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 | 12-35 How many integer pairs $(m, n)$ satisfy $n^{2}+(n+1)^{2}=m^{4}+(m+1)^{4}$?
(China Beijing High School Grade 1 Mathematics Competition, 1988) | (,n)=(0,0),(0,-1),(-1,0),(-1,-1) | 58 | 22 |
math | 10.3. In a row from left to right, all natural numbers from 1 to 37 are written in such an order that each number, starting from the second to the 37th, divides the sum of all numbers to its left: the second divides the first, the third divides the sum of the first and second, and so on, the last divides the sum of the first thirty-six. The number 37 is on the first place from the left, what number is on the third place? | 2 | 108 | 1 |
math | Example 19 (Adapted from the 2002 National High School Competition) Let the quadratic function $f(x)=a x^{2}+b x+c(a, b, c \in$ R) satisfy the following 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 $f(x)$. | f(x)=\frac{1}{4}(x+1)^{2} | 159 | 17 |
math | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
$$ (x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z) $$
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^{i} y^{j} z^{k} \in \mathcal{B}$ for all nonnegative integers $i, j, k$ satisfying $i+j+k \geqslant n$. (Venezuela) Answer: $n=4$. | 4 | 179 | 1 |
math | ## Task 25/66
What is the largest multiple of 11 in which none of the digits occurs more than once? | 9876524130 | 29 | 10 |
math | [ $\quad$ Rebus $\quad]$ [ Arithmetic progression ] Avor: Akumich i.f. To a natural number $A$, three digits were appended on the right. The resulting number turned out to be equal to the sum of all natural numbers from 1 to $A$. Find $A$.
# | 1999 | 68 | 4 |
math | 4. If the complex coefficient equation with respect to $x$
$$
(1+2 \mathrm{i}) x^{2}+m x+1-2 \mathrm{i}=0
$$
has real roots, then the minimum value of the modulus of the complex number $m$ is $\qquad$ | 2 | 65 | 1 |
math | 1. Find all positive integers $n$, such that $3^{n}+n^{2}+2019$ is a perfect square.
（Zou Jin, contributed） | 4 | 39 | 1 |
math | 12 、Place $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex has one number, and all numbers can only be used once), such that the sum of the two numbers on each edge is a prime number. Then the maximum sum of the four numbers on one face is | 18 | 80 | 2 |
math | For how many ordered pairs of positive integers $(x, y)$ is the least common multiple of $x$ and $y$ equal to $1{,}003{,}003{,}001$? | 343 | 50 | 3 |
math | Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$. | 53 | 122 | 2 |
math | Pick out three numbers from $0,1,\cdots,9$, their sum is an even number and not less than $10$. We have________different ways to pick numbers. | 51 | 38 | 2 |
math | Four. (50 points) On a plane, there are $n$ points, no three of which are collinear. Each pair of points is connected by a line segment, and each line segment is colored either red or blue. A triangle with all three sides of the same color is called a "monochromatic triangle." Let the number of monochromatic triangles be $S$.
(1) If $n=6$, for all possible colorings, find the minimum value of $S$;
(2) If $n=2 k$ (integer $k \geqslant 4$), for all possible colorings, find the minimum value of $S$.
保留了源文本的换行和格式。 | \frac{k(k-1)(k-2)}{3} | 152 | 14 |
math | 3. If the sum of the terms of the infinite geometric sequence $\left\{a_{n}\right\}$ is 1, and the sum of the absolute values of the terms is 2, then the value of the first term $a_{1}$ is . $\qquad$ | \frac{4}{3} | 60 | 7 |
math | \section*{Task 20 - V01220}
Calculate the internal dimensions of a cylindrical cast iron pan with a capacity of 20 t, which should exhibit the lowest possible heat losses through appropriate shaping!
Based on experience, it is assumed that the heat losses from the surface of the liquid cast iron (per unit area) are twice the heat losses through the wall and bottom surface. (Weight of liquid cast iron: \(7.2 \mathrm{t} / \mathrm{m}^{3}\) ) | r\approx0.6655\, | 110 | 11 |
math | Determine the smallest rational number $\frac{r}{s}$ such that $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$ whenever $k, m,$ and $n$ are positive integers that satisfy the inequality $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$. | \frac{41}{42} | 87 | 9 |
math | $ABCD$ is a square and $AB = 1$. Equilateral triangles $AYB$ and $CXD$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $XY$? | \sqrt{3} - 1 | 51 | 8 |
math | Example 3-11 6 people attend a meeting, hanging their hats on a coat rack upon entry, and in their haste to leave, they each grab a hat at random. What is the probability that no one gets their own hat? | 0.368 | 50 | 5 |
math | 29(1105). Two cars set off simultaneously from $A$ to $B$ and from $B$ to $A$, and they met after 3 hours. The first car arrived in $B$ 1.1 hours later than the second car arrived in $A$. How many times greater is the speed of the second car compared to the speed of the first car? | 1.2 | 81 | 3 |
math | 2. Let two acute angles $\alpha, \beta$ satisfy
$$
\begin{array}{l}
(\sin \alpha+\cos \alpha)(\sin \beta+\cos \beta)=2 \text {. } \\
\text { Then }(\sin 2 \alpha+\cos 3 \beta)^{2}+(\sin 2 \beta+\cos 3 \alpha)^{2} \\
=
\end{array}
$$ | 3-2 \sqrt{2} | 92 | 8 |
math | 4. If a positive integer is equal to 4 times the sum of its digits, then we call this positive integer a quadnumber. The sum of all quadnumbers is $\qquad$ .
| 120 | 40 | 3 |
math | $ f(x)$ is a continuous function with the piriodicity of $ 2\pi$ and $ c$ is a positive constant number.
Find $ f(x)$ and $ c$ such that $ \int_0^{2\pi} f(t \minus{} x)\sin tdt \equal{} cf(x)$ with $ f(0) \equal{} 1$ for all real numbers $ x$. | f(x) = \cos(x) + \sin(x) | 87 | 14 |
math | 2. When dividing a three-digit number by a two-digit number, we get the sum of the digits of the divisor as the quotient, and the remainder is the number formed by swapping the digits of the divisor. If we multiply this remainder by the quotient and add the divisor, the resulting number consists of the digits of the dividend, but in reverse order. What is the dividend and the divisor? | 957 | 80 | 3 |
math | [ Extremal properties (miscellaneous).]
What is the largest number of points that can be placed on a segment of length 1 so that on any segment of length $d$, contained in this segment, there are no more than $1+1000 d^{2}$ points? | 32 | 61 | 2 |
math | Task 1. We have 1000 balls in 40 different colors, with exactly 25 balls of each color. Determine the smallest value of $n$ with the following property: if you randomly arrange the 1000 balls in a circle, there will always be $n$ balls next to each other where at least 20 different colors occur. | 452 | 79 | 3 |
math | H4. The points $A, B$ and $C$ are the centres of three faces of a cuboid that meet at a vertex. The lengths of the sides of the triangle $A B C$ are 4,5 and 6 .
What is the volume of the cuboid? | 90\sqrt{6} | 61 | 7 |
math | The integers from 1 to 20 are written on the board. We can erase any two numbers, $a$ and $b$, and write the number $a+b-1$ in their place. What number will be written on the board after 19 operations? And if we wrote $a+b+ab$ instead of $a+b-1$?
## Extremum Principle
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 21!-1 | 107 | 5 |
math | 1. The digit at the 2007th position after the decimal point of the irrational number $0.2342343423434342343434342 \cdots$ is $\qquad$ . | 3 | 59 | 1 |
math | 7.5. There are 100 boxes numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The audience can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and, without reading the questions aloud, honestly answers all of them. What is the minimum number of notes that need to be sent to definitely find out where the prize is? | 99 | 105 | 2 |
math | The teacher said to Xu Jun: "Two years ago, my age was three times your age." Xu Jun said to the teacher: "In eight years, your age will be twice mine." Xu Jun is $\qquad$ years old this year. | 12 | 51 | 2 |
math | 9.124. $y=\log _{3}\left(0.64^{2-\log _{\sqrt{2}} x}-1.25^{8-\left(\log _{2} x\right)^{2}}\right)$. | x\in(0;\frac{1}{64})\cup(4;\infty) | 57 | 21 |
math | 5. Positive real numbers $a, b, c$ form a geometric sequence $(q \neq 1), \log _{a} b, \log _{b} c, \log _{c} a$ form an arithmetic sequence. Then the common difference $d=$ | -\frac{3}{2} | 60 | 7 |
math | Problem 2. The number 27 is written on the board. Every minute, the number is erased from the board and replaced with the product of its digits, increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7+12=26$. What will be on the board after an hour? | 14 | 77 | 2 |
math | From the sequence of natural numbers, we remove the perfect squares. In the remaining sequence of numbers, which is the 2001st, and which position does the number 2001 occupy? | 1957 | 43 | 4 |
math | Problem 1. Vasya and Petya ran out from the starting point of a circular running track at the same time and ran in opposite directions. They met at some point on the track. Vasya ran a full lap and, continuing to run in the same direction, reached the place of their previous meeting at the moment when Petya had run a full lap. How many times faster did Vasya run than Petya? | \frac{\sqrt{5}+1}{2} | 92 | 12 |
math | Test $\mathbf{G}$ Calculation:
$$
\frac{\left(2^{4}+\frac{1}{4}\right)\left(4^{4}+\frac{1}{4}\right)\left(6^{4}+\frac{1}{4}\right)\left(8^{4}+\frac{1}{4}\right)\left(10^{4}+\frac{1}{4}\right)}{\left(1^{4}+\frac{1}{4}\right)\left(3^{4}+\frac{1}{4}\right)\left(5^{4}+\frac{1}{4}\right)\left(7^{4}+\frac{1}{4}\right)\left(9^{4}+\frac{1}{4}\right)} .
$$
(1991 Jiangsu Province Junior High School Mathematics Competition Question) | 221 | 184 | 3 |
math | 7. (40 points) To enter Ali Baba's cave, it is necessary to zero out 28 counters, each set to a natural number in the range from 1 to 2017. Treasure hunters are allowed, in one move, to decrease the values of some of the counters by the same number, which they can change from move to move. Indicate the minimum number of moves in which the treasure hunters can, with certainty, zero out the counters (regardless of the initial values) and enter the cave. | 11 | 110 | 2 |
math | 8. In $\triangle A B C$, $\angle C=\frac{\pi}{3}$, let $\angle B A C=\theta$. If there exists a point $M$ on line segment $B C$ (different from points $B$ and $C$), such that when $\triangle B A M$ is folded along line $A M$ to a certain position to get $\triangle B^{\prime} A M$, it satisfies $A B^{\prime} \perp C M$, then the range of $\theta$ is $\qquad$ | (\frac{\pi}{6},\frac{2\pi}{3}) | 116 | 16 |
math | Example 3 Find the number of ordered integer pairs $(a, b)$ such that $x^{2}+a x+b=167 y$ has integer solutions $(x, y)$, where $1 \leqslant$ $a, b \leqslant 2004$.
---
The translation maintains the original text's line breaks and format. | 2020032 | 79 | 7 |
math | 3.355. $\sin ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\sin ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}$. | \frac{3}{2} | 65 | 7 |
math | 8. Given the odd function $f(x)$ satisfies $f(x+2)=f(x)$, when $x \in(0,1)$, $f(x)=2^{x}$, $f\left(\log _{\frac{1}{2}} 23\right)=$ $\qquad$ . | -\frac{23}{16} | 67 | 9 |
math | How many such square columns are there, in which the edge length measured in cm is an integer, and the surface area measured in $\mathrm{cm}^{2}$ is the same as the volume measured in $\mathrm{cm}^{3}$? | 4 | 51 | 1 |
math | Natural numbers $m$ and $n$ are such that $m>n$, $m$ does not divide $n$, and the remainder of $m$ divided by $n$ is the same as the remainder of $m+n$ divided by $m-n$.
Find the ratio $m: n$. | 5:2 | 63 | 3 |
math | Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
[i]Proposed by Ray Li[/i] | 200 | 98 | 3 |
math | $10 \cdot 32$ In the number 3000003, what digits should the hundred's place digit 0 and the ten-thousand's place digit 0 be replaced with, so that the resulting number is divisible by 13?
(Polish Mathematical Competition, 1950) | 3080103,3050203,3020303,3090503,3060603,3030703,3000803 | 69 | 55 |
math | Problem 1. Calculate:
a) $\lim _{x \rightarrow \infty}\left(\sqrt{\frac{1}{4} x^{2}+x+2}+\sqrt{\frac{1}{4} x^{2}+2 x+5}-\sqrt{x^{2}+x+1}\right)$;
b) $\lim _{n \rightarrow \infty} \sin \left(\pi \sqrt{n^{2}+3}\right)$. | 0 | 102 | 1 |
math | 3. In the coordinate plane, there are two regions $M$ and $N, M$ is defined by $\left\{\begin{array}{l}y \geqslant 0, \\ y \leqslant x, \quad N \text { is a region that varies with } t, \\ y \leqslant 2-x,\end{array}\right.$ it is determined by the inequality $t \leqslant x \leqslant t+1$, where the range of $t$ is $0 \leqslant t \leqslant 1$, then the common area of $M$ and $N$ is the function $f(t)=$ $\qquad$ | -^2++\frac{1}{2} | 154 | 11 |
math | 8. There are 10 cards, each card has two different numbers from 1, 2, 3, 4, 5, and no two cards have the same pair of numbers. Place these 10 cards into five boxes labeled 1, 2, 3, 4, 5, with the rule that a card with numbers $i, j$ can only be placed in box $i$ or box $j$. A placement is called "good" if and only if the number of cards in box 1 is more than the number of cards in each of the other boxes. How many good placements are there? | 120 | 134 | 3 |
math | 3. Factorize: $a+(a+b) x+(a+2 b) x^{2}+(a+3 b) x^{3}+3 b x^{4}+2 b x^{5}+b x^{6}$. | (1+x)(1+x^{2})(++^{2}+^{3}) | 53 | 17 |
math | In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won? | 11 | 63 | 2 |
math | (a) Find all positive integers $n$ for which the equation $(a^a)^n = b^b$ has a solution
in positive integers $a,b$ greater than $1$.
(b) Find all positive integers $a, b$ satisfying $(a^a)^5=b^b$ | (a, b) = (4^4, 4^5) | 63 | 17 |
math | 18. (2007 Bulgarian National Mathematical Olympiad) Find all positive integers $x, y$ such that $\left(x y^{2}+2 y\right)$ | $\left(2 x^{2} y+x y^{2}+8 x\right)$. | (,2a)where\in{N}_{+},(3,1),(8,1) | 60 | 22 |
math | 1 Elimination Method
Example 1 Given that $x, y, z$ are real numbers, and $x+2y-z=6, x-y+2z=3$. Then, the minimum value of $x^{2}+y^{2}+z^{2}$ is $\qquad$
(2001, Hope Cup Junior High School Mathematics Competition) | 14 | 79 | 2 |
math | 4. $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ 的最大值为 $\qquad$
The maximum value of $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ is $\qquad$ | \frac{8\sqrt{3}}{9} | 67 | 12 |
math | 7. Let $p$ be a prime number, and $q=4^{p}+p^{4}+4$ is also a prime number. Find the value of $p+q$.
The text above is translated into English, keeping the original text's line breaks and format. | 152 | 61 | 3 |
math | 8. In a rectangular coordinate system, given two points $A(0, a), B(0, b), a>b>0$ on the positive half of the $y$-axis. $C$ is a point on the positive half of the $x$-axis, and makes $\angle A C B$ maximum, then the coordinates of point $C$ are $\qquad$ . | (\sqrt{},0) | 83 | 5 |
math | 1. If we subtract the unit digit from any natural number with at least two digits, we get a number that is one digit "shorter." Find all original numbers that are equal to the absolute value of the difference between the square of the "shorter" number and the square of the removed digit. | 48,100,147 | 62 | 10 |
math | Example 45 Find the maximum positive integer $n$, such that there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying for any $1 \leqslant i<j \leqslant$ $n$, $\left(1+x_{i} x_{j}\right)^{2} \leqslant 0.99\left(1+x_{i}^{2}\right)\left(1+x_{j}^{2}\right)$. | 31 | 112 | 2 |
math | 1. If positive real numbers $a, b$ satisfy
$$
\log _{8} a+\log _{4} b^{2}=5, \log _{8} b+\log _{4} a^{2}=7 \text {, }
$$
then $\log _{4} a+\log _{8} b=$ $\qquad$ | 4 | 79 | 1 |
math | 9.2. On the board, a certain natural number $N$ was written nine times (one under the other). Petya appended a non-zero digit to the left or right of each of the 9 numbers; all the appended digits are different. What is the maximum number of prime numbers that could result among the 9 obtained numbers?
(I. Efremov) | 6 | 77 | 1 |
math | In two rooms of an educational center, lectures were being held. The average age of eight people present in the first room was 20 years, and the average age of twelve people in the second room was 45 years. During the lecture, one participant left, and as a result, the average age of all people in both rooms increased by one year. How old was the participant who left?
(L. Hozová) | 16 | 89 | 2 |
math | Three students named João, Maria, and José took a test with 100 questions, and each of them answered exactly 60 of them correctly. A question is classified as difficult if only one student answered it correctly, and it is classified as easy if all three answered it correctly. We know that each of the 100 questions was solved by at least one student. Are there more difficult or easy questions? Additionally, determine the difference between the number of difficult and easy questions. | 20 | 100 | 2 |
math | 2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 5134315 is symmetric, while 5134415 is not. How many seven-digit symmetric numbers exist such that adding 1100 to them leaves them symmetric? | 810 | 77 | 3 |
math | 3. If $x^{2}+y^{2}+2 x-4 y+5=0$, what is $x^{2000}+2000 y$? | 4001 | 42 | 4 |
math | There are four points $A$, $B$, $C$, $D$ on a straight line, $AB: BC: CD=2: 1: 3$. Circles $\odot O_{1}$ and $\odot O_{2}$ are constructed with $AC$ and $BD$ as diameters, respectively. The two circles intersect at $E$ and $F$. Find $ED: EA$.
(1996, Huanggang Region, Hubei Province, Junior High School Mathematics Competition) | \sqrt{2} | 109 | 5 |
math | 3. Given a natural number $n$. Determine the number of solutions to the equation
$$
x^{2}-\left[x^{2}\right]=(x-[x])^{2}
$$
for which $1 \leq x \leq n$. | n^2-n+1 | 54 | 6 |
math | Let $a$ and $b$ be real numbers such that $ \frac {ab}{a^2 + b^2} = \frac {1}{4} $. Find all possible values of $ \frac {|a^2-b^2|}{a^2+b^2} $. | \frac{\sqrt{3}}{2} | 61 | 10 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-9.5,9.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 380 | 57 | 3 |
math | Find the area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively. | 400 | 42 | 3 |
math | You flip a fair coin which results in heads ($\text{H}$) or tails ($\text{T}$) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$? | \frac{7}{8} | 54 | 7 |
math | 6. If $x \in\left(0, \frac{\pi}{2}\right)$, then the maximum value of the function $f(x)=2 \cos ^{3} x+3 \cos ^{2} x-6 \cos x-2 \cos 3 x$ is
$\qquad$ . | \frac{1}{9} | 70 | 7 |
math | A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising? | \frac{1}{\pi} | 70 | 8 |
math | 10. (9 points) The distance between each pair of consecutive utility poles along the road is 50 meters. Xiao Wang is traveling in a car at a constant speed and sees 41 utility poles within 2 minutes after seeing the first pole. Calculate how many meters the car travels per hour? | 60000 | 63 | 5 |
math | 7. How many three-digit natural numbers have the product of their digits equal to 28? Write them down! | 9 | 24 | 1 |
math | 4. Given positive integers $a$ and $b$ differ by 120, their least common multiple is 105 times their greatest common divisor. The larger of $a$ and $b$ is $\qquad$ | 225 | 49 | 3 |
math | 5. The range of real numbers $x$ that satisfy $\sqrt{1-x^{2}} \geqslant x$ is . $\qquad$ | \left[-1, \frac{\sqrt{2}}{2}\right] | 33 | 17 |
math | Example 9. Find $\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x+1}-2}$. | 24 | 32 | 2 |
math | 146. From a natural number, the sum of its digits was subtracted, and then one digit was erased from the resulting difference. The sum of the remaining digits of the difference is 131. Which digit was erased? | 4 | 49 | 1 |
math | Example 7. Calculate the central moments of a random variable distributed according to the normal law. | \mu_{2}=1\cdot3\ldots(2-1)\sigma^{2} | 19 | 21 |
math | We call an even positive integer $n$ nice if the set $\{1,2, \ldots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of 3. For example, 6 is nice, because the set $\{1,2,3,4,5,6\}$ can be partitioned into subsets $\{1,2\},\{3,6\},\{4,5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$. | 2^{2022} - 1 | 132 | 10 |
math | Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$. | 2009 | 86 | 4 |
math | 405. Find all natural numbers, when divided by 6, the quotient is the same as the remainder. | 7,14,21,28,35 | 24 | 13 |
math | 1. It is known that $\sin x=\frac{3}{2} \sin y-\frac{2}{3} \cos y, \cos x=\frac{3}{2} \cos y-\frac{2}{3} \sin y$. Find $\sin 2 y$. | \frac{61}{72} | 61 | 9 |
math | 5. Let $d_{1}, d_{2}, \ldots, d_{n}$ be all the natural divisors of the number $10!=1 \cdot 2 \cdot \ldots \cdot 10$. Find the sum
$$
\frac{1}{d_{1}+\sqrt{10!}}+\frac{1}{d_{2}+\sqrt{10!}}+\ldots+\frac{1}{d_{n}+\sqrt{10!}}
$$ | \frac{270}{2\sqrt{10!}} | 107 | 15 |
math | Condition of the problem
Calculate the limit of the function:
$\lim _{x \rightarrow-2} \frac{\operatorname{tg} \pi x}{x+2}$ | \pi | 38 | 2 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{1}{3}(x-2) \sqrt{x+1}+\ln (\sqrt{x+1}+1)
$$ | \frac{3x\sqrt{x+1}+3x-\sqrt{x+1}+2}{6\sqrt{x+1}\cdot(\sqrt{x+1}+1)} | 40 | 40 |
math | 5.1. How many triangles with integer sides have a perimeter equal to 2017? (Triangles that differ only in the order of their sides, for example, 17, 1000, 1000 and 1000, 1000, 17, are counted as one triangle.) | 85008 | 74 | 5 |
math | 15.23. In how many ways can the number $n$ be represented as the sum of several addends, each equal to 1 or 2? (Representations differing in the order of the addends are considered different.) | F_{n+1} | 50 | 6 |
math | [ Lagrange Interpolation Polynomial ]
Construct polynomials $f(x)$ of degree not higher than 2 that satisfy the conditions:
a) $f(0)=1, f(1)=3, f(2)=3$
b) $f(-1)=-1, f(0)=2, f(1)=5$
c) $f(-1)=1, f(0)=0, f(2)=4$.
# | -x^{2}+3x+1,3x+2,x^{2} | 92 | 18 |
math | Let $ABC$ be an acute triangle with $\cos B =\frac{1}{3}, \cos C =\frac{1}{4}$, and circumradius $72$. Let $ABC$ have circumcenter $O$, symmedian point $K$, and nine-point center $N$. Consider all non-degenerate hyperbolas $\mathcal H$ with perpendicular asymptotes passing through $A,B,C$. Of these $\mathcal H$, exactly one has the property that there exists a point $P\in \mathcal H$ such that $NP$ is tangent to $\mathcal H$ and $P\in OK$. Let $N'$ be the reflection of $N$ over $BC$. If $AK$ meets $PN'$ at $Q$, then the length of $PQ$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers such that $c$ is not divisible by the square of any prime. Compute $100a+b+c$.
[i]Proposed by Vincent Huang[/i] | 100a + b + c | 226 | 10 |
math | Five, given $P_{n}(x)=\sum_{1 \leqslant 2 k+1 \leq n} C_{n}^{2 k+1} x^{n-2 k-1}\left(x^{2}-1\right)^{k-1}$. $S_{n}$ is the sum of the absolute values of the coefficients of the polynomial $P_{n}(x)$. For any positive integer $n$, find the exponent of 2 in the prime factorization of $S_{n}$. | t_{n}+1 | 114 | 6 |
math | ## Task 4 - 191234
Investigate whether among all tetrahedra $ABCD$ with a given volume $V$ and right angles $\angle BDC, \angle CDA, \angle ADB$, there exists one with the smallest possible sum $AB + AC + AD + BC + BD + CD$.
If this is the case, determine (in dependence on $V$) this smallest possible sum. | 3\cdot(1+\sqrt{2})\cdot\sqrt[3]{6V} | 93 | 20 |
math | 8. $[\mathbf{5}]$ Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that
$$
n^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x+3 x+3 y+3 z-6
$$ | 8 | 79 | 1 |
math | 5. Arrange all positive divisors of 8128 in ascending order as $a_{1}, a_{2}, \cdots, a_{n}$, then $\sum_{k=1}^{n} k a_{k}=$ $\qquad$ . | 211335 | 57 | 6 |
math | In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$. (The incenter is the center of the inscribed circle of triangle .) | 84 | 82 | 2 |
math | 10. (10 points) A $15 \times 15$ grid composed of unit squares, using the grid points as vertices to form squares with integer side lengths, then the number of squares with side length greater than 5 is $\qquad$ . | 393 | 56 | 3 |
math | Problem 8.8. A computer can apply three operations to a number: "increase by 2", "increase by 3", "multiply by 2". The computer was given the number 1 and was made to try all possible combinations of 6 operations (each of these combinations is applied to the initial number 1). After how many of these combinations will the computer end up with an even number? | 486 | 84 | 3 |
math | 3. Let the function be
$$
y(x)=(\sqrt{1+x}+\sqrt{1-x}+2)\left(\sqrt{1-x^{2}}+1\right) \text {, }
$$
where, $x \in[0,1]$. Then the minimum value of $y(x)$ is | 2+\sqrt{2} | 70 | 6 |
math | 80. Cryptarithm with "ham". In the following cryptarithm, each letter stands for a decimal digit (its own for each letter):
$$
7(F R Y H A M)=6(H A M F R Y)^{*}
$$
Determine which digit each letter represents. | FRY=461,HAM=538 | 61 | 12 |
math | (15) Real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, then the maximum value of $\sqrt{2} x y+y z$ is
$\qquad$ . | \frac{\sqrt{3}}{2} | 53 | 10 |
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