Olympiad problems

2012 Today's Calculation Of Integral, 826

Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.

2014 ELMO Shortlist, 8

Tags: function , greatest common divisor , number theory

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

2017 CMIMC Number Theory, 5

Tags: greatest common divisor , number theory

One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write \[\gcd\left(\dfrac ab,\dfrac cd\right) = \dfrac{\gcd(ad,bc)}{bd}.\] For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of positive rational numbers $(m,n)$ with $m<1$ and $n<1$ such that \[\gcd(m,n)=\dfrac{1}{K}.\] What is $f(2017)-f(2016)$?

2014 Tuymaada Olympiad, 1

Tags: least common multiple , modular arithmetic , greatest common divisor , number theory , relatively prime

Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

2007 Baltic Way, 17

Tags: greatest common divisor , inequalities , number theory

Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.

2004 USAMO, 2

Tags: induction , algorithm , greatest common divisor , number theory , relatively prime , vector , integration

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

2020 Princeton University Math Competition, 3

Tags: number theory , least common multiple , greatest common divisor

Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form $2^a3^b$, where a and b are positive integers between $ 1$ and $2020$, inclusive. Each turn, Alice guess a number m, and Bob will tell her either $\gcd (m, n)$ or $lcm (m, n)$ (letting her know that he is saying that $gcd$ or $lcm$), as well as whether any of the respective powers match up in their prime factorization. In particular, if $m = n$, Bob will let Alice know this, and the game is over. Determine the smallest number $k$ so that Alice is always able to find $n$ within $k$ guesses, regardless of Bob’s number or choice of revealing either the $lcm$, or the $gcd$ .

1994 Cono Sur Olympiad, 2

Tags: conic , analytic geometry , quadratic , greatest common divisor , number theory , calculus , integration

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

2012 Thailand Mathematical Olympiad, 7

Tags: gcd , divide , number theory , greatest common divisor

Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

2010 USA Team Selection Test, 6

Tags: greatest common divisor , number theory , combinatorics

Let $T$ be a finite set of positive integers greater than 1. A subset $S$ of $T$ is called [i]good[/i] if for every $t \in T$ there exists some $s \in S$ with $\gcd(s, t) > 1$. Prove that the number of good subsets of $T$ is odd.

2013 Princeton University Math Competition, 4

Tags: greatest common divisor , modular arithmetic

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

1967 Miklós Schweitzer, 1

Tags: superior algebra , algebra , greatest common divisor , polynomial

Let \[ f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) \] and \[ g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) \] be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $ 6$. [i]L. Redei[/i]

2019 Thailand TST, 1

Tags: greatest common divisor , number theory

Let $n$ be a positive integer. Let $S$ be a set of $n$ positive integers such that the greatest common divisors of all nonempty sets of $S$ are distinct. Determine the smallest possible number of distinct prime divisors of the product of the elements of $S$.

2014 Online Math Open Problems, 14

Tags: algorithm , euclidean algorithm , greatest common divisor , number theory

What is the greatest common factor of $12345678987654321$ and $12345654321$? [i]Proposed by Evan Chen[/i]

2013 Online Math Open Problems, 39

Tags: greatest common divisor , relatively prime , number theory

Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on). [i]Victor Wang[/i]

1959 AMC 12/AHSME, 42

Tags: greatest common divisor , relatively prime , least common multiple , number theory

Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. Then, which two of the following statements are true? $ \text{(1)}\ \text{the product MD cannot be less than abc} \qquad$ $\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad$ $\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad$ $\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}$ $\text{ (This means: no two have a common factor greater than 1.)}$ $ \textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $

2017 CMIMC Number Theory, 10

Tags: factorial , greatest common divisor , number theory

For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.

2009 Irish Math Olympiad, 2

Tags: number theory , greatest common divisor

For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$ Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$

2012 Dutch IMO TST, 1

Tags: greatest common divisor , gcd , perfect power , prime , number theory

For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2005 Baltic Way, 16

Tags: greatest common divisor , number theory

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.

Russian TST 2020, P2

Tags: inequalities , greatest common divisor , number theory

Given a natural number $n{}$ find the smallest $\lambda$ such that\[\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda,\] for any positive integers $y{}$ and $x \geqslant y + n$.

2012-2013 SDML (High School), 14

Tags: greatest common divisor , arithmetic sequence , number theory

A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2017 China Team Selection Test, 5

Tags: number theory , greatest common divisor

Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.

2016 Iran Team Selection Test, 6

Tags: function , number theory , greatest common divisor

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

2014 ITAMO, 3

Tags: greatest common divisor , number theory , modular arithmetic

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.