Found problems: 580
2006 Dutch Mathematical Olympiad, 3
$1+2+3+4+5+6=6+7+8$.
What is the smallest number $k$ greater than $6$ for which:
$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?
1954 Moscow Mathematical Olympiad, 281
*. Positive numbers $x_1, x_2, ..., x_{100}$ satisfy the system $$\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\
x_1 + x_2 + ...+ x_{100} < 300 \end{cases}$$
Prove that among these numbers there are three whose sum is greater than $100$.
1982 Polish MO Finals, 5
Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.
2004 Switzerland Team Selection Test, 1
Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$
2015 Dutch IMO TST, 3
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
2017 Irish Math Olympiad, 5
Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies
$$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$.
(a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero.
(b) Find a sequence none of whose terms are zero but which is $2017$-powerful.
2019 Saudi Arabia JBMO TST, 2
Let $a, b, c$ be positive real numbers. Prove that
$$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$
2019 Final Mathematical Cup, 2
Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .
2000 Switzerland Team Selection Test, 4
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.
2018 Abels Math Contest (Norwegian MO) Final, 4
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.
2001 Tuymaada Olympiad, 2
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 $101$?
1985 Austrian-Polish Competition, 3
In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.
1991 Tournament Of Towns, (302) 3
Prove that $$\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1$$
This means
$1/(2+ (1/(3+ (1/(4+(...+1/1991))))))
+1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)$
(G. Galperin, Moscow-Tel Aviv)
1990 Romania Team Selection Test, 2
Prove the following equality for all positive integers $m,n$:
$$\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}$$
2012 NZMOC Camp Selection Problems, 2
Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.
1995 Tuymaada Olympiad, 8
Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.
1997 Abels Math Contest (Norwegian MO), 3a
Each subset of $97$ out of $1997$ given real numbers has positive sum.
Show that the sum of all the $1997$ numbers is positive.
1997 Argentina National Olympiad, 3
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$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$
1973 Chisinau City MO, 67
The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?
1999 Switzerland Team Selection Test, 8
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with
$$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$
2013 India PRMO, 20
What is the sum (in base $10$) of all the natural numbers less than $64$ which have exactly three ones in their base $2$ representation?
2013 Danube Mathematical Competition, 1
Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$
1978 Bundeswettbewerb Mathematik, 3
For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$
1996 Mexico National Olympiad, 5
The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes.
(a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube.
(b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.
1988 Austrian-Polish Competition, 2
If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.