task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | If for the real numbers $x, y, z, k$ the following conditions are valid, $x \neq y \neq z \neq x$ and $x^{3}+y^{3}+k\left(x^{2}+y^{2}\right)=y^{3}+z^{3}+k\left(y^{2}+z^{2}\right)=z^{3}+x^{3}+k\left(z^{2}+x^{2}\right)=2008$, find the product $x y z$. | x y z=1004 | 122 | 8 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x y=500 \\
& x^{\lg y}=25
\end{aligned}
$$ | x_{1}=y_{2}=100x_{2}=y_{1}=5 | 39 | 20 |
math | 7. There is a rectangular iron sheet of size $80 \times 50$. Now, a square of the same size is to be cut from each corner, and then it is to be made into an open box. What should be the side length of the square to be cut so that the volume of this open box is maximized? | 10 | 71 | 2 |
math | 7. A die is rolled twice in succession, and the numbers obtained are $a$ and $b$ respectively. Then the probability $p=$ $\qquad$ that the cubic equation $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$ has three distinct real roots is $\qquad$ .(Answer with a number). | \frac{3}{4} | 84 | 7 |
math | Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value. | \frac{3}{2}e^{-\frac{1}{3}} | 46 | 16 |
math | Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \] | 1 | 68 | 3 |
math | 5. The numbers $1,2,3, \ldots, 99$ are written on the board. Petya and Vasya are playing a game, with Petya starting. Each move involves erasing three numbers that sum to 150. The player who cannot make a move loses. Which player can win, regardless of how the opponent plays? | Petya | 79 | 3 |
math | Sure, here is the translated text:
```
II. (40 points) Find all positive integers $n$, such that for any positive real numbers $a, b, c$ satisfying $a+b+c=1$, we have
$$
a b c\left(a^{n}+b^{n}+c^{n}\right) \leqslant \frac{1}{3^{n+2}}.
$$
``` | n=1, 2 | 92 | 6 |
math | Triangle $ABC$ satisfies $\tan A \cdot \tan B = 3$ and $AB = 5$. Let $G$ and $O$ be the centroid and circumcenter of $ABC$ respectively. The maximum possible area of triangle $CGO$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $a$ and $c$ relatively prime and $b$ not divisible by ... | 100 | 114 | 3 |
math | 4. Let $f(x)=(x+a)(x+b)$ (where $a, b$ are given positive real numbers), and $n \geqslant 2$ be a given integer. For non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}+x_{2}+\cdots+x_{n}=1
$$
find the maximum value of
$$
F=\sum_{1 \leq i<j \leq n} \min \left\{f\left(x_{i}\right), f\left(x_{j}\... | \frac{n-1}{2}\left(\frac{1}{n}+a+b+n a b\right) | 145 | 25 |
math | Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had t... | EDACB | 121 | 4 |
math | Tokarev S.i.
Can one of the factorials be erased from the product $1!\cdot 2!\cdot 3!\cdot \ldots \cdot 100!$ so that the product of the remaining ones is a square of an integer? | 50! | 55 | 3 |
math | 9.207. $\left\{\begin{array}{l}0,2^{\cos x} \leq 1, \\ \frac{x-1}{2-x}+\frac{1}{2}>0 .\end{array}\right.$ | x\in(0;\frac{\pi}{2}] | 56 | 12 |
math | 4. In $\triangle A B C$, $\angle B A C=60^{\circ}$, the angle bisector $A D$ of $\angle B A C$ intersects $B C$ at $D$, and $\overrightarrow{A D}=\frac{1}{4} \overrightarrow{A C}+t \overrightarrow{A B}$. If $A B=8$, then $A D=$ . $\qquad$ | 6\sqrt{3} | 95 | 6 |
math | 11.48 Which is greater: $3^{400}$ or $4^{300} ?$ | 3^{400}>4^{300} | 26 | 12 |
math | Alice is given a rational number $r>1$ and a line with two points $B \neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is p... | r = \frac{q+1}{q} \text{ for some integer } q \text{ with } 1 \leqslant q \leqslant 1010 | 178 | 42 |
math | # Problem No. 6 (10 points)
A pot was filled with $2 \pi$ liters of water, taken at a temperature of $t=0{ }^{\circ} C$, and brought to a boil in 10 minutes. After that, without removing the pot from the stove, ice at a temperature of $t=0{ }^{\circ} \mathrm{C}$ was added. The water began to boil again only after 15 m... | 1.68 | 205 | 4 |
math | 4. A three-digit number, all digits of which are different and non-zero, will be called balanced if it is equal to the sum of all possible two-digit numbers formed from the different digits of this number. Find the smallest balanced number. | 132 | 49 | 3 |
math | A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle. | 24 | 42 | 2 |
math | 8. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is
保留源文本的换行和格式，翻译结果如下：
8. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is | 17 | 62 | 2 |
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 | Example 3 If $a \neq 1, b \neq 1, a>0, b>0$, and satisfy the relation $\log _{a} 2=\log _{\frac{a}{2}} 4=\log _{b} 3$, find the values of $a$ and $b$. | =\frac{1}{2},b=\frac{1}{3} | 71 | 15 |
math | 13. (1993 Putnam Mathematical Competition, 53rd USA) Let $S$ be a set of $n$ distinct real numbers, and let $A_{s}$ be the set of all distinct averages of pairs of elements of $S$. For a given $n \geqslant 2$, what is the least number of elements that $A_{s}$ can have? | 2n-3 | 85 | 4 |
math | 1. How many strikes do the clocks make in a day if they strike once every half hour, and at each hour $1,2,3 \ldots 12$ times? | 180 | 39 | 3 |
math | 5. If for any $x \in[0,1]$, we have
$$
f(x)=k\left(x^{2}-x+1\right)-x^{4}(1-x)^{4} \geqslant 0 \text {, }
$$
then the minimum value of $k$ is $\qquad$ . | \frac{1}{192} | 74 | 9 |
math | 1. If for all $x$ such that $|x| \leqslant 1$, $t+1>(t^2-4)x$ always holds, then the range of values for $t$ is $\qquad$ | (\frac{\sqrt{13}-1}{2},\frac{\sqrt{21}+1}{2}) | 51 | 25 |
math | 6. [6] Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6 -sided die. If she gets a 1 or 2 , she goes to DC. If she rolls a 3,4 , or 5 , she goes to Russia. If she rolls a 6 , she rolls again. What is the probability that she goes to DC? | \frac{2}{5} | 89 | 7 |
math | Example 3.25. Find the gradient of the function $z=x^{2}-x y+y^{3}$ at the point $A(1, -1)$ and the derivative in the direction of the vector $\bar{a}=3 \bar{i}-4 \bar{j}$. | \frac{1}{5} | 61 | 7 |
math | 45th Putnam 1984 Problem B1 Define f(n) = 1! + 2! + ... + n! . Find a recurrence relation f(n + 2) = a(n) f(n + 1) + b(n) f(n), where a(x) and b(x) are polynomials. | f(n+2)=(n+3)f(n+1)-(n+2)f(n) | 70 | 19 |
math | 166. The sum of the reciprocals of three positive integers is equal to one. What are these numbers? | \frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1;\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1;\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1 | 25 | 66 |
math | 8. Let $a_{1}=1, a_{2}=2$, for $n \geqslant 2$ we have
$$
a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1} .
$$
If for all positive integers $n \geqslant m$, we have $a_{n}>2+$ $\frac{2008}{2009}$, then the smallest positive integer $m$ is $\qquad$ . | 4019 | 122 | 4 |
math | Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$ | n = 7 | 68 | 5 |
math | 10. The number of positive integers not exceeding 2012 and having exactly three positive divisors is $\qquad$ . | 14 | 28 | 2 |
math | Sonkin $M$.
Solve the equation $\left(x^{2}-y^{2}\right)^{2}=1+16 y$ in integers. | (\1,0),(\4,3),(\4,5) | 34 | 15 |
math | Let's determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for any $x, y$, $f(x+y) + f(x) f(y) = x^{2} y^{2} + 2 x y$. | f(x)=x^{2}-1 | 58 | 8 |
math | Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
[i]Proposed by Kevin Sun[/i] | 450 | 57 | 3 |
math | Example 5 Let $x, y, z, w$ be four real numbers, not all zero. Find:
$S=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$'s maximum value. | \frac{1}{2}(1+\sqrt{2}) | 62 | 13 |
math | (12) If for all positive real numbers $x, y$, the inequality $\frac{y}{4}-\cos ^{2} x \geqslant a \sin x- \frac{9}{y}$ holds, then the range of the real number $a$ is $\qquad$. | [-3,3] | 65 | 5 |
math | Example 9 Given $x^{2}-y^{2}=16$. Find the minimum and maximum values of the function
$$
f(x, y)=\frac{1}{x^{2}}+\frac{y}{8 x}+1
$$ | \frac{9}{8} \text{ and } \frac{7}{8} | 54 | 19 |
math | 5.10. Given vectors $\bar{a}(6 ;-8 ; 5 \sqrt{2})$ and $\bar{b}(2 ;-4 ; \sqrt{2})$. Find the angle formed by the vector $\bar{a}-\bar{b}$ with the $O z$ axis. | 45 | 65 | 2 |
math | $\begin{aligned} & \text { [ Examples and counterexamples. Constructions ] } \\ & \text { [Numerical inequalities. Comparing numbers.] }\end{aligned}$
The sum of several numbers is 1. Can the sum of their squares be less than 0.1? | 1/11<0.1 | 62 | 8 |
math | 9. (16 points) Given that $f(x)$ is a function defined on the set of real numbers $\mathbf{R}$, $f(0)=2$, and for any $x \in \mathbf{R}$, we have
$$
\begin{array}{l}
f(5+2 x)=f(-5-4 x), \\
f(3 x-2)=f(5-6 x) .
\end{array}
$$
Find the value of $f(2012)$. | 2 | 113 | 1 |
math | ## SUBJECT I
Solve the equation $\frac{x-2}{x}+\frac{x-4}{x}+\frac{x-6}{x}+\ldots . . .+\frac{2}{x}=12$ | 50 | 48 | 2 |
math | Example 4. Calculate the integral
$$
\int_{1-i}^{2+i}\left(3 z^{2}+2 z\right) d z
$$ | 7+19i | 37 | 5 |
math | G3.3 Let $x$ and $y$ be positive real numbers with $x<y$. If $\sqrt{x}+\sqrt{y}=1$ and $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{10}{3}$ and $x<y$, find the value of $y-x$. | \frac{1}{2} | 72 | 7 |
math | In a grid containing 50 points by 50 points, each point has been painted either blue or red. Directly adjacent points, whether horizontally or vertically, that are of the same color are connected by segments of the same color, while points of different colors are connected by black segments. Among the points, 1510 were... | 1976 | 111 | 4 |
math | Exercise 1. N.B. In this exercise, and only this one, an answer without justification is requested.
Let $m>n>p$ be three (positive integer) prime numbers such that $m+n+p=74$ and $m-n-p=44$. Determine $m, n$, and $p$.
(A prime number is an integer strictly greater than one, and whose only divisors are one and itself.... | =59,n=13,p=2 | 88 | 10 |
math | Example 13. Find $\int \frac{d x}{x^{2} \sqrt{1+x^{2}}}$. | C-\frac{\sqrt{1+x^{2}}}{x} | 27 | 14 |
math | 7. Find all values of $\alpha$ such that all terms in the sequence $\cos \alpha, \cos 2 \alpha, \cdots, \cos 2^{n} \alpha \cdots$ are negative. | \alpha=2k\pi\\frac{2\pi}{3},k\in\mathbb{Z} | 48 | 25 |
math | 1. Given the equations
$$
x^{2}-a x+b-4=0 \text { and } y^{2}-b y+a-\frac{1}{4}=0, a, b \in \mathbb{R}
$$
For which values of $a$ and $b$, the roots of the second equation are the reciprocal values of the roots of the first equation? | a_{1}=0,a_{2}=-\frac{7}{4},a_{3}=\frac{9}{4};b_{1}=0,b_{2}=\frac{7}{2},b_{3}=\frac{9}{2} | 82 | 55 |
math | \section*{Problem 6}
Find all integers \(x, y\) satisfying \(x^{2}+x=y^{4}+y^{3}+y^{2}+y\).
| x,y=-1,1;0,-1;-1,0;0,0;-6,2;5,2 | 42 | 26 |
math | Let $S_{n}$ denote the sum of the digits of the number $n$ in the decimal system. Determine all numbers $n$ for which $n=S^{2}(n)-S(n)+1$ | 1,13,43,91,157 | 44 | 14 |
math | 1. Let the function $f(x)=x \sin x(x \in \mathbf{R})$ attain an extremum at $x=x_{0}$, then $\left(1+x_{0}^{2}\right)\left(1+\cos 2 x_{0}\right)=$
$\qquad$ . | 2 | 69 | 1 |
math | 1. A certain 4-digit number was added to the number written with the same digits but in reverse order, and the result was 4983. What numbers were added | 19922991 | 37 | 8 |
math | Let $m=999 \ldots 99$ be the number formed by 77 digits all equal to 9 and let $n=777 \ldots 77$ be the number formed by 99 digits all equal to 7. What is the number of digits of $m \cdot n$? | 176 | 72 | 3 |
math | 6. B. Given that the lengths of the two legs are integers $a$ and $b$ $(b<2011)$. Then the number of right triangles with the hypotenuse length $b+1$ is | 31 | 48 | 2 |
math | # Task 1. (10 points)
How many natural numbers $n$ exist such that the equation $n x-12=3 n$ has an integer solution
# | 6 | 38 | 1 |
math | (1) Given the sets $A=\left\{y \mid y=x^{2}+2 x-3\right\}, B=\left\{y \left\lvert\, y=x+\frac{1}{x}\right., x<\right.$ 0), then $A \cap B=$ . $\qquad$ | [-4,-2] | 73 | 5 |
math | 5. Each pair of numbers $A$ and $B$ is assigned a number $A * B$. Find $2021 * 1999$, if it is known that for any three numbers $A, B, C$ the identities are satisfied: $A * A=0$ and $A *(B * C)=(A * B)+C$. | 22 | 77 | 2 |
math | We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers $n$ such that $n$ has a multiple which is alternating. | 20 \nmid n | 46 | 6 |
math | Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$. | \frac{2}{2024} | 117 | 10 |
math | $8 \cdot 57$ a (finite) sequence of the same non-zero digit can be the ending of a perfect square. Find the maximum length of such a sequence, and the smallest square number whose ending is such a sequence.
(31st Putnam Mathematical Competition, 1970) | 1444 | 64 | 4 |
math | Find the angle between the line of intersection of the planes $2 x-y-3 z+5=0$ and $x+y-2=0$ and the plane passing through the points $M(-2 ; 0 ; 3), N(0 ; 2 ; 2)$ and $K(3 ;-3 ; 1)$.
# | \arcsin\frac{22}{3\sqrt{102}} | 73 | 18 |
math | 2. A team consisting of boys and girls from the Rostov region went to a CS:GO esports tournament. The average number of points scored by the girls turned out to be 22, by the boys - 47, and the average number of points for the entire team - 41. What is the percentage of girls in this team? | 24 | 74 | 2 |
math | 6.081. $\left\{\begin{array}{l}x^{3}+y^{3}=65 \\ x^{2} y+x y^{2}=20\end{array}\right.$ | (4;1)(1;4) | 47 | 9 |
math | 5. Ante, Bruno, Ciprijan, Davor, Emanuel, and Franko need to line up in a row.
a) In how many different ways can the boys line up if Bruno stands to the left of Emanuel?
b) In how many different ways can the boys line up if there is no one standing between Ciprijan and Davor?
The use of a pocket calculator or any re... | 360 | 135 | 3 |
math | 110) Let $p$ be a given odd prime, and let the positive integer $k$ be such that $\sqrt{k^{2}-p k}$ is also a positive integer. Then $k=$ $\qquad$ | \frac{(p+1)^{2}}{4} | 48 | 13 |
math | Test $\mathbf{A}$ Given that $a, b, c, d, e$ are real numbers satisfying
$$
\begin{array}{c}
a+b+c+d+e=8, \\
a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16
\end{array}
$$
determine the maximum value of $e$. | \frac{16}{5} | 87 | 8 |
math | Let's determine the maximum of the function $f(x)=\sqrt{x-2}+\sqrt{2 x-7}+\sqrt{18-3 x}$. | 3\sqrt{3} | 36 | 6 |
math | Find all positive integers $(x, y, n, k)$ such that $x$ and $y$ are coprime and such that
$$
3^{n}=x^{k}+y^{k}
$$ | (x,y,k)=(2,1,3) | 46 | 10 |
math | Task 3. (15 points) The function $f(x)$ satisfies the condition: for any real numbers $a$ and $b$, the equality $f\left(\frac{a+2 b}{3}\right)=\frac{f(a)+2 f(b)}{3}$ holds. Find the value of the function $f(2021)$, if $f(1)=5, f(4)=2$. | -2015 | 92 | 5 |
math | $$
\begin{aligned}
f(x)= & |a \sin x+b \cos x-1|+ \\
& |b \sin x-a \cos x| \quad(a, b \in \mathbf{R})
\end{aligned}
$$
If the maximum value of the function is 11, then $a^{2}+b^{2}=$ $\qquad$ . | 50 | 85 | 2 |
math | 13.366. On a river with a current speed of 5 km/h, there are piers $A, B$, and $C$ in the direction of the current, with $B$ located halfway between $A$ and $C$. From pier $B$, a raft and a boat depart simultaneously in the direction of the current towards pier $C$, and the boat heads towards pier $A$, with the boat's ... | 5<V<15 | 139 | 5 |
math | Tobias downloads $m$ apps. Each app costs $\$ 2.00$ plus $10 \%$ tax. He spends $\$ 52.80$ in total on these $m$ apps. What is the value of $m$ ?
(A) 20
(B) 22
(C) 18
(D) 24
(E) 26 | 24 | 86 | 2 |
math | Let $S$ be a subset of $\{1,2, \ldots, 9\}$, such that the sums formed by adding each unordered pair of distinct numbers from $S$ are all different. For example, the subset $\{1,2,3,5\}$ has this property, but $\{1,2,3,4,5\}$ does not, since the pairs $\{1,4\}$ and $\{2,3\}$ have the same sum, namely 5.
What is the ma... | 5 | 122 | 1 |
math | 2. Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. Let $D, E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $B C$, respectively. If $D E=3$ and $E F=5$, compute the length of $B C$. | 20 | 80 | 2 |
math | 5. Given the function $f(x)$
$=\log _{a}\left(a x^{2}-x+\frac{1}{2}\right)$ is always positive on the interval $[1,2]$, then the range of the real number $a$ is $\qquad$ | (\frac{1}{2},\frac{5}{8})\cup(\frac{3}{2},+\infty) | 61 | 27 |
math | 8. $[\mathbf{6}]$ Let $A:=\mathbb{Q} \backslash\{0,1\}$ denote the set of all rationals other than 0 and 1. A function $f: A \rightarrow \mathbb{R}$ has the property that for all $x \in A$,
$$
f(x)+f\left(1-\frac{1}{x}\right)=\log |x| .
$$
Compute the value of $f(2007)$. | \log(2007/2006) | 112 | 13 |
math | Example 4 (2002 China Western Mathematical Olympiad) Consider a square on the complex plane, whose 4 vertices correspond to the 4 roots of a certain monic quartic equation with integer coefficients $x^{4}+p x^{3}+q x^{2}+r x+s=0$. Find the minimum value of the area of such a square.
| 2 | 79 | 1 |
math | $\square$ Example 1 Let real numbers $a_{1}, a_{2}, \cdots, a_{100}$ satisfy $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0, a_{1}+$ $a_{2} \leqslant 100, a_{3}+a_{4}+\cdots+a_{100} \leqslant 100$, determine the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$, and find the sequence $a_{... | 10000 | 192 | 5 |
math | 5. Determine the angle between the tangents to the parabola $y^{2}=4 x$ at the points of its intersection $\mathrm{s}$
(8) with the line $2 x+y-12=0$. | 45 | 49 | 2 |
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 | ## Zadatak B-4.7.
Odredite sve prirodne brojeve $x$ koji su rješenje nejednadžbe
$$
\log _{x}^{4} 2017+6 \cdot \log _{x}^{2} 2017>4 \cdot \log _{x}^{3} 2017+4 \cdot \log _{x} 2017
$$
| x\in{2,3,4,5,6,\ldots,44} | 107 | 20 |
math | 7. In the tetrahedron $S-ABC$,
$$
SA=SB=SC=\sqrt{21}, BC=6 \text{. }
$$
If the projection of point $A$ onto the plane of the side $SBC$ is exactly the orthocenter of $\triangle SBC$, then the volume of the inscribed sphere of the tetrahedron $S-ABC$ is . $\qquad$ | \frac{4\pi}{3} | 92 | 9 |
math | Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$.
Find the maximum n... | \frac{n(n-1)}{2} | 112 | 10 |
math | Solve the following equation:
$$
\sqrt{4^{x}+\frac{17}{64}}-\sqrt{2 \cdot 4^{x}-\frac{7}{64}}=\sqrt{4^{x}-\frac{1}{16}}
$$ | -\frac{3}{2} | 59 | 7 |
math | You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}
$\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$
Find a general formula for $x\diamond y$, assuming that $y\gex>0$. | (y - x)^x | 105 | 7 |
math | Find all positive integers $ n$ such that $ 20^n \minus{} 13^n \minus{} 7^n$ is divisible by $ 309$. | n = 1 + 6k | 37 | 9 |
math | 8.308. $\cos 3z - \cos^3 z + \frac{3}{4} \sin 2z = 0$. | z_{1}=\frac{\pi}{2}(2k+1);z_{2}=\pin;z_{3}=(-1)^{k}\frac{\pi}{6}+\pi, | 34 | 41 |
math | 3. [4] Three real numbers $x, y$, and $z$ are such that $(x+4) / 2=(y+9) /(z-3)=(x+5) /(z-5)$. Determine the value of $x / y$. | \frac{1}{2} | 57 | 7 |
math | \section*{Problem 2}
Find the smallest positive integer which can be represented as \(36^{\mathrm{m}}-5^{\mathrm{n}}\).
\section*{Answer}
11
| 11 | 45 | 2 |
math | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-1, -2, 1), B(-4, -2, 5), C(-8, -2, 2)$ | \frac{1}{\sqrt{2}} | 59 | 10 |
math | 25. $[\mathbf{1 3}]$ Evaluate the sum
$$
\cos \left(\frac{2 \pi}{18}\right)+\cos \left(\frac{4 \pi}{18}\right)+\cdots+\cos \left(\frac{34 \pi}{18}\right) .
$$ | -1 | 73 | 2 |
math | ## Task Condition
Write the canonical equations of the line.
$$
\begin{aligned}
& 2 x+3 y-2 z+6=0 \\
& x-3 y+z+3=0
\end{aligned}
$$ | \frac{x+3}{-3}=\frac{y}{-4}=\frac{z}{-9} | 51 | 25 |
math | Example 6. Calculate $\int_{1}^{e} \ln x d x$. | 1 | 19 | 1 |
math | Example 10 Given $\left(x^{2}-x+1\right)^{6}=a_{12} x^{12}+$
$$
\begin{array}{l}
a_{11} x^{11}+\cdots+a_{2} x^{2}+a_{1} x+a_{0} \text {. Then } a_{12}+a_{10} \\
+\cdots+a_{2}+a_{0}=\ldots
\end{array}
$$
(Fifth National Partial Provinces Junior High School Mathematics Correspondence Competition) | 365 | 126 | 3 |
math | (8) Let $S(n)$ denote the sum of the digits of the positive integer $n$, then $\sum_{n=1}^{2011} S(n)=$ | 28072 | 39 | 5 |
math | Sergeev I.n.
The sum of the absolute values of the terms of a finite arithmetic progression is 250. If all its terms are increased by 1 or all its terms are increased by 2, then in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 250. What values can the quantit... | \1000 | 112 | 5 |
math | How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 12 | 67 | 2 |
math | 1. [4] Let $A B C$ be a triangle with area 1. Let points $D$ and $E$ lie on $A B$ and $A C$, respectively, such that $D E$ is parallel to $B C$ and $D E / B C=1 / 3$. If $F$ is the reflection of $A$ across $D E$, find the area of triangle $F B C$. | \frac{1}{3} | 93 | 7 |
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