Olympiad problems

1975 Chisinau City MO, 113

Prove that any integer $n$ satisfying the inequality $n <(44 + \sqrt{1975})^100 <n + 1$ is odd.

2008 Switzerland - Final Round, 6

Tags: odd , divide , divisible , number theory

Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

2011 Singapore Junior Math Olympiad, 4

Tags: sequence , odd , number theory

Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.

2010 Belarus Team Selection Test, 4.3

Tags: odd , floor function , infinitely many solutions , number theory

a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$ b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd. (I. Voronovich)

1931 Eotvos Mathematical Competition, 2

Tags: odd , perfect square , number theory

Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.

2012 Belarus Team Selection Test, 2

Tags: floor function , integer , odd , algebra

Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ . (I Voronovich)

2017 Grand Duchy of Lithuania, 4

Tags: factor , number theory , odd , multiple

Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$. (For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)

2013 Saudi Arabia BMO TST, 8

Tags: algebra , sum , odd , number theory

Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.

2017 Balkan MO Shortlist, N5

Tags: arithmetic mean , fractional part , odd , prime , quadratic residue , number theory

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

2019 Auckland Mathematical Olympiad, 3

Tags: odd , number theory

Let $x$ be the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $b$, $c$, $d$ are non-negative integers. Prove that $x$ is odd.

2010 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , divisibility , odd

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

1981 Bundeswettbewerb Mathematik, 1

Tags: floor function , sequence , odd , algebra

A sequence $a_1, a_2, a_3, \ldots $ is defined as follows: $a_1$ is a positive integer and $$a_{n+1} = \left\lfloor \frac{3}{2} a_n \right\rfloor +1$$ for all $n \in \mathbb{N}$. Can $a_1$ be chosen in such a way that the first $100000$ terms of the sequence are even, but the $100001$-th term is odd?

2017 Romania Team Selection Test, P4

Tags: arithmetic mean , fractional part , odd , prime , quadratic residue , number theory

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

2015 Brazil Team Selection Test, 1

Tags: function , odd , even , periodic , algebra

Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.

1987 All Soviet Union Mathematical Olympiad, 448

Tags: geometry , combinatorics , combinatorial geometry , odd , convex , broken line

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

2015 Gulf Math Olympiad, 1

Tags: odd , divide , divisible , prime , number theory

a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.

1999 Estonia National Olympiad, 1

Tags: divisible , odd , prime , number theory

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

2017 May Olympiad, 1

Tags: number theory , digit , odd

To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.

1989 Austrian-Polish Competition, 9

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

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

2021 Saudi Arabia IMO TST, 1

Tags: irrational number , combinatorics , odd

For a non-empty set $T$ denote by $p(T)$ the product of all elements of $T$. Does there exist a set $T$ of $2021$ elements such that for any $a\in T$ one has that $P(T)-a$ is an odd integer? Consider two cases: 1) All elements of $T$ are irrational numbers. 2) At least one element of $T$ is a rational number.