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 | Five, $f$ is a mapping from the set of natural numbers $N$ to set $A$. If for $x$, $y \in \mathbf{N}$, $x-y$ is a prime number, then $f(x) \neq f(y)$. How many elements does $A$ have at least? | 4 | 70 | 1 |
math | Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$. | PQ = 5 | 76 | 5 |
math | Example 1 Find the range of real number $a$ such that for any real number $x$ and any $\theta \in\left[0, \frac{\pi}{2}\right]$, the inequality $(x+$ $3+2 \sin \theta \cdot \cos \theta)^{2}+(x+a \sin \theta+a \cos \theta)^{2} \geqslant \frac{1}{8}$ always holds.
(1996 National High School League Question) | \geqslant\frac{7}{2} | 107 | 12 |
math | 7.3 In a six-digit number, one digit was crossed out to obtain a five-digit number. The five-digit number was subtracted from the original number, and the result was 654321. Find the original number | 727023 | 49 | 6 |
math | 7. (10 points) A natural number that reads the same from left to right as from right to left is called a "palindrome number", for example:
909. Then the average of all three-digit palindrome numbers is. | 550 | 50 | 3 |
math | A rectangular parallelepiped has edges meeting at a vertex in the ratio $1: 2: 3$. How do the lateral surface areas of the right circular cylinders circumscribed around the parallelepiped compare? | P_{1}:P_{2}:P_{3}=\sqrt{13}:2\sqrt{10}:3\sqrt{5} | 44 | 31 |
math | Example 2.8. Calculate the integral $\int_{2}^{7}(x-3)^{2} d x$.
a) Using the substitution $z=x-3$.
b) Using the substitution $z=(x-3)^{2}$. | \frac{65}{3} | 56 | 8 |
math | 68(1178). What digit does the number end with: a) $2^{1000}$; b) $3^{1000}$; c) $7^{1000}$? | 6,1,1 | 49 | 5 |
math | 8. Find all prime numbers $p$ such that there exists an integer-coefficient polynomial
$$
f(x)=x^{p-1}+a_{p-2} x^{p-2}+\cdots+a_{1} x+a_{0} \text {, }
$$
satisfying that $f(x)$ has $p-1$ consecutive positive integer roots, and $p^{2} \mid f(\mathrm{i}) f(-\mathrm{i})$, where $\mathrm{i}$ is the imaginary unit. | p\equiv1(\bmod4) | 111 | 9 |
math | Dominik observed a chairlift. First, he found that one chair passed the lower station every 8 seconds. Then he picked one chair, pressed his stopwatch, and wanted to measure how long it would take for the chair to return to the lower station. After 3 minutes and 28 seconds, the cable car was sped up, so chairs passed t... | 119 | 116 | 3 |
math | ## 163. Math Puzzle $12 / 78$
A retention basin has a capacity of 0.5 million $\mathrm{m}^{3}$ of water. Even during low water, 20 $\mathrm{m}^{3}$ of water flow out per hour.
After a storm, a flood with an average of 120 $\mathrm{m}^{3}$ of water per hour flows in for ten hours.
By what percentage will the basin be... | 0.2 | 103 | 3 |
math | 3. (7p) Given the sets $A=\left\{x / x=11 a-3 ; a \in \mathbb{N}^{*}\right\}$ and $B=\{y / y=103-2 b ; b \in \mathbb{N}\}$. Determine $A \cap B$.
G.M. nr. 11/2015 | {19,41,63,85} | 88 | 13 |
math | One. (20 points) Given the equation in terms of $x$
$$
k x^{2}-\left(k^{2}+6 k+6\right) x+6 k+36=0
$$
the roots of which are the side lengths of a certain isosceles right triangle. Find the value of $k$. | 0, \sqrt{15}-3, \sqrt{9+3 \sqrt{2}}-3, \sqrt{6}+\sqrt{3}-3 | 74 | 35 |
math | 4. In how many ways can seven different items (3 weighing 2 tons each, 4 weighing 1 ton each) be loaded into two trucks with capacities of 6 tons and 5 tons, if the arrangement of the items inside the trucks does not matter? (12 points)
# | 46 | 61 | 2 |
math | In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit? | \frac{\sqrt{2}}{2} \text{ or } \frac{\sqrt{5} + 1}{2} | 55 | 28 |
math | In the following equation, determine $m$ such that one root is twice the other:
$$
2 x^{2}-(2 m+1) x+m^{2}-9 m+39=0
$$ | m_{1}=10,m_{2}=7 | 45 | 11 |
math | 8.241. $\operatorname{tg}^{3} z+\operatorname{ctg}^{3} z-8 \sin ^{-3} 2 z=12$.
8.241. $\tan^{3} z+\cot^{3} z-8 \sin^{-3} 2 z=12$. | (-1)^{k+1}\frac{\pi}{12}+\frac{\pik}{2},k\inZ | 75 | 27 |
math | 2.a) Calculate:
$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{n}+\sqrt{n+1}}$.
b) Determine $n \in \mathbb{N}^{*}$ for which the number:
$\frac{1-a_{1}(1-\sqrt{2})}{a_{1}+1+\sqrt{2}}+\frac{1-a_{2}(\sqrt{2}-\sqrt{3})}{a_{2}+\sqrt{2}+\sqrt{3}}+\cdots+\frac{1-a_{n}(\sqrt{n}-\sqrt{n+1})}{a_{n}+... | k^{2}-1,k\in\mathbb{N}^{*}-{1} | 206 | 19 |
math | 11. A math competition problem: the probabilities of A, B, and C solving this problem independently are $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$, respectively, where $a$, $b$, and $c$ are positive integers less than 10. Now A, B, and C are solving this problem independently. If the probability that exactly one o... | \frac{4}{15} | 114 | 8 |
math | Lenka had a paper flower with eight petals. On each petal, there was exactly one digit, and no digit was repeated on any other petal. When Lenka played with the flower, she noticed several things:
- It was possible to tear off four petals such that the sum of the numbers written on them would be the same as the sum of... | allthedigitsexcept78,orallexcept69 | 165 | 13 |
math | The first term of a certain arithmetic and geometric progression is 5; the second term of the arithmetic progression is 2 less than the second term of the geometric progression; the third term of the geometric progression is equal to the sixth term of the arithmetic progression. What progressions satisfy the conditions... | thegeometricprogressions:5,15,45,\ldots;5,10,20,\ldots\quadthearithmeticprogressions:5,13,21,\ldots;5,8,11,\ldots | 61 | 55 |
math | 9. Given the ellipse $\Gamma: \frac{x^{2}}{9}+\frac{y^{2}}{5}=1$, a line passing through the left focus $F(-2,0)$ of the ellipse $\Gamma$ with a slope of $k_{1}\left(k_{1} \notin\{0\right.$, $\infty\})$ intersects the ellipse $\Gamma$ at points $A$ and $B$. Let point $R(1,0)$, and extend $A R$ and $B R$ to intersect th... | 305 | 199 | 3 |
math | Positive integers $x_{1}, \ldots, x_{m}$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_{1}, \ldots, F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$?
(Here $F_{1}, \ldots, F_{2018}$ are th... | 1009 | 144 | 4 |
math | Example 5 Let $p$ be a given positive integer, $A$ is a subset of $X=\left\{1,2,3,4, \cdots, 2^{p}\right\}$, and has the property: for any $x \in A$, $2 x \notin A$. Find the maximum value of $|A|$. (1991 French Mathematical Olympiad) | \frac{2^{p+1}+(-1)^{p}}{3} | 87 | 19 |
math | 21311 ㅊ Let the three-digit number $n=\overline{a b c}$, if $a, b, c$ can form an isosceles (including equilateral) triangle, find the number of such three-digit numbers $n$.
| 165 | 57 | 3 |
math | 10. To pack books when moving a school library, you can buy small boxes that hold 12 books or large ones that hold 25 books. If all the books are packed in small boxes, 7 books will be left, and if all the books are packed in large boxes, there will be room for 5 more books. The library's collection contains between 50... | 595 | 96 | 3 |
math | 2.254. $\left(\frac{x+2 y}{8 y^{3}\left(x^{2}+2 x y+2 y^{2}\right)}-\frac{(x-2 y): 8 y^{2}}{x^{2}-2 x y+2 y^{2}}\right)+\left(\frac{y^{-2}}{4 x^{2}-8 y^{2}}-\frac{1}{4 x^{2} y^{2}+8 y^{4}}\right)$ $x=\sqrt[4]{6}, \quad y=\sqrt[8]{2}$. | 3 | 131 | 1 |
math | Example 2 Divide a circle into $n$ sectors, and color the sectors using $r$ colors, with each sector being one color, and no two adjacent sectors having the same color. Question: How many coloring methods are there?
| (r-1)^{n}+(r-1)(-1)^{n},forn\geqslant2;\quadr,forn=1 | 48 | 33 |
math | 11.3 A weirdo chose 677 different natural numbers from the list $1,2,3, \ldots, 2022$. He claims that the sum of no two of the chosen numbers is divisible by 6. Did he go too far with his claim? | 676 | 62 | 3 |
math | 7. Let $a$ be a non-zero real number, in the Cartesian coordinate system $x O y$, the focal distance of the quadratic curve $x^{2}+a y^{2}+a^{2}=0$ is 4, then the value of $a$ is $\qquad$ | \frac{1-\sqrt{17}}{2} | 64 | 13 |
math | 1. [4 points] Around a bird feeder, in the same plane as it, a tit and a bullfinch are flying along two circles at the same speed. In the plane, a rectangular coordinate system is introduced, in which the feeder (the common center of the circles) is located at point $O(0 ; 0)$. The tit is moving clockwise, while the bu... | (4\sqrt{3};0),(-2\sqrt{3};6),(-2\sqrt{3};-6) | 157 | 28 |
math | Ten, $1447, 1005, 1231$ have many things in common: they are all four-digit numbers, their highest digit is 1, and they each have exactly two identical digits. How many such numbers are there? | 432 | 56 | 3 |
math | 1. A four-digit number, $n$, is written as ' $A B C D$ ' where $A, B, C$ and $D$ are all different odd digits. It is divisible by each of $A, B, C$ and $D$. Find all the possible numbers for $n$. | 1395,1935,3195,3915,9135,9315 | 65 | 29 |
math | 1. A triangle with all sides as integers, and the longest side being 11, has
$\qquad$ possibilities. | 36 | 27 | 2 |
math | Example 6 Let $m>n \geqslant 1$. Find the smallest $m+n$ such that
$$\text {1000| } 1978^{m}-1978^{n} \text {. }$$ | 106 | 55 | 3 |
math | 7. If the equation $\left(x^{2}-1\right)\left(x^{2}-4\right)=k$ has four non-zero real roots, and the four points corresponding to them on the number line are equally spaced, find the value of $k$. | \frac{7}{4} | 55 | 7 |
math | 3. The set of positive odd numbers $\{1,3,5, \cdots\}$ is grouped in ascending order such that the $n$-th group contains $(2 n-1)$ odd numbers: (first group) $\{1\}$, (second group) $\{3,5,7\}$, (third group) $\{9,11,13,15,17\}, \cdots$ Then 1991 is in the group. | 32 | 106 | 2 |
math | Problem 7.7. In the election for class president, Petya and Vasya competed. Over three hours, 27 students in the class voted for one of the two candidates. In the first two hours, Petya received 9 more votes than Vasya. In the last two hours, Vasya received 9 more votes than Petya. In the end, Petya won. By what maximu... | 9 | 99 | 1 |
math | Example 5 Let $a$ and $d$ be non-negative real numbers, $b$ and $c$ be positive numbers, and $b+c \geqslant a+d$. Find the minimum value of $\frac{b}{c+d}+$ $\frac{c}{a+b}$. | \sqrt{2}-\frac{1}{2} | 63 | 12 |
math | 4. Which is greater: $\sqrt{2016}+\sqrt{2015+\sqrt{2016}}$ or $\sqrt{2015}+\sqrt{2016+\sqrt{2015}}$? | \sqrt{2016}+\sqrt{2015+\sqrt{2016}}>\sqrt{2015}+\sqrt{2016+\sqrt{2015}} | 56 | 46 |
math | 318. When dividing a number by 72, the remainder is 68. How will the quotient change and what will we get as the remainder if we divide the same number by 24? | t_{1}=3+2,remainder20 | 44 | 11 |
math | Example 1 Solve the system of congruences $\left\{\begin{array}{l}x \equiv 1(\bmod 7), \\ x \equiv 1(\bmod 8), \\ x \equiv 3(\bmod 9) .\end{array}\right.$ | 57 | 63 | 2 |
math | Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$? | 10 | 72 | 2 |
math | Determine the real number $a$ having the property that $f(a)=a$ is a relative minimum of $f(x)=x^4-x^3-x^2+ax+1$. | a = 1 | 41 | 5 |
math | 1. Find all the solutions of each of the following diophantine equations
a) $x^{2}+3 y^{2}=4$
b) $x^{2}+5 y^{2}=7$
c) $2 x^{2}+7 y^{2}=30$. | a) x= \pm 2, y=0 ; x= \pm 1, y= \pm 1 \quad b) \text{no solution} \quad c) x= \pm 1, y= \pm 2 | 63 | 53 |
math | 【Question 28】
Place five cards $A, K, Q, J, 10$ randomly into five envelopes labeled $A, K, Q, J, 10$, with one card in each envelope. How many ways are there to place the cards so that each card ends up in the wrong envelope? | 44 | 68 | 2 |
math | Example 3.21. Find the points of discontinuity of the function
$$
z=\frac{x y+1}{x^{2}-y}
$$ | x^{2} | 34 | 4 |
math | Four, (Full marks 20 points) An arithmetic sequence with a common difference of 4 and a finite number of terms, the square of its first term plus the sum of the rest of the terms does not exceed 100. Please answer, how many terms can this arithmetic sequence have at most?
保留源文本的换行和格式，所以翻译结果如下：
Four, (Full marks 20 po... | 8 | 142 | 1 |
math | 3. Solve the system of equations $\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y+z}=-\frac{2}{15} \\ \frac{1}{y}+\frac{1}{x+z}=-\frac{2}{3}, \\ \frac{1}{z}+\frac{1}{x+y}=-\frac{1}{4}\end{array}\right.$, | (5;-1;-2) | 95 | 7 |
math | 48. (SWE 1) Determine all positive roots of the equation $x^{x}=1 / \sqrt{2}$. | x_{1}=1/2x_{2}=1/4 | 29 | 14 |
math | 10.324. A circle is inscribed in a triangle. The lines connecting the center of the circle with the vertices divide the area of the triangle into parts with areas 4, 13, and $15 \mathrm{~cm}^{2}$. Find the sides of the triangle. | \frac{8}{\sqrt{3}},\frac{26}{\sqrt{3}},\frac{30}{\sqrt{3}} | 65 | 32 |
math | Consider a set with $n$ elements.
How many subsets of odd cardinality exist? | 2^{n-1} | 18 | 6 |
math | 9. Given the complex sequence $\left\{z_{n}\right\}$ satisfies: $z_{1}=\frac{\sqrt{3}}{2}, z_{n+1}=\overline{z_{n}}\left(1+z_{n} \mathrm{i}\right)(n=1,2, \cdots)$, where $\mathrm{i}$ is the imaginary unit, find the value of $z_{2021}$. | z_{2021}=\frac{\sqrt{3}}{2}+(\frac{1}{2}+\frac{1}{2^{20200}})\mathrm{i} | 96 | 41 |
math | The five numbers $17$, $98$, $39$, $54$, and $n$ have a mean equal to $n$. Find $n$. | 52 | 37 | 2 |
math | ## Task A-1.1.
If $a$ and $b$ are natural numbers, then $\overline{\overline{a . b}}$ is the decimal number obtained by writing a decimal point after the number $a$ and then the number $b$. For example, if $a=20$ and $b=17$, then $\overline{\overline{a . b}}=20.17$ and $\overline{\overline{b . a}}=17.2$.
Determine al... | (2,5)(5,2) | 154 | 9 |
math | 13. Find the sum of all the real numbers $x$ that satisfy the equation
$$
\left(3^{x}-27\right)^{2}+\left(5^{x}-625\right)^{2}=\left(3^{x}+5^{x}-652\right)^{2} .
$$ | 7 | 75 | 1 |
math | Example 8. Find $\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$. | \frac{}{b} | 28 | 6 |
math | Let $P$ be a polynomial of degree 2008 such that for any integer $k \in \{1, \cdots, 2009\}$, $P(k) = \frac{1}{k}$. Calculate $P(0)$.
## Elementary Symmetric Polynomials
Here we address the relationship between the coefficients and the roots of a polynomial: consider the polynomial
$$
P(X) = a_{n} X^{n} + \cdots + a... | \sum_{\mathrm{k}=1}^{2009}\frac{1}{\mathrm{k}} | 441 | 23 |
math | 1. Two cyclists set off simultaneously from point A to point B. When the first cyclist had covered half the distance, the second cyclist still had 24 km to go, and when the second cyclist had covered half the distance, the first cyclist still had 15 km to go. Find the distance between points A and B. | 40 | 68 | 2 |
math | ## Task B-3.1.
If $\sin x-\cos x=\frac{\sqrt{3}}{3}$, calculate $\sin ^{6} x+\cos ^{6} x$. | \frac{2}{3} | 42 | 7 |
math | ## Task 6 - 190836
A taxi driver was supposed to pick up a guest from the train station at 15:00. With an average speed of $50 \frac{\mathrm{km}}{\mathrm{h}}$, he would have arrived on time. Due to unfavorable traffic conditions, he could only drive at an average speed of $30 \frac{\mathrm{km}}{\mathrm{h}}$ and theref... | 12.5 | 136 | 4 |
math | Example 5 Find the last three digits of $1 \times 3 \times 5 \times \cdots \times 1997$. | 375 | 32 | 3 |
math | Example 5 Calculate $\left[\frac{23 \times 1}{101}\right]+\left[\frac{23 \times 2}{101}\right]+\cdots+$ $\left[\frac{23 \times 100}{101}\right]$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 1100 | 88 | 4 |
math | # Variant 4.
1) Let's draw an arbitrary line $y=k-b$, intersecting the parabola at points $A$ and $B$. Then the abscissas of the endpoints of the chord $A$ and $B$ must satisfy the conditions: $k x - b = x^2$, i.e., $x^2 - k x + b = 0$, $x_A = \frac{k - \sqrt{k^2 - 4b}}{2}$, $x_B = \frac{k + \sqrt{k^2 - 4b}}{2}$. Then... | f(x)=\frac{5}{8x^{2}}-\frac{x^{3}}{8} | 716 | 22 |
math | 6-152 Find all functions $f$ from the set of real numbers to the set of real numbers that satisfy the following conditions:
(1) $f(x)$ is strictly increasing;
(2) For all real numbers $x, f(x) + g(x) = 2x$, where $g(x)$ is the inverse function of $f(x)$. | f(x)=x+ | 77 | 5 |
math | Martin has written a five-digit number with five different digits on a piece of paper, with the following properties:
- by crossing out the second digit from the left (i.e., the digit in the thousands place), he gets a number that is divisible by two,
- by crossing out the third digit from the left, he gets a number t... | 98604 | 152 | 5 |
math | 12. Let the equation $x y=6(x+y)$ have all positive integer solutions $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, then $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=$
$\qquad$ . | 290 | 89 | 3 |
math | Comparing the fractions ${ }^{111110} / 111111,{ }^{222221 / 222223},{ }^{333331} /{ }_{333334}$, arrange them in ascending order.
# | 111110/111111<333331/333334<222221/222223 | 68 | 41 |
math | [ Methods for solving problems with parameters ] [ Quadratic equations. Vieta's theorem ]
It is known that the equation $x^{2}+5 b x+c=0$ has roots $x_{1}$ and $x_{2}, x_{1} \neq x_{2}$, and some number is a root of the equation $y^{2}+2 x_{1} y+2 x_{2}=0$ and a root of the equation $z^{2}+2 x_{2} z+2 x_{1}=0$. Find $... | \frac{1}{10} | 121 | 8 |
math | Problem 10.4. Find all values of the real parameter $a$ such that the number of the solutions of the equation
$$
3\left(5 x^{2}-a^{4}\right)-2 x=2 a^{2}(6 x-1)
$$
does not exceed the number of the solutions of the equation
$$
2 x^{3}+6 x=\left(3^{6 a}-9\right) \sqrt{2^{8 a}-\frac{1}{6}}-(3 a-1)^{2} 12^{x}
$$
Ivan L... | \frac{1}{3} | 131 | 7 |
math | 1. Calculation question: $(20.15+40.3) \times 33+20.15=$ | 2015 | 29 | 4 |
math | 6. (3 points) A number, when it is reduced by 5 times, and then expanded by 20 times, the result is 40. What is this number? | 10 | 39 | 2 |
math | Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold:
[list=a][*]$f(1)=0$,
[*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list]
Find the value of $a^{2013}+b^{2013}+c^{2013}$. | -1 | 124 | 2 |
math | 3. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and the hyperbola $C$ intersects the circle $x^{2}+y^{2}=r^{2}(r>0)$ at a point $P$. If the maximum value of $\frac{\left|P F_{1}\right|+\left|P F_{2}\right|}{r}$ is $4 \sqrt{2}$, th... | 2 \sqrt{2} | 148 | 6 |
math | Putnam 1997 Problem A5 Is the number of ordered 10-tuples of positive integers (a 1 , a 2 , ... , a 10 ) such that 1/a 1 + 1/a 2 + ... + 1/a 10 = 1 even or odd? Solution | odd | 69 | 1 |
math | 5. Given that $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C: \frac{x^{2}}{19}+\frac{y^{2}}{3}=1$, and point $P$ is on the ellipse $C$. If $S_{\triangle P F_{1} F_{2}}=\sqrt{3}$, then $\angle F_{1} P F_{2}=$ $\qquad$ | 60^{\circ} | 100 | 6 |
math | Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than $3$ strea... | 932 | 112 | 3 |
math | ## Problem 4
Find $x>1$ for which $\frac{1}{[x]}+\frac{1}{\{x\}}=2014 x$, where $[x]$ denotes the integer part of $x$, and $\{x\}$ represents the fractional part of $x$.
## Mathematical Gazette
## Note.
All problems are mandatory.
Each problem is worth 7 points.
Working time 3 hours.
Proposers: Prof. Hecser Enik... | n+\frac{1}{2014n}, | 159 | 12 |
math | \section*{Exercise 2 - 081012}
Determine all prime numbers \(p\) with the following property!
If you add the number 50 to \(p\) and subtract the number 50 from \(p\), you get two prime numbers. | 53 | 59 | 2 |
math | $7 \cdot 2$ When two numbers are drawn without replacement from the set $\left\{-3,-\frac{5}{4},-\frac{1}{2}, 0, \frac{1}{3}, 1, \frac{4}{5}, 2\right\}$, find the probability that the two numbers are the slopes of a pair of perpendicular lines. | \frac{3}{28} | 81 | 8 |
math | $A$ and $B$ are chess players who compete under the following conditions: The winner is the one who first reaches (at least) 2 points; if they both reach 2 points at the same time, the match is a draw. (A win is 1 point, a draw is $1 / 2$ point, a loss is 0 points) -
a) What is the expected number of games, if the pro... | 0.315 | 132 | 5 |
math | Which are the two consecutive odd numbers in the number sequence, the sum of whose squares is of the form $\frac{n(n+1)}{2}$, where $n$ is a natural number. | 1^{2}+3^{2}=\frac{4\cdot5}{2} | 41 | 19 |
math | $3-$ Area of a trapezoid $\quad$ K [ Mean proportionals in a right triangle ]
A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid. | 156 | 61 | 3 |
math | 8.2. If $a, b, c, d$ are positive real numbers such that $a b=2$ and $c d=27$, find the minimum value of the expression $E=(a+1)(b+2)(c+3)(d+4)$. | 600 | 61 | 3 |
math | A3 Two circles, each of radius 5 units long, are drawn in the coordinate plane such that their centres $A$ and $C$ have coordinates $(0,0)$ and $(8,0)$ respectively. How many points where both coordinates are integers are within the intersection of these circles (including its boundary)? | 9 | 65 | 1 |
math | Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$ | 376 | 66 | 3 |
math | 17. Person A and Person B start from points $A$ and $B$ respectively at the same time, heading towards each other. When A reaches the midpoint $C$ between $A$ and $B$, B is still 240 meters away from $C$. When B reaches $C$, A has already passed $C$ by 360 meters. Then, when the two meet at point $D$, the distance $CD$... | 144 | 101 | 3 |
math | The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point. | 6 | 96 | 3 |
math | An equilateral $12$-gon has side length $10$ and interior angle measures that alternate between $90^\circ$, $90^\circ$, and $270^\circ$. Compute the area of this $12$-gon.
[i]Proposed by Connor Gordon[/i] | 500 | 65 | 3 |
math | 9. If $|z|=2, u=\left|z^{2}-z+1\right|$, then the minimum value of $u$ is $\qquad$ (here $z \in \mathbf{C}$ ). | \frac{3}{2}\sqrt{3} | 50 | 11 |
math | 1. The numbers $u, v, w$ are roots of the equation $x^{3}-3 x-1=0$. Find $u^{9}+v^{9}+w^{9}$. (12 points) | 246 | 50 | 3 |
math | 4.021. An arithmetic progression has the following property: for any $n$ the sum of its first $n$ terms is equal to $5 n^{2}$. Find the common difference of this progression and its first three terms. | 10;5,15,25 | 51 | 10 |
math | 15. Let $S=\{1,2,3, \cdots, 65\}$. Find the number of 3-element subsets $\left\{a_{1}, a_{2}, a_{3}\right\}$ of $S$ such that $a_{i} \leq a_{i+1}-(i+2)$ for $i=1,2$. | 34220 | 84 | 5 |
math | 4.1. How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference of two squares of integers? | 250 | 31 | 3 |
math | Example 18 Let the quadratic function $f(x)=a x^{2}+b x+c(a \neq 0)$ have values whose absolute values do not exceed 1 on the interval $[0,1]$, find the maximum value of $|a|+|b|+|c|$. | 17 | 66 | 2 |
math | Let $f(x)$ be the polynomial with respect to $x,$ and $g_{n}(x)=n-2n^{2}\left|x-\frac{1}{2}\right|+\left|n-2n^{2}\left|x-\frac{1}{2}\right|\right|.$
Find $\lim_{n\to\infty}\int_{0}^{1}f(x)g_{n}(x)\ dx.$ | f\left( \frac{1}{2} \right) | 94 | 15 |
math | [ | [ Investigation of a quadratic trinomial ] |
| :---: | :---: |
For what values of the parameter $a$ are both roots of the equation $(1+a) x^{2}-3 a x+4 a=0$ greater than 1? | -\frac{16}{7}<-1 | 58 | 10 |
math | 3. A line $l$ passing through the right focus $F$ of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$, and there are exactly 3 such lines, find $\lambda$.
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将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4 | 108 | 1 |
math | 7.2. Let $a$ be the number of six-digit numbers divisible by 13 but not divisible by 17, and $b$ be the number of six-digit numbers divisible by 17 but not divisible by 13.
Find $a-b$. | 16290 | 57 | 5 |
math | ## Task B-1.2.
Determine all natural numbers $n$ for which the fraction $\frac{4 n}{4 n-2019}$ is an integer. | n=504,n=505,n=673 | 38 | 15 |
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