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 | ## 8. Cyclist and Pedestrian
A cyclist and a pedestrian, 1 km apart, start moving towards each other at the same time. The cyclist travels 300 meters per minute, and the pedestrian travels 90 meters per minute. How far apart will they be 1 minute and 48 seconds after they start?
Result: $\quad 298$ | 298 | 81 | 3 |
math | The numbers $p, q, r$, and $t$ satisfy $p<q<r<t$. When these numbers are paired, each pair has a different sum and the four largest sums are 19,22,25, and 28 . What is the the sum of the possible values for $p$ ? | \frac{17}{2} | 68 | 8 |
math | We shuffle a 52-card French deck, then draw cards one by one from the deck until we find a black ace. On which draw is it most likely for the first black ace to appear? | 1 | 41 | 1 |
math | Let $a_0, a_1,\dots, a_{19} \in \mathbb{R}$ and $$P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}.$$ If $P(x)=P(-x)$ for all $x \in \mathbb{R}$, and $$P(k)=k^2,$$ for $k=0, 1, 2, \dots, 9$ then find $$\lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}.$$ | -(9!)^2 + 1 | 138 | 8 |
math | 2. (12 points) Find a natural number $n$ such that the numbers $n+15$ and $n-14$ are squares of other numbers. | 210 | 37 | 3 |
math | A $0,1,4,5,7,9$ digits can form how many 4-digit numbers in which at least one digit is repeated? | 780 | 32 | 3 |
math | Find all sets of real numbers $\{a_1,a_2,\ldots, _{1375}\}$ such that
\[2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},\]
and
\[2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.\] | a_n = n \text{ for all } n \in \{1, 2, \ldots, 1375\} | 122 | 31 |
math | 1. A cyclist covered a certain distance in 1 hour and 24 minutes at a constant speed of $30 \mathrm{~km} / \mathrm{h}$. At what speed did he travel on the return trip if he traveled 12 minutes less? | 35\mathrm{~}/\mathrm{} | 57 | 10 |
math | Example 2.75. Determine the work that needs to be expended to compress an elastic spring by $\Delta h$ units of length. Perform the calculations assuming Hooke's law $F=-k x$ is valid. | -\frac{k\Delta^{2}}{2} | 46 | 11 |
math | [ The ratio in which the bisector divides the side. ] [ Thales' Theorem and the theorem of proportional segments ]
On each side of the rhombus, there is one vertex of a square, the sides of which are parallel to the diagonals of the rhombus.
Find the side of the square if the diagonals of the rhombus are 8 and 12. | 4.8 | 82 | 3 |
math | 17. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of $(1+2 x)^{100}$, where $0 \leq k \leq 100$. Find the number of integers $r: 0 \leq r \leq 99$ such that $a_{r}<a_{r+1}$. | 67 | 84 | 2 |
math | 7. Given the polynomial $P(x)=x^{5}+a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0}$, it is known that $P(2014)=1, P(2015)=2, P(2016)=3, P(2017)=4$, $P(2018)=5$. Find $P(2013)$. | -120 | 112 | 4 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0):$
$f(x)=\left\{\begin{array}{c}\tan\left(x^{3}+x^{2} \sin \left(\frac{2}{x}\right)\right), x \neq 0 \\ 0, x=0\end{array}\right.$ | 0 | 82 | 1 |
math | The tower clock strikes three times in 12 s. How long will it take to strike six times?
# | 30 | 23 | 2 |
math | 5. Given a tetrahedron $ABCD$ such that $AB=\sqrt{3}, AD=$ $BC=\sqrt{10}, AC=CD=BD=\sqrt{7}$. The volume of the tetrahedron is $\qquad$. | \sqrt{2} | 55 | 5 |
math | A sphere of volume $V$ is inscribed in a regular tetrahedron, into which another sphere is inscribed, then a tetrahedron into that sphere, and so on to infinity. First, calculate the sum of the volumes of the spheres, and second, the sum of the volumes of the tetrahedra. | \frac{27}{26}V | 69 | 10 |
math | 9. (10 points) A four-digit number "HaoShiChengShuang" divided by a two-digit number "ChengShuang" has a remainder exactly equal to "HaoShi". If different Chinese characters represent different digits and "HaoShi" and "ChengShuang" are not coprime, then the largest four-digit number "HaoShiChengShuang" is | 7281 | 89 | 4 |
math | 3. For a sequence of real numbers $x_{1}, x_{2}, \cdots, x_{n}$, define its "value" as $\max _{1 \leqslant i \leqslant n}\left\{\left|x_{1}+x_{2}+\cdots+x_{i}\right|\right\}$. Given $n$ real numbers, David and George want to arrange these $n$ numbers into a sequence with low value. On one hand, diligent David examines ... | 2 | 322 | 1 |
math | A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle. | 24 | 42 | 2 |
math | 2. Triangle $A B C$ of area 1 is given. Point $A^{\prime}$ lies on the extension of side $B C$ beyond point $C$ with $B C=C A^{\prime}$. Point $B^{\prime}$ lies on extension of side $C A$ beyond $A$ and $C A=A B^{\prime}$. $C^{\prime}$ lies on extension of $A B$ beyond $B$ with $A B=B C^{\prime}$. Find the area of tria... | 7 | 130 | 1 |
math | Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is [i]ocular[/i] if it starts at $(0, 0)$ and passes through at least one point in $G$. Let $A$ be the set of angle measures of acute angles formed by two distinct ocular ... | \frac{1}{722} | 116 | 9 |
math | Three integers form a geometric progression. If we add 8 to the second number, an arithmetic progression is formed. If we add 64 to the third term of this arithmetic progression, we again get a geometric progression. What are the original numbers? | 4,12,36 | 51 | 7 |
math | 9.167. $\frac{1}{x+1}-\frac{2}{x^{2}-x+1} \leq \frac{1-2 x}{x^{3}+1}$. | x\in(-\infty;-1)\cup(-1;2] | 47 | 16 |
math | 12. The base of a hexagonal prism is a regular hexagon, and its lateral edges are perpendicular to the base. It is known that all vertices of the hexagonal prism lie on the same sphere, and the volume of the hexagonal prism is $\frac{9}{8}$, with the perimeter of the base being 3. Then the volume of this sphere is $\qq... | \frac{4}{3} \pi | 81 | 9 |
math | A number is called [i]6-composite[/i] if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is [i]composite[/i] if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
[i]Ray Li.[/i] | 441 | 76 | 3 |
math | Problem 3. Little kids were eating candies. Each one ate 7 candies less than all the others together, but still more than one candy. How many candies were eaten in total?
[5 points] (A. V. Shapovalov) | 21 | 52 | 2 |
math | ## Task 3 - 260613
The connecting roads of three places $A, B, C$ form a triangle. In the middle of the road from $B$ to $C$ lies another place $D$. The distance from $A$ via $B$ to $C$ is $25 \mathrm{~km}$, from $B$ via $C$ to $A$ is $27 \mathrm{~km}$, and from $C$ via $A$ to $B$ is $28 \mathrm{~km}$.
A pioneer grou... | 19\mathrm{~} | 222 | 7 |
math | ## Task Condition
Write the canonical equations of the line.
$$
\begin{aligned}
& 2 x+3 y-2 z+6=0 \\
& x-3 y+z+3=0
\end{aligned}
$$ | \frac{x+3}{-3}=\frac{y}{-4}=\frac{z}{-9} | 51 | 25 |
math | 2. Given the function $f(x)=a x^{2}-c$, satisfying $-4 \leqslant f(1) \leqslant-1,-1 \leqslant f(2) \leqslant 5, f(3)$'s range of values is $\qquad$ . | -1\leqslantf(3)\leqslant20 | 70 | 17 |
math | 8.121. $\frac{\sin ^{2} x-2}{\sin ^{2} x-4 \cos ^{2} \frac{x}{2}}=\tan ^{2} \frac{x}{2}$. | \frac{\pi}{2}(2k+1),k\inZ | 52 | 16 |
math | 10.1. Kolya wrote a ten-digit number on the board, consisting of different digits. Sasha added one digit so that the resulting number would be divisible by 9. Which digit could Sasha have added? | 0or9 | 45 | 3 |
math | 1.18. The lengths of two sides of a triangle are $a$, and the length of the third side is $b$. Calculate the radius of its circumscribed circle. | \frac{^2}{\sqrt{4a^2-b^2}} | 38 | 17 |
math | 50. How many divisors does the integer
$$
N=a^{\alpha} \cdot b^{\beta} \cdot c^{\gamma} \ldots l^{\lambda}
$$
have, where $a, b, c, \ldots l$ are prime numbers and $\alpha, \beta, \gamma, \ldots \lambda$ are any integers? | (\alpha+1)(\beta+1)(\gamma+1)\cdots(\lambda+1) | 81 | 22 |
math | For what value of $n$ will
$$
2^{11}+2^{8}+2^{n} \text { be a perfect square? }
$$ | 12 | 36 | 2 |
math | 2. Let the real-coefficient quadratic equation $x^{2}+a x+2 b-$ $2=0$ have two distinct real roots, one of which lies in the interval $(0,1)$, and the other in the interval $(1,2)$. Then the range of $\frac{b-4}{a-1}$ is . $\qquad$ | (\frac{1}{2},\frac{3}{2}) | 78 | 14 |
math | Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$. For example, this gives $f(8)=1$, $f(9)=1$, $f(10)=2$, $f(11)=4$, and $f(12)=5$. Determine the value of $f(10^{100})$. | 10^{101} + 1 | 88 | 10 |
math | Three, (25 points) Find all positive integers such that it is 224 times the sum of its digits.
The above text has been translated into English, retaining the original text's line breaks and format. | 2016 | 45 | 4 |
math | 10.21 Write the complex number $z=\sin 36^{\circ}+i \cos 54^{\circ}$ in trigonometric form. | \sqrt{2}\sin36(\cos45+i\sin45) | 37 | 18 |
math | Example 6 Let positive integers $a, b, c, d$ satisfy $\frac{a}{b}+\frac{c}{d} < 1$ and $a+c=20$. Find the maximum value of $\frac{a}{b}+\frac{c}{d}$. | \frac{1385}{1386} | 62 | 13 |
math | 8.4. In the country, there are 15 cities, some of which are connected by roads. Each city is assigned a number equal to the number of roads leading out of it. It turned out that there are no roads between cities with the same number. What is the maximum number of roads that can be in the country? | 85 | 69 | 2 |
math | 3.2.7 * Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=3, a_{2}=8, a_{n+2}=2 a_{n+1}+2 a_{n}, n=1,2, \cdots$. Find the general term $a_{n}$ of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{2+\sqrt{3}}{2\sqrt{3}}(1+\sqrt{3})^{n}+\frac{\sqrt{3}-2}{2\sqrt{3}}(1-\sqrt{3})^{n} | 87 | 55 |
math | 4. (7 points) Two pedestrians set out at dawn. Each walked at a constant speed. One walked from $A$ to $B$, the other from $B$ to $A$. They met at noon (i.e., exactly at 12 o'clock) and, without stopping, arrived: one at $B$ at 4 PM, and the other at $A$ at 9 PM. At what time was dawn that day? | 6 | 93 | 1 |
math | Find the largest positive integer $n$ for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$. | 3 | 74 | 1 |
math | How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3? | 2008 | 36 | 4 |
math | ## Task 4 - 160714
In a cycling race on a circular track of $1 \mathrm{~km}$ in length, at a certain point in time, cyclist $A$ had exactly $500 \mathrm{~m}$ lead over cyclist $B$. $B$ was riding at a speed of $50 \frac{\mathrm{km}}{\mathrm{h}}$, and $A$ at a speed of $45 \frac{\mathrm{km}}{\mathrm{h}}$.
a) After how... | 6 | 211 | 1 |
math | Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and ea... | n = 7 | 90 | 5 |
math | Example 6 Find the number of $n$-digit numbers formed by $1,2,3$, where each of 1, 2, and 3 must appear at least once in the $n$-digit number. | 3^{n}-3\cdot2^{n}+3 | 48 | 13 |
math | 2. Find all rational numbers $r$ for which all solutions of the equation
$$
r x^{2}+(r+1) x+r=1
$$
are integers. | 1 | 39 | 1 |
math | Allowing $x$ to be a real number, what is the largest value that can be obtained by the function $25\sin(4x)-60\cos(4x)?$ | 65 | 40 | 2 |
math | [ Arithmetic operations. Numerical identities ]
In the competition, 50 shooters participated. The first scored 60 points; the second - 80; the third - the arithmetic mean of the points of the first two; the fourth - the arithmetic mean of the points of the first three. Each subsequent shooter scored the arithmetic mea... | 70 | 95 | 2 |
math | 10. Let the function $f:(0,1) \rightarrow \mathbf{R}$ be defined as
$$
f(x)=\left\{\begin{array}{l}
x, \text { when } x \text { is irrational, } \\
\frac{p+1}{q}, \text { when } x=\frac{p}{q},(p, q)=1,0<p<q
\end{array}\right.
$$
Find the maximum value of $f(x)$ on the interval $\left(\frac{7}{8}, \frac{8}{9}\right)$. | \frac{16}{17} | 130 | 9 |
math | 1. Rectangle A has length $6 \mathrm{~cm}$ and area $36 \mathrm{~cm}^{2}$.
Rectangle B has length $12 \mathrm{~cm}$ and area $36 \mathrm{~cm}^{2}$.
Rectangle $\mathrm{C}$ has length $9 \mathrm{~cm}$ and area $36 \mathrm{~cm}^{2}$.
The rectangle with the smallest width has a width of $x \mathrm{~cm}$. What is the value ... | 3 | 116 | 1 |
math | 68. Find two numbers, the ratio of which is 3, and the ratio of the sum of their squares to their sum is 5. | 6,2 | 31 | 3 |
math | A passenger train departs from $A$ to $B$ at 12 noon, traveling at a constant speed of $60 \mathrm{~km} /$ hour. Meanwhile, on the same track, a freight train travels from $B$ to $A$ at a speed of $40 \mathrm{~km} /$ hour. Both trains arrive at their destinations at the same time, and at this moment, they are three tim... | 9^{30} | 117 | 5 |
math | 1 . Find the value of $1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\cdots+$ $n \cdot n!$. | (n+1)!-1 | 38 | 6 |
math | 2. Let $a=\sqrt{x^{2}+x y+y^{2}}, b=p \sqrt{x y}$, $c=x+y$. If for any positive numbers $x$ and $y$, a triangle exists with sides $a$, $b$, and $c$, then the range of the real number $p$ is $\qquad$ | (2-\sqrt{3}, 2+\sqrt{3}) | 73 | 14 |
math | Let's determine the value of the following expression:
$$
100^{2}-99^{2}+98^{2}-97^{2}+\ldots+2^{2}-1^{2}
$$ | 5050 | 47 | 4 |
math | 1. N1 (TWN) ${ }^{1 \mathrm{MO} 04}$ Find all pairs of positive integers $(x, p)$ such that $p$ is a prime, $x \leq 2 p$, and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$. | (1,p),(2,2),(3,3) | 73 | 12 |
math | Example 6 Express $\frac{225}{43}$ as a continued fraction. | [5,4,3,3] | 19 | 9 |
math | Exercise 5. For all integers $n \geqslant 1$, determine all $n$-tuples of real numbers $\left(x_{1}, \ldots, x_{n}\right)$ such that
$$
\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\ldots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+\ldots+x_{n}\right)
$$ | x_{i}=2i^{2} | 107 | 9 |
math | 13.331. Several balls of equal mass for bearings and several piston rings, also of equal mass, were made from a certain grade of metal. If the number expressing the mass of each ball in grams were 2 less than the number of rings made, and the number expressing the mass of each ring in grams were 2 more than the number ... | 25 | 147 | 2 |
math | There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive intege... | 107 | 91 | 3 |
math | Write in the form of a fraction the number
$$
x=0,512341234123412341234123412341234 \ldots
$$ | \frac{51229}{99990} | 54 | 15 |
math | 4・119 Solve the system of equations $\left\{\begin{array}{l}\cos x=2 \cos ^{3} y, \\ \sin x=2 \sin ^{3} y .\end{array}\right.$ | {\begin{pmatrix}2\pi+\frac{k\pi}{2}+\frac{\pi}{4},\\\frac{k\pi}{2}+\frac{\pi}{4}0\end{pmatrix}\quad(,k\in\mathbb{Z}).} | 53 | 59 |
math | 14. If for any non-negative integer $n$, $\cos 2^{n} \alpha<-\frac{1}{3}$ holds, find the real number $\alpha$. | \alpha=2k\pi\\frac{2\pi}{3}(k\in{Z}) | 38 | 22 |
math | ## Task A-4.1.
Given is the sequence $\left(a_{n}\right)$,
$$
a_{1}=1, \quad a_{n}=3 a_{n-1}+2^{n-1}, \text { for } n \geq 2
$$
Express the general term of the sequence $a_{n}$ in terms of $n$. | a_{n}=3^{n}-2^{n} | 81 | 12 |
math | 52. One fountain fills the pool in 2.5 hours, the other in 3.75 hours. How long will it take for both fountains to fill the pool? | 1.5 | 39 | 3 |
math | 27.15. How many digits does the number $2^{100}$ have?
## 27.3. Identities for logarithms | 31 | 33 | 2 |
math | 18th Putnam 1958 Problem A6 Assume that the interest rate is r, so that capital of k becomes k(1 + r) n after n years. How much do we need to invest to be able to withdraw 1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4 and so on (in perpetuity)? | \frac{(1+r)(2+r)}{r^3} | 98 | 14 |
math |
03.4. Let $\mathbb{R}^{*}=\mathbb{R} \backslash\{0\}$ be the set of non-zero real numbers. Find all functions $f: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}$ satisfying
$$
f(x)+f(y)=f(x y f(x+y))
$$
for $x, y \in \mathbb{R}^{*}$ and $x+y \neq 0$.
| f(x)=\frac{1}{x} | 111 | 10 |
math | 1015*. Find a three-digit number such that its product by 6 is a three-digit number with the same sum of digits. List all such solutions. | 117,135 | 34 | 7 |
math | 17th Chinese 2002 Problem A3 18 people play in a tournament of 17 rounds. There are 9 games in each round and each person plays in one game in each round. Each person plays every other person just once in the tournament. What is the largest n such that however the tournament is arranged we can find 4 people amongst who... | 7 | 90 | 1 |
math | # 7.1. (7 points)
Find the value of the expression
$$
\left(1+\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \ldots\left(1+\frac{1}{2 m}\right)\left(1-\frac{1}{2 m+1}\right)
$$ | 1 | 101 | 1 |
math | 10. Let the natural number $n \geqslant 3$, and the real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy $x_{1}+x_{2}+\cdots+x_{n}=n, x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=$ $n^{2}$,
find the minimum value of $\sum_{i=1}^{n} x_{i}^{3}$ and the $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ when the minimum value is achieved. | -n^3+6n^2-4n | 141 | 11 |
math | 27. (POL 2) Determine the maximum value of the sum
$$
\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)
$$
over all $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$, satisfying $x_{i} \geq 0$ and $\sum_{i=1}^{n} x_{i}=1$. | \frac{1}{4} | 100 | 7 |
math | 72. Given three integers $x, y, z$ satisfying $x+y+z=100$, and $x<y<2z$, then the minimum value of $z$ is | 21 | 40 | 2 |
math | 812. Solve in integers $x, y$ and $z$ the equation
$$
5 x^{2} + y^{2} + 3 z^{2} - 2 y z = 30
$$ | (1;5;0),(1;-5;0),(-1;5;0),(-1;-5;0) | 49 | 27 |
math | Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour. | n = 17 | 56 | 6 |
math | Given $\lambda_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, n)$. Try to find all positive real solutions $x_{1}, x_{2}, \cdots, x_{n}$ of the equation
$$
\sum_{i=1}^{n} \frac{\left(x_{i}+\lambda_{i}\right)^{2}}{x_{i+1}}=4 \sum_{i=1}^{n} \lambda_{i}
$$
(Here $x_{n+1}=x_{1}$ ).
(Xiao Zhengang) | x_{k}=\frac{1}{2^{n}-1}\sum_{i=1}^{n}2^{i-1}\lambda_{k-1+i},k=1,2,\cdots,n | 132 | 46 |
math | 23. Find all non-zero integer triples $\{a, b, c\}$, satisfying the conditions:
$$a \equiv b(\bmod |c|), \quad b \equiv c(\bmod |a|), \quad c \equiv a(\bmod |b|)$$ | \{1,1, c\}, \{-1,-1, c\}, \{1,-n, n+1\}, \{2,-(2 n+1), 2 n+3\}, \{-1,1,2\}, \{-1,2,3\}, \{1,-1,1\}, \{-1,1,1\} | 62 | 82 |
math | Leibniz Gottfried Wilhelm
Consider a numerical triangle:
$$
\begin{array}{cccccc}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \ldots & \frac{1}{1993} \\
\frac{1}{2} & \frac{1}{6} & \frac{1}{12} & \ldots & \frac{1}{1992 \cdot 1993} \\
\frac{1}{3} & \frac{1}{12} & \ldots &
\end{array}
$$
(The first row is given, and... | \frac{1}{1993} | 182 | 10 |
math | A.Y. Evnin
2000 people registered on a new website. Each of them invited 1000 people to be their friends. Two people are considered friends if and only if each of them invited the other to be a friend. What is the minimum number of pairs of friends that could have formed? | 1000 | 66 | 4 |
math | What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$?
[i]2016 CCA Math Bonanza Lightning #2.4[/i] | 120 | 47 | 3 |
math | $4 \cdot 78$ Solve the equation $3^{x^{2}+1}+3^{x^{2}-1}-270=0$. | \2 | 36 | 2 |
math | The four-digit base ten number $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$ has all nonzero digits and is a multiple of $99$. Additionally, the two-digit base ten number $\underline{a}\;\underline{b}$ is a divisor of $150$, and the two-digit base ten number $\underline{c}\;\underline{d}$ is a divisor of... | 108 | 151 | 3 |
math | Two regular hexagons of side length $2$ are laid on top of each other such that they share the same center point and one hexagon is rotated $30^\circ$ about the center from the other. Compute the area of the union of the two hexagons. | 48\sqrt{3} - 72 | 58 | 11 |
math | 16. Line $a$ is parallel to line $b$, and there are 5 points and 6 points on lines $a$ and $b$ respectively. Using these points as vertices, $\qquad$ different triangles can be drawn. | 135 | 51 | 3 |
math | Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$? | 141 | 41 | 3 |
math | H2. How many nine-digit integers of the form 'pqrpqrpqr' are multiples of 24 ? (Note that $p, q$ and $r$ need not be different.) | 112 | 42 | 3 |
math | 7. The least common multiple of natural numbers $a$ and $b$ is 140, and their greatest common divisor is 5. Then the maximum value of $a+b$ is $\qquad$ | 145 | 45 | 3 |
math | 3.14 A positive integer is thought of. To its representation, the digit 7 is appended on the right, and from the resulting new number, the square of the thought number is subtracted. The remainder is then reduced by $75\%$ of this remainder, and the thought number is subtracted again. In the final result, zero is obtai... | 7 | 90 | 1 |
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 | 10.2. Find all pairs of natural numbers $x$ and $y$ such that their least common multiple is equal to $1 + 2x + 3y$. | 4,9or10,3 | 38 | 8 |
math | 4・140 Solve the system of equations
$$\left\{\begin{array}{l}
x y z=x+y+z, \\
y z t=y+z+t, \\
z t x=z+t+x, \\
t x y=t+x+y .
\end{array}\right.$$ | (0,0,0,0),(\sqrt{3}, \sqrt{3}, \sqrt{3}, \sqrt{3}),(-\sqrt{3},-\sqrt{3},-\sqrt{3},-\sqrt{3}) | 60 | 50 |
math | 6. If $2n+1, 20n+1 \left(n \in \mathbf{N}_{+}\right)$ are powers of the same positive integer, then all possible values of $n$ are | 4 | 47 | 1 |
math | 8. Let \( a_{k}=\frac{2^{k}}{3^{2^{k}}+1}, k \) be a natural number, and let \( A=a_{0}+a_{1}+\cdots+a_{9}, B=a_{0} a_{1} \cdots a_{9} \), then \( \frac{A}{B}=\) \qquad . | \frac{3^{2^{10}}-2^{11}-1}{2^{47}} | 86 | 23 |
math | $1 \cdot 5 n$ is a fixed positive integer, find the sum of all positive integers that have the following property: in binary, this number has exactly $2 n$ digits, of which $n$ are 1s and $n$ are 0s (the leading digit cannot be $0$). | 2^{2n-1}\timesC_{2n-1}^{n}+(2^{2n-1}-1)\timesC_{2n-2}^{n} | 67 | 39 |
math | ## Task 3 - 130513
The triple of the sum of the numbers 38947 and 12711 is to be divided by the sextuple of the difference of the numbers 9127 and 8004.
What is the quotient? | 23 | 65 | 2 |
math | 11. (10 points) Several people completed the task of planting 2013 trees, with each person planting the same number of trees. If 5 people did not participate in planting, the rest would need to plant 2 more trees each to fail to complete the task, while planting 3 more trees each would exceed the task. How many people ... | 61 | 79 | 2 |
math | Example 5 In an exam with a full score of 100, $A, B, C, D, E$ five people participated in this exam. $A$ said: “I got 94 points.” $B$ said: “I scored the highest among the five of us.” $C$ said: “My score is the average of $A$ and $D$.” $D$ said: “My score is exactly the average of the five of us.” $E$ said: “I scored... | 94, 98, 95, 96, 97 | 162 | 18 |
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