35, then how many of them are 1?
$\qquad$ ... | 4 | 110 | 1 |
math | 4. Find all prime numbers $p$ not exceeding 1000 such that $2p+1$ is a perfect power (i.e., there exist natural numbers $m, n, n \geqslant 2$, such that $2p+1=m^n$). | 13 | 61 | 2 |
math | Three. (20 points) Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=\frac{1}{2}, a_{2}=\frac{1}{3}$. And for any positive integer $n$, we have
$$
a_{n+2}=2(n+2) a_{n+1}-(n+2)(n+1) a_{n}+\frac{n^{2}+3 n+1}{n+3} \text {. }
$$
Try to find the general term formula of the sequence $a_{n}$. | a_{n}=\frac{1}{n+1} | 127 | 13 |
math | 20.2.1 For integers $m, n$ with $n>m \geqslant 2$, if the set $\{m, m+1, \cdots, n\}$ is arbitrarily divided into two
subsets, at least one subset contains $a, b, c$ (not necessarily distinct) such that $a^{b}=c$. Find the minimum value of $n$, $f(m)$. | ^{+2} | 92 | 4 |
math | # 4. CONDITION
The sequence of numbers $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}}, \ldots$ satisfies the relations $\mathrm{a}_{\mathrm{n}}=\mathrm{a}_{\mathrm{n}-1} \cdot \mathrm{a}_{\mathrm{n}-3}$ for $\mathrm{n}=4,5,6, \ldots$ Find $\mathrm{a}_{2019}$, given that $\mathrm{a}_{1... | -1 | 141 | 2 |
math | 168 If the line $l$ passes through point $A\left(\frac{1}{2}, \frac{1}{3}\right)$ and point $B\left(\frac{1}{4}, \frac{1}{5}\right)$, then the nearest lattice point on $l$ to point $A$ is $\qquad$. | (-2,-1) | 74 | 5 |
math | 5. Find the sum of all numbers of the form $x+y$, where $x$ and $y$ are natural number solutions to the equation $5 x+17 y=307$. | 164 | 42 | 3 |
math | 12.41 Find the positive integer solutions to the equation $n!+1=(m!-1)^{2}$.
(Recommended by the Soviet Ministry of Education, 1991) | =3,\quadn=4 | 43 | 7 |
math | Let $A B C D$ be a trapezoid such that $(A B)$ is parallel to $(C D), A B=3 C D=3 D A$ and $\widehat{A D C}=120^{\circ}$. Determine the angle $\widehat{C B A}$ in degrees. | 30 | 68 | 2 |
math | Find the value of the expression
$$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$
assuming $f(x) =\frac{x^2}{1 + x^2}$ . | 1999.5 | 144 | 6 |
math | 9.6. On an infinite strip of paper, all natural numbers whose digits sum to 2018 are written in ascending order. What number is written in the 225th position?
(Method Commission) | 3\underbrace{999\ldots99}_{223nines}8 | 45 | 21 |
math | There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say... | 24 | 167 | 2 |
math | Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=a_{2}=a_{3}=1, a_{4}=-1, a_{5}=0,
$$
and for any $n \in \mathbf{Z}_{+}$, we have
$$
a_{n+5}=3 a_{n+4}-4 a_{n+3}+4 a_{n+2}-3 a_{n+1}+a_{n} \text {. }
$$
Find the general term formula for the sequence $\left\{a_{n}\right\}$. | a_{n}= \frac{-3+7 \mathrm{i}}{8} \mathrm{i}^{n}+\frac{-3-7 \mathrm{i}}{8}(-\mathrm{i})^{n}+ \frac{3 n^{2}-19 n+27}{4} | 134 | 63 |
math | Let $x_{0}+\sqrt{2003} y_{0}$ be the fundamental solution of the Pell equation
$$
x^{2}-2003 y^{2}-1
$$
Find the solution $(x, y)$ of (1), where $x, y$ are positive numbers and all prime factors of $x$ divide $x_{0}$. | x_{0},y_{0} | 81 | 8 |
math | Solve the following equation:
$$
\left(\frac{x}{3}\right)^{3+\log x}=30000
$$ | x_{1}=30,x_{2}=\frac{1}{10000} | 31 | 21 |
math | Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$ | 376 | 66 | 3 |
math | 4. Find all pairs of prime numbers $(p, q)$ such that $p q \mid\left(5^{p}+5^{q}\right)$.
(2009, China Mathematical Olympiad) | (2,3),(2,5),(3,2),(5,2),(5,5),(5,313),(313,5) | 46 | 33 |
math | 40. The difference equals the quotient. Find two numbers whose difference and quotient are both equal to 5. | \frac{25}{4},\quad\frac{5}{4} | 23 | 17 |
math | Example 4 There are $n$ squares arranged in a row, to be painted with red, yellow, and blue. Each square is painted one color, with the requirement that no two adjacent squares are the same color, and the first and last squares are also different colors. How many ways are there to paint them?
(1991, Jiangsu Mathematics... | a_{n}=2^{n}+2(-1)^{n} | 76 | 16 |
math | Find all ordered triples $(x, y, z)$ of integers satisfying the following system of equations:
$$
\begin{aligned}
x^{2}-y^{2} & =z \\
3 x y+(x-y) z & =z^{2}
\end{aligned}
$$ | (0,0,0),(1,0,1),(0,1,-1),(1,2,-3),(2,1,3) | 59 | 31 |
math | ## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\cos ^{2} t \\
y=\operatorname{tg}^{2} t
\end{array}\right.
$$ | \frac{2}{\cos^{6}} | 70 | 10 |
math | Writing successively the natural numbers, we obtain the sequence
$$
12345678910111213141516171819202122 \ldots
$$
What is the digit that is in the $2009^{th}$ position of this sequence? | 0 | 76 | 1 |
math | Example 8. Find the general solution of the equation
$$
y^{\prime \prime}-12 y^{\prime}+36 y=\sin 3 x
$$ | (C_{1}+C_{2}x)e^{6x}+\frac{4}{225}\cos3x+\frac{1}{75}\sin3x | 39 | 38 |
math | Problem 3. (Option 2)
Find the sum of the squares of the roots of the equation $\left(x^{2}+6 x\right)^{2}-1580\left(x^{2}+6 x\right)+1581=0$. | 3232 | 59 | 4 |
math | 1. In a certain triangle, the sum of the tangents of the angles turned out to be 2016. Estimate (at least to the nearest degree) the magnitude of the largest of its angles. | 90 | 44 | 2 |
math | 6.5 Arrange 5 different red beads and 3 different blue beads around a circular plate. How many ways are there to arrange them? If the blue beads are not adjacent, how many ways are there? What if the blue beads are together? | 7!,1440,5!3! | 51 | 11 |
math | 12 Given $\sin (x+\sin x)=\cos (x-\cos x), x \in[0, \pi]$, then $x=$
$\qquad$ . | \frac{\pi}{4} | 39 | 7 |
math | 8. Let the integer sequence $a_{1}, a_{2}, \cdots, a_{10}$ satisfy $a_{10}=3 a_{1}, a_{2}+a_{8}=2 a_{5}$, and
$$
a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}, i=1,2, \cdots, 9 \text {, }
$$
then the number of such sequences is $\qquad$ . | 80 | 112 | 2 |
math | Senderov B.A.
Find all pairs $(a, b)$ of natural numbers such that for any natural $n$, the number $a^{n}+b^{n}$ is a perfect $(n+1)$-th power. | (2,2) | 48 | 5 |
math | Find all functions $f : R \to R$ which satisfy $f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x$
and $f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) $ for all $x, y \in R$, with $y \ne 0$ | f(x) = x + \frac{1}{x} | 97 | 14 |
math | For nine non-negative real numbers $a_{1}, a_{2}, \cdots, a_{9}$ whose sum is 1, let
$$
\begin{array}{l}
S=\min \left\{a_{1}, a_{2}\right\}+2 \min \left\{a_{2}, a_{3}\right\}+\cdots+8 \min \left\{a_{8}, a_{9}\right\}+9 \min \left\{a_{9}, a_{1}\right\}, \\
T=\max \left\{a_{1}, a_{2}\right\}+2 \max \left\{a_{2}, a_{3}\ri... | [\frac{36}{5},\frac{31}{4}] | 274 | 16 |
math | Example 2 Find all positive integers $x>1, y>1, z>1$, such that $1!+2!+\cdots+x!=y^{z}$. | x=y=3, z=2 | 38 | 8 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=\sqrt[5]{e^{7 x-1}}$ | (\frac{7}{5})^{n}\cdot\sqrt[5]{e^{7x-1}} | 29 | 23 |
math | 1. Function
$$
f(x)=\sin ^{4} x+\sin x \cdot \cos x+\cos ^{4} x
$$
The maximum value is $\qquad$. | \frac{9}{8} | 42 | 7 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x+f(y))-f(x)=(x+f(y))^{4}-x^{4}
$$
for all $x, y \in \mathbb{R}$. | f(x)=0f(x)=x^{4}+kforanyrealconstantk | 61 | 18 |
math | 19. Let the quadratic function be
$$
f(x)=a x^{2}+(2 b+1) x-a-2(a, b \in \mathbf{R}, a \neq 0)
$$
have at least one root in $[3,4]$. Find the minimum value of $a^{2}+b^{2}$. | \frac{1}{100} | 78 | 9 |
math | 49. Let $x, y, z \in [0,1]$, and $|x-y| \leqslant \frac{1}{2}, |y-z| \leqslant \frac{1}{2}, |z-x| \leqslant \frac{1}{2}$, find the maximum and minimum values of $W=x+y+z-xy-yz-zx$. (2010 China Southeast Mathematical Olympiad) | \frac{5}{6} | 99 | 7 |
math | What is the value of the sum $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} 2002\right]$? | 17984 | 63 | 5 |
math | Given 12 sticks of the same length. How can you cut them into smaller sticks so that you can form 13 equal triangles, with each of the smaller sticks being a side of one of these triangles?
# | 13 | 45 | 2 |
math | [b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer.
$$x + y = 3$$
$$3xy -z^2 = 9$$
[b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during ... | 0 | 410 | 1 |
math | How many positive integers, less than 2011, are multiples of 3 or 4, but not of 5? | 804 | 28 | 3 |
math | 22. Let $S$ be the set of points whose coordinates $x, y$ and $z$ are integers that satisfy $0 \leq x \leq 2,0 \leq y \leq 3$ and $0 \leq z \leq 4$. Two distinct points are randomly chosen from $S$. Find the probability that the midpoint of the two chosen points also belongs to $S$ ?
設 $S$ 爲座標 $x 、 y 、 z$ 皆臬整數, 且滿足 $0 ... | \frac{23}{177} | 188 | 10 |
math | Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$. | \frac{1}{4} | 39 | 7 |
math | 10. (20 points) Given the function
$$
f(x)=a x \sqrt{x-1}+b x+c \text {, }
$$
where, $x \in[1,+\infty), a, b, c \in \mathbf{R}$, and $a \neq 0, f(x)$ satisfies $0<2 f(5)=3 f(10)=4 f(17) \leqslant 1$. Find the maximum value of the real number $a$. | \frac{3}{200} | 115 | 9 |
math | $14 \cdot 16$ Solve the equation $x^{2}-2 x-3=12 \cdot\left[\frac{x-1}{2}\right]$.
(China Sichuan Province Junior High School Mathematics League, 1990) | 1+2\sqrt{7} | 57 | 8 |
math | Find the locus of the foot of the perpendicular from the center of a rectangular hyperbola to a tangent. Obtain its equation in polar coordinates and sketch it. | r^2 = 2 \sin(2\theta) | 32 | 13 |
math | Let $n,k$ be given natural numbers. Find the smallest possible cardinality of a set $A$ with the following property: There exist subsets $A_1,A_2,\ldots,A_n$ of $A$ such that the union of any $k$ of them is $A$, but the union of any $k-1$ of them is never $A$. | \binom{n}{n-k+1} | 79 | 10 |
math | 12.339. The acute angle of the rhombus at the base of a quadrilateral pyramid is $\alpha$. The ratio of the total surface area of the pyramid to the square of the side of the base is $k$. Find the sine of the angle between the apothem and the height of the pyramid, given that all its lateral faces are equally inclined ... | \frac{\sin\alpha}{k-\sin\alpha};k>2\sin\alpha | 92 | 20 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\text { GCD }(a ; b ; c)=33, \\
\text { LCM }(a ; b ; c)=3^{19} \cdot 11^{15} .
\end{array}\right.
$$ | 9072 | 91 | 4 |
math | Ada is younger than Darwyn. Max is younger than Greta. James is older than Darwyn. Max and James are the same age. Which of the five people is the oldest? | Greta | 38 | 2 |
math | Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}
| (2, 2, 2) | 84 | 10 |
math | The sum
$$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$
$ $ \\
can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$. | 977 | 98 | 3 |
math | 5. On New Year's Day, January 1st, the Elderly Sage was reflecting on his life. He noticed that over the past 5 years, all days of the week had been equally represented, and 10 years ago, the New Year's celebration fell on a Friday. On which day of the week did the sage's somber reflections take place? It is known that... | Thursday | 105 | 1 |
math | 7.214. $9^{x}+6^{x}=2^{2 x+1}$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
7.214. $9^{x}+6^{x}=2^{2 x+1}$. | 0 | 72 | 1 |
math | 6. A farmer presented 6 types of sour cream in barrels of $9,13,17,19,20,38$ liters at the market. On the first day, he sold sour cream from three barrels completely, and on the second day, from two more barrels completely. The volume of sour cream sold on the first day was twice the volume of sour cream sold on the se... | 66 | 117 | 2 |
math | 【Example 2】To place $n+1$ distinct balls into $n$ distinct boxes, how many ways are there such that no box is empty? | C_{n}^{2}\cdotn! | 33 | 10 |
math | 295. Find the coefficient of $x^{50}$ after expanding the brackets and combining like terms in the expressions:
a) $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+\ldots+x^{1000}$;
b) $(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+1000(1+x)^{1000}$. | C_{1001}^{50} | 115 | 11 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=\log _{3}(x+5)$ | y^{(n)}=\frac{(-1)^{n-1}\cdot(n-1)!}{\ln3\cdot(x+5)^{n}} | 26 | 34 |
math | 6. A triangle has side lengths $7,11,14$. Find the length of its inradius. | \frac{3\sqrt{10}}{4} | 24 | 13 |
math | 1. How many ordered (integer) quadruples $(i, j, k, h)$ satisfy $1 \leqslant i<j \leqslant k<h \leqslant n+1$? | C_{n+2}^{4} | 46 | 9 |
math | 15. (6 points) Given a three-digit number $\mathrm{abc}$, and $a(b+c)=33, b(a+c)=40$, then this three-digit number is $\qquad$ | 347 | 44 | 3 |
math | Example 18. In a workshop, two motors are working independently of each other. The probability that the first motor will not require the master's attention during an hour is 0.85, and for the second motor, this probability is 0.8. Find the probability that during the hour neither of the motors will require the master's... | 0.68 | 73 | 4 |
math | How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$? | 3^n | 32 | 2 |
math | Divide into two parts, each part containing an odd number of edges of $P$, then this diagonal is called a "good edge". It is stipulated that each edge of $P$ is a "good edge".
Given 2003 non-intersecting diagonals inside $P$ that partition $P$ into several triangles. How many isosceles triangles with two "good edges" ... | 1003 | 92 | 4 |
math | 3. The 29th Summer Olympic Games were held in Beijing on August 8, 2008, forming a memorable number 20080808. The number of different positive divisors of 20080808 divided by 8 is $\qquad$ | 8 | 65 | 1 |
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