task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | If $E, U, L, S, R, T$ represent $1, 2, 3, 4, 5, 6$ (different letters represent different numbers), and satisfy:
(1) $E+U+L=6$;
(2) $S+R+U+T=18$;
(3) $U \times T=15$;
(4) $S \times L=8$.
Then the six-digit number $\overline{E U L S R T}=$ _. $\qquad$ | 132465 | 119 | 6 |
math | Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$ | 43 | 43 | 2 |
math | 2. Can 2010 be written as the sum of squares of $k$ distinct prime numbers? If so, find the maximum value of $k$; if not, please briefly explain the reason. | 7 | 44 | 1 |
math | Two lines intersect inside a unit square, splitting it into four regions. Find the maximum product of the areas of the four regions.
[i]Proposed by Nathan Ramesh | \frac{1}{256} | 35 | 9 |
math | A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is 1001. What is the sum of the six numbers on the faces? | 31 | 58 | 2 |
math | 1. The range of the function $f(x)=\frac{1+a e^{-x}}{1+e^{-x}}(a \neq 1)$ is | (1,)when>1;(,1)when<1 | 36 | 13 |
math | 61. Another task about a piece of land
- Here's another task,- said the Black Queen. Another farmer had a piece of land. On one third of his land, he grew pumpkins, on one fourth he planted peas, on one fifth he sowed beans, and the remaining twenty-six acres he allocated for corn.
How many acres of land did the farm... | 120 | 100 | 3 |
math | 18 (12 points) In $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and it satisfies $(2 a-c) \cos B=b \cos C, \sin ^{2} A=\sin ^{2} B+\sin ^{2} C-\lambda \sin B \sin C$ $(\lambda \in \mathbf{R})$.
(1) Find the size of angle $B$;
(2) If $\lambda=\sqrt{3}$, determine the shape of $\tr... | (-1,0)\cup(\sqrt{3},2) | 156 | 13 |
math | 3-4. How many pairs of integers $x, y$, lying between 1 and 1000, are there such that $x^{2}+y^{2}$ is divisible by 7. | 20164 | 45 | 5 |
math | 4. (6 points) Define the operation: $A \triangle B=2A+B$, given that $(3 \triangle 2) \triangle x=20, x=$ | 4 | 38 | 1 |
math | Example 1. In a plane, there are $\mathrm{n}$ lines $(\mathrm{n} \in \mathrm{N})$, among which no two are parallel, and no three intersect at the same point. Question: How many regions does these $\mathrm{n}$ lines divide the plane into? | \frac{1}{2}\left(n^{2} + n + 2\right) | 61 | 20 |
math | A2 (1-3, Hungary) Given the quadratic equation in $\cos x$: $a \cos ^{2} x+b \cos x+c$ $=0$. Here, $a, b, c$ are known real numbers. Construct a quadratic equation whose roots are $\cos 2 x$. In the case where $a$ $=4, b=2, c=-1$, compare the given equation with the newly constructed equation. | 4\cos^{2}2x+2\cos2x-1=0 | 93 | 18 |
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 | Kenooarov $\mathbf{P . 5}$.
In quadrilateral $A B C D$, side $A B$ is equal to diagonal $A C$ and is perpendicular to side $A D$, and diagonal $A C$ is perpendicular to side $C D$. A point $K$ is taken on side $A D$ such that $A C=A K$. The bisector of angle $A D C$ intersects $B K$ at point $M$. Find the angle $A C M... | 45 | 107 | 2 |
math | 5.1. For the celebration of the Name Day in the 5th grade parallel, several pizzas were ordered. 14 pizzas were ordered for all the boys, with each boy getting an equal share. Each girl also received an equal share, but half as much as each boy. How many pizzas were ordered if it is known that there are 13 girls in thi... | 15 | 95 | 2 |
math | 3. Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$ Let $s$ $=x^{2}+y^{2}$, then $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}$ is
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{8}{5} | 96 | 7 |
math | 7. We call fractions with a numerator of 1 and a denominator of a natural number greater than 1 unit fractions. If the unit fraction $\frac{1}{6}$ is expressed as the sum of two different unit fractions, then all possible representations are | \frac{1}{6}=\frac{1}{7}+\frac{1}{42}=\frac{1}{8}+\frac{1}{24}=\frac{1}{9}+\frac{1}{18}=\frac{1}{10}+\frac{1}{15} | 52 | 68 |
math | 5.2 .3 * Let $z$, $\omega$, and $\lambda$ be complex numbers, $|\lambda| \neq 1$, solve the equation for $z$, $\bar{z} - \lambda z = \omega$.
| \frac{\bar{\lambda}\omega+\bar{\omega}}{1-|\lambda|^{2}} | 52 | 22 |
math | 2. Given the function
$$
f(x)=\sin x+\sqrt{1+\cos ^{2} x}(x \in \mathbf{R}) \text {. }
$$
Then the range of the function $f(x)$ is $\qquad$ | [0,2] | 56 | 5 |
math | Example 18. Solve the equation
$$
x=(\sqrt{1+x}+1)(\sqrt{10+x}-4)
$$ | -1 | 32 | 2 |
math | 411. Find the curvature of the curve: 1) $x=t^{2}, y=2 t^{3}$ at the point where $t=1$; 2) $y=\cos 2 x$ at the point where $x=\frac{\pi}{2}$. | 4 | 62 | 1 |
math | ** Given in the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, the edge lengths are $A B=B C=3$, $B B_{1}=4$. Connect $B_{1} C$, and draw a perpendicular from point $B$ to $B_{1} C$ intersecting $C C_{1}$ at point $E$ and $B_{1} C$ at point $F$.
(1) Prove: $A_{1} C \perp$ plane $E B D$;
(2) Let $A_{1} C \cap$ pla... | \frac{25\sqrt{34}}{34} | 156 | 15 |
math | 5.19. Given three non-zero vectors $\bar{a}, \bar{b}, \bar{c}$, each pair of which are non-collinear. Find their sum, if $(\bar{a}+\bar{b}) \| \bar{c}$ and $(\bar{b}+\bar{c}) \| \bar{a}$. | \overline{0} | 77 | 6 |
math | 1. During the night shift, four attendants ate a whole barrel of pickles. If assistant Murr had eaten half as much, a tenth of the barrel would have remained. If lab assistant Trotter had eaten half as much, an eighth of the barrel would have remained. If intern Glupp had eaten half as much, a quarter of the barrel wou... | \frac{1}{40} | 95 | 8 |
math | Example 10. At an enterprise, products of a certain type are manufactured on three production lines. The first line produces $30 \%$ of the products from the total production volume, the second line - $25 \%$, and the third line produces the remaining part of the products. Each line is characterized by the following pe... | 0.032 | 105 | 5 |
math | 7. In tetrahedron $ABCD$, one edge has a length of 3, and the other five edges have a length of 2. Then the radius of its circumscribed sphere is $\qquad$ . | \frac{\sqrt{21}}{3} | 47 | 11 |
math | # Task 4.
In modern conditions, digitalization - the conversion of all information into digital code - is considered relevant. Each letter of the alphabet can be assigned a non-negative integer, called the code of the letter. Then, the weight of a word can be defined as the sum of the codes of all the letters in that ... | 10 | 170 | 2 |
math | Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }... | 6 | 132 | 1 |
math | 5. Find all natural numbers $N<10000$ such that $N=26 \cdot S(N)$, where $S(N)$ denotes the sum of the digits of the number $N$. | 234,468 | 45 | 7 |
math | Example 5 Let non-negative numbers $\alpha, \beta, \gamma$ satisfy $\alpha+\beta+\gamma=\frac{\pi}{2}$, find the minimum value of the function
$$f(\alpha, \beta, \gamma)=\frac{\cos \alpha \cos \beta}{\cos \gamma}+\frac{\cos \beta \cos \gamma}{\cos \alpha}+\frac{\cos \gamma \cos \alpha}{\cos \beta}$$ | \frac{5}{2} | 99 | 7 |
math |
5. The side-lengths $a, b, c$ of a triangle $A B C$ are positive integers. Let
$$
T_{n}=(a+b+c)^{2 n}-(a-b+c)^{2 n}-(a+b-c)^{2 n}+(a-b-c)^{2 n}
$$
for any positive integer $n$. If $\frac{T_{2}}{2 T_{1}}=2023$ and $a>b>c$, determine all possible perimeters of the triangle $A B C$.
| 49 | 118 | 2 |
math | 9. Arrange the positive integers (excluding 0) that are neither perfect squares nor perfect cubes in ascending order, resulting in 2, $3,5,6,7,10, \cdots \cdots$, what is the 1000th number in this sequence? | 1039 | 61 | 4 |
math | In how many different ways can 22 be written as the sum of 3 different prime numbers? That is, determine the number of triples $(a, b, c)$ of prime numbers with $1<a<b<c$ and $a+b+c=22$. | 2 | 55 | 1 |
math | 3. (17 points) Find the smallest natural number that is simultaneously twice a perfect square and three times a perfect cube.
# | 648 | 27 | 3 |
math | In math class, $2 / 3$ of the students had a problem set, and $4 / 5$ of them brought a calculator. Among those who brought a calculator, the same proportion did not have a problem set as those who did not bring a calculator. What fraction of the students had both a problem set and a calculator? | \frac{8}{15} | 70 | 8 |
math | 9.3. Inside an isosceles triangle $\mathrm{ABC}$ with equal sides $\mathrm{AB}=\mathrm{BC}$ and an angle of 80 degrees at vertex $\mathrm{B}$, a point $\mathrm{M}$ is taken such that the angle
$\mathrm{MAC}$ is 10 degrees, and the angle $\mathrm{MCA}$ is 30 degrees. Find the measure of angle $\mathrm{AMB}$. | 70 | 96 | 2 |
math | 4.1. On the side $A C$ of triangle $A B C$, points $E$ and $K$ are taken, with point $E$ lying between points $A$ and $K$ and $A E: E K: K C=3: 5: 4$. The median $A D$ intersects segments $B E$ and $B K$ at points $L$ and $M$ respectively. Find the ratio of the areas of triangles $B L M$ and $A B C$. Answer: $\frac{1}{... | \frac{1}{5} | 120 | 7 |
math | \section*{Problem 5 - 330935}
Determine all positive integers \(n\) with the property that the three numbers \(n+1, n+10\) and \(n+55\) have a common divisor greater than 1! | 3m-1 | 57 | 4 |
math | ## Task A-2.2.
Determine all real solutions of the system
$$
\begin{aligned}
& \left(1+4 x^{2}\right) y=4 z^{2} \\
& \left(1+4 y^{2}\right) z=4 x^{2} \\
& \left(1+4 z^{2}\right) x=4 y^{2}
\end{aligned}
$$ | (0,0,0)(\frac{1}{2},\frac{1}{2},\frac{1}{2}) | 92 | 28 |
math | Group Event 9
$B C, C A, A B$ are divided respectively by the points $X, Y, Z$ in the ratio $1: 2$. Let
area of $\triangle A Z Y$ : area of $\triangle A B C=2: a$ and
area of $\triangle A Z Y$ : area of $\triangle X Y Z=2: b$.
G9.1 Find the value of $a$.
G9.2 Find the value of $b$.
G9.3 Find the value of $x$.
G9.4 Fin... | =9,b=3,11,10 | 131 | 11 |
math | 5. Solve the system of equations $\left\{\begin{array}{l}x^{2} y+x y^{2}+3 x+3 y+24=0, \\ x^{3} y-x y^{3}+3 x^{2}-3 y^{2}-48=0 .\end{array}\right.$ | (-3,-1) | 73 | 5 |
math | Let Akbar and Birbal together have $n$ marbles, where $n > 0$.
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.”
Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”
What is the minimum possible value of... | 12 | 101 | 2 |
math | $ABCD$ is a parallelogram, and circle $S$ (with radius $2$) is inscribed insider $ABCD$ such that $S$ is tangent to all four line segments $AB$, $BC$, $CD$, and $DA$. One of the internal angles of the parallelogram is $60^\circ$. What is the maximum possible area of $ABCD$? | \frac{32\sqrt{3}}{3} | 84 | 13 |
math | Let $a, b>0$. Find all functions $f: \mathbb{R}_{+} \rightarrow \mathbb{R}$ such that for all positive real numbers $x, y$:
$$
f(x) f(y)=y^{a} f\left(\frac{x}{2}\right)+x^{b} f\left(\frac{y}{2}\right)
$$ | f(x)=^{} | 83 | 5 |
math | 7. A circle is drawn through two vertices of an equilateral triangle $A B C$ with an area of $21 \sqrt{3} \mathrm{~cm}^{2}$, for which two sides of the triangle are tangents. Find the radius of this circle. | 2\sqrt{7} | 59 | 6 |
math | Let's find the positive integer values of $n$ for which $2^{n}+65$ is a perfect square. | n=4orn=10 | 27 | 7 |
math | We have a pile of 2013 black balls and 2014 white balls, and a reserve of as many white balls as we want. We draw two balls: if they are of the same color, we put back one white ball, otherwise we put back the black ball. We repeat this until only one ball remains. What is its color? | black | 75 | 1 |
math | Gaikoov S.B.
Find all such $a$ and $b$ that $|a|+|b| \geqslant \frac{2}{\sqrt{3}}$ and for all $x$ the inequality $|a \sin x + b \sin 2x| \leq 1$ holds. | =\\frac{4}{3\sqrt{3}},\quadb=\\frac{2}{3\sqrt{3}} | 71 | 27 |
math | (i) Consider two positive integers $a$ and $b$ which are such that $a^a b^b$ is divisible by $2000$. What is the least possible value of $ab$?
(ii) Consider two positive integers $a$ and $b$ which are such that $a^b b^a$ is divisible by $2000$. What is the least possible value of $ab$? | 10 | 90 | 4 |
math | Example 19 consists of forming $n(\geqslant 3)$-digit numbers using the digits $1,2,3$, with the requirement that each of the digits $1,2,3$ must appear at least once in the $n$-digit number. Find the number of such $n$-digit numbers. | 3^{n}-3\cdot2^{n}+3 | 71 | 13 |
math | 2. A cylindrical bucket with a base diameter of $32 \mathrm{~cm}$ is filled with an appropriate amount of water. After placing an iron ball into the water, the ball is completely submerged, and the water level rises by $9 \mathrm{~cm}$ (no water spills out). The surface area of the ball is $\qquad$ $\mathrm{cm}^{2}$. | 576 \pi | 83 | 5 |
math | 1. Given the set $N=\{x \mid a+1 \leqslant x<2 a-1\}$ is a subset of the set $M=\{x \mid-2 \leqslant x \leqslant 5\}$. Then the range of values for $a$ is $\qquad$ . | a \leqslant 3 | 74 | 8 |
math | Suppose we have a convex polygon in which all interior angles are integers when measured in degrees, and the interior angles at every two consecutive vertices differ by exactly $1^{\circ}$. If the greatest and least interior angles in the polygon are $M^{\circ}$ and $m^{\circ}$, what is the maximum possible value of $M... | 18 | 75 | 2 |
math | 21. For each real number $x$, let $f(x)$ be the minimum of the numbers $4 x+1, x+2$ and $-2 x+4$. Determine the maximum value of $6 f(x)+2012$. | 2028 | 54 | 4 |
math | Problem 3. Brothers Petya and Vasya decided to shoot a funny video and post it on the internet. First, they filmed each of them walking from home to school - Vasya walked for 8 minutes, and Petya walked for 5 minutes. Then they came home and sat down at the computer to edit the video: they started Vasya's video from th... | 5 | 184 | 1 |
math | Let's determine the value of the sum $\sum_{n=1}^{\infty} \operatorname{arcctg}\left(2 n^{2}\right)$. | \frac{\pi}{4} | 38 | 7 |
math | 6. Tire economy. The tires on the rear wheels of a truck wear out after 15,000 km of driving, and on the front wheels - after 25,000 km. How many kilometers can the truck travel without replacing the tires, if the front and rear tires are swapped at the right moment? | 18750\mathrm{} | 69 | 8 |
math | 1. Let real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$. If $S=x^{2}+y^{2}$, denote the maximum and minimum values of $S$ as $p$ and $q$, respectively, then $\frac{1}{p}+\frac{1}{q}=$ $\qquad$ | \frac{8}{5} | 81 | 7 |
math | Example 8 Let the complex number $z$ satisfy $\left|z-z_{1}\right|=\lambda\left|z-z_{2}\right|$, where $z_{1}, z_{2}$ are given distinct complex numbers, and $\lambda$ is a positive real constant. Try to discuss the locus of the point corresponding to the complex number $z$ in the complex plane. | |z-\frac{z_{1}-\lambda^{2}z_{2}}{1-\lambda^{2}}|=\sqrt{|\frac{z_{1}-\lambda^{2}z_{2}}{1-\lambda^{2}}|^{2}-\frac{|z_{1}|^{2}-\lambda^{2}|z_{2}|^{2}}{1-\lambda^{} | 81 | 84 |
math | ## 134. Math Puzzle $7 / 76$
For the fencing of a square schoolyard, which is being erected by the pioneers and FDJ members of a school, 992,- Marks were paid to the State Forestry Enterprise.
One meter of the fence costs 4,- Marks. How many hectares is the area of the schoolyard? | 0.3844 | 76 | 6 |
math | 1. The speed of light is 300,000 kilometers per second, and the distance from the Sun to the Earth is 150 million kilometers. Question: How many minutes does it take for light to travel from the Sun to the Earth (round the answer to one decimal place)? | 8.3 | 63 | 3 |
math | Example 1.8.1. Suppose that $a, b, c$ are three positive real numbers satisfying
$$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=13 .$$
Find the minimum value of
$$P=\left(a^{2}+b^{2}+c^{2}\right)\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right) .$$ | 33 | 118 | 2 |
math | Exercise 5. There are 2 ways to place two identical $1 \times 2$ dominoes to cover a $2 \times 2$ chessboard: either by placing both horizontally, or by placing both vertically.
In how many ways can a $2 \times 11$ chessboard be covered with 11 identical $1 \times 2$ dominoes? | 144 | 82 | 3 |
math | A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye? | 10\sqrt{5} | 48 | 7 |
math | $2 \cdot 12$ Let $S=\{1,2, \cdots, 1963\}$, how many numbers can be selected from $S$ at most, so that the sum of any two numbers cannot be divisible by their difference? | 655 | 57 | 3 |
math | ## Problem 1.
Determine all natural numbers $a$ and $b$, where $a < b$, that are two-digit numbers, knowing that the greatest common divisor (gcd) of $a$ and $b$ is a prime number, 20 times smaller than the least common multiple (lcm) of $a$ and $b$. | =12,b=15;=20,b=25;=28,b=35;=44,b=55;=52,b=65;=68,b=85;=76,b=95 | 73 | 55 |
math | Find all rational numbers $a$,for which there exist infinitely many positive rational numbers $q$ such that the equation
$[x^a].{x^a}=q$ has no solution in rational numbers.(A.Vasiliev) | \mathbb{Q} \setminus \left\{ \frac{1}{n} \mid n \in \mathbb{Z}_{\ne 0} \right\} | 49 | 41 |
math | 16. When determining the germination of a batch of seeds, a sample of 1000 units was taken. Out of the selected seeds, 90 did not germinate. What is the relative frequency of the appearance of viable seeds | 0.91 | 51 | 4 |
math | $10 \cdot 42$ Find all three-digit numbers $\overline{a b c}$ that satisfy $\overline{a b c}=(a+b+c)^{3}$.
(China Shanghai Senior High School Mathematics Competition, 1988) | 512 | 56 | 3 |
math | C14 (16-1, USA) $A, B, C$ are playing a game: On 3 cards, integers $p, q, r (0<p<q<r)$ are written. The 3 cards are shuffled and distributed to $A, B, C$, each getting one. Then, according to the number on each person's card, they are given marbles. After that, the cards are collected, but the marbles are kept by each ... | C | 193 | 1 |
math | Example 1 Given $y=f(x)$ is a monotonically increasing odd function, its domain is $[-1,1]$, find the domain and range of the function $g(x)=$ $\sqrt{f\left(x^{2}-3\right)+f(x+1)}$. | 0 | 61 | 1 |
math | 1. Karlson filled a conical glass with lemonade and drank half of it by height (measuring from the surface of the liquid to the apex of the cone), and Little Man drank the second half. How many times more lemonade did Karlson drink compared to Little Man? | 7 | 58 | 1 |
math | # Task 2.
What is the last digit of the value of the sum $2019^{2020}+2020^{2019} ?$ | 1 | 40 | 1 |
math | 16th Putnam 1956 Problem A1 α ≠ 1 is a positive real. Find lim x→∞ ( (α x - 1)/(αx - x) ) 1/x . | \begin{cases}\alpha&if\alpha>1\\1&if\alpha<1\end | 45 | 22 |
math | 5. The range of real numbers $x$ that satisfy $\sqrt{1-x^{2}} \geqslant x$ is $\qquad$ | [-1,\frac{\sqrt{2}}{2}] | 32 | 12 |
math | [ Arithmetic. Mental calculation, etc.]
In the basket, there are 30 russulas and boletus. Among any 12 mushrooms, there is at least one russula, and among any 20 mushrooms, there is at least one boletus. How many russulas and how many boletus are in the basket?
# | 19 | 72 | 2 |
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 | 325. a) Find the smallest possible value of the polynomial
$$
P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}
$$
b) ${ }^{*}$ Prove that this polynomial cannot be represented as a sum of squares of polynomials in the variables $x, y$. | 3 | 82 | 1 |
math | $$
\begin{array}{l}
A=\left\{x \mid \log _{2}(x-1)<1\right\}, \\
B=\{x|| x-a \mid<2\} .
\end{array}
$$
If $A \cap B \neq \varnothing$, then the range of real number $a$ is | (-1,5) | 77 | 5 |
math | In a deck of cards consisting only of red and black cards, there are 2 times as many black cards as red cards. If 4 black cards are added, there are then 3 times as many black cards as red cards. How many cards were in the deck before adding the 4 black cards?
Only a numerical answer is expected here. | 12 | 71 | 2 |
math | Example 8 Given
$$
\begin{array}{l}
\frac{x}{m}+\frac{y}{n}+\frac{z}{p}=1, \\
\frac{m}{x}+\frac{n}{y}+\frac{p}{z}=0,
\end{array}
$$
Calculate the value of $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}+\frac{z^{2}}{p^{2}}$.
(1996, Tianjin City Junior High School Mathematics Competition) | \frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}+\frac{z^{2}}{p^{2}}=1 | 122 | 37 |
math | ## Problem Statement
Find the differential $d y$.
$y=\sqrt{3+x^{2}}-x \ln \left|x+\sqrt{3+x^{2}}\right|$ | -\ln|x+\sqrt{3+x^{2}}|\cdot | 40 | 13 |
math | 5. Given $x \geq 1, y \geq 1$ and $\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}$, then the maximum value of $u=\lg x y$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2+2\sqrt{2} | 94 | 8 |
math | 2. Honza has three cards, each with a different non-zero digit. The sum of all three-digit numbers that can be formed from these cards is a number 6 greater than three times one of them. What digits are on the cards? | 8,2,1 | 50 | 5 |
math | 30. It is known that for pairwise distinct numbers $a, b, c$, the equality param1 holds. What is the smallest value that the expression $a+b+c$ can take?
| param1 | Answer |
| :---: | :---: |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+2\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -2 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b... | -10 | 332 | 3 |
math | ## Task 4 - 280734
Determine all pairs $(p, q)$ of two prime numbers that satisfy the following conditions!
(1) It holds that $q > p + 1$.
(2) The number $s = p + q$ is also a prime number.
(3) The number $p \cdot q \cdot s$ is divisible by 10. | (2,5) | 86 | 5 |
math | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ with the property that $$ f(x-f(y))=f(f(x))-f(y)-1 $$ holds for all $x, y \in \mathbb{Z}$. (Croatia) | f(x) = x + 1 \text{ or } f(x) = -1 | 64 | 19 |
math | 11. Magic Pen (recommended for 8th grade, 1 point). Katya correctly solves a problem with a probability of $4 / 5$, while the magic pen correctly solves a problem without Katya's help with a probability of $1 / 2$. In the test, there are 20 problems, and to get a B, one needs to solve at least 13 of them correctly. How... | 10 | 124 | 2 |
math | 3. Determine the natural numbers $p$ and $q$ such that the roots of the trinomials
$$
x^{2}-p x+q \text { and } x^{2}-q x+p
$$
are also natural numbers. | (5,6),(6,5),(4,4) | 54 | 13 |
math | Example 4 Let the function $y=f(x)$ have the domain $\mathbf{R}$, and when $x>0$, we have $f(x)>1$, and for any $x, y \in \mathbf{R}$, we have $f(x+y)=f(x) f(y)$. Solve the inequality
$$
f(x) \leqslant \frac{1}{f(x+1)} .
$$
Analysis: This problem involves an abstract function, and we can find a prototype function $f(x... | x \leqslant -\frac{1}{2} | 164 | 14 |
math | 5.60 Solve the system of equations
$$
\left\{\begin{array}{l}
\sin x \sin y=0.75 \\
\operatorname{tg} x \operatorname{tg} y=3
\end{array}\right.
$$ | \\frac{\pi}{3}+\pi(k+n),\\frac{\pi}{3}+\pi(n-k),n,k\in\boldsymbol{Z} | 59 | 34 |
math | 9.1 $\quad A_{x}^{2} C_{x}^{x-1}=48$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
9.1 $\quad A_{x}^{2} C_{x}^{x-1}=48$. | 4 | 73 | 1 |
math | Example 3 Given $a \neq b$, the remainders when polynomial $f(x)$ is divided by $x-a$ and $x-b$ are $c$ and $d$ respectively, find the remainder when $f(x)$ is divided by $(x-a)(x-b)$. | \frac{-}{-b}x+\frac{-}{-b} | 61 | 15 |
math | Let's write the following numerical expressions in a simpler form:
(1)
$$
\begin{gathered}
\frac{1}{(1+\sqrt{1+\sqrt{8}})^{4}}+\frac{1}{(1-\sqrt{1+\sqrt{8}})^{4}}+\frac{2}{(1+\sqrt{1+\sqrt{8}})^{3}}+\frac{2}{(1-\sqrt{1+\sqrt{8}})^{3}} \\
\frac{1}{(1+\sqrt{1+\sqrt{2}})^{4}}+\frac{1}{(1-\sqrt{1+\sqrt{2}})^{4}}+\frac{2}... | -1,-1,0 | 296 | 6 |
math | 8. A box contains 5 white balls and 5 black balls. Now, the 10 balls are taken out one by one, ensuring that after each ball is taken out, the number of black balls left in the box is not less than the number of white balls. Assuming balls of the same color are indistinguishable, there are $\qquad$ ways to do this. | 42 | 79 | 2 |
math |
1. Let $n$ be a given natural number greater than 1. Find all pairs of integers $s$ and $t$ for which the equations
$$
x^{n}+s x-2007=0, \quad x^{n}+t x-2008=0
$$
have at least one common root in the domain of the real numbers.
| (,)=(2006,2007)(,)=((-1)^{n}-2007,(-1)^{n}-2008) | 84 | 37 |
math | Example 5 Suppose we have a $4 \times 4$ grid where each cell is colored differently. Each cell is filled with either 0 or 1, such that the product of the numbers in any two adjacent cells is 0. How many different ways are there to fill the grid with numbers?
(2007, Korean Mathematical Olympiad) | 1234 | 74 | 4 |
math | Problem 11.2. In the store, there are 9 headphones, 13 computer mice, and 5 keyboards. In addition, there are 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse". In how many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646. | 646 | 79 | 3 |
math | 10. In the Cartesian coordinate system, let point $A(0,4)$, $B(3,8)$. If point $P(x, 0)$ makes $\angle A P B$ maximum, then $x=$ $\qquad$ | 5 \sqrt{2}-3 | 53 | 7 |
math | 2. [4 points] Find all pairs of real parameters $a$ and $b$, for each of which the system of equations
$$
\left\{\begin{array}{l}
2(a-b) x+6 y=a \\
3 b x+(a-b) b y=1
\end{array}\right.
$$
has infinitely many solutions. | (-1;2),(-2;1),(\frac{3-\sqrt{17}}{2};-\frac{3+\sqrt{17}}{2}),(\frac{3+\sqrt{17}}{2};\frac{\sqrt{17}-3}{2}) | 76 | 61 |
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