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 | Example 3 (38th Austrian Mathematical Olympiad) Find all non-negative integers $a(a<2007)$, such that $x^{2}+a \equiv 0(\bmod 2007)$ has exactly two distinct non-negative integer solutions less than 2007. | 446,1115,1784 | 65 | 13 |
math | 2. Determine for which $m$ there exist exactly $2^{15}$ subsets $\mathrm{X}$ of the set $\{1,2,3, \ldots, 47\}$ with the property: the number $m$ is the smallest element of the set $\mathbf{X}$ and for every $x \in \mathbf{X}$, either $x+m \in \mathbf{X}$ or $x+m>47$.
(R. Kučera) | =6=32 | 106 | 5 |
math | 5. The line $y=k x-2$ intersects the parabola $y^{2}=8 x$ at points $A, B$. If the x-coordinate of the midpoint of segment $A B$ is 2, then the length of segment $A B$ $|A B|=$ $\qquad$ | 2\sqrt{15} | 67 | 7 |
math | 8,9 | |
The perimeter of the rhombus is 48, and the sum of the diagonals is 26. Find the area of the rhombus. | 25 | 39 | 2 |
math | $1 \cdot 6$ Let $n$ be an integer. If the tens digit of $n^{2}$ is 7, what is the units digit of $n^{2}$? | 6 | 41 | 1 |
math | 61. Write the general equation of a sphere whose center coincides with the origin. Write the general equation of a sphere passing through the origin and with its center on the $O x$ axis. | x^{2}-2Rx+y^{2}+z^{2}=0 | 41 | 16 |
math | 11.5. Compare the numbers $X=2019^{\log _{2018} 2017}$ and $Y=2017^{\log _{2019} 2020}$. | X>Y | 56 | 3 |
math | 8. Solve the system $\left\{\begin{array}{l}3^{x} \cdot 2^{y}=972 ; \\ \log _{\sqrt{3}}(x-y)=2 .\end{array}\right.$ | {5;2} | 52 | 5 |
math | 7. (5 points) In a division equation, the dividend is 12, and the divisor is a natural number less than 12. The sum of all possible different remainders is $\qquad$ .
| 15 | 45 | 2 |
math | 45. 18 $k \star$ Find the smallest real number $\lambda$ such that the inequality
$$
5(a b c+a b d+a c d+b c d) \leqslant \lambda a b c d+12
$$
holds for any positive real numbers $a, b, c, d$ satisfying $a+b+c+d=4$. | 8 | 81 | 1 |
math | 2. Determine the geometric progression of real numbers in which the sum of its first four terms is 15, and the sum of their squares is 85. | 1,2,4,8,\ldots8,4,2,1,\frac{1}{2},\ldots | 34 | 27 |
math | 1. The arithmetic sequence $\left\{a_{n}\right\}$ satisfies $a_{2021}=a_{20}+a_{21}=1$, then the value of $a_{1}$ is $\qquad$ | \frac{1981}{4001} | 52 | 13 |
math | Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number.
[i]Proposed by Gabriel Wu[/i] | \frac{3}{4} | 51 | 7 |
math | ## 141. Math Puzzle $2 / 77$
A KAP wants to deliver grain to the collection point by 11:00 AM. If she takes the tractor, the load would only arrive at 11:30 AM. If she takes the truck, it would already be there by 10:45 AM.
How far is the collection point, if both vehicles start at the same departure time, the tracto... | 22.5\mathrm{~} | 130 | 9 |
math | A hotel has 5 distinct rooms, all with single beds for up to 2 people. The hotel has no other guests and 5 friends want to spend the night there. In how many ways can the 5 friends choose their rooms? | 2220 | 49 | 4 |
math | 11. (20 points) It is known that a box contains 100 red and 100 blue cards, each color of cards containing one card labeled with each of the numbers $1, 3, 3^2, \cdots, 3^{99}$. The total sum of the numbers on the cards of both colors is denoted as $s$.
For a given positive integer $n$, if it is possible to pick sever... | 2^{200}-1 | 169 | 7 |
math | Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$3a + 4b \le 5c.$$
When does equality holds? | 3a + 4b \leq 5c | 60 | 14 |
math | 4. Solve the equation $\frac{2 \operatorname{tg}^{4} 6 x+4 \sin 4 x \sin 8 x-\cos 8 x-\cos 16 x+2}{\sqrt{\cos x-\sqrt{3} \sin x}}=0$. | 2\pin,-\frac{\pi}{6}+2\pin,-\frac{\pi}{3}+2\pin,-\frac{\pi}{2}+2\pin,-\frac{2\pi}{3}+2\pin,n\inZ | 64 | 57 |
math | Example + Given that $f(x)$ is an even function defined on $\mathbf{R}$, if $g(x)$ is an odd function, and $g(x)=f(x-1)$, $g(1)=2003$, find the value of $f(2004)$. | 2003 | 65 | 4 |
math | 21. Determine the number of pairs of positive integers $n$ and $m$ such that
$$
1!+2!+3!+\cdots+n!=m^{2} \text {. }
$$ | 2 | 45 | 1 |
math | 4. In $\triangle A B C$, $\sin A: \sin B: \sin C=2: 3: 4$, then $\angle A B C=$ $\qquad$ (the result should be expressed using inverse trigonometric function values). | \arccos\frac{11}{16} | 54 | 13 |
math | 3. Solve the system $\left\{\begin{array}{l}x+3 y+14 \leq 0, \\ x^{4}+2 x^{2} y^{2}+y^{4}+64-20 x^{2}-20 y^{2}=8 x y .\end{array}\right.$ | (-2,-4) | 75 | 5 |
math | Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$. | 32 | 58 | 2 |
math | [ Sequences (other). ] [ Identical transformations ]
## Find the largest term of the sequence $x_{n}=\frac{n-1}{n^{2}+1}$. | x_{2}=x_{3}=0.2 | 39 | 11 |
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 ... | -^2++\frac{1}{2} | 154 | 11 |
math | 420. Using only a compass, it is required to construct a fourth proportional segment to three given segments $a, b, c$. | x | 29 | 1 |
math | 9.4 In the thirtieth kingdom, there are three types of coins in circulation: bronze rubles, silver coins worth 9 rubles, and gold coins worth 81 rubles. From the treasury, which contains an unlimited supply of each type of coin, a certain amount was issued with 23 coins, which is less than 700 rubles. Find this amount,... | 647 | 98 | 3 |
math | Exercise 11. In a classroom, there are ten students. Aline writes ten consecutive integers on the board. Each student chooses one of the ten integers written on the board, such that any two students always choose two different integers. Each student then calculates the sum of the nine integers chosen by the other nine ... | 4 | 115 | 1 |
math | An ant is crawling from the left end of a $4 \mathrm{~m}$ long rubber band towards the right end at a constant speed, covering exactly one meter per minute. After each minute, the horizontally placed rubber band, fixed at the left end, is uniformly stretched by one meter. In which minute does the ant reach the right en... | 7 | 103 | 1 |
math | What is the smallest number of points that can be chosen on a circle of length 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circumference)?
# | 1304 | 59 | 4 |
math | 50. How many five-digit numbers are there that are divisible by 5 and do not have any repeated digits in their representation? | 5712 | 27 | 4 |
math | 4. Let $S$ be a set of $n$ distinct real numbers, and $A_{s}$ be the set of all distinct averages of pairs of elements from $S$. For a given $n \geqslant 2$, what is the minimum number of elements in $A_{s}$?
(1993 Putnam Competition) | 2n-3 | 74 | 4 |
math | 1. Given an integer $n \geqslant 2$, for any pairwise coprime positive integers $a_{1}, a_{2}, \cdots, a_{n}$, let
$A=a_{1}+a_{2}+\cdots+a_{n}$.
For $i=1,2, \cdots, n$, let the greatest common divisor of $A$ and $a_{i}$ be $d_{i}$; the greatest common divisor of the remaining $n-1$ numbers after removing $a_{i}$ from $... | (n-1)^{n} | 176 | 7 |
math | Find all polynomials $P(x)$ with real coefficients such that
$$
(x-2010) P(x+67)=x P(x)
$$
for every integer $x$. | P(x) = c(x-67)(x-2 \cdot 67) \ldots (x-30 \cdot 67) | 41 | 33 |
math | 2. In an equilateral triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen on side $BC$ such that $B A_{1}=A_{1} A_{2}=A_{2} C$. On side $AC$, a point $B_{1}$ is chosen such that $A B_{1}: B_{1}C=1: 2$. Find the sum of the angles $\angle A A_{1} B_{1}+\angle A A_{2} B_{1}$. | 30 | 113 | 2 |
math | 18. A factory produces 2 large products daily, with a production cost of 2000 yuan per item. The probability of a product being of first-grade quality is 0.5; the probability of it being of second-grade quality is 0.4. The factory price of each first-grade product is 10000 yuan, and that of each second-grade product is... | 12200 | 224 | 5 |
math | 10.65 Find all natural numbers $x$ that satisfy the following conditions: the product of the digits of $x$ equals $44x - 86868$, and the sum of the digits is a perfect cube.
(52nd Moscow Mathematical Olympiad, 1989) | 1989 | 66 | 4 |
math | 2. A TV station is going to broadcast a 30-episode TV series. If it is required that the number of episodes aired each day must be different, what is the maximum number of days the TV series can be broadcast?
---
The translation maintains the original format and line breaks as requested. | 7 | 61 | 1 |
math | In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw \( k \) lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the... | 2013 | 106 | 4 |
math | 4・118 Solve the system of equations
$$\left\{\begin{array}{l}
\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}=\frac{7}{\sqrt{x y}} \\
\sqrt[4]{x^{3} y}-\sqrt[4]{x y^{3}}=\sqrt{12}
\end{array}\right.$$ | \left\{\begin{array}{l}x=16, \\ y=9\end{array}\right.} 和 | 84 | 28 |
math | 17 Given that $a+\frac{1}{a+1}=b+\frac{1}{b-1}-2$ and $a-b+2 \neq 0$, find the value of $a b-a+b$. | 2 | 49 | 1 |
math | $$
\begin{array}{l}
78 \times 4 + 488 = \\
1903 - 475 \times 4 = \\
350 \times (12 + 342 \div 9) = \\
480 \div (125 - 117) = \\
(3600 - 18 \times 200) \div 253 = \\
(243 - 162) \div 27 \times 380 =
\end{array}
$$ | 800,3,17500,60,0,1140 | 126 | 21 |
math | 18 4 \% 6 Solve the equation for positive integer solutions:
$$
5^{x}-3^{y}=2
$$ | 1 | 28 | 1 |
math | Example 2 Given the sets
$$
\begin{array}{l}
M=\{(x, y) \mid x(x-1) \leqslant y(1-y)\}, \\
N=\left\{(x, y) \mid x^{2}+y^{2} \leqslant k\right\} .
\end{array}
$$
If $M \subset N$, then the minimum value of $k$ is $\qquad$ .
(2007, Shanghai Jiao Tong University Independent Admission Examination) | 2 | 116 | 1 |
math | 16. In a Cartesian coordinate system, there are 10 different points $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), \cdots, P_{10}\left(x_{10}, y_{10}\right)$. If $x_{i}=x_{j}$ or $y_{i}=y_{j}$, then $P_{i}$ and $P_{j}$ are called a "coordinate pair" (the order of $P_{i}$ and $P_{j}$ does not matter). ... | 4 | 192 | 1 |
math | 9. Gari is seated in a jeep, and at the moment, has one 10 -peso coin, two 5 -peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos? | \frac{37}{42} | 76 | 9 |
math | Example 5 Find all positive integers $x, y$ such that
$$y^{x}=x^{50}$$ | (x, y) = (1,1), \left(2,2^{25}\right), \left(2^{2}, 2^{25}\right), \left(5,5^{10}\right), \left(5^{2}, 5^{4}\right), \left(10,10^{5}\right), (50,50), (100,10) | 26 | 91 |
math | 2. If real numbers $x, y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
(x-1)^{2011}+(x-1)^{2009}+2010 x=4020, \\
(y-1)^{2011}+(y-1)^{2009}+2010 y=0,
\end{array}\right.
$$
then $x+y=$ $\qquad$ . | 2 | 111 | 1 |
math | Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $ 100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks ... | 52 | 112 | 2 |
math | 8. If three dice are thrown at random, the probability that the numbers shown on the three dice can serve as the side lengths of a triangle is $\qquad$ .
| \frac{37}{72} | 35 | 9 |
math | [Theorem of Three Perpendiculars]
The height of a right triangle $ABC$, dropped to the hypotenuse, is 9.6. From the vertex $C$ of the right angle, a perpendicular $CM$ is erected to the plane of triangle $ABC$, with $CM=28$. Find the distance from point $M$ to the hypotenuse $AB$. | 29.6 | 80 | 4 |
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 | 12. Arrange $n$ people in a row, and call two adjacent people "friends". If a subset $A$ of the set of these $n$ people satisfies: no two people in $A$ are "friends", then the subset $A$ is called a "bad subset". Try to find the number of "bad subsets" of the set of $n$ people that contain $k$ people. | C_{n-k+1}^{k} | 86 | 10 |
math | 2.3.11 * $\star$ Let $a, b, c$ be the three sides of $\triangle ABC$, $a \leqslant b \leqslant c$, $R$ and $r$ be the circumradius and inradius of $\triangle ABC$, respectively. Let $f=a+b-2R-2r$, try to determine the sign of $f$ using the size of angle $C$. | C<\frac{\pi}{2} | 92 | 9 |
math | 3. (8 points) In another 12 days, it will be 2016, Hao Hao sighs: I have only experienced 2 leap years so far, and the year I was born is a multiple of 9, so in 2016, Hao Hao is $\qquad$ years old. | 9 | 69 | 1 |
math | Three, someone buys 13 chicken eggs, 5 duck eggs, and 9 goose eggs, spending a total of 9.25 yuan; if they buy 2 chicken eggs, 4 duck eggs, and 3 goose eggs, they spend a total of 3.20 yuan. Try to find: how much would it cost to buy 1 chicken egg, 1 duck egg, and 1 goose egg each? | x+y+z=1.05 | 91 | 8 |
math | 11. Arrange 10 flowers in a row, using red, yellow, and blue flowers (assume there are plenty of each color), and yellow flowers cannot be adjacent. How many different arrangements are there (the 10 flowers can be of one color or two colors)? | 24960 | 57 | 5 |
math | Problem 2. $n$ mushroom pickers went to the forest and brought a total of 338 mushrooms (it is possible that some of them did not bring any mushrooms home). Boy Petya, upon learning this, said: "Some two of them must have brought the same number of mushrooms!" For what smallest $n$ will Petya definitely be right? Don't... | 27 | 86 | 2 |
math | Example 7 In $\triangle A B C$, $\angle A B C=50^{\circ}, \angle A C B=20^{\circ}, N$ is a point inside $\triangle A B C$, $\angle N A B=40^{\circ}$. $\angle N B C=30^{\circ}$. Find the degree measure of $\angle N C B$. | 10 | 83 | 2 |
math | 2.1. Every month Ivan pays a fixed amount from his salary for a mortgage, and the remaining part of the salary is spent on current expenses. In December, Ivan paid $40 \%$ of his salary for the mortgage. In January, Ivan's salary increased by $9 \%$. By what percentage did the amount spent on current expenses increase ... | 15 | 79 | 2 |
math | Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube. | 371 | 52 | 3 |
math | 8. (10 points) There are three two-digit numbers $A$, $B$, and $C$. $A$ is a perfect square, and each of its digits is also a perfect square; $B$ is a prime number, and each of its digits is also a prime number, and the sum of its digits is also a prime number; $C$ is a composite number, and each of its digits is also ... | 120 | 137 | 3 |
math | 1. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
a) $f(x+f(y))=f(x+y)+1$ for all $x, y \in \mathbb{R}$,
b) $f$ is a strictly increasing function. | f(y)=y+1 | 68 | 6 |
math | Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. What possible values can $a^3 + \frac{1}{a^3}$ and $a^5 + \frac{1}{a^5}$ take? | (8\sqrt{11}, 71\sqrt{11}), (-8\sqrt{11}, -71\sqrt{11}) | 64 | 35 |
math | Five, (15 points) Find the smallest positive integer $n$ such that $2^{2005}$ । $\left(161^{n}-1\right)$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
---
Five, (15 points) Find the smallest positive integer $n$ such tha... | 2^{2000} | 109 | 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 (2n-1) odd numbers:
$\{1\}$,
$\{3,5,7\}$,
$\{9,11,13,15,17\}, \cdots$
（First group）
（Second group）
（Third group）
Then 1991 is in the group. | 32 | 106 | 2 |
math | 6.133. Without solving the equation $3 x^{2}-5 x-2=0$, find the sum of the cubes of its roots. | \frac{215}{27} | 33 | 10 |
math | Prove that for all real numbers $ a,b$ with $ ab>0$ we have:
$ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$
and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$
$ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}... | \sqrt[3]{\frac{a^2 b^2 (a+b)^2}{4}} \le \frac{a^2 + ab + b^2}{3} | 149 | 40 |
math | 4. The minimum value of the sum of the squares of the distances from a point in the rectangular coordinate plane to the three lines $x=0, y=0, 4x+3y=12$ is $\qquad$ | \frac{72}{25} | 50 | 9 |
math | Three numbers form a geometric sequence. Their sum is 19; by reducing the last number by 1, we get an arithmetic sequence; which are these three numbers? | 4,6,99,6,4 | 35 | 10 |
math | Example 6. What is the probability that in a randomly chosen two-digit number, the digits are the same? | 0.1 | 23 | 3 |
math | 7. If $\sin ^{2}\left(x+\frac{\pi}{12}\right)-\sin ^{2}\left(x-\frac{\pi}{12}\right)=-\frac{1}{4}$, and $x \in\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right)$, then the value of $\tan x$ is . $\qquad$ | -2-\sqrt{3} | 88 | 7 |
math | On the same side of a street, six adjacent houses will be built. The houses can be brick or wooden, but as a safety measure against fire, two wooden houses cannot be adjacent. In how many ways can the construction of these houses be planned? | 21 | 51 | 2 |
math | There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had... | 217 | 141 | 3 |
math | 4. In the sequence $\left\{a_{n}\right\}$, $a_{k}+a_{k+1}=2 k+1\left(k \in \mathbf{N}^{*}\right)$, then $a_{1}+a_{100}$ equals | 101 | 64 | 3 |
math | Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist
$ d\in\mathbb{Q}$ such that $ P(d)\equal{}r$ | P(x) = ax + b | 54 | 8 |
math | 2.41 For what integer value of $p$ do the equations $3 x^{2}-4 x+p-2=0$ and $x^{2}-2 p x+5=0$ have a common root? Find this root. | 3,1 | 52 | 3 |
math | The sequence $\mathrm{Az}\left(a_{n}\right)$ is defined as follows:
$$
a_{1}=k \text { (positive integer), }
$$
$$
a_{n+1}= \begin{cases}a_{n} / 2, & \text { if } a_{n} \text { is even } \\ a_{n}+5, & \text { if } a_{n} \text { is odd. }\end{cases}
$$
For which positive integers $k$ is it true that 1 appears among t... | 1 | 133 | 1 |
math | 7. Let the line $l$ passing through the fixed point $M(a, 0)$ intersect the parabola $y^{2}=4 x$ at points $P$ and $Q$. If $\frac{1}{|P M|^{2}}+\frac{1}{|Q M|^{2}}$ is a constant, then the value of $a$ is $\qquad$ . | 2 | 85 | 1 |
math | 4. Among the positive integers less than 20, each time three numbers are taken without repetition, so that their sum is divisible by 3. Then the number of different ways to do this is $\qquad$ . | 327 | 46 | 3 |
math | 10.80 Euler's conjecture was disproved by American mathematicians in 1960, who confirmed the existence of a positive integer $n$ such that $133^{5}+110^{5}+84^{5}+27^{5}=n^{5}$. Find the value of $n$.
(7th American Invitational Mathematics Examination, 1989) | 144 | 90 | 3 |
math | 8. A $3 \times 3$ grid with the following properties is called a "T-grid":
(1) Five cells are filled with 1, and four cells are filled with 0;
(2) Among the three rows, three columns, and two diagonals, at most one of these eight lines has three numbers that are pairwise equal.
Then the number of different T-grids is $... | 68 | 86 | 2 |
math | Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$? | 90 | 38 | 2 |
math | Example 15 (Question from the 13th "Hope Cup" Invitational Competition) Given $ab=1000, a>1, b>1$, what is the maximum value of $\sqrt{1+\lg a}+$ $\sqrt{1+\lg b}$? | \sqrt{10} | 61 | 6 |
math | 3.1. For what greatest $a$ is the inequality $\frac{\sqrt[3]{\operatorname{tg} x}-\sqrt[3]{\operatorname{ctg} x}}{\sqrt[3]{\sin x}+\sqrt[3]{\cos x}}>\frac{a}{2}$ satisfied for all permissible $x \in\left(\frac{3 \pi}{2} ; 2 \pi\right)$? Round the answer to the nearest hundredth if necessary. | 4.49 | 106 | 4 |
math | 8.2. In the decimal representation of a natural number $N$, a 0 was inserted between the second and third digits from the right, resulting in $9N$. What can $N$ be? (list all possible values). | 225;450;675 | 49 | 11 |
math | Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$.
[i]Huang Yumin[/i] | \lambda = \frac{n}{\sqrt{3}} | 98 | 13 |
math | 13.2. Calculate the sum
$$
1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\ldots+n \cdot n!
$$ | (n+1)!-1 | 40 | 6 |
math | Three, find all real numbers $k$ such that the inequality
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
holds for all $a, b, c, d \in[-1,+\infty)$.
(Xu Wanyi, problem contributor) | \frac{3}{4} | 80 | 7 |
math | 1. Simplify the fraction $\frac{\sqrt{-x}-\sqrt{-3 y}}{x+3 y+2 \sqrt{3 x y}}$.
Answer. $\frac{1}{\sqrt{-3 y}-\sqrt{-x}}$. | \frac{1}{\sqrt{-3y}-\sqrt{-x}} | 54 | 16 |
math | Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle?
[i]2016 CCA Math Bonanza Individual Round #2[/i] | 65 | 49 | 2 |
math | 108801 topics: $[\quad$ Regular Tetrahedron $\quad]$ [ Angles between lines and planes ]
In a regular tetrahedron, find the angle between an edge and the plane of a face that does not contain this edge.
# | \arccos\frac{1}{\sqrt{3}} | 56 | 14 |
math | Solve the following system of equations:
$$
\begin{aligned}
& \frac{1}{x-y}+\frac{1}{x+y}=a \\
& \frac{1}{x-y}-\frac{1}{x+y}=b
\end{aligned}
$$ | \frac{2a}{^2-b^2},\quad\frac{2b}{^2-b^2} | 59 | 26 |
math | 4. The function $f(x, y)$ satisfies for all non-negative integers $x, y$:
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f[x, f(x+1, y)]$.
Determine $f(4,1981)$. | 2^{(1984)}-3 | 93 | 10 |
math | 7. Find the maximum value of the function $f(x)=\sqrt{5 x}+\sqrt{6-x}$.
| 6 | 26 | 1 |
math | 4. Calculate
$$
\int\left(5 x^{5}+3 x^{3}\right) \sqrt{x^{3}+x} d x, \quad x>0
$$ | \frac{2}{3}(x^{6}+x^{4})\sqrt{x^{3}+x}+\mathcal{C} | 43 | 31 |
math | 2. At the parade of the royal musketeers, seeing that they could not be arranged in rows of 11 musketeers, D'Artagnan decided to arrange the musketeers in rows of 10 musketeers, but it turned out that there was one empty place in the last row. Then he tried to arrange them in rows of 9, 8, 7, 6, 5, 4, 3, 2 musketeers, ... | 5039 | 168 | 4 |
math | $\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$.
[i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b].
[color=#B6D7A8]#1733[/color] | f(x) = x^2 | 154 | 8 |
math | Evaluate the expression
\[
\frac{121 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{
11 \left( \frac{1}{13} - \frac{1}{17} \right)
+ 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \f... | 41 | 183 | 2 |
math | The students of two schools achieved the following results on a test:
The average score of students from the first school is 74 points, with boys scoring an average of 71 and girls 76 points. The average score of students from the second school is 84 points, with boys scoring an average of 81 and girls 90 points. If t... | 84 | 100 | 2 |
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