Found problems: 15460
JOM 2025, 1
Is it possible for Pingu to choose $2025$ positive integers $a_1, ..., a_{2025}$ such that:
1. The sequence $a_i$ is increasing;
2. $\gcd(a_1,a_2)>\gcd(a_2,a_3)>...>\gcd(a_{2024},a_{2025})>\gcd(a_{2025},a_1)>1$?
[i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]
2014 APMO, 4
Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$.
(a) Prove that $8$ is $100$-discerning.
(b) Prove that $9$ is not $100$-discerning.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
1983 Federal Competition For Advanced Students, P2, 5
Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.
2021 Olimphíada, 5
Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.
2019 New Zealand MO, 4
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.
2015 Romania Team Selection Tests, 3
A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.
1917 Eotvos Mathematical Competition, 1
If $a$ and $b$ are integers and if the solutions of the system of equations
$$y - 2x - a = 0$$
$$y^2 - xy + x^2 - b = 0$$
are rational, prove that the solutions are integers.
2021 Taiwan TST Round 3, N
Let $a_1$, $a_2$, $a_3$, $\ldots$ be a sequence of positive integers such that $a_1=2021$ and
$$\sqrt{a_{n+1}-a_n}=\lfloor \sqrt{a_n} \rfloor. $$
Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence.
[i] Proposed by Li4, Tsung-Chen Chen, and Ming Hsiao.[/i]
2014 CentroAmerican, 1
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
2013 IFYM, Sozopol, 8
The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ :
$[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$.
Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?
2000 IMO Shortlist, 2
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.
2016 IFYM, Sozopol, 3
Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.
1993 Spain Mathematical Olympiad, 2
In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above.
$0 \, 1\, 2\, 3 \,4\, ... \,1991 \,1992\, 1993$
$\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985$
$\,\,\,4 \,8 \,12\, .......... \,\,\,7968$
·······································
Prove that the bottom number is a multiple of $1993$.
2008 ISI B.Stat Entrance Exam, 9
Suppose $S$ is the set of all positive integers. For $a,b \in S$, define
\[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\]
For example $8*12=6$.
Show that [b]exactly two[/b] of the following three properties are satisfied:
(i) If $a,b \in S$, then $a*b \in S$.
(ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$.
(iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.
2017 Dutch IMO TST, 2
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.
Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)
2014 Contests, 4
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$.
2012 Indonesia MO, 3
Let $n$ be a positive integer. Show that the equation \[\sqrt{x}+\sqrt{y}=\sqrt{n}\] have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$.
[i]Proposer: Nanang Susyanto[/i]
2012 Peru IMO TST, 6
Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$
[i]Proposed by Romeo Meštrović, Montenegro[/i]
2013 Junior Balkan Team Selection Tests - Moldova, 2
Determine the elements of the sets $A = \{x \in N | x \ne 4a + 7b, a, b \in N\}$, $B = \{x \in N | x\ne 3a + 11b, a, b \in N\}$.
2012 BAMO, 2
Answer the following two questions and justify your answers:
(a) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+5^{2012}$?
(b) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+...+2011^{2012}+2012^{2012}$?
2007 Germany Team Selection Test, 1
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.)
[b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often.
[b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .
2011 India IMO Training Camp, 3
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
2020 March Advanced Contest, 1
In terms of \(a\), \(b\), and a prime \(p\), find an expression which gives the number of \(x \in \{0, 1, \ldots, p-1\}\) such that the remainder of \(ax\) upon division by \(p\) is less than the remainder of \(bx\) upon division by \(p\).
2020 DMO Stage 1, 4.
[b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that:
$$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds.
[i]Proposed by dangerousliri [/i]
2019 Spain Mathematical Olympiad, 4
Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$