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 | ## 264. Math Puzzle $5 / 87$
An analog clock is showing exactly the sixth hour. The hour and minute hands are at an angle of $180^{\circ}$. During the next hour, the clock hands form an angle of $110^{\circ}$ exactly twice.
How many minutes elapse between the two angles of $110^{\circ}$? | 40 | 86 | 2 |
math | 1. Given that the function $f(x)$ is a decreasing function on $\mathbf{R}$, and it is an odd function. If $m, n$ satisfy the inequality system $\left\{\begin{array}{l}f(m)+f(n-2) \leqslant 0, \\ f(m-n-1) \leqslant 0,\end{array}\right.$ then the range of $5 m-n$ is $\qquad$ . | [7,+\infty) | 101 | 7 |
math | How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 12 | 67 | 2 |
math | Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$. | f(x) = x | 88 | 6 |
math | Example 2.61. Calculate the surface area formed by the rotation of the arc of the circle $x^{2}+y^{2}=16(y>0)$ over the segment $-1 \geqslant x \geqslant 1$ around the $O X$ axis. | 16\pi | 65 | 4 |
math | G2.2 If $S_{n}=1-2+3-4+\ldots+(-1)^{n-1} n$, where $n$ is a positive integer, determine the value of $S_{17}+S_{33}+S_{50}$. | 1 | 63 | 1 |
math | 3. The remainder of the division of a natural number n by 2021 is 800 more than the remainder of the division of the number n by 2020. Find the smallest such n. (A. Gолованов) | 2466420 | 55 | 7 |
math | 518. Find the limits:
1) $\lim _{x \rightarrow 0} \frac{x}{\sin x}$; 2) $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 5 x}$; 3) $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$. | 1,\frac{4}{5},\frac{1}{2} | 83 | 15 |
math | Example 8. At a distribution base, there are electric bulbs manufactured by two factories. Among them, $60 \%$ are made by the first factory and $40 \%$ by the second. It is known that out of every 100 bulbs made by the first factory, 95 meet the standard, and out of 100 bulbs made by the second factory, 85 meet the st... | 0.91 | 101 | 4 |
math | V OM - I - Problem 7
In the plane, a line $ p $ and points $ A $ and $ B $ are given. Find a point $ M $ on the line $ p $ such that the sum of the squares $ AM^2 + BM^2 $ is minimized. | M | 61 | 1 |
math | Find all triples $(p,q,n)$ that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where $p,q$ are odd primes and $n$ is an positive integer. | (3, 3, n) | 73 | 9 |
math | 276. $u=\log _{5} \cos 7 x$.
276. $u=\log _{5} \cos 7 x$. | -\frac{7\operatorname{tg}7x}{\ln5} | 37 | 17 |
math | 3. For every 7 boys in a school, there are 8 girls, and for every 9 boys, there is one teacher. In this school, there are a total of 675 students. How many teachers are there in this school? | 35 | 53 | 2 |
math | 16. All the positive integers which are co-prime to 2012 are grouped in an increasing order in such a way that the $n^{\text {th }}$ group has $2 n-1$ numbers. So, the first three groups in this grouping are (1), $(3,5,7),(9,11,13,15,17)$. It is known that 2013 belongs to the $k^{\text {th }}$ group. Find the value of ... | 32 | 135 | 2 |
math | Task 3. Determine all positive integers that cannot be written as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a, b$ positive and integer. | 1theoftheform2^+2with\geq0 | 41 | 14 |
math | [ Triangle Inequality ]
The least distance from a given point to points on a circle is $a$, and the greatest is $b$. Find the radius.
# | \frac{}{2} | 33 | 6 |
math | GS. 2 Let $x \geq 0$ and $y \geq 0$. Given that $x+y=18$. If the maximum value of $\sqrt{x}+\sqrt{y}$ is $d$, find the value of $d$. | 6 | 56 | 1 |
math | 3.159. $\cos \left(2 \alpha+\frac{7}{4} \pi\right)$, if $\operatorname{ctg} \alpha=\frac{2}{3}$. | \frac{7\sqrt{2}}{26} | 45 | 13 |
math |
1. Let $\left(a_{n}\right)_{n=1}^{\infty}$ be an infinite sequence such that for all positive integers $n$ we have
$$
a_{n+1}=\frac{a_{n}^{2}}{a_{n}^{2}-4 a_{n}+6}
$$
a) Find all values $a_{1}$ for which the sequence is constant.
b) Let $a_{1}=5$. Find $\left\lfloor a_{2018}\right\rfloor$.
(Vojtech Bálint)
| 2 | 126 | 1 |
math | 3. Find a three-digit number that, when multiplied by 7, gives a cube of a natural number. | 392 | 23 | 3 |
math | At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then ... | 151 | 112 | 3 |
math | 7. [5] A line in the plane is called strange if it passes through $(a, 0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points? | \frac{50}{3} | 82 | 8 |
math | 7. Given real numbers $a, b, c, d$ satisfy $5^{a}=4,4^{b}=3,3^{c}=2,2^{d}=5$, then $(a b c d)^{2018}=$ | 1 | 54 | 1 |
math | 1. The perimeter of a rectangle is 40 cm, and its area does not exceed $40 \mathrm{~cm}^{2}$. The length and width of the rectangle are expressed as natural numbers. By how much will the area of the rectangle increase if its perimeter is increased by 4 cm? | 2,4,21,36,38\mathrm{~}^{2} | 65 | 20 |
math | Let $n(n \geqslant 2)$ be a given positive integer, and real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy $x_{1}+2 x_{2}+\cdots+n x_{n}=0$. Try to find the smallest real number $\lambda(n)$, such that
$$
\left(x_{1}+x_{2}+\cdots+x_{n}\right)^{2} \leqslant \lambda(n)\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right) \te... | \frac{n^{2}-n}{4n+2} | 142 | 13 |
math | Let $M$ and $N$ , respectively, be the points of the sides $[AB]$ and $[BC]$ of a rectangle $ABCD$ . Let $a,b,c$ be the area of the triangle $AMD,MBN, NCD$ respectively. Express in terms of $a,b$ and $c$ , the area of the triangle $DMN$ . | \sqrt{(a+b+c)^2 - 4ac} | 80 | 14 |
math | Given the parabola $y=-2 x^{2}+x-\frac{1}{8}$ and the point $A\left(\frac{1}{4}, \frac{11}{8}\right)$. Draw any line through the point $F\left(\frac{1}{4},-\frac{1}{8}\right)$, intersecting the parabola at points $B$ and $C$.
(1) Find the equation of the centroid trajectory of $\triangle A B C$, and express it in the f... | \frac{3}{5} | 198 | 7 |
math | 5. In triangle $\mathrm{ABC}$ with sides $\mathrm{AB}=5, \mathrm{BC}=\sqrt{17}$, and $\mathrm{AC}=4$, a point $\mathrm{M}$ is taken on side $\mathrm{AC}$ such that $\mathrm{CM}=1$. Find the distance between the centers of the circumcircles of triangles $\mathrm{ABM}$ and $\mathrm{BCM}$. | 2 | 89 | 1 |
math | Find all pairs of real numbers $x$ and $y$ which satisfy the following equations:
\begin{align*}
x^2 + y^2 - 48x - 29y + 714 & = 0 \\
2xy - 29x - 48y + 756 & = 0
\end{align*} | (31.5, 10.5), (20, 22), (28, 7), (16.5, 18.5) | 83 | 39 |
math | [ Radii of the inscribed, circumscribed, and exscribed circles (other). [ Area of a triangle (using the semiperimeter and the radius of the inscribed or exscribed circle).
A circle is inscribed in a right triangle. One of the legs is divided by the point of tangency into segments of 6 and 10, measured from the vertex... | 240 | 92 | 3 |
math | Example 4. Find $\int \sqrt{a^{2}+x^{2}} d x$. | \frac{1}{2}x\sqrt{^{2}+x^{2}}+\frac{^{2}}{2}\ln|x+\sqrt{^{2}+x^{2}}|+C | 22 | 44 |
math | 4. Find the sum: $S_{n}=2^{3}+5^{3}+8^{3}+\cdots+(3 n-1)^{3}$. | S_{n}=\frac{n}{4}(27n^{3}+18n^{2}-9n-4) | 38 | 28 |
math | Mr. Dolphin and Mr. Shark were skilled fishermen. Once, they caught 70 fish together. Five ninths of the fish caught by Mr. Dolphin were trouts. Two seventeenths of the fish caught by Mr. Shark were carps.
How many fish did Mr. Dolphin catch?
(L. Hozová) | 36 | 69 | 2 |
math | Example 3 Find the Möbius transform $F(n)$ of $\Omega(n)$.
| \frac{1}{2} \Omega(n) \tau(n) | 19 | 15 |
math | 46. (9th grade) Find the sum
$$
\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}
$$ | \frac{n}{2(3n+2)} | 68 | 11 |
math | Example 1.28 Use $m(m \geqslant 2)$ colors to paint a $1 \times n$ chessboard, with each cell painted one color. Let $h(m, n)$ denote the number of ways to paint such that adjacent cells have different colors and all colors are used. Find the counting formula for $h(m, n)$. s is not from the beginning. | (,n)=\sum_{k=2}^{}(-1)^{-k}\binom{}{k}\cdotk(k-1)^{n-1} | 84 | 36 |
math | 4. Let the internal angles $\angle A, \angle B, \angle C$ of $\triangle ABC$ correspond to the sides $a, b, c$ which form a geometric sequence. Then the range of $\frac{\sin A \cdot \cot C+\cos A}{\sin B \cdot \cot C+\cos B}$ is $\qquad$ | (\frac{\sqrt{5}-1}{2},\frac{\sqrt{5}+1}{2}) | 75 | 23 |
math | ## 140. Math Puzzle $1 / 77$
In a 20-liter canister, there is a fuel mixture with a mixing ratio of gasoline to oil $=33.33: 1$. How many liters of oil are in the canister? | 0.582 | 59 | 5 |
math | A group of students consists of 5 boys and 4 girls. We want to form a team of three students chosen from the students in this group.
a) What is the number of possible teams?
b) Is it true that less than $5 \%$ of the possible teams are made up entirely of girls? | 84 | 64 | 2 |
math | Brennan chooses a set $A = \{a, b,c, d, e \}$ of five real numbers with $a \leq b \leq c \leq d \leq e.$ Delaney determines the subsets of $A$ containing three numbers and adds up the numbers in these subsets. She obtains the sums $0, 3; 4, 8; 9, 10, 11, 12, 14, 19.$ What are the five numbers in Brennan's set? | \{-3, -1, 4, 7, 8\} | 120 | 18 |
math | The dividend is six times larger than the divisor, and the divisor is six times larger than the quotient. What are the dividend, divisor, and quotient?
# | 216,36,6 | 32 | 8 |
math | 858. Find the smallest pair of natural numbers $x$ and $y$ that satisfy the equation
$$
5 x^{2}=3 y^{5}
$$ | 675,15 | 36 | 6 |
math | 9. Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube $k$-harmonic if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let... | 2040200 | 132 | 7 |
math | 7.1. A natural number $n$ was multiplied by the sum of the digits of the number $3 n$, and the resulting number was then multiplied by 2. As a result, 2022 was obtained. Find $n$. | 337 | 52 | 3 |
math | 8.34 Write the natural numbers on a blackboard in sequence, with the rule that when a perfect square is encountered, it is skipped and the next natural number is written instead, thus forming the sequence
$$
2,3,5,6,7,8,10,11, \cdots
$$
This sequence starts with the first number being 2, the 4th number being 6, the 8t... | 2037 | 140 | 4 |
math | \section*{Problem \(3-340933=341032\)}
Calculate the number
\(123456785 \cdot 123456787 \cdot 123456788 \cdot 123456796 - 123456782 \cdot 123456790 \cdot 123456791 \cdot 123456793\)
without calculating the values of the two products individually! | 22222222020 | 132 | 11 |
math | Example 11 Let $p(x)$ be a polynomial of degree $3n$, such that $P(0)=P(3) \cdots=P(3n)=2, P(1)=$ $P(4)=\cdots=P(3n-2)=1, P(2)=P(5)=\cdots=P(3n-1)=0, P(3n+1)=730$. Determine $n$.
(13th US Olympiad Problem) | 4 | 106 | 1 |
math |
1. Some objects are in each of four rooms. Let $n \geqslant 2$ be an integer. We move one $n$-th of objects from the first room to the second one. Then we move one $n$-th of (the new number of) objects from the second room to the third one. Then we move similarly objects from the third room to the fourth one and from ... | 5 | 152 | 1 |
math | 11. The faces of a hexahedron and the faces of a regular octahedron are all equilateral triangles with side length $a$. The ratio of the radii of the inscribed spheres of these two polyhedra is a reduced fraction $\frac{m}{n}$. Then, the product $m \cdot n$ is $\qquad$. | 6 | 76 | 1 |
math | [ Classical combinatorics (miscellaneous). ] [ Formulas of abbreviated multiplication (miscellaneous).]
Twenty-five coins are distributed into piles as follows. First, they are arbitrarily divided into two groups. Then any of the existing groups is again divided into two groups, and so on until each group consists of... | 300 | 100 | 3 |
math | Find all pairs of functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, we have $g(f(x+y))=f(x)+2 x+y \cdot g(y)$ | f(x)=-2x,(x)=0 | 61 | 10 |
math | 9.1. Solve the equation $\left|x^{2}-100\right|=2 x+1$. | x_{1}=1+\sqrt{102},x_{2}=9 | 24 | 17 |
math | 7. If real numbers $a, b, c$ make the quadratic function $f(x) = a x^{2} + b x + c$ such that when $0 \leqslant x \leqslant 1$, always $|f(x)| \leqslant 1$. Then the maximum value of $|a| + |b| + |c|$ is $\qquad$ | 17 | 87 | 2 |
math | 1. (5 points) Find the value of $n$ for which the following equality holds:
$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2014
$$ | 4060224 | 88 | 7 |
math | How many positive integers $N$ less than $1000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$? | 412 | 43 | 3 |
math | B1. Find all natural numbers $n$ for which
$$
\frac{100 n^{5}-n^{4}-50 n^{3}+2 n^{2}-290 n-2}{n^{2}-2}
$$
is an integer.
(20 points) | 1,2,3,10 | 65 | 8 |
math | If $\frac{x}{2}-5=9$, what is the value of $\sqrt{7 x}$ ? | 14 | 23 | 2 |
math | Given is the equation $(m, n) +[m, n] =m+n$ where $m, n$ are positive integers and m>n.
a) Prove that n divides m.
b) If $m-n=10$, solve the equation. | (m, n) = (11, 1), (12, 2), (15, 5), (20, 10) | 56 | 36 |
math | 7. The school organizes 1511 people to go on an outing, renting 42-seat coaches and 25-seat minibuses. If it is required that there is exactly one seat per person and one person per seat, then there are $\qquad$ rental options. | 2 | 60 | 1 |
math | $9 \cdot 37$ Find the largest real number $\alpha$ such that for any positive integers $m$ and $n$ satisfying $\frac{m}{n}<\sqrt{7}$, we have
$$\frac{\alpha}{n^{2}} \leqslant 7-\frac{m^{2}}{n^{2}}$$ | 3 | 76 | 1 |
math | 1. Kolya went to the store in the neighboring village on an electric scooter at a speed of 10 km/h. Having traveled exactly one third of the entire distance, he realized that with the previous speed, he would arrive exactly when the store closes, and he doubled his speed. But when he had traveled exactly $2 / 3$ of the... | 6\frac{2}{3} | 109 | 8 |
math | An equilateral triangle is inscribed inside of a circle of radius $R$. Find the side length of the triangle | R\sqrt{3} | 23 | 7 |
math | Example 5. Calculate the coordinates of the center of gravity of the square $0 \leqslant x \leqslant 2,0 \leqslant y \leqslant 2$ with density $\rho=x+y$. | x_{}=\frac{7}{6},y_{}=\frac{7}{6} | 53 | 20 |
math | Example 2 At the end of the 5th century AD, the Chinese mathematician Zhang Qiujian proposed a famous "Hundred Chickens Problem" in the history of world mathematics in his renowned work "The Mathematical Classic": "A rooster is worth five coins, a hen is worth three coins, and three chicks are worth one coin. With a hu... | (0,25,75),(4,18,78),(8,11,81),(12,4,84) | 97 | 33 |
math | An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic looks at the artwork and counts faults: each time two red dots are adjacent is one fault, and each time two blue dots are exactly two apart (that is, they have exactly one dot in between them) is another.
What is the s... | 250 | 83 | 3 |
math | Let $l$ be a line passing the origin on the coordinate plane and has a positive slope. Consider circles $C_1,\ C_2$ determined by the condition (i), (ii), (iii) as below.
(i) The circles $C_1,\ C_2$ are contained in the domain determined by the inequality $x\geq 0,\ y\geq 0.$
(ii) The circles $C_1,\ C_2$ touch the li... | 7 | 207 | 1 |
math | 2.205. $\left(\frac{b x+4+\frac{4}{b x}}{2 b+\left(b^{2}-4\right) x-2 b x^{2}}+\frac{\left(4 x^{2}-b^{2}\right) \cdot \frac{1}{b}}{(b+2 x)^{2}-8 b x}\right) \cdot \frac{b x}{2}$. | \frac{x^{2}-1}{2x-b} | 95 | 12 |
math | Example 9 Express $\frac{11 x^{2}-23 x}{(2 x-1)\left(x^{2}-9\right)}$ as partial fractions. | N=\frac{1}{2 x-1}+\frac{4}{x+3}+\frac{1}{x-3} | 37 | 29 |
math | 2B. In a certain populated place, all telephone numbers consist of six digits arranged in strictly ascending or strictly descending order, with the first digit in the number not being 0. What is the maximum number of telephone numbers that can exist in this place? | 294 | 52 | 3 |
math | 3. If $n$ is a positive integer, and $n^{2}+9 n+98$ is exactly equal to the product of two consecutive positive integers, then all values of $n$ are $\qquad$ | 34,14,7 | 48 | 7 |
math | 10. Given that the largest angle of an isosceles triangle is 4 times the smallest angle, then the difference between the largest and smallest angles is $(\quad)$ degrees. Exam point: Isosceles triangle | 90 | 47 | 2 |
math | The year 2015, which is about to end, has 7 consecutive days whose date numbers sum up to 100. What are the date numbers of these 7 days? $\qquad$
$\qquad$
$\qquad$
$\qquad$
$\qquad$
$\qquad$ - | 29,30,31,1,2,3,4 | 66 | 16 |
math | 3. [4] Find all $y>1$ satisfying $\int_{1}^{y} x \ln x d x=\frac{1}{4}$. | \sqrt{e} | 35 | 5 |
math | Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $ | (x, y, z) = (1, 2, 1) | 51 | 18 |
math | 9. (16 points) For a given positive integer $M$, define $f_{1}(M)$ as the square of the sum of the digits of $M$. When $n>1$ and $n \in \mathbf{N}$, $f_{n}\left(f_{n-1}(M)\right)$ represents the $r_{n}$-th power of the sum of the digits of $f_{n-1}(M)$, where, when $n$ is odd, $r_{n}=2$; when $n$ is even, $r_{n}=3$. Fi... | 729 | 162 | 3 |
math | Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all
numbers $N$ with such property. Find the sum of all possible values of $N$ such th... | 135 | 113 | 3 |
math | 6.081. $\left\{\begin{array}{l}x^{3}+y^{3}=65 \\ x^{2} y+x y^{2}=20\end{array}\right.$ | (4;1)(1;4) | 47 | 9 |
math | 3. Place a regular tetrahedron with a volume of 1 inside a cube, then the minimum volume of this cube is $\qquad$ | 3 | 31 | 1 |
math | 1. $\frac{\text { Hua Cup }}{\text { Shao } \times \text { Jun }+ \text { Jin Tan }+ \text { Lun } \times \text { Shu }}=15$
In the equation above, different Chinese characters represent different digits from $1-9$. When the three-digit number “Hua Cup Sai” reaches its maximum value, please write down one way to make t... | 975 | 93 | 3 |
math | 7.4. Determine the natural prime numbers $a, b, c$, knowing that:
$$
\frac{2 a+4 b-c+7}{a+3 b+c+2}=\frac{5 a+b+3 c-22}{a-2 b+2 c+30}=\frac{c-3 b-6 a-14}{3 b+3 c-90}
$$ | =2,b=3,=11 | 89 | 9 |
math | Let $K$ and $N>K$ be fixed positive integers. Let $n$ be a positive integer and let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct integers. Suppose that whenever $m_{1}, m_{2}, \ldots, m_{n}$ are integers, not all equal to 0, such that $\left|m_{i}\right| \leqslant K$ for each $i$, then the sum
$$
\sum_{i=1}^{n} m_{i} a_{... | n=\left\lfloor\log _{K+1} N\right\rfloor | 146 | 20 |
math | 6.3. Oleg, Igor, and Anya are in the 6th grade. Among them, there is the best mathematician, the best chess player, and the best artist. It is known that:
a) the best artist did not paint their own portrait, but painted Igor's portrait
b) Anya has never lost to boys in chess.
Who in the class is the best mathematicia... | Oleg | 95 | 2 |
math | A certain frog that was placed on a vertex of a convex polygon chose to jump to another vertex, either clockwise skipping one vertex, either counterclockwise skipping two vertexes, and repeated the procedure.
If the number of jumps that the frog made is equal to the number of sides of the polygon, the frog has passed t... | \{ n \in \mathbb{N} \mid 6 \nmid n \text{ or } 30 \mid n \} | 103 | 33 |
math | There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$. Suppose that at some point in time, all three flies meet ... | 501 | 109 | 3 |
math | 5. There are 20 teams participating in the national league. Question: What is the minimum number of matches that must be played so that in any group of three teams, at least two teams have played against each other? | 90 | 46 | 2 |
math | 1. Let $A, B \in M_{2}(\mathbb{C})$ such that: $A B=\left(\begin{array}{cc}10 & 30 \\ 4 & 20\end{array}\right)$ and $B A=\left(\begin{array}{cc}x & 60 \\ 2 & y\end{array}\right)$. Find $x$ and $y$. | 20,10or10,20 | 94 | 11 |
math | 1. $n$ houses with house numbers 1 to $n$ are located on SMO Street, where $n$ is a natural number. The houses with odd numbers are on the left-hand side of the street, the houses with even numbers are on the right-hand side. The letter carrier Quirin wants to deliver a newspaper to each house. How many ways does he ha... | 2\cdot(\frac{n}{2}!)^{2} | 99 | 13 |
math | XXXV OM - I - Problem 9
Three events satisfy the conditions:
a) their probabilities are equal,
b) any two of them are independent,
c) they do not occur simultaneously.
Determine the maximum value of the probability of each of these events. | \frac{1}{2} | 54 | 7 |
math | ## Problem Statement
Find the derivative.
$$
y=2 \frac{\cos x}{\sin ^{4} x}+3 \frac{\cos x}{\sin ^{2} x}
$$ | 3\operatorname{cosec}x-8\operatorname{cosec}^{5}x | 44 | 24 |
math | 6. (2001 National High School Competition) The length of the minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is equal to
保留了源文本的换行和格式。 | \frac{2\sqrt{3}}{3} | 48 | 12 |
math | Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose? | 6 | 77 | 1 |
math | 21.3. The numbers $2^{n}$ and $5^{n}$ start with the digit $a$. What is $a$? | 3 | 31 | 1 |
math | 1. Given that $a$, $b$, and $c$ are non-zero real numbers, satisfying
$$
\frac{b+c-a}{a}=\frac{c+a-b}{b}=\frac{a+b-c}{c} \text {. }
$$
Then the value of $\frac{(a+b)(b+c)(c+a)}{a b c}$ is $\qquad$ | -1 \text{ or } 8 | 83 | 9 |
math | 2. Let real numbers $x, y, z, w$ satisfy $x \geqslant y \geqslant z \geqslant w \geqslant 0$, and $5 x+4 y+3 z+6 w=100$. Denote the maximum value of $x+y+z+w$ as $a$, and the minimum value as $b$. Then $a+b=$ $\qquad$ | 45 | 94 | 2 |
math | $\underline{115625}$ topics: [ Tangent circles [Mean proportionals in a right triangle]
Two circles touch each other externally at point $C$. A line is tangent to the first circle at point $A$ and to the second circle at point $B$. The line $A C$ intersects the second circle at point $D$, different from $C$. Find $B ... | 6 | 100 | 1 |
math | 9. How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement?
(1 mark)
有多少種不同的方法可把「BRILLIANT 」一字中的字母重新排列, 使得排列後沒有兩個相鄰的字母相同？ | 55440 | 71 | 5 |
math | Problem 2. Determine the natural prime numbers $p$ and $q$ knowing that there exist $x, y \in \mathbb{N}^{*}$ such that $p=x^{2}+y^{2}$ and $q=x+y+1$. | p=2,q=3 | 56 | 6 |
math | One, (50 points) Try to find all real-coefficient polynomials $f(x)$, such that for all real numbers $a, b, c$ satisfying $a b + b c + c a = 0$, the following holds:
$$
f(a-b) + f(b-c) + f(c-a) = 2 f(a+b+c).
$$ | f(x)=A x^{4}+B x^{2}(A, B \in \mathbf{R}) | 77 | 25 |
math | Let $x+y=a$ and $xy=b$. The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$. What is this polynomial? | a^6 - 6a^4b + 9a^2b^2 - 2b^3 | 44 | 25 |
math | Given the equation $x^2+ax+1=0$, determine:
a) The interval of possible values for $a$ where the solutions to the previous equation are not real.
b) The loci of the roots of the polynomial, when $a$ is in the previous interval. | a \in (-2, 2) | 60 | 10 |
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