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 | 1.94 Find all quadruples of positive integers $(a, b, c, d)$ such that the product of any three of them, when divided by the remaining one, leaves a remainder of 1.
(China National Team Selection Test, 1994) | (2,3,7,41)or(2,3,11,13) | 57 | 22 |
math | 3. Given the equation about $x$: $x^{3}+(1-a) x^{2}-2 a x+a^{2}$ $=0$ has only one real root. Then the range of the real number $a$ is $\qquad$. | a<-\frac{1}{4} | 54 | 9 |
math | A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at... | 25 | 140 | 2 |
math | $$
\begin{array}{l}
\text { 11. If } a+b-2 \sqrt{a-1}-4 \sqrt{b-2} \\
=3 \sqrt{c-3}-\frac{1}{2} c-5 \text {, }
\end{array}
$$
then $a+b+c=$ . $\qquad$ | 20 | 80 | 2 |
math | Problem 2. Determine for which values of $n$ there exists a convex polygon with $n$ sides whose interior angles, expressed in degrees, are all integers, are in arithmetic progression, and are not all equal. | n\in{3,4,5,6,8,9,10,12,15,16,18} | 45 | 31 |
math | ## Task 1 - 210721
a) A rectangular plot of land is divided into two rectangular fields by a path. The length of the plot, measured parallel to this path, is $105 \mathrm{~m}$. The width of the first part of the field is $270 \mathrm{~m}$, and the width of the second part of the field is $180 \mathrm{~m}$. The path is... | 2254 | 237 | 4 |
math | 10.231. A circle of radius $r$ is inscribed in a rectangular trapezoid. Find the sides of the trapezoid if its smaller base is equal to $4 r / 3$. | 4r,\frac{10r}{3},2r | 48 | 13 |
math | 310 four-digit number 2011 can be decomposed into the sum of squares of 14 positive integers, among which, 13 numbers form an arithmetic sequence. Write down this decomposition. | \begin{array}{l}
2011=16^{2}+\sum_{k=5}^{17} k^{2}, \\
2011=23^{2}+\sum_{k=4}^{16} k^{2}
\end{array} | 44 | 63 |
math | $\left[\begin{array}{ll}{[\text { Decimal numeral system }}\end{array}\right]$
From $A$ to $B 999$ km. Along the road, there are kilometer markers indicating the distances to $A$ and to $B$:
$0|999,1| 998, \ldots, 999 \mid 0$.
How many of them have only two different digits? | 40 | 98 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\cos \left(\frac{x}{\pi}\right)\right)^{1+x}$ | 1 | 41 | 1 |
math | [ Pythagorean Theorem (direct and inverse) $]$ [ Circle, sector, segment, etc. $\quad]$
In the circular sector $OAB$, the central angle of which is $45^{\circ}$, a rectangle $KMPT$ is inscribed. The side $KM$ of the rectangle lies on the radius $OA$, the vertex $P$ is on the arc $AB$, and the vertex $T$ is on the radi... | 3\sqrt{13} | 132 | 7 |
math | Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ . | 135^\circ | 68 | 5 |
math | 10.8. On a circle of length 2013, 2013 points are marked, dividing it into equal arcs. A chip is placed at each marked point. We define the distance between two points as the length of the shorter arc between them. For what largest $n$ can the chips be rearranged so that there is again one chip at each marked point, an... | 670 | 108 | 3 |
math | ## Task 1 - 240731
During the Peace Race, the following race situation occurred on one stage:
Exactly 14 riders, none of whom were from the GDR, had fallen behind the main field. Exactly $90 \%$ of the riders who had not fallen behind formed the main field; some, but not all, of the GDR riders were among them.
The r... | Intheleadinggroup,therewereexactly2GDRdrivers,1Czechoslovakdriver,4Sovietdrivers | 165 | 27 |
math | \section*{Problem 17}
\(\mathrm{S}\) is a set of integers. Its smallest element is 1 and its largest element is 100. Every element of S except 1 is the sum of two distinct members of the set or double a member of the set. What is the smallest possible number of integers in \(\mathrm{S}\) ?
\section*{Answer}
| 9 | 86 | 1 |
math | 4. Let $0<\theta<\pi$, then the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$ is
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{4\sqrt{3}}{9} | 59 | 12 |
math | G1.1 Find the value of $\sin ^{2} 1^{\circ}+\sin ^{2} 2^{\circ}+\ldots+\sin ^{2} 89^{\circ}$. | 44.5 | 49 | 4 |
math | 4. For a quadruple of points $A, B, C, D$ in the plane, no three of which are collinear, let $f(A, B, C, D)$ denote the measure of the largest angle formed by these points (out of a total of 12 such angles). Determine $\min f(A, B, C, D)$, where the minimum is taken over all such quadruples of points. | 90 | 89 | 2 |
math | a) Which of the numbers is larger: $2^{100}+3^{100}$ or $4^{100}$?
b) Let $x$ and $y$ be natural numbers such that
$$
2^{x} \cdot 3^{y}=\left(24^{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{60}}\right) \cdot\left(24^{\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{60}}\right)^{2} \cdot\left(24^{\frac{1}{4... | 3540 | 233 | 4 |
math | 29th IMO 1988 shortlist Problem 20 Find the smallest n such that if {1, 2, ... , n} is divided into two disjoint subsets then we can always find three distinct numbers a, b, c in the same subset with ab = c. | 96 | 60 | 2 |
math | ## 23. Division
Division of integers is performed. If the dividend is increased by 65 and the divisor is increased by 5, both the quotient and the remainder remain unchanged. What is this quotient? | 13 | 45 | 2 |
math | 1. How many natural numbers between 1000 and 3000 can be formed from the digits $1,2,3,4,5$ if:
a) repetition of digits is allowed;
b) repetition of digits is not allowed? | 250 | 54 | 3 |
math | Let's calculate the sum
$$
\sum_{j=0}^{n}\binom{2 n}{2 j}(-3)^{j}
$$ | 2^{2n}\cdot\cos(n\cdot\frac{2\pi}{3}) | 34 | 20 |
math | A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n... | \frac{k}{m} | 133 | 7 |
math | 7. In $\triangle A B C$, the three interior angles $A, B, C$ satisfy: $A=3 B=9 C$, then $\cos A \cos B+\cos B \cos C+\cos C \cos A=$ | -\frac{1}{4} | 51 | 7 |
math | 2. Find the value of the expression
$$
2^{2}+4^{2}+6^{2}+\ldots+2018^{2}+2020^{2}-1^{2}-3^{2}-5^{2}-\ldots-2017^{2}-2019^{2}
$$ | 2041210 | 75 | 7 |
math | 2.154. What is the value of $\sqrt{25-x^{2}}+\sqrt{15-x^{2}}$, given that the difference $\sqrt{25-x^{2}}-\sqrt{15-x^{2}}=2$ (the value of $x$ does not need to be found)? | 5 | 69 | 1 |
math | B1. Simplify the expression: $\frac{2-x}{x^{3}-x^{2}-x+1}:\left(\frac{1}{x-1} \cdot \frac{x}{x+1}-\frac{2}{x+1}\right)$. | \frac{1}{x-1} | 59 | 9 |
math | ## Task 1 - 050811
A student had attached the following overview of participation in the 1st stage of the 4th Olympiad of Young Mathematicians of the GDR to the wall newspaper:
Class 8a: Out of 33 students, 20 participated, which is approximately 60.6 percent.
Class 8b: Out of 32 students, 21 participated, which is ... | 0.31 | 216 | 4 |
math | Example 1. Find $\lim _{x \rightarrow 2}\left(4 x^{2}-6 x+3\right)$. | 7 | 30 | 1 |
math | 31. In $\triangle A B C$ (see below), $A B=A C=\sqrt{3}$ and $D$ is a point on $B C$ such that $A D=1$. Find the value of $B D \cdot D C$. | 2 | 56 | 1 |
math | 1. (BEL) ${ }^{\mathrm{IMO} 4}$ (a) For which values of $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? (b) For which values of $n>2$ is there a unique set having the stated property? | n \geq 4 \text{ and } n=4 | 87 | 14 |
math | Someone says that seven times their birth year, when divided by 13, leaves a remainder of 11, and thirteen times their birth year, when divided by 11, leaves a remainder of 7. In which year of their life will this person be in 1954? | 1868 | 62 | 4 |
math | 11. If the ellipse $x^{2}+4(y-a)^{2}=4$ intersects the parabola $x^{2}=2 y$, then the range of the real number $a$ is $\qquad$ . | [-1,\frac{17}{8}] | 50 | 10 |
math | 6. If the function $y=3 \sin x-4 \cos x$ attains its maximum value at $x_{0}$, then the value of $\tan x_{0}$ is $\qquad$ | -\frac{3}{4} | 45 | 7 |
math | There is a $40\%$ chance of rain on Saturday and a $30\%$ chance 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... | 107 | 92 | 3 |
math | 3. Given the seven-variable polynomial
$$
\begin{array}{l}
Q\left(x_{1}, x_{2}, \cdots, x_{7}\right) \\
=\left(x_{1}+x_{2}+\cdots+x_{7}\right)^{2}+2\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{7}^{2}\right)
\end{array}
$$
it can be expressed as the sum of seven squares of polynomials with non-negative integer coefficients, i.e... | 3 | 302 | 1 |
math | 14th USAMO 1985 Problem 5 0 < a 1 ≤ a 2 ≤ a 3 ≤ ... is an unbounded sequence of integers. Let b n = m if a m is the first member of the sequence to equal or exceed n. Given that a 19 = 85, what is the maximum possible value of a 1 + a 2 + ... + a 19 + b 1 + b 2 + ... + b 85 ? | 1700 | 107 | 4 |
math | 11. Given a convex $n$-sided polygon where the degrees of the $n$ interior angles are all integers and distinct, and the largest interior angle is three times the smallest interior angle, the maximum value that $n$ can take is $\qquad$ . | 20 | 57 | 2 |
math | ## Task 3 - 320713
A water tank is to be filled through two pipes. To fill it using only the first pipe would take 3 hours, and to fill it using only the second pipe would take 2 hours.
In how many minutes will the tank be full if both pipes are used simultaneously? | 72 | 69 | 2 |
math | 6. We have an $8 \times 8$ board. An inner edge is an edge between two $1 \times 1$ fields. We cut the board into $1 \times 2$ dominoes. For an inner edge $k$, $N(k)$ denotes the number of ways to cut the board such that the cut goes along the edge $k$. Calculate the last digit of the sum we get when we add all $N(k)$,... | 0 | 105 | 1 |
math | 4. Little One gave a big box of candies to Karlson. Karlson ate all the candies in three days. On the first day, he ate 0.2 of the entire box and 16 more candies. On the second day, he ate -0.3 of the remainder and 20 more candies. On the third day, he ate -0.75 of the remainder and the last 30 candies. How many candie... | 270 | 98 | 3 |
math | $A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$, then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each.
What is the least va... | 28 | 155 | 2 |
math | 15. [9] A cat is going up a stairwell with ten stairs. However, instead of walking up the stairs one at a time, the cat jumps, going either two or three stairs up at each step (though if necessary, it will just walk the last step). How many different ways can the cat go from the bottom to the top? | 12 | 73 | 2 |
math | For all positive integers $m>10^{2022}$, determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$. | 10 | 47 | 2 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{-1 ; 7 ;-4\}$
$p=\{-1 ; 2 ; 1\}$
$q=\{2 ; 0 ; 3\}$
$r=\{1 ; 1 ;-1\}$ | 2p-q+3r | 73 | 6 |
math | 60.2. If $a$ and $b$ are positive real numbers, what is the minimum value of the expression
$$
\sqrt{a+b}\left(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\right) ?
$$ | 2\sqrt{2} | 61 | 6 |
math | 2. Determine the cardinality of the set $A=\left\{x_{n} \in \mathbb{Q} \left\lvert\, x_{n}=\frac{n^{2}+2}{n^{2}-n+2}\right., n \in \mathbb{N}, 1 \leq n \leq 2015\right\}$. | 2014 | 85 | 4 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 3}\left(\frac{\sin x}{\sin 3}\right)^{\frac{1}{x-3}}$ | e^{\cot3} | 45 | 6 |
math | II. (16 points) Find all natural numbers $n$ such that $2^{8}+2^{11}+2^{n}$ is a perfect square. | 12 | 37 | 2 |
math | 2. Non-zero numbers $a, b$ and $c$ are such that the doubled roots of the quadratic polynomial $x^{2}+a x+b$ are the roots of the polynomial $x^{2}+b x+c$. What can the ratio $a / c$ be? | \frac{1}{8} | 61 | 7 |
math | 8.4. Find the natural number $x$ that satisfies the equation
$$
x^{3}=2011^{2}+2011 \cdot 2012+2012^{2}+2011^{3}
$$ | 2012 | 58 | 4 |
math | 34. The digits of the number 123456789 can be rearranged to form a number that is divisible by 11. For example, 123475869, 459267831 and 987453126 . How many such numbers are there? | 31680 | 77 | 5 |
math | Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of... | 9 | 102 | 1 |
math | $2 \cdot 62$ Given 1990 piles of stones, each consisting of $1, 2, \cdots, 1990$ stones, in each round, you are allowed to pick any number of piles and remove the same number of stones from these piles. How many rounds are needed at minimum to remove all the stones? | 11 | 76 | 2 |
math | 12. Given the parabola $y^{2}=2 p x(p>0)$, with its focus at $F$, a line passing through $F$ with an inclination angle of $\theta$ intersects the parabola at points $A$ and $B$. The maximum area of $\triangle A B O$ is $\qquad$ (where $O$ is the origin). | \frac{p^{2}}{2} | 81 | 10 |
math | 15. (9) Find $q$, for which $x^{2}+x+q=0$ has two distinct real roots satisfying the relation $x_{1}^{4}+2 x_{1} x_{2}^{2}-x_{2}=19$. | -3 | 60 | 2 |
math | Consider the collection of all 5-digit numbers whose sum of digits is 43. One of these numbers is chosen at random. What is the probability that it is a multiple of 11?
# | \frac{1}{5} | 42 | 7 |
math | M3. Three positive integers have sum 25 and product 360 . Find all possible triples of these integers. | 4,6,153,10,12 | 26 | 13 |
math | Starting with a positive integer $n$, a sequence is created satisfying the following rule: each term is obtained from the previous one by subtracting the largest perfect square that is less than or equal to the previous term, until reaching the number zero. For example, if $n=142$, we will have the following sequence o... | 167 | 189 | 3 |
math | 3. [30] Find the number of ordered pairs $(A, B)$ such that the following conditions hold:
- $A$ and $B$ are disjoint subsets of $\{1,2, \ldots, 50\}$.
- $|A|=|B|=25$
- The median of $B$ is 1 more than the median of $A$. | \binom{24}{12}^2 | 81 | 12 |
math | 4 Let $X=\{00,01, \cdots, 98,99\}$ be the set of 100 two-digit numbers, and $A$ be a subset of $X$ such that: in any infinite sequence of digits from 0 to 9, there are two adjacent digits that form a two-digit number in $A$. Find the minimum value of $|A|$. (52nd Moscow Mathematical Olympiad) | 55 | 97 | 2 |
math | [ Generating functions Special polynomials (p) Find the generating functions of the Fibonacci polynomial sequence $F(x, z)=F_{0}(x)+F_{1}(x) z+F_{2}(x) z^{2}$ $+\ldots+F_{n}(x) z^{n}+\ldots$
and the Lucas polynomial sequence $L(x, z)=L_{0}(x)+L_{1}(x) z+L_{2}(x) z^{2}+\ldots+L_{n}(x) z^{n}+\ldots$
Definitions of Fibo... | F(x,z)=z(1-xz-z^{2})^{-1},L(x,z)=(2-xz)(1-xz-z^{2})^{-1} | 134 | 34 |
math | Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers. | 2, 3, 5 | 57 | 7 |
math | 2. [7] Suppose that $a, b, c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of
$$
20\left(a^{2}+b^{2}+c^{2}+d^{2}\right)-\sum_{\text {sym }} a^{3} b,
$$
where the sum is over all 12 symmetric terms. | 112 | 93 | 3 |
math | We denote by $\mathbb{R}_{>0}$ the set of strictly positive real numbers. Find all functions $f: \mathbb{R}_{>0} \mapsto \mathbb{R}_{>0}$ such that
$$
f(x+f(x y))+y=f(x) f(y)+1
$$
for all strictly positive real numbers $x$ and $y$. | f: t \mapsto t + 1 | 82 | 10 |
math | 1. Given are fifty natural numbers, of which half do not exceed 50, and the other half are greater than 50 but less than 100. The difference between any two of the given numbers is not 0 or 50. Find the sum of these numbers. | 2525 | 61 | 4 |
math |
For every integer $n \ge 2$ let $B_n$ denote the set of all binary $n$-nuples of zeroes and ones, and split $B_n$ into equivalence classes by letting two $n$-nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers $n \ge 2$... | n = 2 | 112 | 5 |
math | An isosceles triangle has a perimeter of $468 \mathrm{~cm}$. The height corresponding to the leg is in the ratio of $10: 13$ to the height corresponding to the base. What are the lengths of the sides of the triangle? | =130\mathrm{~},b=169\mathrm{~} | 59 | 19 |
math | a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$.
What is the minimal possible value of the sum of all products of couples of those numbers.
b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$.
What is the minimal possible value of the sum of all products of couples of those nu... | -\frac{n}{2} | 97 | 8 |
math | Let's determine the smallest positive integer \( a \) such that \( 47^n + a \cdot 15^n \) is divisible by 1984 for all odd \( n \). | 1055 | 43 | 4 |
math | Example 7 Given $x \in\left(0, \frac{\pi}{2}\right)$, try to find the maximum value of the function $f(x)=3 \cos x + 4 \sqrt{1+\sin ^{2} x}$, and the corresponding $x$ value. | 5 \sqrt{2} | 64 | 6 |
math | 14. Person A departs from location A to location B, while persons B and C depart from location B to location A. After person A has traveled 50 kilometers, persons B and C start from B simultaneously. As a result, person A meets person B at location C, and person A meets person C at location D. It is known that person A... | 130 | 129 | 3 |
math | 4. Dad is preparing gifts. He distributed 115 candies into bags, with each bag containing a different number of candies. In the three smallest gifts, there are 20 candies, and in the three largest gifts, there are 50 candies. How many bags are the candies distributed into? How many candies are in the smallest gift? | 10 | 72 | 2 |
math | [Mutual Position of Two Circles]
What is the mutual position of two circles if:
a) the distance between the centers is 10, and the radii are 8 and 2;
b) the distance between the centers is 4, and the radii are 11 and 17;
c) the distance between the centers is 12, and the radii are 5 and 3? | 2 | 89 | 1 |
math | Determine all positive integers $n$ with the property that $n = (d(n))^2$. Here $d(n)$ denotes the number of positive divisors of $n$. | 1 \text{ and } 9 | 38 | 8 |
math | 12 (14 points) Given the function $f(x)=-2 x+4$, let
$$
S_{n}=f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f(1)\left(n \in \mathbf{N}^{*}\right) \text {, }
$$
If the inequality $\frac{a^{n}}{S_{n}}<\frac{a^{n+1}}{S_{n+1}}$ always holds, find the range of real number $a$. | (\frac{5}{2},+\infty) | 139 | 11 |
math | Find the value of the expression $\sqrt{1+2011^{2}+\left(\frac{2011}{2012}\right)^{2}}+\frac{2011}{2012}$. | 2012 | 52 | 4 |
math | If for the real numbers $x, y, z, k$ the following conditions are valid, $x \neq y \neq z \neq x$ and $x^{3}+y^{3}+k\left(x^{2}+y^{2}\right)=y^{3}+z^{3}+k\left(y^{2}+z^{2}\right)=z^{3}+x^{3}+k\left(z^{2}+x^{2}\right)=2008$, find the product $x y z$. | x y z=1004 | 122 | 8 |
math | 1. Let the polynomial $f(x)$ satisfy: for any $x \in \mathbf{R}$, we have
$$
f(x+1)+f(x-1)=2 x^{2}-4 x .
$$
Then the minimum value of $f(x)$ is $\qquad$ | -2 | 63 | 2 |
math | Example 8. Find $\int\left(x^{2}+2\right)(\sqrt{x}-3) d x$ | \frac{2}{7}\sqrt{x^{7}}-x^{3}+\frac{4}{3}\sqrt{x^{3}}-6x+C | 27 | 33 |
math | 10. (3 points) The sum of all natural numbers between 1 and 200 that are not multiples of 2 or 5 is | 8000 | 32 | 4 |
math | 8. (10 points) Among the three given phrases “尽心尽力”, “可拔山”, and “山穷水尽”, each Chinese character represents a number between 1 and 8, with the same character representing the same number and different characters representing different numbers. If the sum of the numbers represented by the characters in each phrase is 19, ... | 7 | 102 | 1 |
math | Solve the system of equations
$$
\begin{gathered}
\frac{\sqrt{x+z}+\sqrt{x+y}}{\sqrt{y+z}}+\frac{\sqrt{y+z}+\sqrt{x+y}}{\sqrt{x+z}}=14-4 \sqrt{x+z}-4 \sqrt{y+z} \\
\sqrt{x+z}+\sqrt{x+y}+\sqrt{z+y}=4
\end{gathered}
$$ | 2,2,-1 | 94 | 5 |
math | Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$.
Croatia | \frac{(n+1)(n+3)}{8} | 96 | 14 |
math | G3.3 If $f(n)=a^{n}+b^{n}$, where $n$ is a positive integer and $f(3)=[f(1)]^{3}+f(1)$, find the value of $a \cdot b$. | -\frac{1}{3} | 57 | 7 |
math | 377. It is known that the number $\boldsymbol{e}$ (the base of natural logarithms) is defined as the limit of the expression $\left(1+\frac{1}{n}\right)^{n}$ as $n$ increases without bound:
$$
e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}
$$
But one might reason as follows: if the exponent of a base sl... | 2.71828\ldots | 212 | 10 |
math | 2. (6 points) The agricultural proverb 'counting the nines in winter' refers to the practice of dividing every nine days into a segment starting from the Winter Solstice, sequentially called the first nine, second nine, ... ninth nine, with the Winter Solstice being the first day of the first nine. December 21, 2012, w... | 6,7 | 109 | 3 |
math | Find all $f: \mathbb{N}^{*} \longrightarrow \mathbb{N}^{*}$ such that $\forall m, n$ :
$$
f\left(f(m)^{2}+2 f(n)^{2}\right)=m^{2}+2 n^{2}
$$ | f(n)=n | 66 | 4 |
math | 24. L. N. Tolstoy's Problem. Five brothers divided their father's inheritance equally. The inheritance included three houses. Since the three houses could not be divided into 5 parts, the three older brothers took them, and in return, the younger brothers were given money. Each of the three brothers paid 800 rubles, an... | 2000 | 99 | 4 |
math | 8.3. Find the prime solutions of the equation $\left[\frac{p}{2}\right]+\left[\frac{p}{3}\right]+\left[\frac{p}{6}\right]=q$ (where $[x]$ denotes the greatest integer not exceeding $x$). | p = 3, q = 2; p = 5, q = 3 | 60 | 19 |
math | Let $k$ and $n$ be integers such that $k \geq 2$ and $k \leq n \leq 2 k-1$. Place rectangular tiles, each of size $1 \times k$ or $k \times 1$, on an $n \times n$ chessboard so that each tile covers exactly $k$ cells, and no two tiles overlap. Do this until
no further tile can be placed in this way. For each such $k$ a... | \min (n, 2 n-2 k+2) | 121 | 14 |
math | 1. Given the sum of $n$ positive integers is 2017. Then the maximum value of the product of these $n$ positive integers is $\qquad$ | 2^{2}\times3^{671} | 37 | 11 |
math | 1. Let the circumcenter of $\triangle A B C$ be $O$, and
$$
3 \overrightarrow{O A}+4 \overrightarrow{O B}+5 \overrightarrow{O C}=\mathbf{0} \text {. }
$$
Then the size of $\angle C$ is $\qquad$ | 45 | 72 | 2 |
math | 9. In an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference not equal to 0, $a_{4}=10$, and $a_{3}, a_{6}, a_{10}$ form a geometric sequence, then the general term formula of the sequence $\left\{a_{n}\right\}$ is $\qquad$. | a_{n}=n+6 | 79 | 7 |
math | 3. Three mryak are more expensive than five bryak by 10 rubles. And six mryak are more expensive than eight bryak by 31 rubles. By how many rubles are seven mryak more expensive than nine bryak | 38 | 58 | 2 |
math | Problem 4. There are 10 identical swimming pools and two hoses with different pressures. It is known that the first hose fills a pool 5 times faster than the second. Petya and Vasya each started filling 5 pools, Petya with the first hose, and Vasya with the second. It is known that Petya finished an hour earlier. How l... | 1 | 104 | 1 |
math | For which values of $ k$ and $ d$ has the system $ x^3\plus{}y^3\equal{}2$ and $ y\equal{}kx\plus{}d$ no real solutions $ (x,y)$? | k = -1 | 51 | 5 |
math |
99.2. Consider 7-gons inscribed in a circle such that all sides of the 7-gon are of different length. Determine the maximal number of $120^{\circ}$ angles in this kind of a 7-gon.
| 2 | 55 | 1 |
math | [b]Q6.[/b] At $3:00$ AM, the temperature was $13^o$ below zero. By none it has risen to $32^o$. What is the average hourly increase in temperature ? | 5 \text{ deg/hr} | 50 | 7 |
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