Olympiad problems

2018 Estonia Team Selection Test, 3

Tags: Sum , algebra , inequalities , max , min

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.

1992 Bundeswettbewerb Mathematik, 2

Tags: number theory , Sum , Product

A positive integer $n$ is called [i]good [/i] if they sum up in one and only one way at least of two positive integers whose product also has the value $n$. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: rational , Sum , algebra

Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.

2009 Abels Math Contest (Norwegian MO) Final, 4b

Tags: algebra , inequalities , Sum

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

1995 Romania Team Selection Test, 1

Tags: Sum , Sum of powers , inequalities , algebra

Let $a_1, a_2,...., a_n$ be distinct positive integers. Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.

2013 IFYM, Sozopol, 4

Tags: number theory , algebra , Sum

Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find $\max S=\sum_{i\mid j}a_i a_j $.

VII Soros Olympiad 2000 - 01, 9.3

Tags: number theory , Sum , primes

Write $102$ as the sum of the largest number of distinct primes.

1985 Tournament Of Towns, (104) 1

Tags: convex , diagonal , Sum , perimeter , geometric inequality

We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

1960 Kurschak Competition, 2

Tags: number theory , Integer , Sum

Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.

1946 Moscow Mathematical Olympiad, 122

Tags: Sum , areas , Locus , ratio , geometry

On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

1998 Tuymaada Olympiad, 7

Tags: algebra , arithmetic sequence , Sequence , square , Sum

All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.

1999 Tournament Of Towns, 2

Tags: number theory , prime , Sum of powers , Sum

Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$. (a) May it happen that $d = 2$? (b) May it happen that $d$ is prime? (V Senderov)

2023 Germany Team Selection Test, 3

Tags: Sets , Sum , elements of set , combinatorics

Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$. (a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element. (b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$. (c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.

1986 Polish MO Finals, 3

Tags: number theory , Sum , divisible , prime , 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 $p$.

2005 Abels Math Contest (Norwegian MO), 1a

Tags: number theory , Sum , Perfect Square

A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.

2013 India PRMO, 18

Tags: Sum , maximum , consecutive integers , algebra

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

1986 All Soviet Union Mathematical Olympiad, 440

Tags: geometry , 3D geometry , sphere , tetrahedron , fixed , angles , Sum

Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .

2002 Estonia National Olympiad, 4

Tags: Integers , number theory , Sum , max

Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.

2017 Czech And Slovak Olympiad III A, 4

Tags: combinatorics , Sum , Probabilistic Method

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} $

2013 India PRMO, 3

Tags: algebra , Sum , Integer

It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?

2008 Greece JBMO TST, 4

Tags: number theory , Integers , Diophantine equation , Product , Sum

Product of two integers is $1$ less than three times of their sum. Find those integers.

2010 Saudi Arabia BMO TST, 2

Tags: algebra , Sum

Evaluate the sum $$1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 1 0 + . . . - 2010$$ , where each three consecutive signs $+$ are followed by two signs $-$.

2020 Durer Math Competition Finals, 6

Tags: algebra , Sum

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]

2016 Hanoi Open Mathematics Competitions, 8

Tags: number theory , perfect cubes , Sum , sum of digits

Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

1993 Austrian-Polish Competition, 3

Tags: power of prime , Product , Sum , number theory

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.