Found problems: 15460
2015 Indonesia MO Shortlist, N5
Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*.
Clarification:
* For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$
2015 IMC, 4
Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$
such that~
$$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
1997 China National Olympiad, 3
Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions:
i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$;
ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.
2024-IMOC, N4
Given a set of integers $S$ satisfies that: for any $a,b,c\in S$ ($a,b,c$ can be the same), $ab+c\in S$\\
Find all pairs of integers $(x,y)$ such that if $x,y\in S$, then $S=\mathbb{Z}$.
2001 IMO Shortlist, 6
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?
2005 France Team Selection Test, 4
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
2022 JHMT HS, 4
For an integer $a$ and positive integers $n$ and $k$, let $f_k(a, n)$ be the remainder when $a^k$ is divided by $n$. Find the largest composite integer $n\leq 100$ that guarantees the infinite sequence
\[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \]
to be periodic for all integers $a$ (i.e., for each choice of $a$, there is some positive integer $T$ such that $f_k(a,n) = f_{k+T}(a,n)$ for all $k$).
2010 International Zhautykov Olympiad, 1
Find all primes $p,q$ such that $p^3-q^7=p-q$.
2022-IMOC, N3
Find all positive integer $n$ satifying $$2n+3|n!-1$$
[i]Proposed by ltf0501[/i]
2024 Argentina Iberoamerican TST, 1
Find all positive prime numbers $p$, $q$ that satisfy the equation
$$p(p^4+p^2+10q)=q(q^2+3).$$
2018 PUMaC Number Theory A, 7
Find the smallest positive integer $G$ such that there exist distinct positive integers $a, b, c$ with the following properties:
$\: \bullet \: \gcd(a, b, c) = G$.
$\: \bullet \: \text{lcm}(a, b) = \text{lcm}(a, c) = \text{lcm}(b, c)$.
$\: \bullet \: \frac{1}{a} + \frac{1}{b}, \frac{1}{a} + \frac{1}{c},$ and $\frac{1}{b} + \frac{1}{c}$ are reciprocals of integers.
$\: \bullet \: \gcd(a, b) + \gcd(a, c) + \gcd(b, c) = 16G$.
2019 Tournament Of Towns, 5
Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice.
(Yury Markelov)
1949-56 Chisinau City MO, 3
Prove that the number $N = 10 ...050...01$ (1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.
2019 China Team Selection Test, 4
Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ?
Here $A+B=\{a+b|a\in A, b\in B\}$.
2009 ISI B.Math Entrance Exam, 6
Let $a,b,c,d$ be integers such that $ad-bc$ is non zero. Suppose $b_1,b_2$ are integers both of which are multiples of $ad-bc$. Prove that there exist integers simultaneously satisfying both the equalities $ax+by=b_1, cx+dy=b_2$.
Kvant 2024, M2783
The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a]
[*] the square of this number;
[*]the fourth power of this number,
[/list] given that $k\geqslant 4.$
[i]From the folklore[/i]
1997 Baltic Way, 9
The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?
2007 Spain Mathematical Olympiad, Problem 2
Determine all the possible non-negative integer values that are able to satisfy the expression:
$\frac{(m^2+mn+n^2)}{(mn-1)}$
if $m$ and $n$ are non-negative integers such that $mn \neq 1$.
1988 Canada National Olympiad, 4
Let $x_{n + 1} = 4x_n - x_{n - 1}$, $x_0 = 0$, $x_1 = 1$, and $y_{n + 1} = 4y_n - y_{n - 1}$, $y_0 = 1$, $y_1 = 2$. Show that for all $n \ge 0$ that $y_n^2 = 3x_n^2 + 1$.
2014 China Team Selection Test, 6
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
2013 Singapore MO Open, 3
Let n be a positve integer. prove there exists a positive integer n st $n^{2013}-n^{20}+n^{13}-2013$ has at least N distinct prime factors.
2010 China Western Mathematical Olympiad, 8
Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.
2009 HMNT, 2
You start with a number. Every second, you can add or subtract any number of the form $n!$ to your current number to get a new number. In how many ways can you get from $0$ to $100$ in $4$ seconds?
($n!$ is de ned as $n\times (n -1)\times(n - 2) ... 2\times1$, so $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$, etc.)
2012 Singapore Junior Math Olympiad, 5
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
IV Soros Olympiad 1997 - 98 (Russia), 11.5
Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.