Found problems: 15460
2022 Stars of Mathematics, 1
Find all positive integers $n$, such that there exist positive integers $a,b$, such that $a+2^b=n^{2022}$ and $a^2+4^b=n^{2023}$.
2021 Bolivian Cono Sur TST, 1
Find the sum of all positive integers $n$ such that
$$\frac{n+11}{\sqrt{n-1}}$$
is an integer.
2021 Baltic Way, 19
Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.
2024 Mozambican National MO Selection Test, P3
Find all triples of positive integers $(a,b,c)$ such that:
$a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$
2025 Kyiv City MO Round 2, Problem 2
Find all pairs of positive integers \( a, b \) such that one of the two numbers \( 2(a^2 + b^2) \) and \( (a + b)^2 + 4 \) is divisible by the other.
[i]Proposed by Oleksii Masalitin[/i]
2013 Romania Team Selection Test, 4
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.
2021 Dutch IMO TST, 3
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.
2013 Dutch Mathematical Olympiad, 5
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result?
KoMaL A Problems 2023/2024, A. 882
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
II Soros Olympiad 1995 - 96 (Russia), 11.6
For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?
2007 Pre-Preparation Course Examination, 13
Let $\{a_i\}_{i=1}^{\infty}$ be a sequence of positive integers such that $a_1<a_2<a_3\cdots$ and all of primes are members of this sequence. Prove that for every $n<m$
\[\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N\]
IMSC 2023, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
2011 Saint Petersburg Mathematical Olympiad, 7
Sasha and Serg plays next game with $100$-angled regular polygon . In the beggining Sasha set natural numbers in every angle. Then they make turn by turn, first turn is made by Serg. Serg turn is to take two opposite angles and add $1$ to its numbers. Sasha turn is to take two neigbour angles and add $1$ to its numbers. Serg want to maximize amount of odd numbers. What maximal number of odd numbers can he get no matter how Sasha plays?
2009 China Girls Math Olympiad, 8
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$
2021 Federal Competition For Advanced Students, P2, 6
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers.
Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer.
(Walther Janous)
2006 Taiwan TST Round 1, 2
Let $p,q$ be two distinct odd primes. Calculate
$\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.
2001 AIME Problems, 9
In triangle $ABC$, $AB=13,$ $BC=15$ and $CA=17.$ Point $D$ is on $\overline{AB},$ $E$ is on $\overline{BC},$ and $F$ is on $\overline{CA}.$ Let $AD=p\cdot AB,$ $BE=q\cdot BC,$ and $CF=r\cdot CA,$ where $p,$ $q,$ and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5.$ The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2000 JBMO ShortLists, 10
Prove that there are no integers $x,y,z$ such that
\[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]
1992 IMO, 3
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
V Soros Olympiad 1998 - 99 (Russia), 10.8
In how many ways can you choose several numbers from the numbers $1,2,3,..., 11$ so that among the selected numbers there are not three consecutive numbers?
DMM Individual Rounds, 2002 Tie
[b]p1.[/b] Suppose $a$, $b$ and $c$ are integers such that $c$ divides $a^n + b^n$ for all integers, $n \ge 1$. If the greatest common divisor of $a$ and $b$ is $7$, what is the largest possible value of $c$?
[b]p2.[/b] Consider a sequence of points $\{P_1, P_2,...\}$ on a circle w with the property that $\overline{P_{i+1}P_{i+2}}$ is parallel to the tangent line through $P_i$ for each $i \ge 1$. If $P_5 = P_1$, what is the largest possible angle formed by $P_1P_3P_2$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2016 IMAR Test, 1
Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the fi rst $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?
2015 Singapore Junior Math Olympiad, 1
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.
1982 All Soviet Union Mathematical Olympiad, 331
Once upon a time, three boys visited a library for the first time.
The first decided to visit the library every second day.
The second decided to visit the library every third day.
The third decided to visit the library every fourth day.
The librarian noticed, that the library doesn't work on Wednesdays.
The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases.
They strictly obeyed these rules.
Some Monday later I met them all in that library.
What day of week was when they visited a library for the first time?
2023/2024 Tournament of Towns, 4
4. There are several (at least two) positive integers written along the circle. For any two neighboring integers one is either twice as big as the other or five times as big as the other. Can the sum of all these integers equal 2023 ?
Sergey Dvoryaninov