Found problems: 85335
2022 Brazil EGMO TST, 7
Let $a_1, a_2, \cdots, a_{2n}$ be $2n$ elements of $\{1, 2, 3, \cdots, 2n-1\}$ ($n>3$) with the sum $a_1+a_2+\cdots+a_{2n}=4n$. Prove that exist some numbers $a_i$ with the sum is $2n$.
2019 Dutch IMO TST, 2
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
1993 Tournament Of Towns, (371) 3
Each number in the second, third, and further rows of the following triangle:
[img]https://cdn.artofproblemsolving.com/attachments/1/5/589d9266749477b0f56f0f503d4f18a6e5d695.png[/img]
is equal to the difference of two neighbouring numbers standing above it. Find the last number (at the bottom of the triangle).
(GW Leibnitz,)
2021 AMC 10 Spring, 21
A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C’$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C’D=\frac{1}{3}$. What is the perimeter of $\triangle AEC’$?
[asy]
//Diagram by Samrocksnature
pair A=(0,1);
pair CC=(0.666666666666,1);
pair D=(1,1);
pair F=(1,0.62);
pair C=(1,0);
pair B=(0,0);
pair G=(0,0.25);
pair H=(-0.13,0.41);
pair E=(0,0.5);
dot(A^^CC^^D^^C^^B^^E);
draw(E--A--D--F);
draw(G--B--C--F, dashed);
fill(E--CC--F--G--H--E--CC--cycle, gray);
draw(E--CC--F--G--H--E--CC);
label("A",A,NW);
label("B",B,SW);
label("C",C,SE);
label("D",D,NE);
label("E",E,NW);
label("C'",CC,N);
[/asy]
$\textbf{(A) }2 \qquad \textbf{(B) }1+\frac{2}{3}\sqrt{3} \qquad \textbf{(C) }\frac{13}{6} \qquad \textbf{(D) }1+\frac{3}{4}\sqrt{3} \qquad \textbf{(E) }\frac{7}{3}$
2016 Online Math Open Problems, 27
Let $ABC$ be a triangle with circumradius $2$ and $\angle B-\angle C=15^\circ$. Denote its circumcenter as $O$, orthocenter as $H$, and centroid as $G$. Let the reflection of $H$ over $O$ be $L$, and let lines $AG$ and $AL$ intersect the circumcircle again at $X$ and $Y$, respectively. Define $B_1$ and $C_1$ as the points on the circumcircle of $ABC$ such that $BB_1\parallel AC$ and $CC_1\parallel AB$, and let lines $XY$ and $B_1C_1$ intersect at $Z$. Given that $OZ=2\sqrt 5$, then $AZ^2$ can be expressed in the form $m-\sqrt n$ for positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Michael Ren[/i]
2022 Romania EGMO TST, P4
For every positive integer $N\geq 2$ with prime factorisation $N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ we define \[f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k.\] Let $x_0\geq 2$ be a positive integer. We define the sequence $x_{n+1}=f(x_n)$ for all $n\geq 0.$ Prove that this sequence is eventually periodic and determine its fundamental period.
2011 Morocco National Olympiad, 2
Compute the sum
\[S=1+2+3-4-5+6+7+8-9-10+\dots-2010\]
where every three consecutive $+$ are followed by two $-$.
2007 Mongolian Mathematical Olympiad, Problem 2
For all $n\ge2$, let $a_n$ be the product of all coprime natural numbers less than $n$. Prove that
(a) $n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha$
(b) $n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha$
Here $p$ is an odd prime number and $\alpha\in\mathbb N$.
2020 Brazil National Olympiad, 3
Let $r_A,r_B,r_C$ rays from point $P$. Define circles $w_A,w_B,w_C$ with centers $X,Y,Z$ such that $w_a$ is tangent to $r_B,r_C , w_B$ is tangent to $r_A, r_C$ and $w_C$ is tangent to $r_A,r_B$. Suppose $P$ lies inside triangle $XYZ$, and let $s_A,s_B,s_C$ be the internal tangents to circles $w_B$ and $w_C$; $w_A$ and $w_C$; $w_A$ and $w_B$ that do not contain rays $r_A,r_B,r_C$ respectively. Prove that $s_A, s_B, s_C$ concur at a point $Q$, and also that $P$ and $Q$ are isotomic conjugates.
[b]PS: The rays can be lines and the problem is still true.[/b]
Kvant 2022, M2714
Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$.
[i]From the folklore[/i]
2012 NIMO Problems, 7
In quadrilateral $ABCD$, $AC = BD$ and $\measuredangle B = 60^\circ$. Denote by $M$ and $N$ the midpoints of $\overline{AB}$ and $\overline{CD}$, respectively. If $MN = 12$ and the area of quadrilateral $ABCD$ is 420, then compute $AC$.
[i]Proposed by Aaron Lin[/i]
2019 Jozsef Wildt International Math Competition, W. 7
If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$
2012 ELMO Shortlist, 8
Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).
a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
[i]Victor Wang.[/i]
2011 QEDMO 8th, 7
$9004$ lemmings, including an equal number of both sexes, cross in rank and file the new bridge from Eyjafjallajokull to Katla. The entire column therefore moves equidistantly and at a constant speed about the bridge, whereby this is able to hold exactly half of the lemmings for the present distance. The bridge should fulfill as many lemming dreams as possible and at the same time preserve the species be opened briefly at some point in order to halve the total population. However, the law to prevent gender discrimination requires that exact half is female. Show that these sufficient claims are also can be done.
[hide=original wording]9004 Lemminge, davon gleich viele von beiden Geschlechtern, uberqueren in Reih und Glied die neue Brucke vom Eyjafjallajokull zum Katla. Die gesammte Kolonne bewegt sich also aquidistant und mit konstanter Geschwindigkeit uber die Brucke, wobei diese fur den vorliegenden Abstand genau die Halfte der Lemminge zu fassen vermag. Zur Erfullung moglichst vieler Lemmingtraume und gleichzeitiger Arterhaltung soll die Brucke irgendwann einmalig kurzzeitig aufgeklappt werden, um die Gesamtpopulation zu halbieren. Das Gesetz zur Verhinderung geschlechtsspezifischer Diskriminierung erfordert jedoch, dass davon ex akt die Halfte weiblich ist.
Man zeige, dass diesen Anspruchen auch Genuge getan werden kann.[/hide]
[img]https://cdn.artofproblemsolving.com/attachments/f/8/3bf1ef0f90d3eb3761ca3db04ed48480c8aab5.png[/img]
2021 AMC 12/AHSME Fall, 12
What is the number of terms with rational coefficients among the $1001$ terms of the expression $( x \sqrt[3]{2} + y \sqrt{3})^{1000}$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\
500 \qquad\textbf{(E)}\ 501$
2021 Korea National Olympiad, P4
For a positive integer $n$, there are two countries $A$ and $B$ with $n$ airports each and $n^2-2n+ 2$ airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in $A$ and each airport in $B$, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports $P$ and $Q$, denote by "[i]$(P, Q)$-travel route[/i]" the list of airports $T_0, T_1, \ldots, T_s$ satisfying the following conditions.
[list]
[*] $T_0=P,\ T_s=Q$
[*] $T_0, T_1, \ldots, T_s$ are all distinct.
[*] There exists an airline that operates between the airports $T_i$ and $T_{i+1}$ for all $i = 0, 1, \ldots, s-1$.
[/list]
Prove that there exist two airports $P, Q$ such that there is no or exactly one [i]$(P, Q)$-travel route[/i].
[hide=Graph Wording]Consider a complete bipartite graph $G(A, B)$ with $\vert A \vert = \vert B \vert = n$. Suppose there are $n^2-2n+2$ colors and each edge is colored by one of these colors. Define $(P, Q)-path$ a path from $P$ to $Q$ such that all of the edges in the path are colored the same. Prove that there exist two vertices $P$ and $Q$ such that there is no or only one $(P, Q)-path$. [/hide]
2020 Korea Junior Math Olympiad, 3
The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too.
For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$
Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$
2008 Indonesia TST, 3
$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.
2009 Kazakhstan National Olympiad, 6
Let $P(x)$ be polynomial with integer coefficients.
Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$
2014 Hanoi Open Mathematics Competitions, 6
Let $a,b,c$ be the length sides of a given triangle and $x,y,z$ be the sides length of bisectrices, respectively. Prove the following inequality $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
2003 District Olympiad, 2
Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as
$$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$
[b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $
[b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.
2008 Harvard-MIT Mathematics Tournament, 8
Determine the number of ways to select a sequence of $ 8$ sets $ A_1,A_2,\ldots,A_8$, such that each is a subset (possibly empty) of $ \{1,2\}$ and $ A_m$ contains $ A_n$ if $ m$ divides $ n$.
2008 Iran Team Selection Test, 2
Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.
1993 Denmark MO - Mohr Contest, 1
Three friends A, B and C have a total of $120$ kroner. First, A gives as much money to B as B already has. Next, B gives as many money to C that C already has. In the end, C gives the same amount of money to A as A now has. After these transactions, A, B and C have equal amounts of money. How many money did each of the three companions have originally?
2005 Manhattan Mathematical Olympiad, 2
What is the largest number of Sundays can be in one year? Explain your answer.