Found problems: 580
2006 Tournament of Towns, 1
Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. With the new pair of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)
1995 Swedish Mathematical Competition, 4
The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.
1960 Poland - Second Round, 1
Prove that if the real numbers $ a $ and $ b $ are not both equal to zero, then for every natural $ n $
$$ a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0. $$
2012 Tournament of Towns, 5
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
1994 North Macedonia National Olympiad, 3
a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
2016 Indonesia MO, 7
Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$, denote $r_k$ as the remainder of the division $k^p$ by $p^2$. Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$
2014 Denmark MO - Mohr Contest, 5
Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.
1999 Estonia National Olympiad, 5
The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that
a) there are three consecutive numbers with the sum being at least $15$,
b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.
1974 Chisinau City MO, 79
There are many of the same regular triangles. At the vertices of each of them, the numbers $1, 2, 3$ are written in random order. The triangles were superimposed on one another and found the sum of the numbers that fell into each of the three corners of the stack. Could it be that in each corner the sum is equal to:
a) $25$,
b) $50$?
1998 German National Olympiad, 5
A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$.
Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.
2015 Indonesia MO Shortlist, C7
Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that :
(i) $|A| = 12$
(ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.
1988 Mexico National Olympiad, 7
Two disjoint subsets of the set $\{1,2, ... ,m\}$ have the same sums of elements. Prove that each of the subsets $A,B$ has less than $m / \sqrt2$ elements.
1962 Polish MO Finals, 1
Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then
$$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$
1991 All Soviet Union Mathematical Olympiad, 545
The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$
1978 All Soviet Union Mathematical Olympiad, 252
Let $a_n$ be the closest to $\sqrt n$ integer. Find the sum $$1/a_1 + 1/a_2 + ... + 1/a_{1980}$$
1994 Spain Mathematical Olympiad, 3
A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table:
Region , sunny or rainy , unclassified
$A \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 336 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,29$
$B \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 321 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,44$
$C \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 335 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,30$
$D \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 343 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,22$
$E \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 329 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,36$
$F \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 330 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,35$
Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.
2009 Danube Mathematical Competition, 2
Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .
1990 Swedish Mathematical Competition, 2
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.
2017 India PRMO, 6
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$.
2002 Switzerland Team Selection Test, 3
Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.
Kyiv City MO 1984-93 - geometry, 1993.8.4
The diameter of a circle of radius $R$ is divided into $4$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.
1951 Moscow Mathematical Olympiad, 204
* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.
VMEO III 2006 Shortlist, N9
Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have
$$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$
1998 Chile National Olympiad, 1
Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.
2021 Abels Math Contest (Norwegian MO) Final, 2a
Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$