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 | 3. Compute the greatest common divisor of $4^{8}-1$ and $8^{12}-1$. | 15 | 24 | 2 |
math | 5. Let $x, y, z$ be positive integers such that
$$
\begin{array}{l}
(x+y)(y+z)=2016 \\
(x+y)(z+x)=1080
\end{array}
$$
Determine the smallest possible value for $x+y+z$. | 61 | 66 | 2 |
math | A right, quadrilateral pyramid is constructed around a sphere with radius $r$. What is the volume of this pyramid if the center of the sphere divides the height of the pyramid in the ratio $p: q$? | \frac{4}{3}\frac{r^{3}(q+p)^{2}}{p(q-p)} | 44 | 24 |
math | Example 6 Find all positive integer solutions of the equation $3^{x}-5^{y}=z^{2}$. ${ }^{[5]}$
(2009, Balkan Mathematical Olympiad) | (x, y, z)=(2,1,2) | 44 | 12 |
math | Question 114, Given the equation $\mathrm{x}^{10}+(13 \mathrm{x}-1)^{10}=0$ has 5 pairs of conjugate complex roots $\mathrm{r}_{\mathrm{k}} 、 \overline{\mathrm{r}_{\mathrm{k}}}(1 \leq \mathrm{k} \leq 5)$, try to find the value of $\sum_{\mathrm{k}=1}^{5} \frac{1}{\mathrm{r}_{\mathrm{k}} \cdot \mathrm{r}_{\mathrm{k}}}$. | 850 | 124 | 3 |
math | \section*{Problem 3 - 031033}
Two students are given the task to multiply two numbers \(a\) and \(b\) \((a>0, b>0)\). To check their work, they divide the product by the smaller factor. The first student gets 575 with a remainder of 227. The second student gets 572 with a remainder of 308. Each had forgotten to add a ... | 576,327 | 153 | 7 |
math | 4. Let the $n$ roots of the equation $x^{n}=1$ (where $n$ is an odd number) be $1, x_{1}, x_{2}, \cdots, x_{n-1}$. Then $\sum_{i=1}^{n-1} \frac{1}{1+x_{i}}=$ $\qquad$ . | \frac{n-1}{2} | 80 | 8 |
math | 3. Find the number of pairs of consecutive integers in the set $\{1000,1001,1002, \ldots, 2000\}$ such that no carrying is required when the two integers are added.
(1 mark)
3. 在集合 $\{1000,1001,1002, \ldots, 2000\}$ 中, 有多少對連續整數加起來時不用進位? | 156 | 105 | 3 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2 ; 3 ; 1)$
$M_{2}(4 ; 1 ;-2)$
$M_{3}(6 ; 3 ; 7)$
$M_{0}(-5 ;-4 ; 8)$ | 11 | 89 | 2 |
math | 7.274. $\left\{\begin{array}{l}\left(\log _{a} x+\log _{a} y-2\right) \log _{18} a=1, \\ 2 x+y-20 a=0 .\end{array}\right.$ | (18),(92) | 66 | 7 |
math | Let $S$ be the number of bijective functions $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$ such that $f((m+n)\pmod{17})$ is divisible by $17$ if and only if $f(m)+f(n)$ is divisible by $17$. Compute the largest positive integer $n$ such that $2^n$ divides $S$. | 270 | 98 | 3 |
math | 2.289. Represent the polynomial $x^{8}-16$ as a product of polynomials of the second degree. | (x^{2}-2)(x^{2}+2)(x^{2}-2x+2)(x^{2}+2x+2) | 28 | 32 |
math | 25. [14] A particular coin can land on heads $(\mathrm{H})$, on tails $(\mathrm{T})$, or in the middle $(\mathrm{M})$, each with probability $\frac{1}{3}$. Find the expected number of flips necessary to observe the contiguous sequence HMMTHMMT...HMMT, where the sequence HMMT is repeated 2016 times. | \frac{3^{8068}-81}{80} | 88 | 16 |
math | ## Task A-2.2. (4 points)
Given are two quadratic equations
$$
x^{2}+a x+1=0, \quad x^{2}+x+a=0
$$
Determine all values of the parameter $a$ for which these equations have at least one common solution. | =-2=1 | 67 | 4 |
math | Example 43 Let $T$ be the set of all positive divisors of $60^{100}$. $S$ is a subset of $T$ in which no number is a multiple of another. Find the maximum value of $|S|$.
| 10201 | 56 | 5 |
math | Task 3. Represent the number 1000000 as a product of two numbers, in whose notation not a single zero appears. | 1000000=15625\cdot64 | 31 | 17 |
math | 2. In $\triangle A B C$, $A B=4, C A: C B=5: 3$. Try to find $\left(S_{\triangle A B C}\right)_{\max }$. | 7.5 | 46 | 3 |
math | ## Subject I
Determine $x, y, z \geq 0$ such that:
$$
\left\{\begin{array}{c}
\sqrt{x}+\sqrt{y}+\sqrt{z}=3 \sqrt{2} \\
2^{x^{2}+y}+2^{y^{2}+z}+2^{z^{2}+x}=192
\end{array}\right.
$$ | 2 | 95 | 1 |
math | (1) In a non-obtuse $\triangle A B C$, $A B>A C, \angle B=45^{\circ}, O$ and $I$ are the circumcenter and incenter of $\triangle A B C$, respectively, and $\sqrt{2} O I=A B-A C$. Find $\sin A$. | \sinA=\frac{\sqrt{2}}{2} | 71 | 13 |
math | 1. In the geometric sequence $\left\{a_{n}\right\}$, $a_{2}=\sqrt{2}, a_{3}=\sqrt[3]{3}$, then the value of $\frac{a_{1}+a_{2011}}{a_{7}+a_{2017}}$ is | \frac{8}{9} | 74 | 7 |
math | Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$. | 1997 | 66 | 4 |
math | 3. Find (1) the smallest positive period of $f(x)=\sin \frac{1}{2} x+\cos \frac{1}{3} x$;
(2) the smallest positive period of $g(x)=\sin 3 x \cdot \cos 4 x$. | 2 \pi | 62 | 3 |
math | 11. (3 points) Xiaopang used two stopwatches to measure the speed of a train. He found that this train took 40 seconds to pass over a 660-meter-long bridge, and it took 10 seconds to pass by him at the same speed. Please calculate the length of the train based on the data provided by Xiaopang. $\qquad$ meters. | 220 | 84 | 3 |
math | Example 5. Find the radius of curvature of the cycloid $x=a(t-\sin t), y=$ $=a(1-\cos t)$ at any point of it. | 4|\sin\frac{}{2}| | 38 | 9 |
math | One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$.
What integer was removed ? | 61 | 66 | 2 |
math | ## Task B-4.3.
Determine the values of the real parameter $a$, if the coefficient of the linear term in the expansion of the binomial $\left(x+\frac{1}{a x^{2}}\right)^{7}$ is equal to $\frac{7}{3}$? | \3 | 63 | 2 |
math | 1166. Find the integrals:
1) $\int\left(5 x^{3}-4 x^{2}+2\right) d x$
2) $\int \frac{d x}{x-2}$
3) $\int \frac{5 d x}{(x+3)^{7}}$
4) $\int \frac{2 x-5}{x^{2}+4 x+8} d x$ | \begin{aligned}1)&\quad\frac{5}{4}x^{4}-\frac{4}{3}x^{3}+2x+C\\2)&\quad\ln|x-2|+C\\3)&\quad-\frac{5}{6(x+3)^{6}}+C\\4)&\quad\ln(x^{2}+4x+8)-\frac | 95 | 87 |
math | 18. [12] Triangle $A B C$ has side lengths $A B=19, B C=20$, and $C A=21$. Points $X$ and $Y$ are selected on sides $A B$ and $A C$, respectively, such that $A Y=X Y$ and $X Y$ is tangent to the incircle of $\triangle A B C$. If the length of segment $A X$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relative... | 6710 | 129 | 4 |
math | A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$. | 89 | 62 | 2 |
math | G2.1 If $\frac{137}{a}=0.1 \dot{2} 3 \dot{4}$, find the value of $a$.
G2.2 If $b=1999 \times 19981998-1998 \times 19991999+1$, find the value of $b$.
G2.3 If the parametric equation $\left\{\begin{array}{l}x=\sqrt{3-t^{2}} \\ y=t-3\end{array}\right.$ can be transformed into $x^{2}+y^{2}+c x+d y+6=0$, find the values of... | =1110,b=1,=0,=6 | 163 | 14 |
math | 7. (5 points) It's the New Year, and the students are going to make some handicrafts to give to the elderly in the nursing home. At the beginning, the students in the art group work for one day, then 15 more students join them and they work together for two more days, just completing the task. Assuming each student has... | 10 | 107 | 2 |
math | Dudeney, Amusements in Mathematics Problem 16 Mr Morgan G Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resol... | 823543,117649,16807,2401,343,49,7,1 | 241 | 35 |
math | 6. (10 points) A convoy of trucks is delivering supplies to a disaster victim resettlement point. Each truck has a carrying capacity of 10 tons. If each tent is allocated 1.5 tons of supplies, there will be less than one truck's worth of supplies left over. If each tent is allocated 1.6 tons of supplies, there will be ... | 213 | 104 | 3 |
math | Example 2 If $a+b+c=a b c \neq 0$, find the value of $\frac{\left(1-a^{2}\right)\left(1-b^{2}\right)}{a b}+\frac{\left(1-b^{2}\right)\left(1-c^{2}\right)}{b c}+$ $\frac{\left(1-c^{2}\right)\left(1-a^{2}\right)}{a c}$.
(1990, Wuhan City Mathematics Competition) | 4 | 110 | 1 |
math | 5. (Dongtai City) In the tetrahedron $S-ABC$, $SA$ is perpendicular to the base $ABC$, and the lateral faces $SAB$ and $SBC$ form a right dihedral angle. If $\angle BSC = 45^{\circ}, SB = a$, find the volume of the circumscribed sphere of this tetrahedron. | \frac{\sqrt{2}}{3} \pi a^{3} | 85 | 16 |
math | 6. A class held a math competition with a total of 10 questions, each worth 10 points. $\frac{3}{19}$ of the class got all questions correct, $\frac{13}{19}$ of the class averaged 5 correct answers, and the rest of the class got all questions wrong. The average score for this math competition in the class is points. | 50 | 82 | 2 |
math | Find all triples $(a, b, c)$ of non-negative integers satisfying $a \geqslant b \geqslant c$ and $1 \cdot a^{3}+9 \cdot b^{2}+9 \cdot c+7=1997$. | (10,10,10) | 60 | 10 |
math | Az $x^{2}-(3 a+1) x+\left(2 a^{2}-3 a-2\right)=0$ egyenletben határozzuk meg $a$ értékét úgy, hogy a gyökök valósak legyenek és négyzetösszegük minimális legyen.
| -9+6\sqrt{2} | 77 | 9 |
math | In a single-round-robin tournament, 10 chess players are participating. What is the minimum number of rounds after which a sole winner can be determined prematurely? (In each round, the participants are paired.
Win - 1 point, draw - 0.5 points, loss - 0).
# | 7 | 63 | 1 |
math | 1. Compute $2023!\cdot\left(S_{2022}-1\right)$, if $S_{n}=\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}$. | -1 | 60 | 2 |
math | Solve the following inequalities:
$$
\frac{x^{2}-1}{x^{2}+1}<\frac{x^{3}-1}{x^{3}+1}
$$ | x>1\quador\quadx<0 | 39 | 11 |
math | 19. (6 points) Xiao Ming puts 127 Go stones into several bags. No matter how many stones a child wants (not exceeding 127 stones), Xiao Ming can satisfy the request by taking out a few bags. How many bags does Xiao Ming need to prepare at least? | 7 | 62 | 1 |
math | Example 4 Given a positive integer $n$. Find the largest constant $\lambda$, such that for all positive real numbers $x_{1}, x_{2}, \cdots, x_{2 n}$ satisfying
$$
\frac{1}{2 n} \sum_{i=1}^{2 n}\left(x_{i}+2\right)^{n} \geqslant \prod_{i=1}^{2 n} x_{i}
$$
we have
$$
\frac{1}{2 n} \sum_{i=1}^{2 n}\left(x_{i}+1\right)^{n... | \frac{3^{n}}{2^{2n}} | 163 | 13 |
math | 8. If $x$ is a real number, find the smallest value of $\sqrt{x^{2}+4 x+5}+\sqrt{x^{2}-8 x+25}$.
(1 mark)若 $x$ 是實數, 求 $\sqrt{x^{2}+4 x+5}+\sqrt{x^{2}-8 x+25}$ 的最小值。 | 2\sqrt{13} | 84 | 7 |
math | 6. If $a, b, c$ are positive integers, satisfying $c=(a+b \mathrm{i})^{3}-107 \mathrm{i}$, then $c=$ | 198 | 39 | 3 |
math | Group Event 7
$O A B C$ is a tetrahedron with $O A, O B$ and $O C$ being mutually perpendicular. Given that $O A=O B=O C=6 x$.
G7.1 If the volume of $O A B C$ is $a x^{3}$, find $a$.
G7.2 If the area of $\triangle A B C$ is $b \sqrt{3} x^{2}$, find $b$.
G7.3 If the distance from $O$ to $\triangle A B C$ is $c \sqrt{3} ... | =36,b=18,=2,=6 | 203 | 13 |
math | ## Task 4 - 341234
Determine the smallest natural number $n$ with $n \geq 2$ and the following property ( $\left.{ }^{*}\right)$ :
$(*)$ In every set of $n$ natural numbers, there are (at least) two numbers whose sum or whose difference is divisible by 7. | 5 | 78 | 1 |
math | Problem 50. Two circles with radii $R$ and $r$ touch externally at point $A$. A secant is drawn through point $A$, intersecting the first circle at point $B$ and the second circle at point $C$. A tangent $B D$ is drawn from point $B$ to the second circle. Find the length of $B D$, if $B A=a$. | BD=\sqrt{\frac{R+r}{R}} | 85 | 11 |
math | ## Subject III. (20 points)
Raluca received on her birthday a sum of money equal to the arithmetic mean of the three-digit natural numbers which, when divided by 5, give a remainder of 2, when divided by 7, give a remainder of 5, and when divided by 8, give a remainder of 1. How much more money does Raluca need to buy... | 1262 | 110 | 4 |
math | $11 \cdot 90$ For any positive integer $k$, try to find the smallest positive integer $f(k)$, such that there exist 5 sets $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$, satisfying the following conditions:
(1) $\left|S_{i}\right|=k, i=1,2,3,4,5$;
(2) $S_{i} \cap S_{i+1}=\varnothing\left(S_{6}=S_{1}\right), i=1,2,3,4,5$;
(3) $\l... | f(k)=2k+1+[\frac{k-1}{}] | 187 | 15 |
math | Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is ... | 1116 | 108 | 4 |
math | \[
\left[\begin{array}{l}
\text { Equations in integers } \\
{[\text { Case enumeration }}]
\end{array}\right]
\]
Solve the system in natural numbers
\[
\begin{aligned}
& x+y=z t, \\
& z+t=x y .
\end{aligned}
\] | (1,5,2,3),(5,1,2,3),(1,5,3,2),(5,1,3,2),(2,3,1,5),(2,3,5,1),(3,2,1,5),(3,2,5,1),(2,2,2,2) | 72 | 73 |
math | Exercise 4. Find all positive integers $x$ and $y$ such that $x^{2}-2 \times y!=2021$. | (45,2) | 32 | 6 |
math | Find all quadruplets of integers $(a, b, c, d) \in \mathbb{N}_{\geqslant 2}$ such that $F_{a}+F_{b}=F_{c}+F_{d}$. | (,b,,b),(,b,b,),(,-3,-1,-1),(-3,,-1,-1),(-1,-1,,-3),(-1,-1,-3,) | 54 | 41 |
math | For any sequence of real numbers $A=\left\{a_{1}, a_{2}, \ldots\right\}$, we define $\Delta A$ as the sequence $\left\{a_{2}-a_{1}, a_{3}-a_{2}, \ldots\right\}$.
We assume that all terms of the sequence $\Delta(\Delta A)$ are 1 and that $a_{19}=a_{92}=0$.
Determine $a_{1}$. | 819 | 106 | 3 |
math | 7. The equation $\sum_{k=1}^{2 n+1} \cos ^{2} k x=n\left(n \in \mathbf{N}_{+}\right)$ has $\qquad$ solutions in $[0,2 \pi)$. | 8n+2 | 57 | 4 |
math | Example 7 Solve the system of equations: $\left\{\begin{array}{l}x^{4}+y^{2}+4=5 y z, \\ y^{4}+z^{2}+4=5 z x, \\ z^{4}+x^{2}+4=5 x y .\end{array}\right.$ $[6]$
(61st Czech and Slovak Mathematical Olympiad) | (\\sqrt{2},\\sqrt{2},\\sqrt{2}) | 92 | 16 |
math | 10. (3 points) There are three people, $A$, $B$, and $C$, who are a worker, a teacher, and an engineer, respectively. $A$ is older than the worker, $C$ is a different age from the teacher, and the teacher is younger than $B$. Therefore, the engineer is $\qquad$ . | B | 75 | 1 |
math | 3.220
$$
\frac{1+\cos (2 \alpha-2 \pi)+\cos (4 \alpha+2 \pi)-\cos (6 \alpha-\pi)}{\cos (2 \pi-2 \alpha)+2 \cos ^{2}(2 \alpha+\pi)-1}=2 \cos 2 \alpha
$$ | 2\cos2\alpha | 78 | 6 |
math | 5. (15 points) A massive vertical plate is fixed on a car moving at a speed of $5 \mathrm{M} / \mathrm{c}$. A ball is flying towards it at a speed of $6 \mathrm{m} / \mathrm{s}$ relative to the Earth. Determine the speed of the ball relative to the Earth after a perfectly elastic normal collision. | 16\mathrm{~}/\mathrm{} | 79 | 10 |
math | 3. If the function $y=\frac{k x+5}{k x^{2}+4 k x+3}$ has the domain of all real numbers, then the range of real number $k$ is $\qquad$ | 0 \leqslant k < \frac{3}{4} | 48 | 15 |
math | Example 5 When $x \leqslant y \leqslant z$, find the positive integer solutions of the equation
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{8}
$$
(2007, Taiyuan Junior High School Mathematics Competition) | (2,3,24) \text{ and } (2,4,8) | 75 | 20 |
math | 6. Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $a_{X}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $a_{X}$ is $\qquad$ . | 1001 | 76 | 4 |
math | ## Task 4 - 240714
(a) It is assumed about the measurements in centimeters of the side lengths of a triangle:
(1) These measurements are three consecutive natural numbers.
(2) The perimeter of the triangle is 25 cm longer than the shortest side of the triangle. Determine the three side lengths from these assumptions... | nisanoddn\geq7 | 177 | 8 |
math | Three, (25 points) Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+2 b^{2}+3 c^{2}=2008$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (14,30,2),(10,30,6),(22,6,22),(4,30,8) | 69 | 32 |
math | 16th Putnam 1956 Problem A1 α ≠ 1 is a positive real. Find lim x→∞ ( (α x - 1)/(αx - x) ) 1/x . | \begin{cases}\alpha&if\alpha>1\\1&if\alpha<1\end | 45 | 22 |
math | Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$.
$\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$ | 668 | 124 | 3 |
math | 122. The sum of several consecutive positive integers is 1000. Find these numbers. | =1000,k=0;=108,k=4;=28,k=24;=55,k=15 | 22 | 32 |
math | The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$? | 859 | 64 | 3 |
math | 2. $I=\sum_{k=1}^{10}\left(\cos \frac{k \pi}{11}\right)^{8}=$ | \frac{257}{128} | 33 | 11 |
math | 4. Tourists from the USA, when traveling to Europe, often use an approximate formula to convert temperatures in degrees Celsius $C$ to the familiar degrees Fahrenheit $F$: $F=2 C+30$. Indicate the range of temperatures (in degrees Celsius) for which the deviation of the temperature in degrees Fahrenheit, obtained using... | 1\frac{11}{29}\leqC\leq32\frac{8}{11} | 115 | 26 |
math | Problem 1. We call a 5-tuple of integers arrangeable if its elements can be labeled $a$, $b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all 2017-tuples of integers $n_{1}, n_{2}, \ldots, n_{2017}$ such that if we place them in a circle in clockwise order, then any 5 -tuple of numbers in consecutive positi... | n_{1}=\cdots=n_{2017}=29 | 111 | 16 |
math | 2. Solve the equation $\cos 8 x=\frac{14}{3}(\cos 2 x-\sin 2 x)^{2}-1$. In your answer, specify the number equal to the sum of the roots of the equation that belong to the interval $\left[\frac{11 \pi}{2} ; \frac{13 \pi}{2}\right]$, rounding this number to two decimal places if necessary. | 36.91 | 92 | 5 |
math | Example 14 Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following two conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$.
Try to find: $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$'s maximum value, and explain t... | 189548 | 170 | 6 |
math | 1. (10 points) $3 \frac{4}{5}+5 \frac{6}{7}+7 \frac{8}{9}+9 \frac{10}{11}=$ | 27\frac{1577}{3465} | 46 | 15 |
math | Example 3 Does there exist a positive integer $n$ that satisfies the following two conditions simultaneously:
(1) $n$ can be decomposed into the sum of 1990 consecutive positive integers.
(2) $n$ has exactly 1990 ways to be decomposed into the sum of several (at least two) consecutive positive integers.
(31st IMO Short... | n=5^{180} \times 199^{10}, n=5^{10} \times 199^{180} | 83 | 36 |
math | ## Task Condition
Find the derivative.
$$
y=\sqrt{x^{2}-8 x+17} \cdot \operatorname{arctg}(x-4)-\ln \left(x-4+\sqrt{x^{2}-8 x+17}\right)
$$ | \frac{x-4}{\sqrt{x^{2}-8x+17}}\cdot\operatorname{arctg}(x-4) | 59 | 32 |
math | 6. $a, b, c, d, e$ are 5 elements randomly selected from the set $\{1,2,3,4,5\}$ (repetition allowed). Then the probability that $a b c d + e$ is odd is $\qquad$ | \frac{1794}{3125} | 59 | 13 |
math | 6. Given sets:
$$
\begin{array}{l}
A=\left\{x \mid x^{2}+x-6>0\right\}, \\
B=\left\{x \mid x^{2}-2 a x+3 \leqslant 0\right\} .
\end{array}
$$
If $a>0$, and $A \cap B$ contains exactly two integers, then the range of values for $a$ is $\qquad$ . | 2.375\leqslant2.8 | 106 | 13 |
math | 2. Given $\triangle A B C$ with angles $A, B, C$ opposite to sides $a, b, c$ respectively, and $a^{2}+b^{2}=c^{2}+\frac{2}{3} a b$. If the circumradius of $\triangle A B C$ is $\frac{3 \sqrt{2}}{2}$, then the maximum area of $\triangle A B C$ is $\qquad$. | 4\sqrt{2} | 96 | 6 |
math | 3. Find all primes that can be written both as a sum of two primes and as a difference of two primes. | 5 | 24 | 1 |
math | 8.085. $4 \sin x \cos \left(\frac{\pi}{2}-x\right)+4 \sin (\pi+x) \cos x+2 \sin \left(\frac{3}{2} \pi-x\right) \cos (\pi+x)=1$. | x_{1}=\operatorname{arctg}\frac{1}{3}+\pik;x_{2}=\frac{\pi}{4}+\pin,k,n\inZ | 63 | 39 |
math | 16. The pedestrian named to the traffic inspector the number of the car whose driver had grossly violated the traffic rules. This number is expressed as a four-digit number, the unit digit of which is the same as the tens digit, and the hundreds digit is the same as the thousands digit. Moreover, this number is a perfe... | 7744 | 74 | 4 |
math | 10th Chinese 1995 Problem B1 Four solid balls radii 3, 3, 2, 2 touch each other. What is the radius of the solid ball which touches all four balls? | \frac{6}{11} | 46 | 8 |
math | Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it i... | 52 | 180 | 2 |
math | 1. Two cyclists set off simultaneously from point $A$ to point $B$. When the first cyclist had covered half the distance, the second cyclist had 24 km left to travel, and when the second cyclist had covered half the distance, the first cyclist had 15 km left to travel. Find the distance between points $A$ and $B$. | 40 | 74 | 2 |
math | Problem 9.5. Buratino has many coins of 5 and 6 soldi, more than 10 of each type. Coming to the store and buying a book for $N$ soldi, he realized that he could not pay for it without receiving change. What is the greatest value that the natural number $N$ can take if it is no more than 50? | 19 | 82 | 2 |
math | 4. Let $s=\sum_{k=1}^{2015} k 2^{k}$. Then the remainder when $s$ is divided by 100 is | 6 | 40 | 1 |
math | 4. Find all positive integers $m, n$, such that $\left|12^{m}-5^{n}\right|=7$.
| =n=1 | 30 | 3 |
math | Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that
(a) $a_1 + a_2 + .. + a_n \ge n^2$ and
(b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$ | a_i = n \text{ for all } 1 \le i \le n | 93 | 18 |
math | In a right-angled triangle, the height drawn to the hypotenuse is $12 \mathrm{~cm}$; the difference between the two segments of the hypotenuse is $7 \mathrm{~cm}$. What are the lengths of the sides of the triangle? | 15 | 58 | 2 |
math | ## Subject 1.
$$
\text { a. } \begin{aligned}
\mathrm{a} & =2^{2 n+2} \cdot 7^{n+3}+11 \cdot 2^{2 n+5} \cdot 7^{n}+2^{2 n} \cdot 7^{n+3}-27 \cdot 2^{2 n+1} \cdot 7^{n} \\
& =2^{2 n} \cdot 2^{2} \cdot 7^{n} \cdot 7^{3}+11 \cdot 2^{2 n} \cdot 2^{5} \cdot 7^{n}+2^{2 n} \cdot 7^{n} \cdot 7^{3}-27 \cdot 2^{2 n} \cdot 2^{1}... | 1 | 504 | 1 |
math | Suppose $ n$ is a positive integer and 3 arbitrary numbers numbers are chosen from the set $ 1,2,3,...,3n+1$ with their sum equal to $ 3n+1$. What is the largest possible product of those 3 numbers? | n^3 + n^2 | 58 | 7 |
math | 29.1. Find the three-digit number $\overline{a b c}$, if it is known that
$$
\begin{gathered}
8 a+5 b+c=100 \\
a+b+c=20
\end{gathered}
$$ | 866 | 58 | 3 |
math | 72*. Two hunters $A$ and $B$ went duck hunting. Suppose each of them hits a duck they encounter as often as they miss. Hunter $A$ encountered 50 ducks during the hunt, and hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch will exceed hunter $A$'s catch? | \frac{1}{2} | 77 | 7 |
math | \section*{Problem 4 - 131224}
Determine all pairs \((x, y)\) of real numbers that are solutions to the system of equations:
\[
\begin{aligned}
& x^{3}+y^{2}+x+1=0 \\
& y^{3}+x^{2}+y+1=0
\end{aligned}
\] | (-1,-1) | 87 | 5 |
math | 11. If $3^{k} \mid 1000!$ and $3^{k+1} \nmid 1000!$, find $k$. | 498 | 40 | 3 |
math | M1. The positive integer $N$ has five digits.
The six-digit integer $P$ is formed by appending the digit 2 to the front of $N$. The six-digit integer $Q$ is formed by appending the digit 2 to the end of $N$. Given that $Q=3 P$, what values of $N$ are possible? | 85714 | 74 | 5 |
math | 4A. Determine all natural numbers $n$ for which there exist three consecutive coefficients in the expansion of $(a+b)^{n}$ that form an arithmetic progression. | n=^{2}-2,for=3,4,5,\ldots | 34 | 17 |
math | G3.2 There are $R$ zeros at the end of $\underbrace{99 \ldots 9}_{2009 \text { of }} \times \underbrace{99 \ldots 9}_{2009 \text { of } 9^{\prime} s}+1 \underbrace{99 \ldots 9}_{2009 \text { of } 9^{\prime} s}$, find the value of $R$. | 4018 | 105 | 4 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.