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 | 8. (5 points) Anya and Kolya were collecting apples. It turned out that Anya collected as many apples as the percentage of the total number of apples collected by Kolya, and Kolya collected an odd number of apples. How many apples did Anya and Kolya collect together? | 25,300,525,1900,9900 | 65 | 20 |
math | ## Task A-1.4.
Determine the smallest natural number whose sum of digits is divisible by 7 and has the property that the sum of the digits of its successor is also divisible by 7. | 69999 | 43 | 5 |
math | 4. Given that $\alpha, \beta$ are both acute angles, and
$$
(1+\tan \alpha)(1+\tan \beta)=2 \text{.}
$$
Then $\alpha+\beta=$ $\qquad$ | \frac{\pi}{4} | 50 | 7 |
math | Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane.
Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis. | \frac{128\pi}{105} | 96 | 13 |
math | Sérgio chooses two positive integers $a$ and $b$. He writes 4 numbers in his notebook: $a, a+2, b$ and $b+2$. Then, all 6 products of two of these numbers are written on the board. Let $Q$ be the number of perfect squares written on it, determine the maximum value of $Q$.
# | 2 | 80 | 1 |
math | 6. Solve the system
$$
\left\{\begin{array}{l}
\operatorname{tg}^{3} x+\operatorname{tg}^{3} y+\operatorname{tg}^{3} z=36 \\
\operatorname{tg}^{2} x+\operatorname{tg}^{2} y+\operatorname{tg}^{2} z=14 \\
\left(\operatorname{tg}^{2} x+\operatorname{tg} y\right)(\operatorname{tg} x+\operatorname{tg} z)(\operatorname{tg} ... | 4 | 177 | 1 |
math | 9. (10 points) On a plane, there are 5 different lines, and these 5 lines form $m$ intersection points. How many different values can $m$ have?
保留源文本的换行和格式，翻译结果如下：
9. (10 points) On a plane, there are 5 different lines, and these 5 lines form $m$ intersection points. How many different values can $m$ have? | 9 | 93 | 1 |
math | Example 9 Given real numbers $x, y, z$ satisfy $x+y=5$, $z^{2}=x y+y-9$. Then, $x+2 y+3 z=$ $\qquad$
(1995, Zu Chongzhi Cup Junior High School Mathematics Invitational Competition) | 8 | 66 | 1 |
math | 6.228. $\left\{\begin{array}{l}x \sqrt{y}+y \sqrt{x}=30 \\ x \sqrt{x}+y \sqrt{y}=35\end{array}\right.$ | (4,9),(9,4) | 52 | 9 |
math | ## Task 36/75
Choose any two-digit prime number with a cross sum of 10 and subtract the number 18 from it as many times as necessary until the difference is between 10 and 20. Quadruple this difference! Place the difference in front of this product!
How many "starting numbers" are there for the calculation, and why i... | 1976 | 83 | 4 |
math | 12.153. A circle of radius $r$ is inscribed in a sector of radius $R$. Find the perimeter of the sector. | 2R(1+\arcsin\frac{r}{R-r}) | 32 | 16 |
math | Find all integer solutions to the following equation:
$$
x^{2}+12=y^{4}
$$ | 2,\pm2or-2,\pm2 | 23 | 10 |
math | 2. In the set of natural numbers, solve the equation
$$
x^{5-x}=(6-x)^{1-x}
$$ | 1,5,9 | 29 | 5 |
math | 4. (3 points) On the side $AC$ of triangle $ABC$, a circle with a radius of 10 cm is constructed with $AC$ as its diameter. This circle intersects sides $AB$ and $BC$ at points $X$ and $Y$ respectively. Find $AX \cdot AB + CY \cdot BC$. | 400 | 71 | 3 |
math | 1. From each of the following groups of numbers, take one number, multiply them, then the sum of all such products is $\qquad$ .
First group: $\frac{3}{4}, 0.15$; Second group: $4, \frac{2}{3}$; Third group: $\frac{3}{5}, 1.2$ | 7.56 | 77 | 4 |
math | A1. Evaluate $\left(1+\frac{1}{1^{2}}\right)\left(2+\frac{1}{2^{2}}\right)\left(3+\frac{1}{3^{2}}\right)$. | 14 | 51 | 2 |
math |
4. Let $A B C$ be a triangle. An interior point $P$ of $A B C$ is said to be good if we can find exactly 27 rays emanating from $P$ intersecting the sides of the triangle $A B C$ such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given tri... | 325 | 92 | 3 |
math | 3. Let $x, y, z$ be positive real numbers such that $x+y+z=3$. Find the maximum value reached by
$$
\sqrt{x}+\sqrt{2 y+2}+\sqrt{3 z+6}
$$
For what values of $x, y, z$ is this maximum achieved? | 6 | 70 | 1 |
math | ## Task B-3.3.
If $f(x)=4 \sin ^{2} \frac{3 x}{2}-4 \cos ^{2} \frac{3 x}{2}$, determine $f\left(\frac{2020 \pi}{9}+2021 k \pi\right)$ depending on the integer $k$. | 2 | 79 | 1 |
math | 10. Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=\sqrt{3}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}$, find the general term formula for $a_{n}$. | a_{n}={\begin{pmatrix}\frac{3n-3+2\sqrt{3}}{2},&n\text{isodd}\\\frac{3n-3+\sqrt{3}}{2},&n\text{iseven}\end{pmatrix}.} | 70 | 64 |
math | For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$ | 363 | 73 | 3 |
math | 1. If the inequality $\sqrt{x\left(x^{2}+8\right)(8-x)}<\lambda(x+$ 1) holds for all real numbers $x \in(0,2)$, then the range of the real number $\lambda$ is $\qquad$ . | [4,+\infty) | 61 | 7 |
math | 5. In $\triangle A B C$, $A B=A C=a$, a regular triangle $B C D$ is constructed outward on side $B C, A=$
$\qquad$ when, the area of quadrilateral $A B C D$ is maximized. | 150 | 57 | 3 |
math | 11. (This question is worth 20 points) Let $F_{1}, F_{2}$ be the left and right foci of the ellipse $\frac{x^{2}}{2}+y^{2}=1$, respectively. Suppose a line $l$ that does not pass through the focus $F_{1}$ intersects the ellipse at two distinct points $A, B$. The distance from the focus $F_{2}$ to the line $l$ is $d$. I... | (\sqrt{3},2) | 130 | 7 |
math | 108. a) $2^{-3}$; b) $\left(\frac{1}{3}\right)^{-2}$; c) $\left(\frac{2}{3}\right)^{-4} ;$ d) $(-0.2)^{-3}$. | \frac{1}{8},9,\frac{81}{16},-125 | 59 | 21 |
math | 114. Calculate $\lim _{x \rightarrow 1} \frac{2 x^{3}+4 x-3}{x+4}$. | \frac{3}{5} | 34 | 7 |
math | 1st CIS 1992 Problem 20 Find all integers k > 1 such that for some distinct positive integers a, b, the number k a + 1 can be obtained from k b + 1 by reversing the order of its (decimal) digits. | 3 | 57 | 1 |
math | Kirienko D:
Sasha and Masha each thought of a natural number and told them to Vasya. Vasya wrote the sum of the numbers on one sheet of paper and their product on another, then hid one of the sheets and showed the other (which had the number 2002 written on it) to Sasha and Masha. Seeing this number, Sasha said that h... | 1001 | 119 | 4 |
math | $9 \cdot 23$ Find all real numbers $\alpha$ such that
$\cos \alpha, \cos 2 \alpha, \cos 4 \alpha, \cdots, \cos 2^{n} \alpha, \cdots$
are all negative. | \alpha=\\frac{2\pi}{3}+2k\pi,k\in\mathbb{Z} | 59 | 26 |
math | 1. The positive integer $n$ is divisible by 1990, and $n$ has exactly 12 positive divisors (including 1 and $n$), find $n$. | 2^{2}\cdot5\cdot199;2\cdot5^{2}\cdot199;2\cdot5\cdot199^{2} | 42 | 36 |
math | 13.113. A material particle entered the pipe through an opening, and 6.8 minutes later, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 m/min, while the second particle covered 3 m in the... | 17 | 108 | 2 |
math | 4. In the stands of the hockey arena, there are several rows with 168 seats in each row. For the final match, 2016 students from several sports schools were invited as spectators, with no more than 40 from each school. Students from any school must be seated in one row. What is the minimum number of rows that must be i... | 15 | 87 | 2 |
math | Example 5 The license plates of motor vehicles in a city are consecutively numbered from "10000" to "99999". Among these 90000 license plates, the number of plates that contain at least one digit 9 and have the sum of their digits as a multiple of 9 is $\qquad$.
| 4168 | 75 | 4 |
math | Determine all functions $f: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ such that for all $x \geqslant 0$, we have $f(f(x)) +$ $f(x)=6 x$. | f(x)=2x | 55 | 5 |
math | Given a rectangle $ABCD$, side $AB$ is longer than side $BC$. Find all the points $P$ of the side line $AB$ from which the sides $AD$ and $DC$ are seen from the point $P$ at an equal angle (i.e. $\angle APD = \angle DPC$) | x = n \pm \sqrt{n^2 - m^2} | 70 | 16 |
math | 1. Three pieces of fabric have a total length of $34 \mathrm{~m}$. If we cut off a quarter of the length of the first piece, half of the length of the second piece, and one-sixth of the length of the third piece, the remaining parts of the fabric will have equal lengths. What is the length of each piece of fabric befor... | 10,15,9 | 79 | 7 |
math | 1. How many times do the second hand and the minute hand form a $45^{\circ}$ angle between 0 and 12 o'clock? | 1416 | 33 | 4 |
math | Subject (1). Determine the rational numbers $x$ and $y$ that satisfy
$$
\sqrt{\frac{2}{3}}=\frac{(|2 x+y|-5) \cdot \sqrt{5}+\sqrt{2}}{|3 y-3| \cdot \sqrt{5}+\sqrt{3}}
$$[^2] | (x,y)\in{(-3,1),(2,1)} | 73 | 14 |
math | 10.1. Find the smallest solution of the inequality
$$
\frac{-\log _{2}(120-2 x \sqrt{32-2 x})^{2}+\left|\log _{2} \frac{120-2 x \sqrt{32-2 x}}{\left(x^{2}-2 x+8\right)^{3}}\right|}{5 \log _{7}(71-2 x \sqrt{32-2 x})-2 \log _{2}(120-2 x \sqrt{32-2 x})} \geqslant 0
$$ | -13-\sqrt{57}\approx-20.55 | 142 | 16 |
math | 7. Let the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, where $a_{1}=p, b_{1}=q$. It is known that $a_{n}=p a_{n-1}, b_{n}=q a_{n-1}+r b_{n-1}(p, q, r$ are constants, and $q>0, p>r>0, n \geqslant 2)$, then the general term formula of the sequence $\left\{b_{n}\right\}$ is $b_{n}=$ $\qquad$ | \frac{q(p^{n}-r^{n})}{p-r} | 138 | 16 |
math | 289. $y=\sqrt{x}\left(x^{2}+2 x-5\right)$.
289. $y=\sqrt{x}\left(x^{2}+2 x-5\right)$. | 2.5x\sqrt{x}+3\sqrt{x}-\frac{5}{2\sqrt{x}} | 48 | 24 |
math | At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions l... | 2^{10} - 1 = 1023 | 124 | 16 |
math | Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?
[asy]
size(150);
defaultpen(linewidth(... | 540 | 157 | 3 |
math | $$
\begin{array}{l}
\cos 10^{\circ} \cdot \cos 50^{\circ} \cdot \cos 70^{\circ}+\sin 10^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ} \\
=\quad .
\end{array}
$$ | \frac{1+\sqrt{3}}{8} | 81 | 12 |
math | Exercise 9. Alexie and Baptiste each own a building. Each floor of Alexie's building has 3 bathrooms and 2 bedrooms. Baptiste, on the other hand, has 4 bathrooms and 3 bedrooms per floor. There are a total of 25 bathrooms and 18 bedrooms. Find the number of floors in Alexie's and Baptiste's buildings.
Only a numerical... | =3,b=4 | 87 | 5 |
math | 4. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(f(x)+f(y))=f(x)+y, \text { for all } x, y \in \mathbb{R} \text {. }
$$ | f(x)=x | 62 | 4 |
math | In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ? | 6 | 101 | 1 |
math | Between the ports of Mumraj and Chaos, two ships are plying the same route. They spend a negligible amount of time in the ports, immediately turn around, and continue their journey. One morning, at the same moment, a blue ship sets sail from the port of Mumraj and a green ship from the port of Chaos. For the first time... | 35\mathrm{~} | 164 | 7 |
math | 2. Given that $n$ is a positive integer, such that there exist positive integers $x_{1}$, $x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1} x_{2} \cdots x_{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right)=100 n .
$$
Find the maximum possible value of $n$.
(Lin Jin, problem contributor) | 9702 | 99 | 4 |
math | Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.
[i]Proposed by Evan Chen[/i] | 2306 | 79 | 4 |
math | Solve the system of equations in real numbers:
\[
\begin{cases*}
(x - 1)(y - 1)(z - 1) = xyz - 1,\\
(x - 2)(y - 2)(z - 2) = xyz - 2.\\
\end{cases*}
\]
[i]Vladimir Bragin[/i] | x = y = z = 1 | 83 | 8 |
math | 8.072. $(\cos 6 x-1) \operatorname{ctg} 3 x=\sin 3 x$.
8.072. $(\cos 6 x-1) \cot 3 x=\sin 3 x$. | \\frac{2}{9}\pi+\frac{2}{3}\pik,k\inZ | 58 | 21 |
math | ## Task 1 - 240811
At a school, a waste collection campaign is being conducted in grades 5 to 10. During the subsequent evaluation for a competition between the grade levels, the following is observed:
The students of grade 9 collected waste worth $42 \mathrm{M}$; the students of grade 10 collected the same amount. T... | 1200\mathrm{M} | 168 | 9 |
math | $f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of
$$\int_0^1(x-f(x))^{2016}dx$$ | 0 | 61 | 1 |
math | Find all real solutions of the system of equations $\frac{2 x_{1}^{2}}{1+x_{1}^{2}}=x_{2}, \frac{2 x_{2}^{2}}{1+x_{2}^{2}}=x_{3}, \frac{2 x_{3}^{2}}{1+x_{3}^{2}}=x_{1}$. | (0,0,0)(1,1,1) | 85 | 13 |
math | Adam and Bettie are playing a game. They take turns generating a random number between $0$ and $127$ inclusive. The numbers they generate are scored as follows:
$\bullet$ If the number is zero, it receives no points.
$\bullet$ If the number is odd, it receives one more point than the number one less than it.
$\bullet$ ... | 429 | 137 | 3 |
math | Determine all real solutions $x, y, z$ of the following system of equations: $\begin{cases}
x^3 - 3x = 4 - y \\
2y^3 - 6y = 6 - z \\
3z^3 - 9z = 8 - x\end{cases}$ | x = y = z = 2 | 69 | 9 |
math | 4. A positive number, if its fractional part, integer part, and the number itself form a geometric sequence, then the number is . $\qquad$ | \frac{1+\sqrt{5}}{2} | 32 | 12 |
math | Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$ | \sqrt{3} | 72 | 5 |
math | ## Task 4
Check the following statements:
a) The product of the smallest two-digit number and the smallest three-digit number is equal to the smallest four-digit number.
a) The product of the largest two-digit number and the largest three-digit number is equal to the largest four-digit number. | 1000 | 60 | 4 |
math | 7.2. Ellie and Toto painted daisies in the field. On the first day, Ellie painted one fourteenth of the entire field. On the second day, she painted twice as much as on the first day, and on the third day, she painted twice as much as on the second day. Toto, in total, painted 7000 daisies. How many daisies were there ... | 14000 | 114 | 5 |
math | A segment of length $1$ is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region. | \frac{5\pi}{6} + \frac{\sqrt{3}}{4} | 34 | 20 |
math | 3. A person adds the page numbers of a book in the order $1,2,3, \cdots$, with one page number being added an extra time, resulting in an incorrect total sum of 2005. Then the page number that was added extra is $\qquad$ . | 52 | 62 | 2 |
math | 10. Let $n$ be a positive integer. If in the $2 n+1$ consecutive natural numbers including 2009, the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers, then the value of $n$ is $\qquad$. | 31 | 73 | 2 |
math | 2. On an island, there live liars and knights, a total of 2021 people. Knights always tell the truth, and liars always lie. Every resident of the island knows whether each person is a knight or a liar. One fine day, all the residents of the island lined up. After that, each resident of the island stated: "The number of... | 1010 | 121 | 4 |
math | ## 13. Math Puzzle $6 / 66$
A stenotypist has a long manuscript to transcribe. For the first half, she types 15 pages per day, and for the second half, she types 25 pages per day.
What is her average daily output overall? | 18.75 | 64 | 5 |
math | ## Zadatak B-2.1.
Koliko je $1+z^{2}+z^{4}+\cdots+z^{2 \cdot 2019}$, ako je $z=\frac{1+\sqrt{3} i}{2}$ ?
| 1 | 58 | 1 |
math | 5. Given $f(x)=\left(x^{2}+3 x+2\right)^{\cos \pi x}$. Then the sum of all $n$ that satisfy the equation
$$
\left|\sum_{k=1}^{n} \log _{10} f(k)\right|=1
$$
is | 21 | 72 | 2 |
math | [
For all real $x$ and $y$, the equality $f\left(x^{2}+y\right)=f(x)+f\left(y^{2}\right)$ holds. Find $f(-1)$.
# | 0 | 49 | 1 |
math | 81. Solve the equation $y^{\prime}+y \tan x=\cos ^{2} x$. | (\sinx+C)\cosx | 25 | 7 |
math | Given a continuous function $ f(x)$ such that $ \int_0^{2\pi} f(x)\ dx \equal{} 0$.
Let $ S(x) \equal{} A_0 \plus{} A_1\cos x \plus{} B_1\sin x$, find constant numbers $ A_0,\ A_1$ and $ B_1$ for which $ \int_0^{2\pi} \{f(x) \minus{} S(x)\}^2\ dx$ is minimized. | A_0 = 0, A_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos x \, dx, B_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin x \, dx | 111 | 69 |
math | 4. Integers $a, b, c$ satisfy $a+b+c=2$, and
$$
S=(2 a+b c)(2 b+c a)(2 c+a b)>200 \text {. }
$$
Then the minimum value of $S$ is $\qquad$. | 256 | 62 | 3 |
math | 9. (12 points) Three people, A, B, and C, depart from location $A$ to location $B$. A departs at 8:00, B at 8:20, and C at 8:30. They all travel at the same speed. 10 minutes after C departs, the distance from A to $B$ is exactly half the distance from B to $B$. At this moment, C is 2015 meters away from $B$. Therefore... | 2418 | 127 | 4 |
math | 29.12. Calculate: a) $\int e^{a x} \sin b x d x$; b) $\int e^{a x} \cos b x d x$. | \inte^{}\sin=\frac{e^{}}{^{2}+b^{2}}(\sin-b\cos)+C | 41 | 27 |
math | Let $a$ and $c$ be positive integers, and let $b$ be a digit. Determine all triples of numbers $(a, b, c)$ that satisfy the following conditions:
(1) $(a, b b b \ldots)^{2}=c, 777 \ldots$ (infinite decimal fractions);
(2) $\frac{c+a}{c-a}$ is an integer! | (1,6,2) | 87 | 7 |
math | Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square? | 16 | 48 | 2 |
math | $4 \cdot 3$ Given the equation $2 x^{2}-9 x+8=0$, find a quadratic equation whose one root is the reciprocal of the sum of the roots of the original equation, and the other root is the square of the difference of the roots of the original equation. | 36 x^{2}-161 x+34=0 | 62 | 15 |
math | Laura is putting together the following list: $a_0, a_1, a_2, a_3, a_4, ..., a_n$, where $a_0 = 3$ and $a_1 = 4$.
She knows that the following equality holds for any value of $n$ integer greater than or equal to $1$:
$$a_n^2-2a_{n-1}a_{n+1} =(-2)^n.$$Laura calculates the value of $a_4$. What value does it get? | \frac{179}{128} | 117 | 11 |
math | ## Task Condition
Derive the equation of the tangent line to the given curve at the point with abscissa $x_{0}$.
$$
y=\frac{x^{2}}{10}+3, x_{0}=2
$$ | \frac{2}{5}\cdotx+\frac{13}{5} | 52 | 17 |
math | How many "quadratic residues" are there $\bmod p q?$ And $\bmod p^{n}$? You can use the fact that $\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)^{*}$ is cyclic. | \frac{(p-1)(q-1)}{4} | 56 | 14 |
math | To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$. Solve the equation $x*36 = 216$. | 7776 | 66 | 4 |
math | Example. Calculate the definite integral
$$
\int_{0}^{2 \pi} \sin ^{4} 3 x \cos ^{4} 3 x d x
$$ | \frac{3\pi}{64} | 41 | 10 |
math | Task A-3.2. (4 points)
What is the maximum possible ratio of a three-digit number to the sum of its digits? | 100 | 29 | 3 |
math | 10. (10 points) $n$ pirates divide gold coins. The 1st pirate takes 1 coin first, then takes $1 \%$ of the remaining coins; then, the 2nd pirate takes 2 coins, then takes $1 \%$ of the remaining coins; the 3rd pirate takes 3 coins, then takes $1 \%$ of the remaining coins; ... the $n$-th pirate takes $n$ coins, then ta... | 9801 | 142 | 4 |
math | Example 7 Let $p$ be a given positive integer, try to determine the minimum positive value of $(2 p)^{2 m}-(2 p-1)^{n}$, where $m, n$ are any positive integers. | 4p-1 | 50 | 4 |
math | Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3, b_{k+1}=a_{k}+b_{k}(k=1,2, \cdots)$. Find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. | 2^{n}+2n-1 | 119 | 9 |
math | In a tournament with 10 teams, each of them plays against each other exactly once. Additionally, there are no ties, and each team has a $50 \%$ chance of winning any match. What is the probability that, after tallying the scores from $\frac{10 \cdot 9}{2}=45$ games, no two players have the same number of wins? | \frac{10!}{2^{45}} | 81 | 12 |
math | Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism. | \sqrt{\frac{K^3}{54}} | 57 | 13 |
math | 24th Putnam 1963 Problem B1 Find all integers n for which x 2 - x + n divides x 13 + x + 90. Solution | 2 | 39 | 1 |
math | Example 9. A radio tube may belong to one of three batches with probabilities: $p_{1}=0.2, p_{2}=0.3, p_{3}=0.5$. The probability that the tube will work for a given number of hours, for these batches, is respectively: 0.9; 0.8; 0.7. Determine the probability that the radio tube will work for the given number of hours. | 0.77 | 94 | 4 |
math | 1. Let $O$ be a point inside $\triangle A B C$, and $\overrightarrow{O A}+2 \overrightarrow{O B}$ $+3 \overrightarrow{O C}=\mathbf{0}$, then the ratio of the area of $\triangle A O C$ to the area of $\triangle B O C$ is $\qquad$ | 2:1 | 78 | 3 |
math | 453. Find the sample mean for the given distribution of sample size $n=10$ :
| $x_{i}$ | 1250 | 1270 | 1280 |
| :--- | :---: | :---: | :---: |
| $n_{i}$ | 2 | 5 | 3 | | 1269 | 79 | 4 |
math | ## Task B-4.5.
If a natural number $n$ when divided by 5 gives a remainder of 2, what remainder does $n^{7}$ give when divided by 5? | 3 | 42 | 1 |
math | Problem 9.8. Young entomologist Dima is observing two grasshoppers. He noticed that when a grasshopper starts jumping, it jumps 1 cm, then after a second, 2 cm, then another second, 3 cm, and so on.
Initially, both grasshoppers were in the same place. One of them started jumping, and after a few seconds, the second on... | 10,15,30 | 163 | 8 |
math | This questions comprises two independent parts.
(i) Let \(g:\mathbb{R}\to\mathbb{R}\) be continuous and such that \(g(0)=0\) and \(g(x)g(-x)>0\) for any \(x > 0\). Find all solutions \(f : \mathbb{R}\to\mathbb{R}\) to the functional equation
\[g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}\]
(ii) Find all continuousl... | f(x) = 0 | 212 | 7 |
math | 9. A chemistry student conducted an experiment: from a bottle filled with syrup solution, he poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again and refilled the bottle with water. As a result, the percentage of syrup decreased from 9 to 4 percent. Determine the vol... | 3 | 74 | 1 |
math | 50.2. Solve the equation
$$
\operatorname{tg}^{2} x=\frac{1-\cos x}{1-\sin x}
$$ | x_{1}=\frac{\pi}{2}+2\pik,x_{2}=\frac{\pi}{4}+\pik | 35 | 30 |
math | 10. There is a sequence of numbers +1 and -1 of length $n$. It is known that the sum of every 10 neighbouring numbers in the sequence is 0 and that the sum of every 12 neighbouring numbers in the sequence is not zero.
What is the maximal value of $n$ ? | 15 | 66 | 2 |
math | Subject (4). For each non-empty subset $A=\left\{a_{1}, a_{2}, \ldots, a_{k}\right\}$ of the set $\{1,2, \ldots, 10\}, k=1,2, \ldots, 10$, consider the sum
$$
S(A)=a_{1}-a_{1} a_{2}+a_{1} a_{2} a_{3}-\cdots-(-1)^{k} a_{1} a_{2} \cdots a_{k}
$$
where $a_{1}<a_{2}<\cdots<a_{k}$. Determine the sum of all these sums. | 512 | 153 | 3 |
math | Consider the situation such that $ n$ circles with radius $ r$ are contained between 2 concentric circles with radius $ 1,\ 1\minus{}2r$ without intersecting one anoyher.Let $ r$ vary with being fixed $ n\geq 2.$
(1) Show that $ r$ should be satisfied $ 0<r\leq \frac{\sin \frac{\pi}{n}}{1\plus{}\sin \frac{\pi}{n}}.$... | r = \frac{2}{n + 4} | 134 | 12 |
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