Found problems: 85335
2011 Cuba MO, 6
Let $ABC$ be a triangle with circumcenter $O$. Let $\omega (O_1)$ be the circumference which passes through $A$ and $B$ and is tangent to $BC$ at $B$. $\omega (O_2)$ the circle that passes through $A$ and $C$ and is tangent to $BC$ at $C$. Let $M$ the midpoint of $O_1O_2$ and $D$ the symmetric point of $O$ with respect to $A$. Prove that $\angle O_1DM = \angle ODO_2$.
1990 Hungary-Israel Binational, 2
Let $ ABC$ be a triangle where $ \angle ACB\equal{}90^{\circ}$. Let $ D$ be the midpoint of $ BC$ and let $ E$, and $ F$ be points on $ AC$ such that $ CF\equal{}FE\equal{}EA$. The altitude from $ C$ to the hypotenuse $ AB$ is $ CG$, and the circumcentre of triangle $ AEG$ is $ H$. Prove that the triangles $ ABC$ and $ HDF$ are similar.
2010 Sharygin Geometry Olympiad, 21
A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that
\[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]
VII Soros Olympiad 2000 - 01, 11.2
For all valid values ​​of $a, b$, and $c$, solve the equation
$$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$
2024 Israel National Olympiad (Gillis), P1
Solve the following system (over the real numbers):
\[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\
3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]
2001 Mexico National Olympiad, 5
$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$. $D$ is the point on $AC$ such that $CD = AB$. Let L be the line through $B$ parallel to $AC$. Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$. Show that $MD = ND$.
2012 USAMTS Problems, 4
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. Let $m$ be a positive integer, $m\geq 3$. For every integer $i$ with $1\leq i\leq m$, let \[S_{m,i}=\left\{\left\lfloor\dfrac{2^m-1}{2^{i-1}}n-2^{m-i}+1\right\rfloor\,:\,n=1,2,3,\ldots\right\}.\] For example, for $m=3$,
\begin{align*}S_{3,1}&=\{\lfloor 7n-3\rfloor\,:\,n=1,2,3,\ldots\}
\\&=\{4,11,18,\ldots\},
\\S_{3,2}&=\left\{\left\lfloor\dfrac72n-1\right\rfloor\,:\,n=1,2,3,\ldots\right\}
\\&=\{2,6,9,\ldots\},
\\S_{3,3}&=\left\{\left\lfloor\dfrac74n\right\rfloor\,:\,n=1,2,3,\ldots\right\}
\\&=\{1,3,5,\ldots\}.\end{align*}
Prove that for all $m\geq 3$, each positive integer occurs in exactly one of the sets $S_{m,i}$.
2010 LMT, 12
Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?
2009 Junior Balkan MO, 2
Solve in non-negative integers the equation $ 2^{a}3^{b} \plus{} 9 \equal{} c^{2}$
2006 IMO Shortlist, 4
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
2019 Silk Road, 3
Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $
($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)
2007 VJIMC, Problem 3
Let $f:[0,1]\to\mathbb R$ be a continuous function such that $f(0)=f(1)=0$. Prove that the set
$$A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}$$is Lebesgue measureable and has Lebesgue measure at least $\frac12$.
2000 Mediterranean Mathematics Olympiad, 1
Let $F=\{1,2,...,100\}$ and let $G$ be any $10$-element subset of $F$. Prove that there exist two disjoint nonempty subsets $S$ and $T$ of $G$ with the same sum of elements.
2003 Greece Junior Math Olympiad, 4
Find all positive integers which can be written in the form $(mn+1)/(m+n)$, where $m,n$
are positive integers.
2020/2021 Tournament of Towns, P1
There were $n{}$ positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer.
[i]Boris Frenkin[/i]
1959 AMC 12/AHSME, 45
If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals:
$ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $
2017 ELMO Problems, 2
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
1989 IMO Longlists, 84
Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which
\[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\]
where $ x_0, x_1, \ldots , x_n$ are real variables.
2020 Grand Duchy of Lithuania, 2
There are $100$ cities in Matland. Every road in Matland connects two cities, does not pass through any other city and does not form crossroads with other roads (although roads can go through tunnels one after the other). Driving in Matlandia by road, it is possible to get from any city to any other. Prove that that it is possible to repair some of the roads of Matlandia so that from an odd number of repaired roads would go in each city.
1974 Bundeswettbewerb Mathematik, 3
A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.
2016-2017 SDML (Middle School), 8
Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$.
2006 Mexico National Olympiad, 2
Let $ABC$ be a right triangle with a right angle at $A$, such that $AB < AC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$. Let $E$ be the intersection of the parallel to $AC$ that passes through $M$, with the perpendicular on $BD$ passing through $B$. Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$.
2020 Purple Comet Problems, 25
A deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of five cards has cards numbered $1, 2, 3, 4, 5$, in that order. The two decks are riffle-shuffled together to form a deck with $13$ cards with the cards from each deck in the same order as they were originally. Thus, numbers on the cards might end up in the order $1122334455678$ or $1234512345678$ but not $1223144553678$. Find the number of possible sequences of the $13$ numbers.
2006 Baltic Way, 3
Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x),\ldots , P_{m}(x)$ with real coefficients such that
\[P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}\]
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.