Olympiad problems

1991 Tournament Of Towns, (281) 1

$N$ integers are given. Prove that the sum of their squares is divisible by $N$ if it is known that the difference between the product of any $N - 1$ of them and the last one is divisible by $N$. (D. Fomin, Leningrad)

1987 Tournament Of Towns, (141) 1

Tags: number theory , sum of squares , perfect square , odd

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

2010 Contests, 1

Tags: digit , number theory , sum of squares , algebra

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

2015 Balkan MO Shortlist, N1

Tags: number theory , sum of squares , combinatorics , square

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

2011 QEDMO 8th, 2

Tags: number theory , sum of squares

Let $n$ be an integer. Show that a natural number $k$ can be found for which, the following applies with a suitable choice of signs: $$n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2$$

2018 Romania Team Selection Tests, 2

Tags: number theory , quadratic reciprocity , sum of squares , composite number

Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.

2006 Kazakhstan National Olympiad, 7

Tags: sum of squares , number theory

Prove that if a natural number $ N $ can be represented in the form the sum of three squares of integers divisible by $3$, then it is also is represented as the sum of three squares of integers that are not divisible by $3$.

2014 India PRMO, 5

Tags: algebra , sum of squares

If real numbers $a, b, c, d, e$ satisfy $a + 1 = b + 2 = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3$, what is the value of $a^2 + b^2 + c^2 + d^2 + e^2$ ?

1952 Moscow Mathematical Olympiad, 216

Tags: integer sequence , sequence , sum of squares , digit , number theory

A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

2013 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , sum of squares , divisible

It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

2021 Poland - Second Round, 3

Tags: number theory , sum of squares

Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that $$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$

2016 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: game strategy , combinatorics , sum of squares , game

Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.

1996 Tournament Of Towns, (504) 1

Tags: number theory , consecutive , algebra , sum of squares

Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers? (Inspired by a diagram in an old text book)

2010 Greece JBMO TST, 1

Tags: digit , number theory , sum of squares , algebra

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

1989 All Soviet Union Mathematical Olympiad, 494

Tags: sum of squares , number theory , 5-digit , algebra , subset

Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.

2006 All-Russian Olympiad Regional Round, 11.7

Tags: perfect square , sum of squares , number theory

Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.

2017 Regional Competition For Advanced Students, 4

Tags: number theory , sum of squares , divisibility , divisor

Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

2016 India Regional Mathematical Olympiad, 5

Tags: number theory , sum of squares

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2000 IMO Shortlist, 6

Tags: additive number theory , sum of squares , perfect square , number theory

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

2006 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , sum of squares , perfect square

A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

1977 Spain Mathematical Olympiad, 4

Tags: perfect square , sum of squares , consecutive , number theory

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

2016 Polish MO Finals, 5

Tags: algebra , real number , sum of squares

There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions: $1$. $a< \frac{p}{q} < \frac{r}{s} < b$. $2.$ $p^2+q^2=r^2+s^2$.

1978 IMO Longlists, 17

Tags: quadratic , number theory , prime , sum of squares

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.

1976 Dutch Mathematical Olympiad, 1

Tags: number theory , sum of squares

Prove that there is no natural $n$ such that $8n + 7$ is the sum of three squares.