Found problems: 85335
1985 IberoAmerican, 1
Find all the triples of integers $ (a, b,c)$ such that:
\[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]
Kyiv City MO 1984-93 - geometry, 1985.9.5
Outside the parallelogram $ABCD$ on its sides $AB$ and $BC$ are constructed equilateral triangles $ABK$, and $BCM$. Prove that the triangle $KMD$ is equilateral.
1980 Bulgaria National Olympiad, Problem 6
Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.
2013 Cuba MO, 1
Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.
The Golden Digits 2024, P2
Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$.
[i]Proposed by Andrei Vila[/i]
2018 CIIM, Problem 4
Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$. Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$, let $p(x)$ the product of the smallest and largest root of $f(x)-s$. Determine the smallest possible value that $p(s)$ for $s\in S$.
2024 Belarusian National Olympiad, 9.1
Find all triples $(x,y,z)$ of positive real numbers such that
$$
\begin{cases}
2x^2+y^3=3 \\
3y^2+z^3=4 \\
4z^2+x^3=5 \\
\end{cases}
$$
[i]M. Zorka[/i]
1967 IMO Longlists, 54
Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?
1987 IMO Longlists, 11
Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$.
We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?
1940 Putnam, B7
Which is greater
$$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$
where $n>8?$
2015 Math Prize for Girls Problems, 10
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
2010 Sharygin Geometry Olympiad, 5
Let $BH$ be an altitude of a right-angled triangle $ABC$ ($\angle B = 90^o$). The incircle of triangle $ABH$ touches $AB,AH$ in points $H_1, B_1$, the incircle of triangle $CBH$ touches $CB,CH$ in points $H_2, B_2$, point $O$ is the circumcenter of triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.
1971 IMO Longlists, 29
A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.
2021 Purple Comet Problems, 19
For some integers $u$,$ v$, and $w$, the equation
$$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$
holds for all real numbers a, b, and c that satisfy
$$au + bv + cw = 0$$
Find the minimum possible value of $u^2 + v^2 + w^2$.
2024 Bulgarian Winter Tournament, 12.1
Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?
2011 VTRMC, Problem 6
Let $S$ be a set with an asymmetric relation $<$; this means that if $a,b\in S$ and $a<b$, then we do not have $b<a$. Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a,b\in S$, then $a<b$ if and only if $a\prec b$, and if $x,y\in T$ with $x\prec y$, then there exists $t\in T$ such that $x\prec t\prec y$.
2014 Cezar Ivănescu, 3
Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that
$$ A^2+BA+C=0=D^2+BD+C. $$
Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $
2014 Bundeswettbewerb Mathematik, 2
The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$.
Note: In all the triangles the three vertices do not lie on a straight line.
2023 ELMO Shortlist, N2
Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]
[i]Proposed by Holden Mui[/i]
2016 Fall CHMMC, 2
Alice and Bob find themselves on a coordinate plane at time $t=0$ at $A(1,0)$ and $B(-1,0)$ respectively. They have no sense of direction, but they want to find each other. They each pick a direction independently and with uniform random probability. Both Alice and Bob travel at a constant speed of $1 \frac{unit}{min}$ in their chosen directions. They continue on their straight line paths forever, each hoping to catch sight of the other. They both have a $1$ unit radius of view; they can see something if and only if its distance from them is at most $1$ unit. What is the probability they never see each other?
2019 Auckland Mathematical Olympiad, 1
Given a convex quadrilateral $ABCD$ in which $\angle BAC = 20^o$, $\angle CAD = 60^o$, $\angle ADB = 50^o$ , and $\angle BDC = 10^o$. Find $\angle ACB$.
2022 HMNT, 1
Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.
2012 Romania National Olympiad, 1
Let $P$ be a point inside the square $ABCD$ and $PA = 1$, $PB = \sqrt2$ and $PC =\sqrt3$.
a) Determine the length of segment $[PD]$.
b) Determine the angle $\angle APB$.
2005 Alexandru Myller, 2
Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$.
[i]Dan Popescu[/i]
2022 Stanford Mathematics Tournament, 8
Let $\Gamma$ and $\Omega$ be circles that are internally tangent at a point $P$ such that $\Gamma$ is contained entirely in $\Omega$. Let $A,B$ be points on $\Omega$ such that the lines $PB$ and $PA$ intersect the circle $\Gamma$ at $Y$ and $X$ respectively, where $X,Y\neq P$. Let $O_1$ be the circle with diameter $AB$ and $O_2$ be the circle with diameter $XY$. Let $F$ be the foot of $Y$ on $XP$. Let $T$ and $M$ be points on $O_1$ and $O_2$ respectively such that $TM$ is a common tangent to $O_1$ and $O_2$. Let $H$ be the orthocenter of $\triangle ABP$. Given that $PF=12$, $FX=15$, $TM=18$, $PB=50$, find the length of $AH$.