This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 54

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?

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.

2016 Saint Petersburg Mathematical Olympiad, 6
Tags: combinatorics , combinatorial geometry , odd , polygon , clo
The circle contains a closed $100$-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by $720$. Prove that this broken line has an odd number of self-intersection points.

1974 Bundeswettbewerb Mathematik, 4
Tags: number theory , divisor , odd , remainder
Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

2001 Dutch Mathematical Olympiad, 5
Tags: combinatorics , subset , even , odd
If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property: If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even). Prove this.

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)

1939 Moscow Mathematical Olympiad, 049
Tags: algebra , polynomial , integer polynomial , even , odd
Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.

2011 QEDMO 10th, 2
Tags: combinatorics , odd , even , sum
Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$

2016 Dutch BxMO TST, 1
Tags: number theory , divisor , odd , equation
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

2005 Thailand Mathematical Olympiad, 13
Tags: diophantine , diophantine equation , number theory , odd
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.

2000 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , odd , multiple , last digit
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.

2000 Abels Math Contest (Norwegian MO), 1a
Tags: odd , number theory , perfect square
Show that any odd number can be written as the difference between two perfect squares.

2016 Balkan MO Shortlist, N2
Tags: divisor , maximum , number theory , odd
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

2011 Singapore Junior Math Olympiad, 4
Tags: number theory , sequence , odd
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 $a_{2011}$.

2011 Belarus Team Selection Test, 1
Tags: number theory , odd , divisible , prime
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with $a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$). Prove that a) $(a-1)\vdots p_i$ for some $i=1,..,n$ b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$? I. Bliznets

1999 Estonia National Olympiad, 4
Tags: odd , combinatorics , chessboard
Let us put pieces on some squares of $2n \times 2n$ chessboard in such a way that on every horizontal and vertical line there is an odd number of pieces. Prove that the whole number of pieces on the black squares is even.

2002 Singapore Team Selection Test, 2
Tags: sequence , combinatorics , odd , even
Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$. If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$

2017 Romania Team Selection Test, P4
Tags: quadratic residue , number theory , arithmetic mean , fractional part , odd , prime
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 $p$ .

1979 Chisinau City MO, 174
Tags: number theory , odd , divide , divisible
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2016 Dutch BxMO TST, 1
Tags: divisor , odd , equation , number theory
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

2015 Thailand Mathematical Olympiad, 8
Tags: perfect square , odd , number theory
Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

1999 Tournament Of Towns, 2
Tags: composite , number theory , odd
Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

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.

2003 Junior Balkan Team Selection Tests - Romania, 2
Tags: odd , number theory , decimal representation , power
Let $a$ be a positive integer such that the number $a^n$ has an odd number of digits in the decimal representation for all $n > 0$. Prove that the number $a$ is an even power of $10$.

2004 Estonia National Olympiad, 4
Tags: divide , divisible , odd , number theory
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.