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 | 1. (6 points) Calculate $10.37 \times 3.4 + 1.7 \times 19.26=$ | 68 | 33 | 2 |
math | 1. Given $a+b+c=0, a>b>c$. Then the range of $\frac{c}{a}$ is $\qquad$ . | -2<\frac{c}{a}<-\frac{1}{2} | 31 | 17 |
math | Four consecutive even numbers are removed from the set \[A=\{ 1, 2, 3, \cdots, n \}.\] If the arithmetic mean of the remaining numbers is $51.5625$, which four numbers were removed? | 22, 24, 26, 28 | 56 | 14 |
math | 6・76 Given $\left|x_{i}\right|<1, i=1,2, \cdots, n$. Also,
$$\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=19+\left|x_{1}+x_{2}+\cdots+x_{n}\right|$$
What is the minimum value of the integer $n$? | 20 | 94 | 2 |
math | 7. Suppose $A B C D$ is an isosceles trapezoid in which $\overline{A B} \| \overline{C D}$. Two mutually externally tangent circles $\omega_{1}$ and $\omega_{2}$ are inscribed in $A B C D$ such that $\omega_{1}$ is tangent to $\overline{A B}, \overline{B C}$, and $\overline{C D}$ while $\omega_{2}$ is tangent to $\over... | \frac{3}{7} | 149 | 7 |
math | In a culturing of bacteria, there are two species of them: red and blue bacteria.
When two red bacteria meet, they transform into one blue bacterium.
When two blue bacteria meet, they transform into four red bacteria.
When a red and a blue bacteria meet, they transform into three red bacteria.
Find, in function of th... | \{(n, 0), (n-2, 1), (n-4, 2), \ldots\} | 104 | 29 |
math | Find the least positive integer $n$ such that $15$ divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers $a_1, a_2, \dots, a_{15}$. | 4 | 79 | 1 |
math | Inverting 7. Given a square with side length 1. Try to find the largest and the smallest area of an inscribed equilateral triangle within this square, and calculate these two areas (You must prove your argument).
(1978, National High School League) | 2 \sqrt{3}-3 \text{ and } \frac{\sqrt{3}}{4} | 57 | 22 |
math | 3. In $\triangle A B C$, $A B=2 A C$, and $S_{\triangle A B C}=1$. Then the minimum value of $B C$ is
In $\triangle A B C$, $A B=2 A C$, and $S_{\triangle A B C}=1$. Then the minimum value of $B C$ is | \sqrt{3} | 77 | 5 |
math | 1. (6 points) $1.25 \times 67.875 + 125 \times 6.7875 + 1250 \times 0.053375$. | 1000 | 52 | 4 |
math | 12. (10 points) A certain natural number minus 39 is a perfect square, and minus 144 is also a perfect square. Find this natural number.
| 160,208,400,2848 | 38 | 16 |
math | When dividing a certain number $m$ by 13 and 15, the same quotient was obtained, but the first division had a remainder of 8, while the second division had no remainder.
Find the number $m$. | 60 | 48 | 2 |
math | The expression $\frac{k^{2}}{1.001^{k}}$ reaches its maximum value with which natural number $k$? | 2001 | 30 | 4 |
math | 2. The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, then in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity $n^{2} d$... | 400 | 108 | 3 |
math | $A$ and $B$ run a 5000-meter race. In the first attempt, $A$ gives $B$ a 1 km head start and finishes 1 minute earlier. In the second attempt, $A$ gives an 8-minute head start and is still 1 km away from the finish line when $B$ reaches the finish. How many minutes does it take for $A$ and how many for $B$ to complete ... | A:15 | 122 | 4 |
math | Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$. | \frac{1}{16} | 33 | 8 |
math | Let's determine $x$ such that
$$
\begin{gathered}
1^{4}+2^{4}+\cdots+x^{4}=1+2 \cdots+x+1^{2}+2^{2}+\cdots+x^{2}+ \\
+1^{3}+2^{3}+\cdots+x^{3}
\end{gathered}
$$ | 2 | 83 | 1 |
math | 258. $\left(a^{m}\right)^{\frac{1}{n}} ;\left(a^{\frac{1}{n}}\right)^{\frac{n}{m}} ;\left(a^{n} b\right)^{\frac{1}{n}} ;\left(a^{n} b^{m}\right)^{\frac{1}{m n}} ;\left(\frac{a^{n}}{b^{m}}\right)^{\frac{1}{m n}}$.
Problems from an anonymous Italian manuscript of the 14th century. | ^{\frac{}{n}};^{\frac{1}{}};^{\frac{1}{n}};^{\frac{1}{}}\cdotb^{\frac{1}{n}};\frac{^{\frac{1}{}}}{b^{\frac{1}{n}}} | 122 | 63 |
math | 19. A random number generator gives outputs of $1,2,3,4$ and 5 with equal probability. The values of $a, b$ and $c$ are each chosen by running the generator once.
The probability that $a \times b+c$ is even can be written as a fraction in its lowest terms as $\frac{N}{D}$.
What is the value of $10 N+D$ ? | 715 | 91 | 3 |
math | Illustrate a semicircle with diameter $A G$. The arc of the semicircle is divided into six equal parts by points $B$, $C$, $D$, $E$, and $F$. $D F$ and $C G$ are both straight line segments. Given that the area of the semicircle is $60 \mathrm{~cm}^{2}$, what is the area of the shaded part in $\mathrm{cm}^{2}$? | 20 | 99 | 2 |
math | For any positive integer $n$, let $D_{n}$ be the set of all positive divisors of $n$, and $f_{i}(n)$ be the number of elements in the set
$$
F_{i}(n)=\left\{a \in D_{n} \mid a \equiv i(\bmod 4)\right\}
$$
Find the smallest positive integer $m$ such that
$$
2 f_{1}(m)-f_{2}(m)=2017 \text {. }{ }^{[1]}
$$
(14th China So... | 2\times5^{2016} | 130 | 10 |
math | Let $f(n)$ be the number of ways to write $n$ as a sum of powers of 2, where we keep track of the order of the summation. For example, $f(4)=6$ because 4 can be written as $4, 2+2, 2+1+1, 1+2+1, 1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than 2013 for which $f(n)$ is odd. | 2047 | 114 | 4 |
math | 33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at lea... | 1890 | 84 | 4 |
math | Example 6 In $\triangle A B C$, find the value of $a^{3} \sin (B-C)+b^{3} \sin (C-A)+c^{3} \sin (A-B)$. | 0 | 46 | 1 |
math | 1. If the function $f(x)=\lg \left(a x^{2}-4 x+a-3\right)$ has a range of $\mathbf{R}$, then the range of real number $a$ is
$\qquad$ | 0\leqslant\leqslant4 | 52 | 12 |
math | 7. There are 10 chess players participating in a round-robin tournament (i.e., each pair of players competes in one match). The rules state that a win earns 2 points, a draw earns 1 point for each player, and a loss earns 0 points. After the tournament, it is found that each player's score is unique, and the second-pla... | 16 | 115 | 2 |
math | Example 6 Light passes through point $A(2,3)$, hits the line $l: x+y+1=0$, and after reflection passes through point $B(1,1)$, find the equations of the incident and reflected rays. | 5x-4y+2=0,4x-5y+1=0 | 52 | 19 |
math | Evokimov
A natural number is written on the board. If you erase the last digit (in the units place), the remaining non-zero number will be divisible by 20, and if you erase the first digit, it will be divisible by 21. What is the smallest number that can be written on the board if its second digit is not equal to 0?
... | 1609 | 80 | 4 |
math | Find the sum of all primes that can be written both as a sum of two primes and as a difference of two primes.
[i]Anonymous Proposal[/i] | 5 | 32 | 1 |
math | We take turns rolling a fair die until the sum $S$ of the numbers obtained exceeds 100. What is the most likely value of $S$? | 101 | 34 | 3 |
math | 2. In how many ways can we choose two different integers between -100 and 100 inclusive, so that their sum is greater than their product? | 10199 | 34 | 5 |
math | 5.1. A grenade lying on the ground explodes into a multitude of small identical fragments, which scatter in a radius of $L=90$ m. Determine the time interval (in seconds) between the moments of impact on the ground of the first and the last fragment, if such a grenade explodes in the air at a height of $H=10 \mathrm{m}... | 6 | 108 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x+x^{5}}
$$ | 0 | 45 | 1 |
math | 31 In a triangle $A B C$, the length of the altitudes $A D$ and $B E$ are 4 and 12 respectively. Find the largest possible integer value for the length of the third altitude $C F$. | 5 | 51 | 1 |
math | Group Event 10
$A B C D$ is a square of side length $20 \sqrt{5} x . P, Q$ are midpoints of $D C$ and $B C$ respectively.
G10.1 If $A P=a x$, find $a$.
G10.2 If $P Q=b \sqrt{10} x$, find $b$.
G10.3 If the distance from $A$ to $P Q$ is $c \sqrt{10} x$, find $c$.
G10.4 If $\sin \theta=\frac{d}{100}$, find $d$. | =50,b=10,=15,=60 | 144 | 15 |
math | Three, (20 points) (1) Given $a+\log _{2}(2 a+6)=11$ and $b+2^{b-1}=14$. Find the value of $a+b$.
(2) Given $f(x)=\frac{2}{2^{x-2}+1}$. Find
$$
f(-1)+f(0)+f(1)+f(2)+f(3)+f(4)+f(5)
$$
the value. | 11 | 109 | 2 |
math | Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number. | n = 5 \cdot 13 \cdot 31 = 2015 | 55 | 21 |
math | 33. How many six-digit numbers are there in which all digits are odd? | 15625 | 17 | 5 |
math | Integers $x$ and $y$, with $x>y$, satisfy $x+y=7$ and $x y=12$.
Integers $m$ and $n$, with $m>n$, satisty $m+n=13$ and $m^{2}+n^{2}=97$.
If $A=x-y$ and $B=m-n$, determine the value of $A-B$. | -4 | 88 | 2 |
math | For example, in $\triangle ABC$, the measures of the three interior angles satisfy
$$
\frac{\angle A}{\angle B}=\frac{\angle B}{\angle C}=\frac{1}{3} \text {. }
$$
Find the value of $T=\cos A+\cos B+\cos C$. ${ }^{[6]}$
(2011, National High School Mathematics League Shanxi Province Preliminary Contest) | \frac{1+\sqrt{13}}{4} | 93 | 13 |
math | 21st VMO 1983 Problem A1 For which positive integers m, n with n > 1 does 2 n - 1 divides 2 m + 1? | (,n)=(k,1)or(2k+1,2) | 40 | 17 |
math | Example 2 (2006 National High School Mathematics Competition Problem) Express 2006 as the sum of 5 positive integers $x_{1}, x_{2}, x_{3}, x_{4}$, $x_{5}$. Let $S=\sum_{1 \leqslant i<j \leqslant 5} x_{i} x_{j}$. Questions:
(1) For what values of $x_{1}, x_{2}, x_{3}, x_{4}$, $x_{5}$ does $S$ attain its maximum value;
(... | x_{1}=402, x_{2}=x_{3}=x_{4}=x_{5}=401 \text{ for maximum } S; x_{1}=x_{2}=x_{3}=402, x_{4}=x_{5}=400 \text{ for minimum } S | 207 | 69 |
math | [ [ CaseAnalysis $\quad$]
Find all odd natural numbers greater than 500 but less than 1000, for each of which the sum of the last digits of all divisors (including 1 and the number itself) is 33.
# | 729 | 57 | 3 |
math | 10. Given the set $M=\left\{x \left\lvert\, x=\lim _{n \rightarrow \infty} \frac{2^{n+1}-2}{\lambda^{n}+2^{n}}\right.\right.$, $\lambda$ is a constant, and $\lambda+2 \neq 0\}$. Then the sum of all elements of $M$ is $\qquad$ . | 3 | 95 | 1 |
math | The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 199 | 50 | 3 |
math | 6. (8 points) On the board, 32 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 32 minutes? | 496 | 69 | 3 |
math | Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and $QC=100$. If $AP$ and $AQ$ trisect $\angle A$, then $\tfrac{AB}{AC}$ can be written uniquely as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are relatively prime positive integers and $q$ is a posi... | 92 | 137 | 2 |
math | $7 \cdot 117$ Let $S=\{1,2,3,4\} ; n$ terms of the sequence $a_{1}, a_{2}, \cdots a_{n}$ have the following property: for any non-empty subset $B$ of $S$ (the number of elements in set $B$ is denoted as $|B|$ ), there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of the ... | 8 | 113 | 1 |
math | Given an integer $n \geq 3$, determine the maximum value of product of $n$ non-negative real numbers $x_1,x_2, \ldots , x_n$ when subjected to the condition
\begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*} | \frac{1}{(n-1)^n} | 79 | 12 |
math | The focus of the parabola $y^{2}=7 x$ is crossed by a line which intersects the $Y$ axis at (-1). What is the area of the parabolic segment thus obtained? | 66.87 | 43 | 5 |
math | 2. Given a large regular tetrahedron with an edge length of 6, a smaller regular tetrahedron is placed inside it. If the smaller tetrahedron can rotate freely within the larger one, the maximum edge length of the smaller tetrahedron is . $\qquad$ | 2 | 62 | 1 |
math | 14. (15 points) The distance between location A and location B is 360 kilometers. A truck loaded with 6 boxes of medicine is driving from location A to location B, while at the same time, a motorcycle starts from location B and heads towards the truck. The truck's speed is 40 kilometers/hour, and the motorcycle's speed... | 8\frac{2}{3} | 175 | 8 |
math | 14. Given $\sin \alpha+\sin \beta=1$, find the range of $\cos \alpha+\cos \beta$. | [-\sqrt{3},\sqrt{3}] | 28 | 11 |
math | Find all natural two digit numbers such that when you substract by seven times the sum of its digit
from the number you get a prime number. | 10, 31, 52, 73, 94 | 30 | 18 |
math | 4. Miha has experimented with writing various numbers using only the digit 1 and the addition sign. For example, he found that there are only two natural numbers $n$ (13 and 4) for which the number 13 can be written using $n$ ones and the addition sign, since the number 13 can be written as the sum of thirteen ones or ... | 14 | 150 | 2 |
math | ## 44. Bag of Balls
Children are dividing a bag of balls among themselves. The first child took one ball and a tenth of the remaining balls, then the second took 2 balls and a tenth of the remaining, then the third took 3 balls and a tenth of the remaining, and so on, until the last child took all that was left.
How ... | 9,N=81,n=9 | 99 | 8 |
math | 3. (6 points) Define new operations: $a \triangle b=(a+b)+2, a \bigcirc b=a \times 3+b$, when $(X \triangle 24) \bigcirc 18=60$, $X$ $=$ . $\qquad$ | -12 | 62 | 3 |
math | Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$, what is the area of the trapezoid? | 100\sqrt{3} | 70 | 8 |
math | Example 4 When point $P$ moves along the line $y=2 x+6$, and point $Q$ moves on the ellipse $\frac{x^{2}}{6}+\frac{y^{2}}{4}=1$, the minimum length of segment $P Q$ is $\qquad$ | \frac{6 \sqrt{5}-2 \sqrt{35}}{5} | 64 | 19 |
math | [^0]A company of several friends communicated in such a way that each email was received by everyone except the sender.
Each person wrote the same number of emails, resulting in a total of 440 emails received by everyone.
How many people could have been in this company?
# | 2,5,11 | 58 | 6 |
math | Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 point... | \ln 2 | 160 | 5 |
math | 8. If $a>0, b>0$, and $a+2 b=6$, then the maximum value of $\lg a+2 \lg b$ is $\qquad$ | 3\lg2 | 40 | 4 |
math | 70(1004). From two settlements $A$ and $B$, two tourists set out towards each other at the same time. Upon meeting, it turns out that the tourist who left from $A$ has walked 2 km more than the second tourist. Continuing their movement at the same speed, the first tourist arrives in $B$ after 1 hour 36 minutes, and the... | 18 | 536 | 2 |
math | ## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=7 p-2 q$
$b=p+3 q$
$|p|=\frac{1}{2}$
$|q|=2$
$(\widehat{p, q})=\frac{\pi}{2}$ | 23 | 71 | 2 |
math | 5. The increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, \cdots$ satisfies $a_{n+2}=a_{n}+a_{n+1}(n \geqslant 1)$. If $a_{7}=120$, then $a_{8}$ equals | 194 | 71 | 3 |
math | 33.6. To compute the value of the polynomial $P(x)=a_{n} x^{n}+$ $+a_{n-1} x^{n-1}+\ldots+a_{0}$ at $x=x_{0}$, one can compute $a_{n} x_{0}^{n}$, $a_{n-1} x_{0}^{n-1}, \ldots, a_{1} x_{0}$, and then add all the obtained numbers and $a_{0}$. This requires $2 n-1$ multiplications (computing $x_{0}^{k}$ for $k=2,3, \ldots... | b_{n}=P(x_{0}) | 192 | 9 |
math | 4. If $m, n$, and $p$ are three different natural numbers, each between 2 and 9 , what then are all the possible integer value(s) of the expression, $\frac{m+n+p}{m+n}$ ? | 2 | 51 | 1 |
math | Find all integers $k$ such that there exist 2017 integers $a_{1}, a_{2}, \cdots, a_{2017}$, satisfying that for any positive integer $n$ not divisible by the prime 2017, we have
$$
\frac{n+a_{1}^{n}+a_{2}^{n}+\cdots+a_{2017}^{n}}{n+k} \in \mathbf{Z} .
$$ | k = 0, 1, \cdots, 2017 | 107 | 17 |
math | 2. Find the real solution to the equation $\sqrt[3]{x(3+\sqrt{8 x-3})-1}+\sqrt[3]{x(3-\sqrt{8 x-3})-1}=1$.
untranslated text:
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。
translated text:
Find the real solution to the equation $\sqrt[3]{x(3+\sqrt{8 x-3})-1}+\sqrt[3]{x(3-\sqrt{8 x-3})-1}=1$.
Note: The note at the end is not... | x\geqslant\frac{3}{8} | 147 | 13 |
math | Let $M$ be the number of multiples of 5 between 1 to 2020 inclusive and $N$ be the number of multiples of 20 between 1 and 2020 inclusive. What is the value of $10 M \div N$.
## | 40 | 61 | 2 |
math | The 63rd question: For positive integers $n \geq 2$, for complex numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ satisfying $\left|\alpha_{i}\right| \leq 1$ (i=1, 2, \ldots, n), find the minimum possible value of $\sum_{\mathrm{i}=1}^{\mathrm{n}}\left|1+\alpha_{i}\right|+\left|1+\prod_{\mathrm{i}=1}^{\mathrm{n}} \... | 1+(-1)^{\mathrm{n}} | 127 | 9 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2} \frac{\ln \left(9-2 x^{2}\right)}{\sin 2 \pi x}$ | -\frac{4}{\pi} | 45 | 8 |
math | 2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \left(-\frac{3}{7}\right)$, and the numbers $\frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z}$ also form an arithmetic progression in the given order. Find $\cos ^{2} y$. | \frac{10}{13} | 95 | 9 |
math | 10.3. Solve the system of equations in natural numbers
$$
\left\{\begin{array}{l}
a b=c+d \\
c d=a+b
\end{array}\right.
$$ | (1;5;2;3),(1;5;3;2),(5;1;2;3),(5;1;3;2),(2;2;2;2),(2;3;1;5),(2;3;5;1),(3;2;1;5),(3;2;5;1) | 44 | 73 |
math | 3. Let the set $X=\{1,2, \cdots, 20\}, A$ be a subset of $X$, the number of elements in $A$ is at least 2, and all elements of $A$ can be arranged as consecutive positive integers. Then the number of such sets $A$ is $\qquad$. | 190 | 74 | 3 |
math | 5. A drawer contains red and blue socks, with a total number not exceeding 2016. If two socks are randomly drawn, the probability that they are the same color is $\frac{1}{2}$. Then the maximum number of red socks in the drawer is _. $\qquad$ | 990 | 62 | 3 |
math | 20. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the set of points on the surface of the cube that are a distance of $\frac{2 \sqrt{3}}{3}$ from point $A$ forms a curve (this curve may not lie in a single plane). Then the length of this curve is $\qquad$ | \frac{5 \sqrt{3} \pi}{6} | 90 | 14 |
math | 1A. Find all natural numbers $n$ for which $n^{3}-n=n!$. | 5 | 21 | 1 |
math | Example 2 Given $a_{1}=1, a_{n}=\frac{2}{3} a_{n-1}+n^{2}-15(n \geqslant 2)$, find $a_{n}$. | a_{n}=25(\frac{2}{3})^{n-1}+3n^{2}-12n-15 | 52 | 30 |
math | Amy has divided a square up into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \( x \), the sum... | 2.5 | 99 | 3 |
math | 10、Ants A, B, and C crawl at a speed ratio of $8: 6: 5$. They crawl along a circle from the same point and in the same direction at the same time. When they first return to the starting point together, the crawling ends. How many times does Ant A catch up with Ant B (including the end moment)? | 2 | 76 | 1 |
math | A rectangle has a perimeter of $124 \mathrm{~cm}$. The perimeter of the rhombus determined by the midpoints of the sides is $100 \mathrm{~cm}$. What are the lengths of the sides of the rectangle? | 48\mathrm{~},14\mathrm{~} | 55 | 14 |
math | Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$. | 403791300 | 65 | 9 |
math | 5. From 30 people with distinct ages, select two groups, the first with 12 people and the second with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. How many ways are there to select these groups? | 4060 | 61 | 4 |
math | 4. Given real numbers $x, y, z$ satisfy $x^{2}+2 y^{2}+3 z^{2}=24$.
Then the minimum value of $x+2 y+3 z$ is $\qquad$ . | -12 | 54 | 3 |
math | 1269. Calculate the integrals:
1) $\int_{0}^{1} x e^{-x} d x$
2) $\int_{1}^{2} x \log _{2} x d x$
3) $\int_{1}^{e} \ln ^{2} x d x$ | 1-\frac{2}{e},2-\frac{3}{4\ln2},e-2 | 69 | 22 |
math | Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $ | A = 180^\circ - 3C, B = 2C, C = C | 53 | 22 |
math | 2. Problem: Find all triples $(a, b, c)$ of real numbers such that
$$
a^{2}+b^{2}+c^{2}=1 \quad \text { and } \quad a(2 b-2 a-c) \geq \frac{1}{2}
$$ | (,b,)=(\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}},\frac{-1}{\sqrt{6}})\text{}(\frac{-1}{\sqrt{6}},\frac{-2}{\sqrt{6}},\frac{1}{\sqrt{6}}) | 66 | 69 |
math | 41. (8th grade) Find the smallest natural number that has the following properties: a) its representation in the decimal system ends with the digit 6; b) if the last digit 6 is erased and this digit 6 is written in front of the remaining digits, the resulting number is four times the original number. | 153846 | 68 | 6 |
math | [ The inscribed angle is half the central angle ]
In an acute-angled triangle $A B C$, altitudes $C H$ and $A H_{1}$ are drawn. It is known that $A C=2$, and the area of the circle circumscribed around triangle $H B H_{1}$ is $\pi / 3$. Find the angle between the altitude $C H$ and the side $B C$.
# | 30 | 92 | 2 |
math | Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya sequentially called out natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice and wrote down the obtained values on the board from left to right. It turned out that the ... | 50 | 98 | 2 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{3+6+9+\ldots+3 n}{n^{2}+4}$ | \frac{3}{2} | 45 | 7 |
math | (1) $a_{1}, a_{2}, a_{3}, \cdots$ is an arithmetic sequence, where $a_{1}>0, S_{n}$ represents the sum of the first $n$ terms. If $S_{3}=S_{11}$, and among $S_{1}, S_{2}, S_{3}, \cdots$, the maximum number is $S_{k}$, then $k=$ $\qquad$ . | 7 | 98 | 1 |
math | 9. Given the hyperbola $x^{2}-y^{2}=2$ with left and right foci $F_{1}, F_{2}$, a moving line through point $F_{2}$ intersects the hyperbola at points $A, B$.
(1) If a moving point $M$ satisfies $\overrightarrow{F_{1} M}=\overrightarrow{F_{1} A}+\overrightarrow{F_{1} B}+\overrightarrow{F_{1} O}$ (where $O$ is the origi... | (x-6)^{2}-y^{2}=4 | 190 | 12 |
math | It is known that a polynomial $P$ with integer coefficients has degree $2022$. What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots a_n$ with $P(a_i)=i$ for all $1\le i\le n$?
[Extra: What happens if $P \in \mathbb{Q}[X]$ and $a_i\in \mathbb{Q}$ instead?] | 2022 | 100 | 4 |
math | 16(!). Solve the equation \(x+y=x y\) in integers. | 2or0 | 16 | 3 |
math | 2. $50 N$ is an integer, its base $b$ representation is 777, find the smallest positive integer $b$, such that $N$ is an integer to the fourth power. | 18 | 44 | 2 |
math | 13. (ROM) Let $P$ be a polynomial of degree $n$ satisfying
$$ P(k)=\binom{n+1}{k}^{-1} \quad \text { for } k=0,1, \ldots, n $$
Determine $P(n+1)$. | P(n+1)= \begin{cases}1, & 2 \mid n ; \\ 0, & 2 \nmid n .\end{cases} | 65 | 36 |
math | 25. [15] Fran writes the numbers $1,2,3, \ldots, 20$ on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number $n$ uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of $n$ that are still on the chalkboard (... | \frac{131}{10} | 115 | 10 |
math | 1. Given that $a$, $b$, and $c$ are three distinct odd prime numbers, the equation $(b+c) x^{2}+(a+1) \sqrt{5} x+225=0$ has two equal real roots.
(1) Find the minimum value of $a$;
(2) When $a$ reaches its minimum value, solve this equation. | x=-\frac{3}{2} \sqrt{5} | 83 | 14 |
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