Olympiad problems

2016 Polish MO Finals, 5

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

2017 Regional Competition For Advanced Students, 4

Tags: number theory , Sum of Squares , Divisors , Divisibility

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]

2015 Bosnia Herzegovina Team Selection Test, 4

Tags: combinatorics , Sum of Squares , set

Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.

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 $3$.

2019 Tournament Of Towns, 4

Tags: number theory , consecutive , multiple , circle , Sum of Squares

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

2010 IMO Shortlist, 3

Tags: algebra , polynomial , number theory , Sum of Squares , IMO Shortlist

Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\] [i]Proposed by Mariusz Skałba, Poland[/i]

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 $3$.

2018 Romania National Olympiad, 1

Tags: number theory , Perfect Squares , Perfect Square , Sum of Squares

Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.

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: $$n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2$$

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.

2010 Saudi Arabia Pre-TST, 2.2

Tags: prime , consecutive , number theory , Sum of Squares

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

1976 Czech and Slovak Olympiad III A, 1

Tags: number theory , Diophantine equation , Perfect Square , Sum of Squares

Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]

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.

2010 Contests, 1

Tags: number theory , Sum of Squares , algebra , Digits , Digit

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.

1991 Mexico National Olympiad, 5

Tags: consecutive , Sum of Squares , number theory

The sum of squares of two consecutive integers can be a square, as in $3^2+4^2 =5^2$. Prove that the sum of squares of $m$ consecutive integers cannot be a square for $m = 3$ or $6$ and find an example of $11$ consecutive integers the sum of whose squares is a square.

1981 Spain Mathematical Olympiad, 8

Tags: number theory , Sum of Squares , divisible

If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.

1978 IMO Longlists, 17

Tags: quadratics , number theory , prime , Sum of Squares , IMO Shortlist

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.

2000 IMO Shortlist, 6

Tags: number theory , Additive Number Theory , Sum of Squares , Perfect Square , IMO Shortlist

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

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)

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 $a^2 + b^2 + c^2 + d^2 + e^2$ ?

1978 IMO Shortlist, 17