Found problems: 85335
Durer Math Competition CD Finals - geometry, 2008.C3
We divided a regular octagon into parallelograms. Prove that there are at least $2$ rectangles between the parallelograms.
2010 Iran Team Selection Test, 11
Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.
2015 CCA Math Bonanza, L5.1
What is the integer closest to $\pi^{\pi}$? (No calculator allowed!)
[i]2015 CCA Math Bonanza Lightning Round #5.1[/i]
1977 IMO Longlists, 51
Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.
2019 Teodor Topan, 2
Let $ P $ be a point on the side $ AB $ of the triangle $ ABC. $ The parallels through $ P $ of the medians $ AA_1,BB_1 $ intersect $ BC,AC $ at $ R,Q, $ respectively. Show that $ P, $ the middlepoint of $ RQ $ and the centroid of $ ABC $ are collinear.
1982 Putnam, B4
Let $n_1,n_2,\ldots,n_s$ be distinct integers such that
$$(n_1+k)(n_2+k)\cdots(n_s+k)$$is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. For each of the following assertions give a proof or a counterexample:
$(\text a)$ $|n_i|=1$ for some $i$
$(\text b)$ If further all $n_i$ are positive, then
$$\{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}.$$
2022 Argentina National Olympiad, 3
Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$.
Note: The set of points that satisfy a property is called a locus.
2013 Chile TST Ibero, 1
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.
2018 Dutch IMO TST, 3
Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satis es $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.
2021 CCA Math Bonanza, L3.2
A frog is standing in a center of a $3 \times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with their current square uniformly at random, and jumps onto the lilypad on their chosen square. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected number of times the frog jumps?
[i]2021 CCA Math Bonanza Lightning Round #3.2[/i]
2005 National Olympiad First Round, 17
Construct outer squares $ABMN$, $BCKL$, $ACPQ$ on sides $[AB]$, $[BC]$, $[CA]$ of triangle $ABC$, respectively. Construct squares $NQZT$ and $KPYX$ on segments $[NQ]$ and $[KP]$. If $Area(ABMN) - Area(BCKL)=1$, what is $Area(NQZT)-Area(KPYX)$?
$
\textbf{(A)}\ \dfrac 34
\qquad\textbf{(B)}\ \dfrac 53
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
2014 German National Olympiad, 2
For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$
1899 Eotvos Mathematical Competition, 2
Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show that $x^3_1$ and $x^3_2$ are the roots of $$y^3-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3 =0.$$
2018 IMO, 5
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$
is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.
[i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]
2024 Indonesia TST, C
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.
[i]Thanasin Nampaisarn, Thailand[/i]
2006 Costa Rica - Final Round, 3
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2013 Princeton University Math Competition, 6
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
2024 AMC 12/AHSME, 18
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
2021 Bulgaria National Olympiad, 3
Find all $f:R^+ \rightarrow R^+$ such that
$f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$
@below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url]
[quote]Feel free to start individual threads for the problems as usual[/quote]
2021 Latvia TST, 2.5
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2014 Contests, 1
The diagram below shows a circle with center $F$. The angles are related with $\angle BFC = 2\angle AFB$, $\angle CFD = 3\angle AFB$, $\angle DFE = 4\angle AFB$, and $\angle EFA = 5\angle AFB$. Find the degree measure of $\angle BFC$.
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
dotfactor=4;
draw(unitcircle);
pair A,B,C,D,E,F;
A=dir(90);
B=dir(66);
C=dir(18);
D=dir(282);
E=dir(210);
F=origin;
dot("$F$",F,NW);
dot("$A$",A,dir(90));
dot("$B$",B,dir(66));
dot("$C$",C,dir(18));
dot("$D$",D,dir(306));
dot("$E$",E,dir(210));
draw(F--E^^F--D^^F--C^^F--B^^F--A);
[/asy]
1995 China National Olympiad, 3
Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow:
$x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$;
$ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $
$i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$.
Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.
2016 Purple Comet Problems, 6
The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots that form a 45 degree angle with a side of the square are
drawn as shown. The area of the shaded region is 75. Find the area of the original square.
For diagram go to http://www.purplecomet.org/welcome/practice
2021 Balkan MO Shortlist, N1
Let $n \geq 2$ be an integer and let \[M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}\] be the set of the arithmetic means of the elements of all non-empty subsets of $\{1, 2, ..., n\}$. Find \[\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.\]
2006 Pre-Preparation Course Examination, 3
Show that if $f: [0,1]\rightarrow [0,1]$ is a continous function and it has topological transitivity then periodic points of $f$ are dense in $[0,1]$.
Topological transitivity means there for every open sets $U$ and $V$ there is $n>0$ such that $f^n(U)\cap V\neq \emptyset$.