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 | 9,10,11 |
| | Product Rule | |
| | Cooperative Algorithms | |
| | Evaluation + Example | |
Authors: $\underline{\text { Knop K.A., }}$ Leontyeva O.
A magician and an assistant are going to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The assistant of the magician covers t... | 101 | 139 | 3 |
math | 11. (20 points) The ellipse $C_{1}$ and the hyperbola $C_{2}$ have common foci $( \pm c, 0)(c>0), C_{1}$ and $C_{2}$ have an eccentricity difference of no more than 1, and $C_{2}$ has an asymptote with a slope of no less than $\frac{4}{3}$. $C_{1}$ and $C_{2}$ intersect the positive x-axis at points $A$ and $B$, respec... | \frac{9 \sqrt{5}}{50} c^{2} | 176 | 17 |
math | 1. (10 points) In a class of 36 people, a total of 50 pencils were bought. Some people bought 1 pencil, some bought 2 pencils, and some bought 3 pencils. If the number of people who bought 1 pencil is twice the number of the rest, then the number of people who bought 2 pencils is $\qquad$ | 10 | 79 | 2 |
math | Mi, (Full marks 20 points) Find the range of real values of $a$ such that the inequality
$$
\begin{array}{l}
\sin 2 \theta-(2 \sqrt{2}+a \sqrt{2}) \sin \left(\theta+\frac{\pi}{4}\right) \\
-\frac{2 \sqrt{2}}{\cos \left(\theta-\frac{\pi}{4}\right)}>-3-2 a
\end{array}
$$
holds for all $\theta \in\left[0, \frac{\pi}{2}\r... | a>3 | 129 | 3 |
math | Example 1 Try to determine all six-digit numbers $p$ with the following properties:
(1) $p, 2p, 3p, 4p, 5p, 6p$ are still six-digit numbers;
(2) the six digits of each six-digit number are still the six digits of the number $p$ (only the order of arrangement is different). | 142857 | 81 | 6 |
math | Assume integer $m \geq 2.$ There are $3m$ people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "$n$-interesting", only if there exists $n(n\leq 3m-1)$ people of them, the time everyone of whom shakes hands with other $3m-1$ people is exactly $1,2,\cdots,n,$ respectivel... | 2m+1 | 128 | 6 |
math | Denote by $\mathbb{Q}^{+}$ the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \in \mathbb{Q}^{+}$: $$ f\left(f(x)^{2} y\right)=x^{3} f(x y) $$ (Switzerland) Answer. The only such function is $f(x)=\frac{1}... | f(x)=\frac{1}{x} | 117 | 10 |
math | Task B-1.5. If we add the sum of the digits of a two-digit number to the square of that sum, we will get the two-digit number again. Determine all such two-digit numbers. | 12,42,90 | 43 | 8 |
math | 1. Solve the equation in natural numbers
$$
2 y^{2}-x y-x^{2}+2 y+7 x-84=0
$$ | (1;6),(14;13) | 35 | 11 |
math | 一、(Full marks 20 points) Find all positive real numbers $a$ such that the equation $x^{2}-a x+4 a=0$ has only integer roots.
---
Translation:
I. (Full marks 20 points) Find all positive real numbers $a$ such that the equation $x^{2} - a x + 4 a = 0$ has only integer roots. | a=25 \text{ or } 18 \text{ or } 16 | 86 | 20 |
math | Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime. | 10 | 32 | 2 |
math | (F-M) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $x+f(x)=f(f(x))$ determine the solutions of $f(f(x))=0$. | 0 | 45 | 1 |
math | (5) The sum of the radii of all circles passing through point $A(1505,1008)$ and tangent to the lines $l_{1}: y=0$ and $l_{2}: y=\frac{4}{3} x$ is $\qquad$ . | 2009 | 64 | 4 |
math | 8.354. $\operatorname{tg}\left(\frac{3 \pi}{2}-x\right)+\frac{\cos \left(\frac{7 \pi}{2}+x\right)}{1+\cos x}=2$. | (-1)^{\}\frac{\pi}{6}+\pi,\inZ | 54 | 16 |
math | [ [MT - line or segment]
Two wheels of radii $r_{1}$ and $r_{2}$ roll along a line $l$. Find the set of points of intersection $M$ of their common internal tangents.
# | \frac{2r_{1}r_{2}}{r_{1}+r_{2}} | 49 | 22 |
math | Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same c... | k \geq 5 | 103 | 7 |
math | 2. Find all functions from the positive integers to the positive integers such that for all $x, y$ we have:
$$
2 y f\left(f\left(x^{2}\right)+x\right)=f(x+1) f(2 x y) \text {. }
$$ | f(x)=x | 61 | 4 |
math | Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$. | (2, 2), (10, 14), (14, 10) | 43 | 22 |
math | 7. (1995 China Mathematical Olympiad) Let $\mathbf{N}$ be the set of natural numbers. Let $f: \mathbf{N} \rightarrow \mathbf{N}$ satisfy the conditions: $f(1)=1$, and for any natural number $n$,
$$
\left\{\begin{array}{l}
3 f(n) f(2 n+1)=f(2 n)(1+3 f(n)), \\
f(2 n)<6 f(n) .
\end{array}\right.
$$
Find all solutions to ... | \begin{pmatrix}{\begin{pmatrix}k=5,\\=47;\end{pmatrix}.\\{\begin{pmatrix}k=13,\\=39;\end{pmatrix}.\\{\begin{pmatrix}k=7,\\=45;\end{pmatrix}.\\{\begin} | 141 | 73 |
math | 8. Solve the equation $\sqrt{15 x^{2}-52 x+45} \cdot(3-\sqrt{5 x-9}-\sqrt{3 x-5})=1$. | 2 | 44 | 1 |
math | ## Task 1.
Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all $a, b \in \mathbb{Z}$, the following holds:
$$
f(f(a)+f(b))=a+b-1
$$ | f(n)=-n+1 | 63 | 7 |
math | 4. A positive integer $n$ has exactly 4 positive divisors (including 1 and $n$). It is known that the sum of two of these divisors is six times the sum of the other two. Then $n=$ $\qquad$ . | 287 \text{ or } 143 | 55 | 12 |
math | 19th USAMO 1990 Problem 1 A license plate has six digits from 0 to 9 and may have leading zeros. If two plates must always differ in at least two places, what is the largest number of plates that is possible? Solution | 10^5 | 56 | 4 |
math | 4.7. [3+5] Before Alex, there are 100 closed boxes, each containing a red or blue ball. Alex has one pound in his account. He chooses one of the closed boxes and one of the two colors and bets some amount not exceeding the amount in his account (there is no requirement to bet whole pennies). The box is opened, and Alex... | )\frac{2^{100}}{100},b)\frac{2^{100}}{C_{n}^{100}} | 207 | 34 |
math | Let $k$ be the smallest positive real number such that for all positive real numbers $x$, we have
$$
\sqrt[3]{x} \leq k(x+1)
$$
What is the value of $k^{3}$? | \frac{4}{27} | 53 | 8 |
math | 9.6. Waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing the watermelon on one scale pan with one or two weights on the other pan (possibly identical). In the process, the seller wrote down on a piece of paper the weights he used. What is the... | 6 | 109 | 1 |
math | 11. In a free-throw test, as long as a person makes 3 shots, they are qualified and do not need to shoot anymore. However, each person can shoot at most 5 times. The probability that a player with a shooting accuracy of $\frac{2}{3}$ qualifies in the test is $\qquad$ . | \frac{64}{81} | 70 | 9 |
math | 18. For any positive integer $n$, let $f(n)=70+n^{2}$ and $g(n)$ be the H.C.F. of $f(n)$ and $f(n+1)$. Find the greatest possible value of $g(n)$.
(2 marks)
對任何正整數 $n$, 設 $f(n)=70+n^{2}$, 而 $g(n)$ 則是 $f(n)$ 與 $f(n+1)$ 的最大公因數。求 $g(n)$ 的最大可能值。 | 281 | 122 | 3 |
math | ## Task 4 - 150714
In the plane $\epsilon$, there are 50 different points arranged such that no line exists that contains three of these 50 points. Each of these 50 points is to be connected to every other point by a line segment.
a) Determine the number of connecting line segments!
b) Suppose the 50 points are the ... | 1225 | 106 | 4 |
math | Example 7 Let $n$ be a positive integer. How many positive integer solutions does the equation $\frac{x y}{x+y}=n$ have?
(21st Putnam Mathematical Competition) | (2\alpha_{1}+1)(2\alpha_{2}+1)\cdots(2\alpha_{k}+1) | 41 | 31 |
math | 10th Balkan 1993 Problem 1 Given reals a 1 ≤ a 2 ≤ a 3 ≤ a 4 ≤ a 5 ≤ a 6 satisfying a 1 + a 2 + a 3 + a 4 + a 5 + a 6 = 10 and (a 1 - 1) 2 + (a 2 - 1) 2 + (a 3 - 1) 2 + (a 4 - 1) 2 + (a 5 - 1) 2 + (a 6 - 1) 2 = 6, what is the largest possible a 6 ? Solution | 3\frac{1}{3} | 147 | 8 |
math | 4. Given the sequence $a_{1}=1, a_{n}=\frac{n}{n^{2}-1}(n \geqslant 2)$, then $\sum_{k=1}^{n}\left(a_{1} a_{2} \cdots a_{k}\right)=$ | 2[1-\frac{1}{(n+1)!}] | 66 | 14 |
math | 3.1. [5-6.3 (20 points)] In three flasks, there is concentrated acid: 10 g in the first, 20 g in the second, and 30 g in the third. There is also a fourth flask with water. If some amount of water from the fourth flask is added to the first flask, and the remaining water is poured into the second flask, then in the fir... | \frac{21}{200}=0.105 | 155 | 15 |
math | 4.41 Given that $n$ is a positive integer, determine the number of solutions in ordered pairs of positive integers $(x, y)$ for the equation $\frac{x y}{x+y}=n$.
(21st Putnam Mathematical Competition, 1960) | (2\alpha_{1}+1)(2\alpha_{2}+1)\cdots(2\alpha_{k}+1) | 59 | 31 |
math | In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$. | BC = 2\sqrt{13} | 48 | 11 |
math | Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number. | 101 | 49 | 3 |
math | [ Coordinate method in space ] [ Parametric equations of a line ] Through the point $M(-2 ; 0 ; 3)$, draw a line that intersects the lines
$$
\left\{\begin{array} { l }
{ x = 2 - t } \\
{ y = 3 } \\
{ z = - 2 + t }
\end{array} \left\{\begin{array}{l}
2 x-2 y-z-4=0 \\
x+3 y+2 z+1=0
\end{array}\right.\right.
$$ | -2+13,-3,3-12 | 124 | 12 |
math | 241. The device consists of 10 independently operating elements: The probability of failure of each element over time $T$ is 0.05. Using Chebyshev's inequality, estimate the probability that the absolute value of the difference between the number of failed elements and the average (expected) number of failures over tim... | 0.880.12 | 88 | 8 |
math | 8.037. $\sin 3 x \cos 3 x=\sin 2 x$.
8.037. $\sin 3 x \cos 3 x=\sin 2 x$. | x_{1}=\frac{\pin}{2};x_{2,3}=\frac{\pi}{12}(6k\1),n,k\inZ | 45 | 35 |
math | 1. Given a natural number $n \geq 3$. For an $n$-element set of real numbers $S$, let $A(S)$ denote the set of all strictly increasing three-term arithmetic progressions composed of elements of $S$. How many elements can the set $A(S)$ have at most? | [\frac{n-1}{2}][\frac{n}{2}] | 66 | 15 |
math | G10.2 The medians $A L, B M, C N$ of $\triangle A B C$ meet at $G$. If the area of $\triangle A B C$ is $54 \mathrm{~cm}^{2}$ and the area of $\triangle A N G$ is $x \mathrm{~cm}^{2}$. Find $x$. | 9 | 80 | 1 |
math | 25. (Northern) Let $x, y, z \in [0,1], |x-y| \leqslant \frac{1}{2}, |z-y| \leqslant \frac{1}{2}, |x-z| \leqslant \frac{1}{2}$, find
$$W(x, y, z)=x+y+z-xy-yz-zx$$
the maximum and minimum values. | \frac{5}{6} | 96 | 7 |
math | 5. In $\triangle A B C$, $A B=3, A C=4, B C=5, I$ is the incenter of $\triangle A B C$, and $P$ is a point within $\triangle I B C$ (including the boundary). If $\overrightarrow{A P}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}(\lambda ; \mu \in \mathbf{R})$, then the minimum value of $\lambda+\mu$ is | \frac{7}{12} | 108 | 8 |
math | Question 128, Define the sequence $\left\{a_{n}\right\}: a_{0}=a, a_{n+1}=2 a_{n}-n^{2}(n \geq 0)$, try to find all real numbers $a$, such that for any non-negative integer $\mathrm{n}$, we have $\mathrm{a}_{\mathrm{n}} \geq 0$. | \in[3,+\infty) | 87 | 9 |
math | Example 3-18 For the problem
$$
\begin{array}{c}
x_{1}+x_{2}+x_{3}=15 \\
0 \leqslant x_{1} \leqslant 5, \quad 0 \leqslant x_{2} \leqslant 6, \quad 0 \leqslant x_{3} \leqslant 7
\end{array}
$$
Find the number of integer solutions. | 10 | 110 | 2 |
math | How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$) | 110 | 106 | 3 |
math | 3. $A$ is a point on the parabola $x=-\frac{2}{7} y^{2}$, $F$ is the focus, $|A F|=14 \frac{7}{8}$, the equation of the line $l$ passing through point $F$ and perpendicular to $O A$ is $\qquad$ . | 3.8 x-4 y+7=0 \text{ or } 8 x+4 y+7=0 | 77 | 26 |
math | Let's calculate the value of the following integral:
$$
\int_{0}^{\pi / 2^{n+1}} \sin x \cdot \cos x \cdot \cos 2 x \cdot \cos 2^{2} x \cdot \ldots \cdot \cos 2^{n-1} x d x
$$ | \frac{1}{2^{2n}} | 74 | 10 |
math | 9. $\triangle A B C$ is isosceles with $A B=A C . P$ is a point inside $\triangle A B C$ such that $\angle B C P=$ $30^{\circ}, \angle A P B=150^{\circ}$ and $\angle C A P=39^{\circ}$. Find $\angle B A P$.
(1 mark)
$A B C$ 是等腰三角形, 其中 $A B=A C \circ P$ 是 $\triangle A B C$ 内的一點, 使得 $\angle B C P=$ $30^{\circ} 、 \angle A ... | 13 | 174 | 2 |
math | In triangle $A B C$, a point $M$ is selected in its interior so that $\angle M A B=10^{\circ}$, $\angle M B A=20^{\circ}, \angle M C A=30^{\circ}$ and $\angle M A C=40^{\circ}$. Determine the value of $\angle M B C$. | 60 | 80 | 2 |
math | II. (50 points) Let real numbers $a, b$ be such that the equation $a x^{3}-x^{2}+b x-1=0$ has three positive real roots. For all real numbers $a, b$ that satisfy the condition, find the minimum value of $P=\frac{5 a^{2}-3 a b+2}{a^{2}(b-a)}$.
---
The translation maintains the original text's formatting and line break... | 12\sqrt{3} | 101 | 7 |
math | 5. [4] Let $f(x)=\sin ^{6}\left(\frac{x}{4}\right)+\cos ^{6}\left(\frac{x}{4}\right)$ for all real numbers $x$. Determine $f^{(2008)}(0)$ (i.e., $f$ differentiated 2008 times and then evaluated at $x=0$ ). | \frac{3}{8} | 84 | 7 |
math | Three. (25 points) Given that $\sqrt{1-|x|}+\sqrt{2-2|y|}$ is meaningful, $M=\sqrt{x^{2}+2 x y+y^{2}}+\sqrt{x^{2}+2 x+1}+|2 x-y-4|$ has maximum and minimum values of $a$ and $b$ respectively. If the five-digit integer $1 a b c d$ is divisible by 99, find the values of $c$ and $d$.
---
Given that $\sqrt{1-|x|}+\sqrt{2... | c=2, d=5 | 219 | 7 |
math | 10. (20 points) Given the ellipse $\frac{x^{2}}{2}+y^{2}=1$ with its upper vertex $B$ and right focus $F$, a line $l$ intersects the ellipse at points $M$ and $N$. Determine whether there exists a line $l$ such that point $F$ is the (1) centroid; (2) orthocenter of $\triangle B M N$. If it exists, find the equation of ... | x-\frac{4}{3} | 114 | 8 |
math | 7.4.1. (12 points) How many integer roots of the equation
$$
\cos 2 \pi x + \cos \pi x = \sin 3 \pi x + \sin \pi x
$$
lie between the roots of the equation $x^{2} + 10 x - 17 = 0$? | 7 | 77 | 1 |
math | 4. The product of all natural numbers from 1 to $\mathrm{n}$ is denoted by $n$! (read as “ $n$ - factorial”). Which of the numbers is greater: 200! or $100^{200}$? | 200!<100^{200} | 57 | 12 |
math | 6. (20 points) Dasha added 158 numbers and got 1580. Then Seryozha tripled the largest of these numbers and decreased another number by 20. The resulting sum did not change. Find the smallest of the original numbers.
# | 10 | 60 | 2 |
math | 23. Given $n \in \mathbf{N}$ and $a \in[0, n]$, under the condition $\sum_{i=1}^{n} \sin ^{2} x_{i}=a$, find the maximum value of $\sum_{i=1}^{n} \sin 2 x_{i} \mid$. (1983 Czechoslovak Mathematical Olympiad Problem) | 2 \sqrt{a(n-a)} | 89 | 8 |
math | Example 2.80. Calculate the change in entropy $S$ of an ideal gas of constant mass $m$ when: a) $P=$ const and the volume changes from $V_{1}$ to $V_{2}$; b) $V=$ const and the pressure changes from $P_{1}$ to $P_{2}$; c) $T=$ const and the volume changes from $V_{1}$ to $V_{2}$. | S_{1}-S_{2}=\frac{}{\mu}C_{p}\ln\frac{V_{2}}{V_{1}}(),\frac{}{\mu}C_{v}\ln\frac{P_{2}}{P_{1}}(b),\frac{}{\mu}R\ln\frac{V_{2}}{V_{1}}\text | 96 | 81 |
math | 2. Let the function $f(x)=\frac{x^{2}+x+16}{x}(2 \leqslant x \leqslant a)$, where the real number $a>2$, if the range of $f(x)$ is $[9,11]$, then the range of values for $a$ is $\qquad$ . | [4,8] | 79 | 5 |
math | ## Task 13/69
The five smallest, non-single-digit consecutive numbers $z_{i}$ (with $\mathrm{i}=1 ; 2 ; 3 ; 4 ; 5$) are sought, for which the following holds: $z_{i}$ is divisible by $i+4$ and has the last digit $i+4$. | 2525,2526,2527,2528,2529 | 76 | 24 |
math | What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors ? | 43 | 34 | 2 |
math | The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$.
Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions. | 7 | 84 | 1 |
math | ## Task 4 - 270614
Kerstin cuts a rectangular piece of paper so that the cut runs from the midpoint of one rectangular side to the midpoint of the opposite side. This results in two smaller rectangles; Kerstin places them exactly on top of each other so that only a smaller rectangle is visible, consisting of two layer... | 160 | 203 | 3 |
math | 14) Given the function $f(x)=\ln (x+1)+\frac{2}{x+1}+a x-2$ (where $a>0$).
(1) When $a=1$, find the minimum value of $f(x)$;
(2) If $x \in[0,2]$, $f(x) \geqslant 0$ always holds, find the range of the real number $a$. | [1,+\infty) | 98 | 7 |
math | One, (20 points) Given the equation in terms of $x$
$$
x^{2}+2(a+2 b+3) x+\left(a^{2}+4 b^{2}+99\right)=0
$$
has no distinct real roots. How many ordered pairs of positive integers $(a, b)$ satisfy this condition? | 16 | 76 | 2 |
math | 21. [12] Let $A B C D$ be a quadrilateral inscribed in a circle with center $O$. Let $P$ denote the intersection of $A C$ and $B D$. Let $M$ and $N$ denote the midpoints of $A D$ and $B C$. If $A P=1, B P=3, D P=\sqrt{3}$, and $A C$ is perpendicular to $B D$, find the area of triangle $M O N$. | \frac{3}{4} | 109 | 7 |
math | ## Task 5
A pedestrian walks $5 \mathrm{~km}$ in one hour, a cyclist rides four times as fast, and a moped rider rides three times as fast as a cyclist.
How many more kilometers does a moped rider travel in one hour compared to a cyclist? | 40\mathrm{~} | 60 | 7 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{3 ; 1 ; 8\}$
$p=\{0 ; 1 ; 3\}$
$q=\{1 ; 2 ;-1\}$
$r=\{2 ; 0 ;-1\}$ | 3p-q+2r | 73 | 6 |
math | 1. A three-digit number has its middle digit three times smaller than the sum of the other two, and the sum of the last two digits is half of the first digit. If the digits in the tens and units places are swapped, the resulting number is 18 less than the given number. What is that number? | 831 | 66 | 3 |
math | Three, (50 points) Find all positive integers $a, b, m, n$ such that $m$ and $n$ are coprime and $\left(a^{2}+b^{2}\right)^{m}=(a b)^{n}$.
| (,b,,n)=(2^{r},2^{r},2r,2r+1) | 58 | 22 |
math | Calculate $3 \times 37$ then $27 \times 37$, then give a criterion for divisibility by 37. | \sum_{k=0}^{3}(n_{3k}+10n_{3k+1}-11n_{3k+2}) | 31 | 34 |
math | Solve the equation in natural numbers: $x^{3}+y^{3}+1=3 x y$.
# | (1,1) | 27 | 5 |
math | 10. If there is a square $A B C D$ in a plane and $M$ is any point in the plane, then the minimum value of $\frac{M A+M C}{M B+M D}$ is | \frac{\sqrt{2}}{2} | 49 | 10 |
math | Example 7. In a cylindrical vessel with a volume of $V_{0}$, atmospheric air is adiabatically (without heat exchange with the environment) compressed to a volume of $V_{1}$. Calculate the work of compression. | W_{1}=\frac{p_{0}V_{0}}{k-1}[(\frac{V_{0}}{V_{1}})^{k-1}-1] | 50 | 41 |
math | Example 4 Given that the real number $x$ and the acute angle $\theta$ satisfy
$$
\begin{array}{l}
x^{2}+2 x \cos \theta=\sin \theta-\frac{5}{4} . \\
\text { Find the value of } \frac{x+\operatorname{tg} \theta}{x-\operatorname{tg} \theta} \text { . }
\end{array}
$$ | \frac{1}{5} | 94 | 7 |
math | L11 12 * Find real numbers $x, y$ such that $\left\{\begin{array}{l}4^{-x}+27^{-y}=\frac{5}{6}, \\ \log _{27} y-\log _{4} x \geqslant \frac{1}{6} \\ 27^{y}-4^{x} \leqslant 1\end{array}\right.$ holds. | \frac{1}{2},\frac{1}{3} | 99 | 14 |
math | Example 8 Given non-zero real numbers $a, b, c$ that are not all equal and satisfy
$$
\frac{a^{2}}{2 a^{2}+b c}+\frac{b^{2}}{2 b^{2}+c a}+\frac{c^{2}}{2 c^{2}+a b}=1 .
$$
Find the value of $a+b+c$. | +b+=0 | 89 | 3 |
math | P r o b l e m 6. Let's find a four-digit number that is a perfect square, if its first two digits and last two digits are equal. | 7744=88^{2} | 35 | 10 |
math | the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered
\[1=d_1<d_2<\cdots<d_k=n\]
Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$. | 28 | 79 | 2 |
math | Example 2: A rope of length 2009 is operated as follows: first, it is divided into two ropes of positive integer lengths, and the lengths of the two ropes are recorded, then the above operation is repeated on one of the ropes, ... until 2009 ropes of length 1 are obtained. If the lengths of the two ropes obtained in a ... | 7 | 141 | 1 |
math | \section*{Problem 16 - V01116}
A trench with a parabolic cross-section is to be excavated. Its width is 3 meters, and its depth is \(b\) meters.
Calculate the cross-sectional area of the trench! | 2b | 55 | 2 |
math | ## 3. Three-digit Ending
Sum all odd four-digit numbers for which the sum of the digits is equal to 4. What are the last three digits of this sum, starting with the hundreds digit?
Result: $\quad 449$ | 449 | 52 | 3 |
math | 1. The sides of a square $A B C D$ with a side length of 10 are reflective on the inside. A light ray enters the square through the vertex $A$ and travels to the point $P$ on $C D$ with $C P=3$ and $P D=7$. At $P$, it naturally reflects off the side $C D$. The light ray can only leave the square through one of the vert... | 10\sqrt{149} | 130 | 9 |
math | Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes. | 2002 | 31 | 4 |
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 $a_{i+1} \in\left\{1+a_{i}, 2+\right.$ $\left.a_{i}\right\}, i=1,2, \cdots, 9$, then the number of such sequences is $\qquad$. | 80 | 108 | 2 |
math | 32nd CanMO 2000 Problem 5 A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms? | 10000 | 71 | 5 |
math | 7. In space, four non-collinear vectors $\overrightarrow{O A} 、 \overrightarrow{O B}$ 、 $\overrightarrow{O C} 、 \overrightarrow{O D}$ have pairwise angles of $\alpha$. Then the size of $\alpha$ is $\qquad$ | \arccos(-\frac{1}{3}) | 64 | 12 |
math | Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $. | 0 | 94 | 1 |
math | 5. Given an integer $n \geqslant 3$. Find the minimum value that $\sum_{i=1}^{n}\left(\frac{1}{x_{i}}-x_{i}\right)$ can achieve, where $x_{1}, x_{2}, \cdots, x_{n}$ are positive real numbers satisfying $\sum_{i=1}^{n} \frac{x_{i}}{x_{i}+n-1}=1$. Also find the values of $x_{i}$ when the minimum value is achieved. | 0 | 118 | 1 |
math | 5. On the website of the football club "Rostov," a poll is being conducted to determine which of the $m$ football players the website visitors consider the best at the end of the season. Each visitor votes once for one player. The website displays the rating of each player, which is the percentage of votes cast for the... | 11 | 107 | 2 |
math | 12.118 Find all prime numbers \( p, q \) and \( r \) that satisfy the equation \( p^{q} + q^{p} = r \). (53rd Moscow Mathematical Olympiad, 1990) | q=2,p=3,r=17 | 54 | 10 |
math | 52. When the number POTOП was taken as an addend 99999 times, the resulting number had the last three digits 285. What number is denoted by the word POTOП? (Identical letters represent identical digits.) | 51715 | 56 | 5 |
math | 9. If $P(x, y)$ is a point on the hyperbola $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$, then the minimum value of $|x-y|$ is . $\qquad$ | 2 | 54 | 1 |
math | A2 Let $x$ be the average of the following six numbers: $\{12,412,812,1212,1612,2012\}$. Determine the value of $x$. | 1012 | 52 | 4 |
math | 10th Irish 1997 Problem B4 How many 1000-digit positive integers have all digits odd, and are such that any two adjacent digits differ by 2? | 8\cdot3^{499} | 40 | 9 |
math | Problem 5. Solve the system of equations
$$
\left\{\begin{aligned}
x^{2}+x y+y^{2} & =37 \\
x^{4}+x^{2} y^{2}+y^{4} & =481
\end{aligned}\right.
$$ | (-4,-3),(-3,-4),(3,4),(4,3) | 68 | 18 |
math | 3. For a positive integer $n$, its decimal representation consists only of the digits 0 and 1, and it is divisible by 225. Find the minimum value of $n$.
| 11111111100 | 42 | 11 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}(\cos \sqrt{x})^{\frac{1}{x}}$ | \frac{1}{\sqrt{e}} | 36 | 10 |
math | 2. Let $a$ and $b$ be integers, and the equation
$$
a x^{2}+b x+1=0
$$
has two distinct positive roots both less than 1. Then the minimum value of $a$ is | 5 | 54 | 1 |
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