Found problems: 15460
2024 AMC 10, 11
How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) } \text{Infinitely many} \qquad
$
1992 IMTS, 3
In a mathematical version of baseball, the umpire chooses a positive integer $m$, $m \leq n$, and you guess positive integers to obtain information about $m$. If your guess is smaller than the umpire's $m$, he calls it a "ball"; if it is greater than or equal to $m$, he calls it a "strike." To "hit" it you must state the the correct value of $m$ after the 3rd strike or the 6th guess, whichever comes first. What is the largest $n$ so that there exists a strategy that will allow you to bat 1.000, i.e. always state $m$ correctly? Describe your strategy in detail.
2000 AIME Problems, 3
In the expansion of $(ax+b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a+b.$
2021 Baltic Way, 18
Find all integer triples $(a, b, c)$ satisfying the equation
$$
5 a^2 + 9 b^2 = 13 c^2.
$$
2017 Iran Team Selection Test, 4
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.
Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.
Is there always a number $x$ that satisfies all the equations?
[i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]
2021 Iran MO (2nd Round), 2
Call a positive integer $n$ "Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of $n$ . ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.
2019 Kosovo National Mathematical Olympiad, 5
Find all positive integers $x,y$ such that $2^x+19^y$ is a perfect cube.
1982 USAMO, 2
Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$.
2017-IMOC, N2
On the blackboard, there are $K$ blanks. Alice decides $N$ values of blanks $(0-9)$ and then Bob determines the remaining digits. Find the largest possible integer $N$ such that Bob can guarantee to make the final number isn't a power of an integer.
2018 Iran MO (3rd Round), 1
$n\ge 2 $ is an integer.Prove that the number of natural numbers $m$ so that $0 \le m \le n^2-1,x^n+y^n \equiv m (mod n^2)$ has no solutions is at least $\binom{n}{2}$
2019 Saudi Arabia JBMO TST, 5
Let non-integer real numbers $a, b,c,d$ are given, such that the sum of each $3$ of them is integer. May it happen that $ab + cd$ is an integer.
1989 IMO Shortlist, 25
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
2019-IMOC, N5
Initially, Alice is given a positive integer $a_0$. At time $i$, Alice has two choices,
$$\begin{cases}a_i\mapsto\frac1{a_{i-1}}\\a_i\mapsto2a_{i-1}+1\end{cases}$$
Note that it is dangerous to perform the first operation, so Alice cannot choose this operation in two consecutive turns. However, if $x>8763$, then Alice could only perform the first operation. Determine all $a_0$ so that $\{i\in\mathbb N\mid a_i\in\mathbb N\}$ is an infinite set.
1967 IMO Shortlist, 1
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that
\[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\]
is divisible by the product $c_1c_2\ldots c_n$.
2022 Polish Junior Math Olympiad First Round, 3.
Let $n\geq 1$ be an integer. Show that there exists an integer between $\sqrt{2n}$ and $\sqrt{5n}$, exclusive.
VII Soros Olympiad 2000 - 01, 9.3
Write $102$ as the sum of the largest number of distinct primes.
2025 Taiwan Mathematics Olympiad, 3
For any pair of coprime positive integers $a$ and $b$, define $f(a, b)$ to be the smallest nonnegative integer $k$ such that $b \mid ak+1$. Prove that if a and b are coprime positive integers satisfying
$$f(a, b) - f(b, a) = 2,$$
then there exists a prime number $p$ such that $p^2\mid a + b$.
[i]Proposed by usjl[/i]
2001 Estonia National Olympiad, 4
We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.
2009 ISI B.Stat Entrance Exam, 10
Let $x_n$ be the $n$-th non-square positive integer. Thus $x_1=2, x_2=3, x_3=5, x_4=6,$ etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$. If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle =1, \langle 2.8 \rangle =3, \langle 3.5\rangle =3$. Show that $x_n=n+\langle \sqrt{n}\rangle$
2011 Regional Olympiad of Mexico Center Zone, 3
We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $.
2007 Estonia Math Open Senior Contests, 3
Let $ b$ be an even positive integer for which there exists a natural number n such
that $ n>1$ and $ \frac{b^n\minus{}1}{b\minus{}1}$ is a perfect square. Prove that $ b$ is divisible by 8.
2009 Postal Coaching, 6
Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$
2012 China Western Mathematical Olympiad, 4
Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition:
$p|n^{ n+1}+(n+1)^n.$
(September 29, 2012, Hohhot)
1969 Kurschak Competition, 1
Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).
2021 Korea Winter Program Practice Test, 4
Find all $f(x)\in \mathbb Z (x)$ that satisfies the following condition, with the lowest degree.
[b]Condition[/b]: There exists $g(x),h(x)\in \mathbb Z (x)$ such that $$f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)$$.