Found problems: 85335
1976 Spain Mathematical Olympiad, 4
Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.
2015 Chile TST Ibero, 1
Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$:
\[
f(g(n)) = n + 2015,
\]
\[
g(f(n)) = n^2 + 2015.
\]
2021 Francophone Mathematical Olympiad, 3
Every point in the plane was colored in red or blue. Prove that one the two following statements is true:
$\bullet$ there exist two red points at distance $1$ from each other;
$\bullet$ there exist four blue points $B_1$, $B_2$, $B_3$, $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i - j|$ from each other, for all integers $i $ and $j$ such as $1 \le i \le 4$ and $1 \le j \le 4$.
2020 AMC 12/AHSME, 23
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that
$$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$
then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$
2020 New Zealand MO, 5
A sequence of $A$s and $B$s is called [i]antipalindromic [/i] if writing it backwards, then turning all the $A$s into $B$s and vice versa, produces the original sequence. For example $ABBAAB$ is antipalindromic. For any sequence of $A$s and $B$s we define the cost of the sequence to be the product of the positions of the $A$s. For example, the string $ABBAAB$ has cost $1\cdot 4 \cdot 5 = 20$. Find the sum of the costs of all antipalindromic sequences of length $2020$.
2014 AMC 12/AHSME, 23
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
$\textbf{(A) }874\qquad
\textbf{(B) }883\qquad
\textbf{(C) }887\qquad
\textbf{(D) }891\qquad
\textbf{(E) }892\qquad$
2015 India IMO Training Camp, 2
For a composite number $n$, let $d_n$ denote its largest proper divisor. Show that there are infinitely many $n$ for which $d_n +d_{n+1}$ is a perfect square.
1926 Eotvos Mathematical Competition, 3
The circle $k'$ rolls along the inside of circle $k$, the radius of $k$ is twice the radius of $k'$. Describe the path of a point on $k$..
2006 CentroAmerican, 3
For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions.
2015 AMC 8, 22
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
$
\textbf{(A) } 21 \qquad
\textbf{(B) } 30 \qquad
\textbf{(C) } 60 \qquad
\textbf{(D) } 90 \qquad
\textbf{(E) } 1080
$
2016 NIMO Summer Contest, 4
Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Michael Tang[/i]
2006 MOP Homework, 5
Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in
the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that
$\angle AOB +\angle COD = 180$.
1999 Balkan MO, 2
Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.
2018 India IMO Training Camp, 3
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
1989 APMO, 1
Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let
\[ S = x_1 + x_2 + \cdots + x_n. \]
Prove that
\[ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \]
2015 South East Mathematical Olympiad, 2
Given a sequence $\{ a_n\}_{n\in \mathbb{Z}^+}$ defined by $a_1=1$ and $a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}$ for all positive integer $k$.
Prove that, for any positive integer $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.
2024 ELMO Shortlist, C7
Let $n\ge 2$ be a positive integer, and consider an $n\times n$ grid of $n^2$ equilateral triangles. Two triangles are adjacent if they share at least one vertex. Each triangle is colored red or blue, splitting the grid into regions.
Find, with proof, the minimum number of triangles in the largest region.
[i]Rohan Bodke[/i]
2007 China Second Round Olympiad, 2
In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.
2013 Online Math Open Problems, 16
Al has the cards $1,2,\dots,10$ in a row in increasing order. He first chooses the cards labeled $1$, $2$, and $3$, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled $2$, $3$, and $4$, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence $3,2,1,4,5,\dots,$ and his second move could have rearranged that to $2,4,1,3,5,\dots$.) He continues this process until he has rearranged the cards with labels $8$, $9$, $10$. Determine the number of possible orderings of cards he can end up with.
[i]Proposed by Ray Li[/i]
1994 Tournament Of Towns, (413) 1
Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that
$$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$
(NB Vassiliev)
2012 Centers of Excellency of Suceava, 2
Calculate $ \lim_{n\to\infty } \frac{f(1)+(f(2))^2+\cdots +(f(n))^n}{(f(n))^n} , $ where $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ is an unbounded and nondecreasing function.
[i]Dan Popescu[/i]
2022 Assara - South Russian Girl's MO, 5
Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.
2007 Nicolae Păun, 2
Prove that the real and imaginary part of the number $ \prod_{j=1}^n (j^3+\sqrt{-1}) $ is positive, for any natural numbers $ n. $
[i]Nicolae Mușuroia[/i]
2018 India Regional Mathematical Olympiad, 1
Let $ABC$ be a triangle with integer sides in which $AB<AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB,AC)>1$.
2011 Iran MO (3rd Round), 3
We have connected four metal pieces to each other such that they have formed a tetragon in space and also the angle between two connected metal pieces can vary.
In the case that the tetragon can't be put in the plane, we've marked a point on each of the pieces such that they are all on a plane. Prove that as the tetragon varies, that four points remain on a plane.
[i]proposed by Erfan Salavati[/i]