Olympiad problems

1970 All Soviet Union Mathematical Olympiad, 137

Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

1991 ITAMO, 3

Tags: number theory , sum , sum of squares

We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)

2007 Thailand Mathematical Olympiad, 14

Tags: binomial coefficient , sum , combination

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 $(a, b)$

2015 Indonesia MO Shortlist, C7

Tags: combinatorics , subset , sum

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 $2015$.

1998 Austrian-Polish Competition, 4

Tags: inequalities , sum , algebra

For positive integers $m, n$, denote $$S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]$$ Prove that $S_m(n) \le n + m (\sqrt[4]{2^m}-1)$

1977 All Soviet Union Mathematical Olympiad, 248

Tags: algebra , combinatorics , sum

Given natural numbers $x_1,x_2,...,x_n,y_1,y_2,...,y_m$. The following condition is valid: $$(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn \,\,\,\, (*)$$ Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.

1945 Moscow Mathematical Olympiad, 095

Tags: geometry , tangent , sum , tangent circle

Two circles are tangent externally at one point. Common external tangents are drawn to them and the tangent points are connected. Prove that the sum of the lengths of the opposite sides of the quadrilateral obtained are equal.

2016 Bundeswettbewerb Mathematik, 1

Tags: consecutive , sum , summation , number theory

There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers. For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?

2007 Estonia Math Open Junior Contests, 10

Tags: number theory , sum , perfect square , combinatorics

Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.

1981 Spain Mathematical Olympiad, 1

Tags: algebra , sum

Calculate the sum of $n$ addends $$7 + 77 + 777 +...+ 7... 7.$$

1945 Moscow Mathematical Olympiad, 099

Tags: combinatorics , number theory , algebra , sum , 4-digit

Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).

2016 JBMO Shortlist, 2

Tags: combinatorics , sum , prime

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

1993 Poland - Second Round, 5

Tags: geometry , inradius , sum , incircle

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose that the inradii of the triangles $AEF,BFD,CDE$ are all equal to $r_1$. If $r_2$ and $r$ are the inradii of triangles $DEF$ and $ABC$ respectively, prove that $r_1 +r_2 =r$.

2017 Irish Math Olympiad, 3

Tags: geometry , sum

A line segment $B_0B_n$ is divided into $n$ equal parts at points $B_1,B_2,...,B_{n-1} $ and $A$ is a point such that $\angle B_0AB_n$ is a right angle. Prove that : $$\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2$$

2019 Istmo Centroamericano MO, 4

Tags: algebra , sum

Let $x, y, z$ be nonzero real numbers such that $ x + y + z = 0$ and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1 -xyz + \frac{1}{xyz}.$$ Determine the value of the expression ' $$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$

2012 Danube Mathematical Competition, 4

Tags: combinatorics , sum , subset , set

Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$. Show that there are two distinct elements of $A$, having the same sum of their elements.

2001 Switzerland Team Selection Test, 4

Tags: number theory , divisor , sum

For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.

2015 Bundeswettbewerb Mathematik Germany, 2

Tags: number theory , sum , integer , positive

A sum of $335$ pairwise distinct positive integers equals $100000$. a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?

2002 Estonia National Olympiad, 3

Tags: number theory , odd , sum , product

John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.

1999 Switzerland Team Selection Test, 8

Tags: algebra , sum , system of equations

Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with $$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$

1998 Tuymaada Olympiad, 1

Tags: algebra , sum , reciprocal sum

Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.

1996 Singapore Senior Math Olympiad, 2

Tags: geometry , trigonometry , angle , max , semicircle , triangle area , sum

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

2009 Chile National Olympiad, 3

Tags: number theory , sum

Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?

2021 Abels Math Contest (Norwegian MO) Final, 2a

Tags: number theory , sum

Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$

2006 Tournament of Towns, 1

Tags: combinatorics , sum

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)