Found problems: 85335
1994 IberoAmerican, 3
In each square of an $n\times{n}$ grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viceversa). At the begin, all the lamps are off. Show that lways is possible, with an appropriated sequence of touches, that all the the lamps on the board end on and find, in function of $n$ the minimal number of touches that are necessary to turn on every lamp.
2009 Peru IMO TST, 5
Let $\mathcal{C}$ be the circumference inscribed in the triangle $ABC,$ which is tangent to sides $BC, AC, AB$ at the points $A' , B' , C' ,$ respectively. The distinct points $K$ and $L$ are taken on $\mathcal{C}$ such that $$\angle AKB'+\angle BKA' =\angle ALB'+\angle BLA'=180^{\circ}.$$
Prove that the points $A', B', C'$ are equidistant from the line $KL.$
2022 Sharygin Geometry Olympiad, 10.2
Let $ABCD$ be a convex quadrilateral. The common external tangents to circles $(ABC)$ and $(ACD)$ meet at point $E$, the common external tangents to circles $(ABD)$ and $(BCD)$ meet at point $F$. Let $F$ lie on $AC$, prove that $E$ lies on $BD$.
1995 National High School Mathematics League, 1
Give a family of curves $2(2\sin\theta-\cos\theta+3)x^2-(8\sin\theta+\cos\theta+1)=0$, where $\theta$ is a parameter. Find the maximum value of the length of the chord that $y=2x$ intersects the curve.
2014 Tuymaada Olympiad, 8
There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number.
The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.
[i](K. Kokhas)[/i]
2022-2023 OMMC, 12
Initially five variables are defined: $a_1=1, a_2=0, a_3=0, a_4=0, a_5=0.$ On a turn, Evan can choose an integer $2 \le i \le 5.$ Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2,$ then now $a_1=1, a_2=0+1=1, a_3=0, a_4=0, a_5=0.$ Find the minimum number of turns Evan needs to make $a_5$ exceed $1,000,000.$
2016 Greece Team Selection Test, 4
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2013 Saudi Arabia GMO TST, 2
Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.
1972 Miklós Schweitzer, 4
Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite.
[i]J. Pelikan[/i]
2015 Spain Mathematical Olympiad, 2
In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$.
Prove that:
[b]a)[/b] The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively.
[b]b)[/b] The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.
2000 IMO Shortlist, 4
Let $ A_1A_2 \ldots A_n$ be a convex polygon, $ n \geq 4.$ Prove that $ A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $ A_j$ one can assign a pair $ (b_j, c_j)$ of real numbers, $ j = 1, 2, \ldots, n,$ so that $ A_iA_j = b_jc_i - b_ic_j$ for all $ i, j$ with $ 1 \leq i < j \leq n.$
2013 NIMO Summer Contest, 15
\begin{quote}
Ted quite likes haikus, \\
poems with five-seven-five, \\
but Ted knows few words.
He knows $2n$ words \\
that contain $n$ syllables \\
for every int $n$.
Ted can only write \\
$N$ distinct haikus. Find $N$. \\
Take mod one hundred.
\end{quote}
Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$.
[i]Proposed by Lewis Chen[/i]
2022 Durer Math Competition Finals, 13
Write some positive integers in the following table such that
$\cdot$ there is at most one number in each field
$\cdot$ each number is equal to how many numbers there are in edge-adjacent fields,
$\cdot$ edge-adjacent fields cannot have equal numbers.
What is the sum of numbers in the resulting table?
[img]https://cdn.artofproblemsolving.com/attachments/a/9/63a9c38762a4c895688fff049ed08c96b2c22c.png[/img]
1955 AMC 12/AHSME, 50
In order to pass $ B$ going $ 40$ mph on a two-lane highway, $ A$, going $ 50$ mph, must gain $ 30$ feet. Meantime, $ C$, $ 210$ feet from $ A$, is headed toward him at $ 50$ mph. If $ B$ and $ C$ maintain their speeds, then, in order to pass safely, $ A$ must increase his speed by:
$ \textbf{(A)}\ \text{30 mph} \qquad
\textbf{(B)}\ \text{10 mph} \qquad
\textbf{(C)}\ \text{5 mph} \qquad
\textbf{(D)}\ \text{15 mph} \qquad
\textbf{(E)}\ \text{3 mph}$
2021 USMCA, 18
Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integer(s). He first rolls the die and if the number showing on top is $k,$ he then uniformly and randomly takes a $k$-element subset from his list and calculates the sum of the numbers in his subset. Given that the expected value of this sum is $2020,$ compute the sum of all possible values of $n.$
2025 Bangladesh Mathematical Olympiad, P2
Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.
2006 Team Selection Test For CSMO, 4
All the squares of a board of $(n+1)\times(n-1)$ squares
are painted with [b]three colors[/b] such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of $n$.
2003 Cuba MO, 4
Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).
2017 AMC 12/AHSME, 19
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2004 Turkey Team Selection Test, 2
Show that
\[
\min \{ |PA|, |PB|, |PC| \} + |PA| + |PB| + |PC| < |AB|+|BC|+|CA|
\]
if $P$ is a point inside $\triangle ABC$.
2008 Bulgaria Team Selection Test, 2
In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.
2000 Miklós Schweitzer, 9
Let $M$ be a closed, orientable $3$-dimensional differentiable manifold, and let $G$ be a finite group of orientation preserving diffeomorphisms of $M$. Let $P$ and $Q$ denote the set of those points of $M$ whose stabilizer is nontrivial (that is, contains a nonidentity element of $G$) and noncyclic, respectively. Let $\chi (P)$ denote the Euler characteristic of $P$. Prove that the order of $G$ divides $\chi (P)$, and $Q$ is the union of $-2\frac{\chi(P)}{|G|}$ orbits of $G$.
2014 India Regional Mathematical Olympiad, 6
Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$. Let $r_j$ and $c_k$ denote the product of all numbers in $j$-th row and $k$-th column respectively, $1 \le j, k \le n$. Prove that
$$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$
2010 Junior Balkan Team Selection Tests - Romania, 1
We consider on a circle a finite number of real numbers with the sum strictly greater than $0$. Of all the sums that have as terms numbers on consecutive positions on the circle, let $S$ be the largest sum and $s$ the smallest sum. Show that $S + s> 0$.
1989 IMO Longlists, 63
Let $ l_i,$ $ i \equal{} 1,2,3$ be three non-collinear straight lines in the plane, which build a triangle, and $ f_i$ the axial reflections in $ l_i$. Prove that for each point $ P$ in the plane there exists finite interconnections (compositions) of the reflections of $ f_i$ which carries $ P$ into the triangle built by the straight lines $ l_i,$ i.e. maps that point to a point interior to the triangle.