Found problems: 85335
2015 Czech and Slovak Olympiad III A, 2
Let $A=[0,0]$ and $B=[n,n]$. In how many ways can we go from $A$ to $B$, if we always want to go from lattice point to its neighbour (i.e. point with one coordinate the same and one smaller or bigger by one), we never want to visit the same point twice and we want our path to have length $2n+2$?
(For example, path $[0,0],[0,1],[-1,1],[-1,2],[0,2],[1,2],[2,2],[2,3],[3,3]$ is one of the paths for $n=3$)
2014 Indonesia MO, 1
A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
1990 Bulgaria National Olympiad, Problem 6
The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?
2017 China Northern MO, 1
Define sequence $(a_n):a_1=\text{e},a_2=\text{e}^3,\text{e}^{1-k}a_n^{k+2}=a_{n+1}a_{n-1}^{2k}$ for all $n\geq2$, where $k$ is a positive real number. Find $\prod_{i=1}^{2017}a_i$.
2016 Novosibirsk Oral Olympiad in Geometry, 4
The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]
1982 Bulgaria National Olympiad, Problem 1
Find all pairs of natural numbers $(n,k)$ for which
$(n+1)^{k}-1 = n!$.
2010 Thailand Mathematical Olympiad, 2
The Ministry of Education selects $2010$ students from $5$ regions of Thailand to participate in a debate tournament, where each pair of students will debate in one of the three topics: politics, economics, and societal problems. Show that there are $3$ students who were born in the same month, come from the same region, are of the same gender , and whose pairwise debates are on the same topic.
2015 Peru MO (ONEM), 1
If $C$ is a set of $n$ points in the plane that has the following property: For each point $P$ of $C$, there are four points of $C$, each one distinct from $P$ , which are the vertices of a square. Find the smallest possible value of $n$.
2014 IFYM, Sozopol, 3
In an acute $\Delta ABC$, $AH_a$ and $BH_b$ are altitudes and $M$ is the middle point of $AB$. The circumscribed circles of $\Delta AMH_a$ and $\Delta BMH_b$ intersect for a second time in $P$. Prove that point $P$ lies on the circumscribed circle of $\Delta ABC$.
Estonia Open Senior - geometry, 2017.1.5
On the sides $BC, CA$ and $AB$ of triangle $ABC$, respectively, points $D, E$ and $F$ are chosen. Prove that
$\frac12 (BC + CA + AB)<AD + BE + CF<\frac 32 (BC + CA + AB)$.
2020 Belarusian National Olympiad, 11.6
Functions $f(x)$ and $g(x)$ are defined on the set of real numbers and take real values. It is known that $g(x)$ takes all real values, $g(0)=0$, and for all $x,y \in \mathbb{R}$ the following equality holds
$$f(x+f(y))=f(x)+g(y)$$
Prove that $g(x+y)=g(x)+g(y)$ for all $x,y \in \mathbb{R}$.
2024 LMT Fall, 10
Find the sum of all positive integers $n\le 2024$ such that all pairs of distinct positive integers $(a,b)$ that satisfy $ab=n$ have a sum that is a perfect square.
2020 Iranian Our MO, 4
In a school there are $n$ classes and $k$ student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than $k$ students. If $k-1$ is not a perfect square, prove that there exist a student that has attended in at least $\sqrt k$ classes.
[i]Proposed by Mohammad Moshtaghi Far, Kian Shamsaie[/i] [b]Rated 4[/b]
1967 IMO Longlists, 31
An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.
2007 France Team Selection Test, 3
Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel.
Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.
2021 Azerbaijan Senior NMO, 2
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
2011 Romania National Olympiad, 2
Prove that any natural number smaller or equal than the factorial of a natural number $ n $ is the sum of at most $ n $ distinct divisors of the factorial of $ n. $
2015 EGMO, 1
Let $\triangle ABC$ be an acute-angled triangle, and let $D$ be the foot of the altitude from $C.$ The angle bisector of $\angle ABC$ intersects $CD$ at $E$ and meets the circumcircle $\omega$ of triangle $\triangle ADE$ again at $F.$
If $\angle ADF = 45^{\circ}$, show that $CF$ is tangent to $\omega .$
2022 Singapore MO Open, Q3
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
2008 Princeton University Math Competition, B1
Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange?
2010 Postal Coaching, 3
Determine the smallest odd integer $n \ge 3$, for which there exist $n$ rational numbers $x_1 , x_2 , . . . , x_n$ with the following properties:
$(a)$ \[\sum_{i=1}^{n} x_i =0 , \ \sum_{i=1}^{n} x_i^2 = 1.\]
$(b)$ \[x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.\]
2013 Purple Comet Problems, 13
Find $n$ so that $4^{4^{4^2}}=2^{8^n}$.
2008 District Olympiad, 2
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that:
[b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic.
[b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $
2018 IMAR Test, 3
Let $S$ be a finite set and let $\mathcal{P}(S)$ be its power set, i.e., the set of all subsets of $S$, the empty set and $S$, inclusive. If $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S),$ let \[\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.\] Given a non-negative integer $n\leqslant |S|,$ determine the minimal size $\mathcal{A}\vee \mathcal{B}$ may have, where $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S)$ such that $|\mathcal{A}|+|\mathcal{B}|>2^n$.
[i]Amer. Math. Monthly[/i]