Found problems: 85335
2021 Ukraine National Mathematical Olympiad, 6
Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$.
(Nazar Serdyuk)
2000 VJIMC, Problem 3
Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?
2018 Iranian Geometry Olympiad, 1
As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is to minimize the total length of the straight cuts. How to achieve this goal, and what is that minimized length? Show the correct cuts and write the final answer. There is no need to prove the answer.
[i]Proposed by Morteza Saghafian[/i]
2011 Federal Competition For Advanced Students, Part 2, 2
Let $k$ and $n$ be positive integers.
Show that if $x_j$ ($1\leqslant j\leqslant n$) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$, then
\[\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}\]
2006 Junior Balkan Team Selection Tests - Moldova, 1
Five segments have lengths such that any three of them can be sides of a - possibly degenerate - triangle. Also, the lengths of these segments are nonzero and pairwisely different. Prove that there exists at least one acute-angled triangle among these triangles.
2015 Bosnia Herzegovina Team Selection Test, 3
Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.
1985 AMC 8, 21
Mr. Green receives a $ 10 \%$ raise every year. His salary after four such raises has gone up by what percent?
\[ \textbf{(A)}\ \text{less than }40 \% \qquad
\textbf{(B)}\ 40 \% \qquad
\textbf{(C)}\ 44 \% \qquad
\textbf{(D)}\ 45 \% \qquad
\textbf{(E)}\ \text{More than }45 \%
\]
2016 Germany National Olympiad (4th Round), 1
Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]
2017-IMOC, A5
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.
2016 Sharygin Geometry Olympiad, P3
Let $AH_1$, $BH_2$ be two altitudes of an acute-angled triangle $ABC$ , $D$ be the projection of $H_1$ to $AC$, $E$ be the projection of $D$ to $AB$, $F$ be the common point of $ED$ and $AH_1$.
Prove that $H_2F \parallel BC$.
[i](Proposed by E.Diomidov)[/i]
2000 USAMO, 6
Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that
\[
\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.
\]
1968 Putnam, A3
Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that
(1) $A_1 =\emptyset$.
(2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$
2003 Romania National Olympiad, 1
Find positive integers $ a,b$ if for every $ x,y\in[a,b]$, $ \frac1x\plus{}\frac1y\in[a,b]$.
2006 All-Russian Olympiad, 5
Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.
2014 Saudi Arabia BMO TST, 5
Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that
[list]
[*] [b](a)[/b] the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$;
the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$;
[*] [b](b)[/b] the board is [i]totally non-symmetric[/i]: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.[/list]
2019 USEMO, 4
Prove that for any prime $p,$ there exists a positive integer $n$ such that
\[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\]
[i]Robin Son[/i]
2008 Romanian Master of Mathematics, 2
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
2007 AMC 12/AHSME, 8
Tom's age is $ T$ years, which is also the sum of the ages of his three children. His age $ N$ years ago was twice the sum of their ages then. What is $ \frac {T}{N}$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
2004 Junior Balkan Team Selection Tests - Moldova, 1
Determine all triplets of integers $(x, y, z)$ that validate the inequality $x^2 + y^2 + z^2 <xy + 3y + 2z$.
2024 Bangladesh Mathematical Olympiad, P7
Let $ABCD$ be a square. $E$ and $F$ lie on sides $AB$ and $BC$, respectively, such that $BE = BF$. The line perpendicular to $CE$, which passes through $B$, intersects $CE$ and $AD$ at points $G$ and $H$, respectively. The lines $FH$ and $CE$ intersect at point $P$ and the lines $GF$ and $CD$ intersect at point $Q$. Prove that the line $DP$ is perpendicular to the line $BQ$.
2012 Tournament of Towns, 1
Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each positive integer $k$ there exists a positive integer $t = t(k)$ such that $a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each positive integer k?
1994 Romania TST for IMO, 1:
Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$.
We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$).
A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd).
(a)Show that the number of even sets equals the number of odd sets.
(b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets.
(c)Compute the sum of the measures of the odd sets.
2010 Contests, 1
The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex $B$. If the side length of the field is $60$ meters, and each one of the three regions has the same area, how far must the points $M$ and $N$ be from $D$?
Note: the area of each region includes the area the well occupies.
[asy]
pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));[/asy]
1957 Poland - Second Round, 2
Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.
2013 Online Math Open Problems, 29
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$, and let $d(n)$ denote the number of positive integer divisors of $n$. For example, $\phi(6) = 2$ and $d(6) = 4$. Find the sum of all odd integers $n \le 5000$ such that $n \mid \phi(n) d(n)$.
[i]Alex Zhu[/i]