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 | 2. Write in digits the number equal to the sum of 22 million, 22 thousand, 22 hundred, and 22 units. | 22024222 | 33 | 8 |
math | 12.103 Find the natural number solutions to the equation $2^{x}+3^{y}=5^{z}$.
(All-Soviet Union Mathematical Winter Camp, 1991) | (1,1,1),(4,2,2) | 45 | 13 |
math | 2. It is known that bag A contains two red balls and three white balls, and bag B contains three red balls and three white balls. If a ball is randomly drawn from bag A and placed into bag B, and then a ball is randomly drawn from bag B and placed into bag A, the number of white balls in bag A at this point is denoted ... | \frac{102}{35} | 92 | 10 |
math | 7.189. $\log _{x} m \cdot \log _{\sqrt{m}} \frac{m}{\sqrt{2 m-x}}=1$. | m | 38 | 1 |
math | 2B. If each root of $x^{2}+3 x-7=0$ is increased by the reciprocal of the other root, the resulting roots are those of the equation $2 x^{2}+a x+b=0$. Determine $a$ and $b$. | \frac{36}{7},-\frac{72}{7} | 59 | 16 |
math | Problem 2. Find all pairs $(P, Q)$ of polynomials with real coefficients such that
$$
\frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}
$$
for infinitely many $x \in \mathbb{R}$.
Nikolai Nikolov, Oleg Mushkarov | Q(x)=x(x+1)R(x)\text{}P(x)=(\frac{1}{2}+x+(x+1))R(x) | 85 | 33 |
math | $5 \cdot 21$ Find all $x \in Z$, such that the polynomial
$$2 x^{2}-x-36$$
is the square of some prime number. | x=5 \text{ and } x=13 | 41 | 12 |
math | Task 1 - 110831 A vessel (without a drain) with a capacity of 1000 liters was initially filled with exactly 30 liters of water per second at a constant flow rate, and from a later time $t$ onwards, exactly 15 liters of water per second. After exactly $40 \mathrm{~s}$, measured from the beginning, the vessel was full.
... | \frac{4}{5} | 106 | 7 |
math | Determine the integers $m \geqslant 2, n \geqslant 2$ and $k \geqslant 3$ having the following property: $m$ and $n$ each have $k$ positive divisors, and if we denote $d_{1}<\ldots<d_{k}$ the positive divisors of $m$ (with $d_{1}=1$ and $d_{k}=m$) and $d_{1}^{\prime}<\ldots<d_{k}^{\prime}$ the positive divisors of $n$ ... | (4,9,3)(8,15,4) | 204 | 14 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x+2 \sqrt{y}=2 \\
& 2 \sqrt{x}+y=2
\end{aligned}
$$ | 4-2\sqrt{3} | 44 | 8 |
math | Example 2. The random variable $X$ is uniformly distributed on the interval $[\alpha, \beta]$. Find the probability of its values falling into the interval $(\gamma, \delta)$, which belongs to the interval $[\alpha, \beta]$. | P(\gamma<X<\delta)=\frac{\delta-\gamma}{\beta-\alpha} | 55 | 20 |
math | Exercise 1. Calculate the number
$$
\frac{4^{8}}{8^{4}}
$$
Only a numerical answer is expected here. | 16 | 32 | 2 |
math | 27. [17] Suppose that $\left(a_{1}, \ldots, a_{20}\right)$ and $\left(b_{1}, \ldots, b_{20}\right)$ are two sequences of integers such that the sequence $\left(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20}\right)$ contains each of the numbers $1, \ldots, 40$ exactly once. What is the maximum possible value of the sum
$$... | 5530 | 152 | 4 |
math | The five-digit number $9 A 65 B$ is divisible by 36 , where $A$ and $B$ are digits. Find all possible values of $A$ and $B$. | A=5,B=2orA=1,B=6 | 42 | 13 |
math | 8.235. $\operatorname{ctg}^{4} x=\cos ^{3} 2 x+1$.
8.235. $\cot^{4} x=\cos ^{3} 2 x+1$. | x_{1}=\frac{\pi}{4}(2k+1);x_{2}=\frac{\pi}{2}(2n+1),k,n\inZ | 54 | 37 |
math | Task 1. Which of the numbers 723, 732, and 273 will decrease the most and by how much if in each of them the digits 7 and 3 swap places? | 396 | 46 | 3 |
math | 3-ча 1. Determine the ratio of two numbers if the ratio of their arithmetic mean to geometric mean is $25: 24$. | x:16:9or9:16 | 31 | 11 |
math | 7. In the Cartesian coordinate plane, the 4 points $A(1,2)$, $B(3,1)$, $C(2,3)$, $D(4,0)$ have a sum of the squares of their distances to the line $y=kx$ denoted as $S$. When $k$ varies, the minimum value of $S$ is $\qquad$ | 22-\sqrt{185} | 84 | 9 |
math |
Problem 4. The number 1 is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a-1$ and $a+1$; if the number 0 occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on t... | \binom{n}{\lfloor\frac{n}{2}\rfloor} | 148 | 17 |
math | 17. The Capricious Princess (recommended for 9th grade, 4 points). A capricious princess is choosing a groom. 100 grooms are courting her, each better than the last, and there are no two equal among them. However, they court her in a random order. We will call a groom prominent if he pleases the princess more than all ... | 5.187 | 152 | 5 |
math | Example 5 Given that when $x \in[0,1]$, the inequality
$$
x^{2} \cos \theta-x(1-x)+(1-x)^{2} \sin \theta>0
$$
always holds. Try to find the range of $\theta$.
| 2 k \pi+\frac{\pi}{12}<\theta<2 k \pi+\frac{5 \pi}{12}(k \in \mathbf{Z}) | 62 | 38 |
math | Example 3. Integrate the differential equation $y^{\prime}=\frac{y}{2 y \ln y+y-x}$. | y\lny+\frac{C}{y} | 29 | 11 |
math | Example 12. Solve the equation
$$
8^{2 / x}-2^{(3 x+3) / x}+12=0
$$ | x_{1}=3\log_{6}2,x_{2}=3 | 35 | 16 |
math | Example 1. Find the Lagrange interpolation polynomial that takes the values $y_{0}=-5, y_{1}=-11, y_{2}=10$ at the points $x_{0}=-3, x_{1}=-1, x_{2}=2$. | 2x^{2}+5x-8 | 61 | 10 |
math | In triangle $A B C$, it is known that $A B=c, B C=a, \angle B=120^{\circ}$. Find the distance between the bases of the altitudes drawn from vertices $A$ and $C$.
# | \frac{1}{2}\sqrt{^{2}+^{2}+} | 54 | 18 |
math | 18.4.3 $\star \star$ A positive integer $n$ is not divisible by $2$ or $3$, and there do not exist non-negative integers $a$, $b$ such that $\left|2^{a}-3^{b}\right|=n$. Find the minimum value of $n$. | 35 | 67 | 2 |
math | 12・19 Find all natural numbers $p$ and $q$ such that the roots of the equation $x^{2}-p q x+p+q=0$ are integers.
(17th All-Russian Mathematical Olympiad, 1991) | (2,3),(3,2),(2,2),(1,5),(5,1) | 56 | 21 |
math | Example 1. Let $n$ be an integer, calculate the following expression:
$$
\begin{array}{l}
{\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{2^{2}}\right]+\left[\frac{n+2^{2}}{2^{3}}\right]} \\
+\cdots .
\end{array}
$$
where the symbol $[x]$ denotes the greatest integer not exceeding $x$.
(IMO-10)
Analysis: In the expressio... | n | 250 | 1 |
math | ## Problem II - 3
Two spheres of radius $r$ are externally tangent. Three spheres of radius $R$ are externally tangent to each other, each tangent to the other two. Each of these spheres is also externally tangent to the first two.
Find the relationship between $R$ and $r$. | 6r | 63 | 2 |
math | Example 4. Team A and Team B each send out 7 players to participate in a Go competition according to a pre-arranged order. Both sides start with Player 1 competing, the loser is eliminated, and the winner then competes with the next (Player 2) of the losing side, $\cdots \cdots \cdots$, until one side is completely eli... | 2 C_{13}^{6} | 110 | 9 |
math | 5. Given the function $f(x)=\ln (2+3 x)-\frac{3}{2} x^{2}$, if for any $x \in\left[\frac{1}{6}, \frac{1}{3}\right]$, the inequality $|a-\ln x|+$ $\ln \left[f^{\prime}(x)+3 x\right]>0$ always holds, then the range of the real number $a$ is $\qquad$. | {\lvert\,\neq\ln\frac{1}{3}.,\in{R}} | 101 | 22 |
math | 15. (12 points) $F(1,0)$ is a fixed point, $P(0, b)$ is a moving point on the $y$-axis, and point $M(a, 0)$ satisfies $\overrightarrow{P M} \cdot \overrightarrow{P F} = 0$. If point $N$ satisfies $2 \overrightarrow{P N} + \overrightarrow{N M} = 0$, find:
(1) The equation of the trajectory curve $C$ of point $N$;
(2) Th... | x=-1 | 138 | 3 |
math | 1-159 Find positive integer pairs $a, b$ that satisfy:
(1) $a b(a+b)$ is not divisible by 7;
(2) $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. Verify your answer. | =18,b=1 | 64 | 6 |
math | 17. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of $(1+2 x)^{100}$, where $0 \leq k \leq 100$. Find the number of integers $r: 0 \leq r \leq 99$ such that $a_{r}<a_{r+1}$. | 67 | 84 | 2 |
math | Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$$:$ {$1,2,...n$}$\rightarrow${$1,2,...,n$} and a binary function $C:$ {$1,2,...,n$}$\rightarrow${$0,1$} "revengeful" if it satisfies the two following conditions:
$1)$For every $i$ $\in$ {$1,2,...,n$}, there exist $j$ $\in$ $S_{i}=${$... | n! | 263 | 3 |
math | 19 Find all positive real numbers $a$, such that for any positive real numbers $t_{1}, t_{2}, t_{3}, t_{4}$ satisfying $t_{1} \cdot t_{2} \cdot t_{3} \cdot t_{4}=a^{4}$, we have
$$
\frac{1}{\sqrt{1+t_{1}}}+\frac{1}{\sqrt{1+t_{2}}}+\frac{1}{\sqrt{1+t_{3}}}+\frac{1}{\sqrt{1+t_{4}}} \leqslant \frac{4}{\sqrt{1+a}} \text {.... | (0,\frac{7}{9}) | 141 | 9 |
math | Let $X=\{1,2, \ldots, 100\}$. How many functions $f: X \rightarrow X$ satisfy $f(b)<f(a)+(b-a)$ for all $1 \leq a<b \leq 100$ ? | (\begin{pmatrix}199\\100\end{pmatrix}) | 60 | 19 |
math | positive integer values) is given by
$$
\begin{aligned}
f(1) & =1, \quad f(3)=3, \\
f(2 n) & =f(n), \\
f(4 n+1) & =2 f(2 n+1)-f(n), \\
f(4 n+3) & =3 f(2 n+1)-2 f(n),
\end{aligned}
$$
for all positive integers $n$. Determine with proof the number of positive integers less than or equal to 1988 for which $f(n)=n$. | 92 | 125 | 2 |
math | Exercise 14. Determine the integers $m \geqslant 2, n \geqslant 2$ and $k \geqslant 3$ having the following property: $m$ and $n$ each have $k$ positive divisors and, if we denote $d_{1}<\ldots<d_{k}$ the positive divisors of $m$ (with $d_{1}=1$ and $d_{k}=m$) and $d_{1}^{\prime}<\ldots<d_{k}^{\prime}$ the positive div... | (4,9,3)(8,15,4) | 198 | 14 |
math | 11. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{9}{4}, 2 a_{n+1} a_{n}-7 a_{n+1}-3 a_{n}+12=0\left(n \in \mathbf{N}_{+}\right)$.
(1) Let $c_{n}=a_{n}-2$, find the general term formula of the sequence $\left\{c_{n}\right\}$;
(2) Let $b_{n}=\frac{n^{2}}{n+1} a_{n}$, find the maximum positive integer ... | 45 | 201 | 2 |
math | Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how ... | 21 | 85 | 2 |
math | Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by
$$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$
Show that $F$ is injective and find the set of values of $F$. | (\ln(2), \infty) | 66 | 10 |
math | 1. Find all values of $x$, for each of which one of the three given numbers $\log _{x}\left(x-\frac{5}{2}\right)$, $\log _{x-\frac{5}{2}}(x-4)$, and $\log _{x-4} x$ is equal to the product of the other two. | \frac{9}{2},2+\sqrt{5} | 76 | 13 |
math | 10.084. The perimeter of a rhombus is 2 m, the lengths of its diagonals are in the ratio $3: 4$. Find the area of the rhombus. | 0.24\mathrm{~}^{2} | 44 | 12 |
math | ## Task Condition
Derive the equation of the tangent line to the given curve at the point with abscissa \( x_{0} \).
$$
y=\frac{-2\left(x^{8}+2\right)}{3\left(x^{4}+1\right)}, x_{0}=1
$$ | -\frac{2}{3}\cdotx-\frac{1}{3} | 68 | 16 |
math | 4. (1) Given the equation $x+y+z=15$, find the number of natural number solutions.
(2) The equation $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+$ $x_{7}+x_{8}+x_{9}+x_{10}=3$ has how many non-negative integer solutions? | 174 | 92 | 3 |
math | 12. At a party, 9 celebrities performed $n$ "trio dance" programs. If in these programs, any two people have collaborated exactly once, then $n=$ $\qquad$ | 12 | 42 | 2 |
math | Example 2. Find the general integral of the homogeneous equation
$$
\left(x^{2}-y^{2}\right) d y-2 y x d x=0
$$ | x^{2}+y^{2}=Cy | 39 | 10 |
math | 28. [14] Determine the value of
$$
\sum_{k=1}^{2011} \frac{k-1}{k!(2011-k)!} .
$$ | \frac{2009(2^{2010})+1}{2011!} | 44 | 24 |
math | 367. Find all natural $n$, for which the number $n^{4}+4$ is composite. | n\neq1 | 25 | 5 |
math | 9. For any real number sequence $A=\left(a_{1}, a_{2}, a_{3}, \cdots\right)$, define $\Delta A$ as the sequence $\left(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-\right.$ $\left.a_{3}, \cdots\right)$, where its $n$-th term is $a_{n+1}-a_{n}$. Assume that all terms of $\Delta(\Delta A)$ are 1, and $a_{19}=a_{92}$ $=0$, try to find... | 819 | 137 | 3 |
math | 3Ask1 * The sequence $a_{1}, a_{2}, \cdots$ is defined as follows: $a_{n}=2^{n}+3^{n}+6^{n}-1(n=1,2,3, \cdots)$. Find all positive integers that are coprime to every term of this sequence. | 1 | 73 | 1 |
math | 1 Let $a_{1}, a_{2}, \cdots, a_{n}$ be given real numbers, not all zero, and $r_{1}, r_{2}, \cdots, r_{n}$ be real numbers. If the inequality
$$
\begin{array}{l}
r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant \\
\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}-\sqr... | r_{k}=\frac{a_{k}}{\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}},k=1,2,\cdots,n | 224 | 49 |
math | Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that
$$
f(n)=f\left(\frac{n}{p}\right)-f(p)
$$
Given that $f(2001)=1$, what is the value of $f(2002)$? | 2 | 89 | 1 |
math | [b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$. | \frac{225}{2} | 97 | 9 |
math | Find the number of subsets of $\{1,2,3,\ldots,10\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\{\mathbf{1},\mathbf{2},5\}$ and $\{1,3,\mathbf{6},\mathbf{7},10\}.$ | 235 | 74 | 3 |
math | Paris has two million inhabitants. A human being has, at most, 600000 hairs on their head. What is the largest number of Parisians that we can hope to find who have exactly the same number of hairs on their head? | 4 | 52 | 1 |
math | A13 (8-5, Czechoslovakia) Solve the system of equations:
$$
\left\{\begin{array}{l}
\left|a_{1}-a_{2}\right| x_{2}+\left|a_{1}-a_{3}\right| x_{3}+\left|a_{1}-a_{4}\right| x_{4}=1, \\
\left|a_{2}-a_{1}\right| x_{1}+\left|a_{2}-a_{3}\right| x_{3}+\left|a_{2}-a_{4}\right| x_{4}=1, \\
\left|a_{3}-a_{1}\right| x_{1}+\left|a... | x_{1}=x_{4}=\frac{1}{a_{1}-a_{4}},x_{2}=x_{3}=0 | 279 | 30 |
math | 12. Let $0<\theta<\pi$, the complex numbers $z_{1}=1-\cos \theta+i \sin \theta, z_{2}=a^{2}+a i, u \in \mathbf{R}, z_{1} z_{2}$ is a pure imaginary number, and $\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}$. Then $\theta=$ $\qquad$ when it is a negative real number. | \frac{\pi}{2} | 114 | 7 |
math | 3.363. $\operatorname{tg}\left(\frac{5 \pi}{4}+x\right)+\operatorname{tg}\left(\frac{5 \pi}{4}-x\right)$, if $\operatorname{tg}\left(\frac{3 \pi}{2}+x\right)=\frac{3}{4}$. | -\frac{50}{7} | 78 | 8 |
math | Let $t$ be TNYWR.
In a magic square, the sum of the numbers in each column, the sum of the numbers in each row, and the sum of the numbers on each diagonal are all the same. In the magic square shown, what is the value of $N$ ?
| | $3 t-2$ | $4 t-6$ | |
| :---: | :---: | :---: | :---: |
| $4 t-1$ | $2 t+12$ | $t+16... | 23 | 162 | 2 |
math | Find the number of 6-digit sequences whose sum is 10. | \binom{15}{5}-6 | 15 | 10 |
math | Find all periodic sequences $x_1,x_2,\dots$ of strictly positive real numbers such that $\forall n \geq 1$ we have $$x_{n+2}=\frac{1}{2} \left( \frac{1}{x_{n+1}}+x_n \right)$$ | a, \frac{1}{a}, a, \frac{1}{a}, \ldots | 67 | 22 |
math | [ Sphere touching the edges or sides of the pyramid ]
The height of the pyramid is 5, and the base is a triangle with sides 7, 8, and 9. A sphere touches the planes of all the lateral faces of the pyramid at points lying on the sides of the base. Find the radius of the sphere.
# | \sqrt{6} | 69 | 5 |
math | 3. Given that $\triangle A B C$ is an equilateral triangle, the ellipse $\Gamma$ has one focus at $A$, and the other focus $F$ lies on the line segment $B C$. If the ellipse $\Gamma$ passes exactly through points $B$ and $C$, then its eccentricity is | \frac{\sqrt{3}}{3} | 66 | 10 |
math | In the multiplication shown, the letters $A, B, C$ and $K(A<B)$ represent different
\begin{tabular}{lll}
& $A$ & $C$ \\
$\times)$ & $B$ & $C$ \\
\hline$K$ & $K$ & $K$
\end{tabular}
integers from 1 to 9 .
(Hint: $K K K=K \times 111$.)
G8.1 Find $A$.
G8.2 Find $B$.
G8.3 Find $C$.
G8.4 Find $K$. | A=2,B=3,C=7,K=9 | 135 | 12 |
math | Problem 8.2. (15 points) Real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ are such that
$$
\left\{\begin{array}{l}
x_{1}+x_{2} \geqslant 12 \\
x_{1}+x_{3} \geqslant 13 \\
x_{1}+x_{4} \geqslant 14 \\
x_{3}+x_{4} \geqslant 22 \\
x_{2}+x_{3} \geqslant 23 \\
x_{2}+x_{4} \geq 24
\end{array}\right.
$$
What is the smallest value t... | 37 | 192 | 2 |
math | 2. let $n$ be a natural number. Determine the number of subsets $A \subset\{1,2, \ldots, 2 n\}$ such that $x+y=2 n+1$ does not hold for any two elements $x, y \in A$.
## Solution: | 3^n | 65 | 2 |
math | Example 13. Calculate the circulation of the vector field given in spherical coordinates: $2=r \mathbf{e}_{r}+(R+r) \sin \theta \mathbf{e}_{\varphi}$, along the circle $L:\{r=$ $\left.R, \theta=\frac{\pi}{2}\right\}$ in the direction of increasing angle $\varphi$ directly and using Stokes' theorem. | 4\piR^{2} | 89 | 7 |
math | Example 11 Let $x_{1}, x_{2}, \cdots, x_{19}$ all be positive integers, and satisfy $x_{1}+x_{2}+\cdots+x_{19}=95$. Find the maximum value of $x_{1}^{2}+x_{2}^{2}+\cdots+$ $x_{19}^{2}$.
(1995, Hebei Province Junior High School Mathematics Competition) | 5947 | 101 | 4 |
math | 6.153. $(2 x+a)^{5}-(2 x-a)^{5}=242 a^{5}$. | x_{1,2}=\ | 30 | 7 |
math | The function $f(x)$ satisfies for any positive numbers $x, y$: $f(x y)=f(x)+f(y)$, and $f(2)=1$, then the value of $f\left(\frac{1}{64}\right)$ is $\qquad$ . | -6 | 60 | 2 |
math | 4. After adding another tug to the one pushing the barge, they started pushing the barge with double the force. How will the power spent on movement change if the water resistance is proportional to the square of the barge's speed? | 2\sqrt{2} | 49 | 6 |
math | Solve the following system of equations:
$$
\begin{aligned}
x^{(\lg y)^{\lg \lg x}} & =10^{y^{2}} \\
y^{(\lg x)^{\lg \lg y}} & =y^{y}
\end{aligned}
$$ | 10^{10^{10}},10^{10} | 61 | 15 |
math | (4) The smallest positive integer $a$ that makes the inequality $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}<a-2007 \frac{1}{3}$ hold for all positive integers $n$ is $\qquad$. | 2009 | 71 | 4 |
math | Calculate the following indefinite integral.
[1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$
[2] $\int \sin x\cos 3x dx$
[3] $\int \sin 2x\sin 3x dx$
[4] $\int \frac{dx}{4x^2-12x+9}$
[5] $\int \cos ^4 x dx$ | \frac{3}{8} x + \frac{1}{4} \sin 2x + \frac{1}{32} \sin 4x + C | 99 | 37 |
math | 9. $M \in \mathbf{Z}, M^{2}+M+7$ is a perfect square, then $M=$ | -7,-2,1,6 | 30 | 8 |
math | 1. Find all positive integers $n$ such that the equation in $x, y$
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{n}
$$
has exactly 2011 solutions in positive integers $(x, y)$ with $x \leqslant y$. | n=p^{2010} | 70 | 8 |
math | 3. Given the function
$$
f(x)=\sin \left(\omega x+\frac{\pi}{4}\right)(\omega>0)
$$
has a maximum value but no minimum value in the interval $\left(\frac{\pi}{12}, \frac{\pi}{3}\right)$. Then the range of $\omega$ is . $\qquad$ | (\frac{3}{4},3) | 78 | 9 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=35 \\
\operatorname{LCM}(a ; b ; c)=5^{18} \cdot 7^{16}
\end{array}\right.
$$ | 9180 | 90 | 4 |
math | 9. Let $O A B C$ be a regular tetrahedron with edge length 1, and let $E$ and $F$ be the midpoints of $A B$ and $O C$, respectively. Then the distance between the skew lines $\mathrm{OE}$ and $\mathrm{BF}$ is $\qquad$ | \frac{\sqrt{10}}{10} | 70 | 12 |
math | 5. (BUL 5) Solve the system $$ \begin{aligned} & x^{2}+x-1=y \\ & y^{2}+y-1=z \\ & z^{2}+z-1=x \end{aligned} $$ | (-1,-1,-1) \text{ and } (1,1,1) | 56 | 19 |
math | Three. (25 points) Does there exist a four-digit number $\overline{a b c d}$, such that the last four digits of its square are also $\overline{a b c d}$? Does there exist a five-digit number $\overline{a b c d e}$, such that the last five digits of its square are also $\overline{a b c d e}$? If they exist, find all of ... | 9376 | 100 | 4 |
math | 10,11
The radius of the base of the cylinder is equal to $r$, and the height is equal to $5 r$. A parallelepiped is circumscribed around the cylinder, the ratio of the volume of which to the volume of the cylinder is $\frac{\tilde{5}}{\pi}$. Find the length of the segment of the larger diagonal of the parallelepiped l... | 3r | 88 | 2 |
math | Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\
x^2- 2bx + c = 0 \\
x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers. | (1, 1, 1) | 72 | 10 |
math | Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$. | \alpha \ge \frac{3\sqrt{3}}{2} | 76 | 16 |
math | ## Task 18/77
Given is a cylindrical beaker of height $H$. The (homogeneous) mantle has the mass $M$, the mass of the bottom is negligible. The glass is filled with a (homogeneous) liquid of mass $m_{H}$ to the brim, but the liquid drips out through an opening in the bottom until it is completely empty.
Required is a... | s_{E}=f(h_{E})=\frac{HM}{m_{H}}(\sqrt{1+\frac{m_{H}}{M}}-1) | 119 | 35 |
math | (i) Find the number of odd subsets and even subsets of $X$;
(ii) Find the sum of the sums of elements of all odd subsets of $X$. | 2^{(n-3)}n(n+1) | 34 | 12 |
math | 8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 200 people in the fa... | \frac{1}{4\cdot3^{65}} | 99 | 13 |
math | ## Task A-2.1. (4 points)
Determine the sum of all positive divisors of the number 2010. | 4896 | 30 | 4 |
math | Example 1. Find $\frac{d u}{d x}$, if $u=e^{z-2 y}$, where $z=\sin x, y=x^{2}$. | e^{\sinx-2x^{2}}(\cosx-4x) | 39 | 18 |
math | How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol? | 175 | 53 | 5 |
math | 1. Solve the equation: $\cos ^{2} x+\cos ^{2} 2 x=1$. | \frac{\pi}{2}+\pik,\\frac{\pi}{6}+\pik | 25 | 21 |
math | 24. Provide the formula for the probability $P_{n}$ that among thirteen cards drawn from a full deck of 52 cards, $n$ cards will be of the spades suit. | P_{n}=\frac{C_{13}^{n}C_{39}^{13-n}}{C_{52}^{13}} | 41 | 35 |
math | 2. Determine all pairs $(x, y)$ of real numbers that satisfy the inequality
$$
(x+y)\left(\frac{1}{x}+\frac{1}{y}\right) \geqq\left(\frac{x}{y}+\frac{y}{x}\right)^{2} .
$$ | y\neq0 | 65 | 5 |
math | 14. Given $y=5 \sin \left(\frac{2 m+1}{3} \pi x+\frac{\pi}{3}\right)$ (where $m \in \mathbf{N}$), for any real number $a$, the value of $y$ is $\frac{5}{4}$ no less than 4 times and no more than 8 times in the interval $[a, a+3]$. Find the value of $m$, | 2or3 | 100 | 3 |
math | Let $\{a, b, c, d, e, f, g, h\}$ be a permutation of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that $\overline{abc} +\overline{def}$ is even? | \frac{3}{7} | 72 | 7 |
math | ## Task B-3.2.
How many solutions does the equation $\frac{1}{\cos ^{2} \frac{x}{2}}+\frac{1}{\sin ^{2} \frac{x}{2}}=\frac{16}{\operatorname{tg} x}$ have on the interval $\langle 0,2019 \pi\rangle$? | 4038 | 81 | 4 |
math | 11th Balkan 1994 Problem 3 What is the maximum value f(n) of |s 1 - s 2 | + |s 2 - s 3 | + ... + |s n-1 - s n | over all permutations s 1 , s 2 , ... , s n of 1, 2, ... , n? Solution | f(2m)=2m^2-1,\,f(2m+1)=2m^2+2m-1 | 80 | 29 |
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 |
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