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 | W6 Find all natural numbers $x$ that satisfy the following conditions: the product of the digits of $x$ equals $44x - 86868$, and the sum of the digits is a cube number. | 1989 | 48 | 4 |
math | Problem 4. A row of 11 numbers is written such that the sum of any three consecutive numbers is 18. Additionally, the sum of all the numbers is 64. Find the central number. | 8 | 45 | 1 |
math | ## 136. Math Puzzle $9 / 76$
In a residential area, the number of retirees makes up $40 \%$ of the eligible voting population and $25 \%$ of the total population.
How are the population groups of retirees, other adults, and children and youth who are not yet eligible to vote distributed in the residential area in per... | r=25;e=37.5;k=37.5 | 78 | 17 |
math | 2.035. $\left(\frac{a+2}{\sqrt{2 a}}-\frac{a}{\sqrt{2 a}+2}+\frac{2}{a-\sqrt{2 a}}\right) \cdot \frac{\sqrt{a}-\sqrt{2}}{a+2}$. | \frac{1}{\sqrt{}+\sqrt{2}} | 70 | 13 |
math | Solve the equation $x^{2}+y^{2}=z-16$ in the set of positive prime numbers. | 2,3,29 | 27 | 6 |
math | Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that
\[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\] | 16 | 71 | 2 |
math | To be calculated $\lim _{n \rightarrow \infty}\left(\sqrt[3^{2}]{3} \cdot \sqrt[3^{3}]{3^{2}} \cdot \sqrt[3^{4}]{3^{3}} \ldots \sqrt[3^{n}]{3^{n-1}}\right)$. | \sqrt[4]{3} | 74 | 7 |
math | 3.361. $\frac{\sin 22^{\circ} \cos 8^{\circ}+\cos 158^{\circ} \cos 98^{\circ}}{\sin 23^{\circ} \cos 7^{\circ}+\cos 157^{\circ} \cos 97^{\circ}}$. | 1 | 81 | 1 |
math | 13. Given $f(x)=\frac{x}{1+x}$. Find the value of the following expression:
$$
\begin{array}{l}
f\left(\frac{1}{2004}\right)+f\left(\frac{1}{2003}\right)+\cdots+f\left(\frac{1}{2}\right)+f(1)+ \\
f(0)+f(1)+f(2)+\cdots+f(2003)+f(2004) .
\end{array}
$$ | 2004 | 119 | 4 |
math | 1. Try to design a method to divide a square into 8 smaller squares without repetition or omission (the sizes of the smaller squares can be different); also, how to divide a square into 31 smaller squares under the same requirements? | 31 | 49 | 2 |
math | Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$.
The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac{m}{n}$, where $m$ and $n$
are relatively prime positive integers. Find $m + n$. A parallelepiped is a sol... | 125 | 222 | 3 |
math | 3.8. Find the equation of the line passing through the point $P(2, -3)$ and the intersection point of the lines $3x + 2y - 4 = 0$ and $x - y + 5 = 0$. | 3.4x+1.6y-2=0 | 55 | 13 |
math | 9. (16 points) Given the function
$$
f(x)=a \cos x+b \cos 2 x+c \cos 3 x,
$$
and $f(x) \geqslant-1$ always holds. Find the maximum value of $a-b+c$. | 1 | 61 | 1 |
math | 9 - 59 Let $x_{i} \geqslant 0, i=1,2, \cdots, n, n \geqslant 2$ and $\sum_{i=1}^{n} x_{i}=1$, find the maximum value of $\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}+x_{j}\right)$. | \frac{1}{4} | 101 | 7 |
math | 8.1. (12 points) In triangle $ABC$, the bisector $BL$ is drawn. Find the area of the triangle if it is known that $AL=2, BL=3\sqrt{10}$, and $CL=3$. | \frac{15\sqrt{15}}{4} | 55 | 14 |
math | Example 3 Let the quadratic function $f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)$ satisfy the conditions:
(1) For $x \in \mathbf{R}$, $f(x-4)=f(2-x)$, and $f(x) \geqslant x$;
(2) For $x \in(0,2)$, $f(x) \leqslant\left(\frac{x+1}{2}\right)^{2}$;
(3) The minimum value of $f(x)$ on $\mathbf{R}$ is 0.
Find the l... | 9 | 210 | 1 |
math | For a composite number $n$, let $f(n)$ denote the sum of its smallest three positive divisors, and $g(n)$ denote the sum of its largest two positive divisors. Find all composite numbers $n$ such that $g(n)$ is a positive integer power of $f(n)$. | n=4 \times 6^{l}\left(l \in \mathbf{Z}_{+}\right) | 63 | 24 |
math | 7.5. One hundred non-zero integers are written in a circle such that each number is greater than the product of the two numbers following it in a clockwise direction. What is the maximum number of positive numbers that can be among these 100 written numbers? | 50 | 54 | 2 |
math | 9. Bags in the Shed. A truck delivered 4 bags of cement. They are stacked in the truck bed. The worker can carry one bag at a time from the truck to the gate, or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it where needed, and placing it on top of the st... | \frac{1}{8} | 208 | 7 |
math | ## 5. Cards
There were seven cards in a box with numbers from 3 to 9 written on them (each card had one number). Mirko randomly took three cards from the box, and Slavko took two cards, while two cards remained in the box. Mirko looked at his cards and said to Slavko: "I know that the sum of the numbers on your cards ... | 192 | 105 | 3 |
math | 2. Find for all natural $n>1$ the positive solutions of the system
$$
\left\{\begin{array}{l}
x_{1}+2 x_{2}+\cdots+n x_{n}=3 \\
\frac{1}{x_{1}}+\frac{1}{2 x_{2}}+\cdots+\frac{1}{n x_{n}}=3
\end{array}\right.
$$ | x_{1}=\frac{3\\sqrt{5}}{2},x_{2}=\frac{3\\sqrt{5}}{4}forn=2;\quadx_{1}=1,x_{2}=1/2,x_{3}=1/3forn=3 | 93 | 61 |
math | 6. In a regular tetrahedron with a base edge length of 4 and a height of 2, first place a sphere \( B_{1} \) that is tangent to the base and all three side faces. Then, for \( n=2,3, \cdots \), place spheres \( B_{n} \) in the regular tetrahedron such that they are externally tangent to \( B_{n-1} \) and tangent to all... | \frac{16\pi}{39} | 118 | 11 |
math |
Problem 9a. 1 Given the equation $\frac{1}{|x-2|}=\frac{1}{|x-52 a|}$, where $a$ is a parameter.
a) Solve the equation.
b) If $a$ is the square of a prime number prove that the equation has a solution which is a compose integer.
| 26a+1 | 76 | 5 |
math | 3. In $\triangle A B C$, $\angle C=90^{\circ}, \angle A$ and $\angle B$ are bisected and intersect at point $P$, and $P E \perp A B$ at point $E$. If $B C=2, A C=3$,
then $A E \cdot E B=$ $\qquad$ | 3 | 79 | 1 |
math | What is the area of a regular dodecagon inscribed in a circle, if the radius of the circle is $r$? | 3r^2 | 28 | 4 |
math | 13.429 First, the motorboat traveled $a$ km across the lake, and then half of this distance up the river flowing into the lake. The entire trip lasted 1 hour. Find the own speed of the motorboat if the speed of the river current is $c$ km/h. | \frac{3+2+\sqrt{9^{2}-4+4^{2}}}{4} | 64 | 22 |
math | Let $a>1$ be a positive integer, and let $d>1$ be a positive integer coprime to $a$. Let $x_{1}=1$ and, for $k \geqslant 1$, define
$$ x_{k+1}= \begin{cases}x_{k}+d & \text { if } a \text { doesn't divide } x_{k} \\ x_{k} / a & \text { if } a \text { divides } x_{k}\end{cases} $$
Find the greatest positive integer ... | n | 174 | 1 |
math | 45. Simpler than it seems. Calculate the root
$$
\left(\frac{1 \cdot 2 \cdot 4+2 \cdot 4 \cdot 8+3 \cdot 6 \cdot 12+\ldots}{1 \cdot 3 \cdot 9+2 \cdot 6 \cdot 18+3 \cdot 9 \cdot 27+\ldots}\right)^{\frac{1}{3}}
$$ | \frac{2}{3} | 99 | 7 |
math | 8.7 For an integer $n>3$, we use $n$ ? to denote the product of all primes less than $n$ (called “$n$-question mark”). Solve the equation $n ?=2 n+16$.
| 7 | 53 | 1 |
math | ## Task A-1.5.
On the board, there are 2023 different real numbers. If each number on the board (simultaneously) is replaced by the sum of all the other numbers, the 2023 numbers on the board will be the same as at the beginning.
What values can the product of all the numbers on the board take at some point? | 0 | 81 | 1 |
math | 2nd Mexico 1988 Problem B4 Calculate the volume of an octahedron which has an inscribed sphere of radius 1. | 4\sqrt{3} | 31 | 6 |
math | Find all solutions to the puzzle: ARKA + RKA + KA + A = 2014. (Different letters correspond to different digits, and the same letters correspond to the same digits.)
# | 1471+471+71+1=2014 | 42 | 18 |
math | Solve the following equation:
$$
\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=\frac{3}{2} \sqrt{\frac{x}{x+\sqrt{x}}}
$$ | \frac{25}{16} | 42 | 9 |
math | ## Task Condition
Find the derivative.
$y=\ln \cos \frac{2 x+3}{2 x+1}$ | \frac{4\operatorname{tg}\frac{2x+3}{2x+1}}{(2x+1)^{2}} | 27 | 31 |
math | 6. (15 points) A metal bar with a temperature of $20^{\circ} \mathrm{C}$ was submerged into water taken at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium was established, the temperature was found to be $80^{\circ} \mathrm{C}$. After this, without removing the first bar from the water, another ide... | 68\mathrm{C} | 125 | 7 |
math | Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number:
\[ A=17^{2006n}+4.17^{2n}+7.19^{5n} \]
can be represented as the product of $k$ consecutive positive integers. | (n, k) = (0, 2) | 81 | 13 |
math | In the fraction below and its decimal notation (with period of length $ 4$) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction:
$ \frac{ADA}{KOK}\equal{}0.SNELSNELSNELSNEL...$
$ Note.$ Ada Kok is a famous ... | \frac{242}{303} = 0.798679867986... | 95 | 28 |
math | 5. In triangle $\mathrm{ABC}$, point $\mathrm{M}$ is the midpoint of $\mathrm{AC}$, $\mathrm{MD}$ and $\mathrm{ME}$ are the angle bisectors of triangles $\mathrm{ABM}$ and $\mathrm{CBM}$, respectively. Segments $\mathrm{BM}$ and $\mathrm{DE}$ intersect at point $\mathrm{F}$. Find $\mathrm{MF}$, if $\mathrm{DE}=7$. | 3.5 | 97 | 3 |
math | 5,6
Try to find two consecutive numbers; the sum of the digits of the first one is 8, and the second one is divisible by 8.
# | 7172 | 35 | 4 |
math | The Matini company released a special album with the flags of the $ 12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is $50\%$. | 3 | 66 | 3 |
math | When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$. | 283 | 71 | 3 |
math | Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}$ be non-negative numbers whose sum is 1. Let $M$ denote the maximum of the quantities $a_{1}+a_{2}+a_{3}, a_{2}+a_{3}+a_{4}, a_{3}+a_{4}+a_{5}$, $a_{4}+a_{5}+a_{6}, a_{5}+a_{6}+a_{7}$.
How small can $M$ be? | \frac{1}{3} | 132 | 7 |
math | 6. (20 points) Let a sequence of non-negative integers be given
$$
k, k+1, k+2, \ldots, k+n
$$
Find the smallest $k$, for which the sum of all numbers in the sequence is equal to 100.
# | 9 | 63 | 1 |
math | 12. If to a certain three-digit number, first the digit 7 is appended on the left, and then on the right, the first of the resulting four-digit numbers will be 3555 more than the second. Find this three-digit number. | 382 | 54 | 3 |
math | A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions:
(1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ ,
(2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ ,
(3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A... | 3 | 213 | 1 |
math | 1. Solve the equation
$$
\sqrt{x^{2}+x}+\sqrt{1+\frac{1}{x^{2}}}=\sqrt{x+3}
$$
in the set of real numbers. | -1 | 45 | 2 |
math | \section*{Problem 1 - 311011}
Determine the number of all pairs \((x ; y)\) of positive natural numbers \(x\) and \(y\) for which the following inequality (1) holds:
\[
x+y<1991
\] | 1979055 | 63 | 7 |
math | Let the tangent of the circle with diameter $AB$ be $AT$. Determine the point $M$ on the circle for which the sum of the distances from the lines $AB$ and $AT$ is $l$, a given value. Investigate for which values of $l$ there is a solution and how many there are. | \leqR(\sqrt{2}+1) | 68 | 12 |
math | 10. Point $P\left(x_{0}, y_{0}\right)$ is any point on the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{2}=1$, and the line $y=k x+m(k m \neq 0)$ intersects the ellipse $C_{1}: \frac{x^{2}}{4}+y^{2}=1$ to form a chord $M N$ which is bisected by the line $O P$ and satisfies $m y_{0}>-1$. The area of $\triangle P M N$ is 1. Det... | 1 | 167 | 1 |
math | Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations:
a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$;
b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.
| f(n) = 2n + 1 | 81 | 10 |
math | 1. Let $n$ ($n<100$) be a positive integer, and there exists a positive integer $k$, such that $1 \leqslant k \leqslant n-1$, satisfying
$$
\frac{4 k+1}{2 n}+\frac{1-2 k^{2}}{n^{2}}=\frac{1}{2} \text {. }
$$
How many values of $n$ satisfy the condition? Prove your conclusion. | 8 | 106 | 1 |
math | $7.23 \quad 7^{\lg x}-5^{\lg x+1}=3 \cdot 5^{\lg x-1}-13 \cdot 7^{\lg x-1}$ | 100 | 47 | 3 |
math | Find all integers $n$ such that $\frac{n^{3}-n+5}{n^{2}+1}$ is an integer.
Initial 241 | n=0 | 34 | 3 |
math | In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer. | 11 | 84 | 2 |
math | 14. Find all positive numbers $a$ such that the quadratic equation $\left(a^{2}+1\right) x^{2}+2 a x+\left(a^{2}-1\right)=0$ has both roots as integers. | a=1 | 52 | 3 |
math | (The 3rd problem of the general competition:)
Determine the number $A B C C$ (written in the decimal system) if
$$
A B C C=(D D-E) \cdot 100+D D \cdot E
$$
where $A, B, C, D$ and $E$ are distinct digits. | 1966 | 74 | 4 |
math | Example 3 Let $n$ be a fixed integer, $n \geqslant 2$.
(1) - Determine the smallest constant $c$ such that the inequality $\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{1 \leqslant i \leqslant n} x_{i}\right)^{4}$ holds for all non-negative numbers;
(2) - For this... | \frac{1}{8} | 148 | 7 |
math | Let $ f ( x ) \in \mathbb { Z } [ x ] $ be a polynomial with integer coefficients such that $ f ( 1 ) = - 1 , f ( 4 ) = 2 $ and $f ( 8 ) = 34 $. Suppose $n\in\mathbb{Z}$ is an integer such that $ f ( n ) = n ^ { 2 } - 4 n - 18 $. Determine all possible values for $n$. | 6, 3 | 103 | 4 |
math | Task 4. Given a natural number $n$, we define $\tau(n)$ as the number of natural numbers that divide $n$, and we define $\sigma(n)$ as the sum of these divisors. Find all natural numbers $n$ for which
$$
\sigma(n)=\tau(n) \cdot\lceil\sqrt{n}\rceil
$$
For a real number $x$, the notation $\lceil x\rceil$ means the smal... | 1,3,5,6 | 105 | 7 |
math | 7. Let $f(x)$ be a function defined on the set of natural numbers $\mathrm{N}$, taking values in $\mathrm{N}$, and for $x, y \in \mathrm{N}$, we have $f[f(x)+f(y)]=x+y$. Find $f(1988)$. | 1988 | 70 | 4 |
math | A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers... | 61 | 97 | 2 |
math | 1. [4 points] Around a flower, in the same plane as it, a bumblebee and a bee are flying along two circles. The speed of the bee is one and a half times the speed of the bumblebee. In the specified plane, a rectangular coordinate system is introduced, in which the flower (the common center of the circles) is at the poi... | (\sqrt{3};1),(-1;\sqrt{3}),(-\sqrt{3};-1),(1;-\sqrt{3}) | 182 | 30 |
math | Example 9 Find the smallest positive integer $n(n>1)$, such that $\sqrt{\frac{1}{n} \sum_{i=1}^{n} i^{2}}$ is an integer.
(2017, Singapore Mathematical Olympiad) | 337 | 56 | 3 |
math | Let $\lambda$ be a real number. Suppose that if \(ABCD\) is any convex cyclic quadrilateral such that \(AC=4\), \(BD=5\), and \(\overline{AB}\perp\overline{CD}\), then the area of $ABCD$ is at least $\lambda$. Then the greatest possible value of $\lambda$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $... | 100m + n = 100 \times 90 + 41 = 9041 | 123 | 28 |
math | Example 4 Find all real numbers $k$ such that $a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)$, for all $a, b, c, d \in[-1,+\infty)$: (2004 China Western Mathematical Olympiad) | \frac{3}{4} | 77 | 7 |
math | 100 Given $\triangle A B C$, the coordinates of vertex $A$ are $(2,-4) . B D, C E$ are the angle bisectors of $\angle B, \angle C$ respectively, and their line equations are $x+y-2=0, x-3 y-6=0$, then the equation of the line on which side $B C$ lies is $\qquad$ . | 100x+7y-6=0 | 88 | 11 |
math | Determine um valor de $n$ para o qual o numero $2^{8}+2^{11}+2^{n}$ seja um quadrado perfeito. | 12 | 36 | 2 |
math | 3. Find the value: $\cos 10^{\circ} \cdot \sin 20^{\circ} \cdot \sin 40^{\circ}=$ | \frac{\sqrt{3}}{8} | 38 | 10 |
math | Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the ar... | 6 | 79 | 1 |
math | In triangle $A B C$, it is known that $A B=10, B C=24$, and the median $B D$ is 13. The circles inscribed in triangles $A B D$ and $B D C$ touch the median $B D$ at points $M$ and $N$ respectively. Find $M N$. | 7 | 76 | 1 |
math | 6. Determine $x, y, z$ if
$$
\frac{a y+b x}{x y}=\frac{b z+c y}{y z}=\frac{c x+a z}{z x}=\frac{4 a^{2}+4 b^{2}+4 c^{2}}{x^{2}+y^{2}+z^{2}}, \quad a, b, c \in \mathbb{R}
$$ | 2a,2b,2c | 98 | 8 |
math | Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of int... | k = 3 | 134 | 5 |
math | Three, (17 points) The quadratic trinomial $x^{2}-x-2 n$ can be factored into the product of two linear factors with integer coefficients.
(1) If $1 \leqslant n \leqslant 30$, and $n$ is an integer, how many such $n$ are there?
(2) When $n \leqslant 2005$, find the largest integer $n$.
| 1953 | 100 | 4 |
math | Find the sum of all distinct possible values of $x^2-4x+100$, where $x$ is an integer between 1 and 100, inclusive.
[i]Proposed by Robin Park[/i] | 328053 | 49 | 6 |
math | (2) Draw two tangent lines to the circle $x^{2}+y^{2}=1$ through the point $(1,2)$. The area of the quadrilateral formed by these two tangent lines with the $x$-axis and $y$-axis is $\qquad$ . | \frac{13}{8} | 62 | 8 |
math | 21. A segment of constant length "slides" with its ends along two mutually perpendicular intersecting lines. What line does the midpoint of the segment describe? | |OM|=\frac{1}{2}|A^{\}B^{\}| | 32 | 18 |
math | 11. The members of a tribe have ten fingers on their hands and nine toes on their feet, and therefore count indifferently in base 10 or 19. In their mathematical culture, a positive integer is called "sacred" if it is written with the same two digits (between 1 and 9) in both bases. How many sacred numbers are there? | 4 | 80 | 1 |
math | 2. (HUN) For which real numbers $x$ does the following inequality hold:
$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9 \text { ? }
$$ | -\frac{1}{2}\leqx<\frac{45}{8}x\neq0 | 53 | 23 |
math | In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$?
[i]Proposed by Evan Chen[/i] | 5 | 62 | 1 |
math | Find the largest integer $k(k \geq 2)$, for which there exists an integer $n(n \geq k)$ such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:
1. each chosen number is not divisible by 6, by 7 and by 8;
2. the positive difference of ... | 108 | 105 | 3 |
math | 930. How many solutions in integers $x$ and $y$ does the inequality
$$
|x|+|y|<10 ?
$$
have? | 181 | 36 | 3 |
math | 6.001. $\frac{x^{2}+1}{x-4}-\frac{x^{2}-1}{x+3}=23$. | x_{1}=-\frac{55}{16},x_{2}=5 | 34 | 19 |
math | 4. For the equation $x^{3}-a x^{2}-2 a x+a^{2}-1=0$ with respect to $x$, if it has only one real root, then the range of values for $a$ is $\qquad$ | a<\frac{3}{4} | 54 | 9 |
math | 4. (7 points) On the board, 39 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 39 minutes? | 741 | 69 | 3 |
math | There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks.
The committee members have been provided with keys in such a way that every six members can open the safe, but no fi ve members can do that.
What is the smallest possible number of locks, and how many keys are ... | 2772 | 77 | 4 |
math | 6. Using $1,2, \cdots, n$ to form an $n$-digit number without repeating digits, where 2 cannot be adjacent to 1 or 3, a total of 2400 different $n$-digit numbers are obtained. Then $n=$ $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation ... | 7 | 91 | 1 |
math | SG. 3 Let $P$ and $P+2$ be both prime numbers satisfying $P(P+2) \leq 2007$. If $S$ represents the sum of such possible values of $P$, find the value of $S$. | 106 | 56 | 3 |
math | 36. $y=\sqrt{2 x^{2}-6 x}$. | x\leqslant0,x\geqslant3 | 17 | 14 |
math | One, (20 points) Find the integer part of $\left(\frac{1+\sqrt{5}}{2}\right)^{19}$. | 9349 | 33 | 4 |
math | Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.
[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
... | 44 | 244 | 2 |
math | 1. (5 points) Find the value of $n$ for which the following equality holds:
$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2015
$$ | 4064255 | 88 | 7 |
math | Problem 13. In triangle $A B C$, side $A C=$ $=5 \text{ cm}, B C-A B=2 \text{ cm}, \angle A: \angle C=2$. Find the lengths of sides $A B$ and $B C$. | AB=4 | 60 | 3 |
math | Example 2 (2000 National High School Competition Question) If: (1) $a, b, c, d$ all belong to $\{1,2,3,4\}$; (2) $a \neq b$, $b \neq c, c \neq d, d \neq a$; (3) $a$ is the smallest value among $a, b, c, d$. Then the number of different four-digit numbers $\overline{a b c d}$ that can be formed is $\qquad$ | 28 | 119 | 2 |
math | 7.5. A steamship from Gorky to Astrakhan takes 5 days, while from Astrakhan to Gorky it takes 7 days. How many days will it take for driftwood to float from Gorky to Astrakhan? | 35 | 55 | 2 |
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 | # Task 9.1
For which natural numbers $n$ is the expression $n^{2}-4 n+11$ a square of a natural number?
## Number of points 7 | 5 | 41 | 1 |
math | 4. (3 points) Arrange $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}$ and the average of these 6 fractions in ascending order, then this average value is in the $\qquad$th position. | 5 | 75 | 1 |
math | Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$. | 1:1 | 79 | 5 |
math | Example 2 uses $P^{*}$ to denote the set of all odd prime numbers less than 10000. Let $p$ be a number in $P^{*}$, satisfying: for any subset $S=\left\{p_{1}, p_{2}, \cdots, p_{k}\right\}$ of $P^{*}$ that does not contain $p$ and where $k \geqslant 2$, there exists $q \in \mathbf{P}^{*} \backslash S$ such that
$$(q+1) ... | \{3,7,31,127,8191\} | 183 | 19 |
math | In square $ABCD$ with $AB = 10$, point $P, Q$ are chosen on side $CD$ and $AD$ respectively such that $BQ \perp AP,$ and $R$ lies on $CD$ such that $RQ \parallel PA.$ $BC$ and $AP$ intersect at $X,$ and $XQ$ intersects the circumcircle of $PQD$ at $Y$. Given that $\angle PYR = 105^{\circ},$ $AQ$ can be expressed in sim... | 23 | 145 | 2 |
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