Found problems: 580
2020 Bundeswettbewerb Mathematik, 4
In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number.
Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.
2004 Estonia National Olympiad, 4
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.
2010 Singapore Junior Math Olympiad, 3
Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?
1991 Chile National Olympiad, 5
The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$
2018 Hanoi Open Mathematics Competitions, 9
Each of the thirty squares in the diagram below contains a number $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ of which each number is used exactly three times. The sum of three numbers in three squares on each of the thirteen line segments is equal to $S$. [img]https://cdn.artofproblemsolving.com/attachments/8/0/3e056ebc252aee9ade1f45fd337cc6a2f84302.png[/img]
1992 Czech And Slovak Olympiad IIIA, 3
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$
2011 Argentina National Olympiad, 1
For $k=1,2,\ldots ,2011$ we denote $S_k=\frac{1}{k}+\frac{1}{k+1}+\cdots +\frac{1}{2011}$.
Compute the sum $S_1+S_1^2+S_2^2+\cdots +S_{2011}^2$.
1985 All Soviet Union Mathematical Olympiad, 401
In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$?
j
h i
e f g
a b c d
2007 Thailand Mathematical Olympiad, 14
The sum $$\sum_{k=84}^{8000}{k \choose 84}{{8084 - k} \choose 84}$$
can be written as a binomial coefficient $a \choose b$ for integers $a, b$. Find a possible pair $(a, b)$
1986 Tournament Of Towns, (119) 1
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.
2019 Nigerian Senior MO Round 3, 3
Show that $$5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}$$
2002 Estonia National Olympiad, 5
The teacher writes numbers $1$ at both ends of the blackboard. The first student adds a $2$ in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers $1, 3, 2, 3, 1$ on the blackboard after the second student, $1, 4, 3, 5, 2, 5, 3, 4, 1$ after the third student etc.) Find the sum of all numbers on the blackboard after the $n$-th student.
1985 Greece National Olympiad, 4
Consider function $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=\frac{4^x}{4^x+2},$ for any $x\in \mathbb{R}$
a) Prove that $f(x)+f(1-x)=1,$
b) Claculate the sum $$f\left(\frac{1}{1986} \right)+f\left(\frac{2}{1986} \right)+\cdots f\left(\frac{1986}{1986} \right).$$
2000 Tournament Of Towns, 4
Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon.
(G Galperin)
2003 Abels Math Contest (Norwegian MO), 1b
Let $x_1,x_2,...,x_n$ be real numbers in an interval $[m,M]$ such that $\sum_{i=1}^n x_i = 0$. Show that $\sum_{i=1}^n x_i ^2 \le -nmM$
2019 Durer Math Competition Finals, 11
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
2003 Greece JBMO TST, 2
Calculate if $n\in N$ with $n>2$ the value of
$$B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}} $$
2004 Junior Tuymaada Olympiad, 2
For which natural $ n \geq 3 $ numbers from 1 to $ n $ can be arranged by a circle so that each number does not exceed $60$ % of the sum of its two neighbors?
2022 IFYM, Sozopol, 5
Prove that
$\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.
2005 Thailand Mathematical Olympiad, 16
Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.
2016 Dutch Mathematical Olympiad, 2
For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together.
For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$.
Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have.
[i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]
1979 Yugoslav Team Selection Test, Problem 1
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
2006 Singapore Junior Math Olympiad, 2
The fraction $\frac23$ can be eypressed as a sum of two distinct unit fractions: $\frac12 + \frac16$ .
Show that the fraction $\frac{p-1}{p}$ where $p\ge 5$ is a prime cannot be expressed as a sum of two distinct unit fractions.
2020 Argentina National Olympiad, 5
Determine the highest possible value of:
$$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$
where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$.
Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.
2018 Singapore Junior Math Olympiad, 4
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.