Olympiad problems

2021 Science ON grade VIII, 1

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

2010 Bulgaria National Olympiad, 1

Tags: modular arithmetic , number theory

Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$

2009 Middle European Mathematical Olympiad, 4

Tags: quadratic , modular arithmetic , number theory

Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.

1999 Italy TST, 1

Tags: modular arithmetic , number theory

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

2014 Junior Balkan MO, 4

Tags: floor function , modular arithmetic , limit , combinatorics , logarithm

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

1970 IMO Longlists, 18

Tags: modular arithmetic , number theory , product , partition

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

PEN J Problems, 20

Tags: divisor function , modular arithmetic

Show that $\sigma (n) -d(m)$ is even for all positive integers $m$ and $n$ where $m$ is the largest odd divisor of $n$.

2011 Balkan MO, 3

Tags: ratio , floor function , modular arithmetic , number theory

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

2007 Iran Team Selection Test, 2

Tags: ceiling function , pigeonhole principle , modular arithmetic , search , combinatorics

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]

2006 China Team Selection Test, 2

Tags: perfect power , modular arithmetic , order

Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

2004 Iran Team Selection Test, 1

Tags: modular arithmetic , quadratic , special factorizations , number theory

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2005 Uzbekistan National Olympiad, 3

Tags: modular arithmetic , number theory

Find the last five digits of $1^{100}+2^{100}+3^{100}+...+999999^{100}$

2015 AMC 12/AHSME, 1

Tags: modular arithmetic , function

What is the value of $2-(-2)^{-2}$? $ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $

2013 China Team Selection Test, 3

Tags: induction , modular arithmetic , algebra

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

2001 Romania Team Selection Test, 3

Tags: modular arithmetic , number theory , combinatorics , additive number theory , additive combinatorics , frobenius

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

2003 National Olympiad First Round, 10

Tags: modular arithmetic , geometry , 3d geometry , algebra , factorization , sum of cubes

Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ \text{None of the preceding} $

1987 IMO Longlists, 78

Tags: modular arithmetic , algebra , polynomial , irrational number , number theory

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

2012 International Zhautykov Olympiad, 3

Tags: modular arithmetic , greatest common divisor , diophantine equation , number theory

Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.

1996 IMO Shortlist, 9

Tags: floor function , modular arithmetic , algebra , sequence , recurrence relation

Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$ \[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\] 1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained. 2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?

2010 National Olympiad First Round, 34

Tags: modular arithmetic

Which one divides $2^{2^{2010}}+2^{2^{2009}}+1$? $ \textbf{(A)}\ 19 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ \text{None} $

2013 Baltic Way, 17

Tags: modular arithmetic , number theory

Let $c$ and $n > c$ be positive integers. Mary's teacher writes $n$ positive integers on a blackboard. Is it true that for all $n$ and $c$ Mary can always label the numbers written by the teacher by $a_1,\ldots, a_n$ in such an order that the cyclic product $(a_1-a_2)\cdot(a_2-a_3)\cdots(a_{n-1}-a_n)\cdot(a_n-a_1)$ would be congruent to either $0$ or $c$ modulo $n$?

PEN M Problems, 17

Tags: modular arithmetic , recursive sequences

A sequence of integers, $\{a_{n}\}_{n \ge 1}$ with $a_{1}>0$, is defined by \[a_{n+1}=\frac{a_{n}}{2}\;\;\; \text{if}\;\; n \equiv 0 \;\; \pmod{4},\] \[a_{n+1}=3 a_{n}+1 \;\;\; \text{if}\;\; n \equiv 1 \; \pmod{4},\] \[a_{n+1}=2 a_{n}-1 \;\;\; \text{if}\;\; n \equiv 2 \; \pmod{4},\] \[a_{n+1}=\frac{a_{n}+1}{4}\;\;\; \text{if}\;\; n \equiv 3 \; \pmod{4}.\] Prove that there is an integer $m$ such that $a_{m}=1$.

2013 Stanford Mathematics Tournament, 10

Tags: modular arithmetic

Consider a sequence given by $a_n=a_{n-1}+3a_{n-2}+a_{n-3}$, where $a_0=a_1=a_2=1$. What is the remainder of $a_{2013}$ divided by $7$?

1956 AMC 12/AHSME, 31

Tags: modular arithmetic , number theory

In our number system the base is ten. If the base were changed to four you would count as follows: $ 1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be: $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 44 \qquad\textbf{(D)}\ 104 \qquad\textbf{(E)}\ 110$

1993 Baltic Way, 2

Tags: modular arithmetic , number theory

Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?