Found problems: 85335
2021 239 Open Mathematical Olympiad, 4
Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.
1977 Putnam, B2
Given a convex quadrilateral $ABCD$ and a point $O$ not in the plane $ABCD$, locate point $A'$ on line $OA,$ point $B'$ on the line $OB$, point $C'$ on line $OC,$ and point $D'$ on line $OD$ so that $A'B'C'D'$ is a parallelogram.
2023 Princeton University Math Competition, B2
Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$
2011 BAMO, 4
Three circles $k_1, k_2$, and $k_3$ intersect in point $O$. Let $A, B$, and $C$ be the second intersection points (other than $O$) of $k_2$ and $k_3, k_1$ and $k_3$, and $k_1$ and $k_2$, respectively. Assume that $O$ lies inside of the triangle $ABC$. Let lines $AO,BO$, and $CO$ intersect circles $k_1, k_2$, and $k_3$ for a second time at points $A', B'$, and $C'$, respectively. If $|XY|$ denotes the length of segment $XY$, prove that $\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1$
2000 France Team Selection Test, 3
$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.
2007 Italy TST, 1
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?
2002 AMC 10, 24
Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is
$ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$
1977 Swedish Mathematical Competition, 4
Show that if
\[
\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1
\]
then
\[
\frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1
\]
1979 Putnam, B1
Prove or disprove: there is at least one straight line normal to the graph of $y=\cosh x$ at a point $(a,\cosh a)$ and also normal to the graph of $y=$ $\sinh x$ at a point $(c,\sinh c).$
2016 Switzerland - Final Round, 4
There are $2016$ different points in the plane. Show that between these points at least $45$ different distances occur.
2009 Pan African, 3
Let $x$ be a real number with the following property: for each positive integer $q$, there exists an integer $p$, such that
\[\left|x-\frac{p}{q} \right|<\frac{1}{3q}. \]
Prove that $x$ is an integer.
2017 Stars of Mathematics, 3
Let
$$ 2^{-n_1}+2^{-n_2}+2^{-n_3}+\cdots,\quad1\le n_1\le n_2\le n_3\le\cdots $$
be the binary representation of the golden ratio minus one.
Prove that $ n_k\le 2^{k-1}-2, $ for all integers $ k\ge 4. $
[i]American Mathematical Monthly[/i]
2014 Ukraine Team Selection Test, 8
The quadrilateral $ABCD$ is inscribed in the circle $\omega$ with the center $O$. Suppose that the angles $B$ and $C$ are obtuse and lines $AD$ and $BC$ are not parallel. Lines $AB$ and $CD$ intersect at point $E$. Let $P$ and $R$ be the feet of the perpendiculars from the point $E$ on the lines $BC$ and $AD$ respectively. $Q$ is the intersection point of $EP$ and $AD, S$ is the intersection point of $ER$ and $BC$. Let K be the midpoint of the segment $QS$ . Prove that the points $E, K$, and $O$ are collinear.
2016 Online Math Open Problems, 10
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
[i]Proposed by Yannick Yao[/i]
2006 Romania National Olympiad, 2
Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that \[ n = \sum_{1\leq i < j \leq k } a_ia_j . \]
2023 Turkey Team Selection Test, 8
Initially the equation
$$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$
is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?
2021 Indonesia TST, G
Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?
1992 AMC 8, 10
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is
[asy]
for (int a=0; a <= 3; ++a)
{
for (int b=0; b <= 3-a; ++b)
{
fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey);
}
}
for (int c=0; c <= 3; ++c)
{
draw((c,0)--(c,4-c),linewidth(1));
draw((0,c)--(4-c,c),linewidth(1));
draw((c+1,0)--(0,c+1),linewidth(1));
}
label("$8$",(2,0),S);
label("$8$",(0,2),W);
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 64$
1973 All Soviet Union Mathematical Olympiad, 182
Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are constructed on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB_1C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.)
a) Prove that the circles circumscribed around the outer triangles intersect in one point.
b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point
1991 India Regional Mathematical Olympiad, 2
If $a,b,c,d$ be any four positive real numbers, then prove that \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4. \]
2015 Mathematical Talent Reward Programme, MCQ: P 7
How many $x$ are there such that $x,[x],\{x\}$ are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here $[x]$ is the largest integer less than equal to $x$ and $\{x\}=x-[ x]$ )
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2025 Azerbaijan Senior NMO, 2
Find all the positive reals $x,y,z$ satisfying the following equations: $$y=\frac6{(2x-1)^2}$$ $$z=\frac6{(2y-1)^2}$$ $$x=\frac6{(2z-1)^2}$$
2009 Bundeswettbewerb Mathematik, 4
How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?
1994 AIME Problems, 4
Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)
2018 Regional Olympiad of Mexico Southeast, 3
Let $ABC$ a triangle with circumcircle $\Gamma$ and $R$ a point inside $ABC$ such that $\angle ABR=\angle RBC$. Let $\Gamma_1$ and $\Gamma_2$ the circumcircles of triangles $ARB$ and $CRB$ respectly. The parallel to $AC$ that pass through $R$, intersect $\Gamma$ in $D$ and $E$, with $D$ on the same side of $BR$ that $A$ and $E$ on the same side of $BR$ that $C$. $AD$ intersect $\Gamma_1$ in $P$ and $CE$ intersect $\Gamma_2$ in $Q$. Prove that $APQC$ is cyclic if and only if $AB=BC$