This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 594

2018 Estonia Team Selection Test, 3
Tags: algebra , sum , max , min , inequalities
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

1990 Tournament Of Towns, (265) 3
Tags: number theory , divide , sum
Find $10$ different positive integers such that each of them is a divisor of their sum (S Fomin, Leningrad)

2004 Estonia National Olympiad, 2
Tags: sum , independent , fixed , angle bisector , geometry
Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.

2000 Tournament Of Towns, 3
Tags: least common multiple , lcm , sum , product , divide , divisible , number theory
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

2012 Korea Junior Math Olympiad, 7
Tags: inequalities , algebra , maximum , positive real , sum
If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)

2004 Estonia National Olympiad, 4
Tags: combinatorics , lattice , sum
In the beginning, number $1$ has been written to point $(0,0)$ and $0$ has been written to any other point of integral coordinates. After every second, all numbers are replaced with the sum of the numbers in four neighbouring points at the previous second. Find the sum of numbers in all points of integral coordinates after $n$ seconds.

2014 India PRMO, 12
Tags: sum , right angle , inradius , geometry
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?

2005 BAMO, 1
Tags: number theory , sum , combinatorics
An integer is called [i]formidable[/i] if it can be written as a sum of distinct powers of $4$, and [i]successful [/i] if it can be written as a sum of distinct powers of $6$. Can $2005$ be written as a sum of a [i]formidable [/i] number and a [i]successful [/i] number? Prove your answer.

2001 Tuymaada Olympiad, 2
Tags: divisible , infinite chessboard , sum , number theory
Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?

2020 Durer Math Competition Finals, 6
Tags: sum , algebra
We build a modified version of Pascal’s triangle as follows: in the first row we write a $2$ and a $3$, and in the further rows, every number is the sum of the two numbers directly above it (and rows always begin with a $2$ and end with a $3$). In the $13$th row, what is the $5$th number from the left? [img]https://cdn.artofproblemsolving.com/attachments/7/2/58e1a9f43fa7c304bfd285fc1b73bed883e9a6.png[/img]

2003 Austrian-Polish Competition, 8
Tags: inequalities , algebra , sum
Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$, show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$.

2018 Thailand Mathematical Olympiad, 8
Tags: max , algebra , sum , maximum value
There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$, but the sum of any $n$ ticket numbers is at most $1165$. What is the maximum value of $n$?

2015 Ukraine Team Selection Test, 4
Tags: sum , product , modulo , number theory
A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

2004 Abels Math Contest (Norwegian MO), 1a
Tags: consecutive , sum , number theory
If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.

2017 Purple Comet Problems, 28
Tags: sum , algebra
Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The in finite series $$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$ has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2000 Abels Math Contest (Norwegian MO), 2a
Tags: algebra , sum
Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.

2015 Argentina National Olympiad, 1
Tags: algebra , sum
Express the sum of $99$ terms$$\frac{1\cdot 4}{2\cdot 5}+\frac{2\cdot 7}{5\cdot 8}+\ldots +\frac{k(3k+1 )}{(3k-1)(3k+2)}+\ldots +\frac{99\cdot 298}{296\cdot 299}$$ as an irreducible fraction.

2016 IFYM, Sozopol, 8
Tags: algebra , sum , power
Let $a_i$, $i=1,2,…2016$, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples $x_1,x_2…x_{2016}$ of natural numbers, for which the sum $\sum_{i=1}^{2016}{a_i x_i^i}$ is a 2017-th power of a natural number.

1964 Vietnam National Olympiad, 1
Tags: trigonometry , sum , geometry
Given an arbitrary angle $\alpha$, compute $cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ . Generalize this result and justify your answer.

1986 Polish MO Finals, 3
Tags: prime , divisible , number theory , sum , prime divisor
$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.

1977 Bundeswettbewerb Mathematik, 1
Tags: algebra , number theory , integer , sum , divisible
Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

1935 Moscow Mathematical Olympiad, 018
Tags: sum , sum of cubes , algebra
Evaluate the sum: $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$.

2004 Tournament Of Towns, 1
Tags: arithmetic progression , sum , power of 2 , number theory
The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.

1997 Argentina National Olympiad, 3
Tags: algebra , sum , inequalities
Let $x_1,x_2,x_3,\ldots ,x_{100}$ be one hundred real numbers greater than or equal to $0$ and less than or equal to $1$. Find the maximum possible value of the sum$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$

2017 Czech And Slovak Olympiad III A, 4
Tags: sum , probabilistic method , combinatorics
For each sequence of $n$ zeros and $n$ units, we assign a number that is a number sections of the same digits in it. (For example, sequence $00111001$ has $4$ such sections $00, 111,00, 1$.) For a given $n$ we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to $(n+1){2n \choose n} $