Found problems: 85335
1989 Chile National Olympiad, 3
In a right triangle with legs $a$, $b$ and hypotenuse $c$, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values $X,Y$ . Calculate $X + Y$.
[img]https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png[/img]
2011 Croatia Team Selection Test, 3
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
2022 Germany Team Selection Test, 2
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
2011 Peru IMO TST, 1
Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $
2008 Singapore MO Open, 2
in the acute triangle $\triangle ABC$.
M is a point in the interior of the segment AC and N is a point on the extension of segment AC such that MN=AC.
let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively
prove that the orthocentre of $\triangle ABC$ lies on circumcircle of $\triangle BED$
2014 Harvard-MIT Mathematics Tournament, 16
Suppose that $x$ and $y$ are positive real numbers such that $x^2-xy+2y^2=8$. Find the maximum possible value of $x^2+xy+2y^2$.
1998 Argentina National Olympiad, 1
Jorge writes a list with an even number of integers, not all equal to $0$ (there may be repeated numbers). Show that Martin can cross out a number from the list, of his choice, so that it is impossible for Jorge to separate the remaining numbers into two groups in such a way that the sum of all the numbers in one group is equal to the sum of all the others. numbers from the other group.
1993 AMC 12/AHSME, 14
The convex pentagon $ABCDE$ has $\angle A=\angle B=120^{\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$?
[asy]
draw((0,0)--(1,0)--(1.5,sqrt(3)/2)--(0.5,3sqrt(3)/2)--(-0.5,sqrt(3)/2)--cycle);
dot((0,0));
dot((1,0));
dot((1.5,sqrt(3)/2));
dot((0.5,3sqrt(3)/2));
dot((-0.5,sqrt(3)/2));
label("A", (0,0), SW);
label("B", (1,0), SE);
label("C", (1.5,sqrt(3)/2), E);
label("D", (0.5,3sqrt(3)/2), N);
label("E", (-.5, sqrt(3)/2), W);
[/asy]
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 7\sqrt{3} \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 9\sqrt{3} \qquad\textbf{(E)}\ 12\sqrt{5} $
1998 Brazil National Olympiad, 3
Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses.
For each $(n, d)$ which player has a winning strategy?
2002 AIME Problems, 11
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}.$ The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n},$ where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n.$
2024 District Olympiad, P3
Let $a,b,c\in\mathbb{C}\setminus\left\{0\right\}$ such that $|a|=|b|=|c|$ and $A=a+b+c$ respectively $B=abc$ are both real numbers. Prove that $ C_n=a^n+b^n+c^n$ is also a real number$,$ $(\forall)n\in\mathbb{N}.$
2007 Princeton University Math Competition, 8
What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?
1986 Bundeswettbewerb Mathematik, 3
Let $d_n$ be the last digit, distinct from 0, in the decimal expansion of $n!$. Prove that the sequence $d_1,d_2,d_3, \ldots$ is not periodic.
1993 All-Russian Olympiad, 1
For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.
2018 Taiwan TST Round 2, 5
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
1994 Irish Math Olympiad, 1
Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$
2004 Balkan MO, 4
The plane is partitioned into regions by a finite number of lines no three of which are concurrent. Two regions are called "neighbors" if the intersection of their boundaries is a segment, or half-line or a line (a point is not a segment). An integer is to be assigned to each region in such a way that:
i) the product of the integers assigned to any two neighbors is less than their sum;
ii) for each of the given lines, and each of the half-planes determined by it, the sum of the integers, assigned to all of the regions lying on this half-plane equal to zero.
Prove that this is possible if and only if not all of the lines are parallel.
1998 All-Russian Olympiad, 4
A maze is an $8 \times 8$ board with some adjacent squares separated by walls, so that any two squares can be connected by a path not meeting any wall. Given a command LEFT, RIGHT, UP, DOWN, a pawn makes a step in the corresponding direction unless it encounters a wall or an edge of the chessboard. God writes a program consisting of a finite sequence of commands and gives it to the Devil, who then constructs a maze and places the pawn on one of the squares. Can God write a program which guarantees the pawn will visit every square despite the Devil's efforts?
2015 Princeton University Math Competition, B3
Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?
2024 LMT Fall, 31
Let $ABC$ be a triangle with circumradius $12$, and denote the orthocenter and circumcenter as $H$ and $O$ respectively. Define $H_A \neq A$ to be the intersection of line $AH$ and the circumcircle of $ABC$. Given that $\overline{OH} \parallel \overline{BC}$ and $\overline{AO} \parallel \overline{BH_A}$, find $AH_A$.
2018 Mediterranean Mathematics OIympiad, 4
Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties:
$*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering.
$*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$.
$*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$.
(Proposed by Gerhard Woeginger, Austria)
2007 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a triangle with $BC = a, AC = b$ and $AB = c$. A point $P$ inside the triangle has the property that for any line passing through $P$ and intersects the lines $AB$ and $AC$ in the distinct points $E$ and $F$ we have the relation $\frac{1}{AE} +\frac{1}{AF} =\frac{a + b + c}{bc}$. Prove that the point $P$ is the center of the circle inscribed in the triangle $ABC$.
2006 National Olympiad First Round, 36
In an exam with $n$ problems where $n$ is a positive integer, each problem was answered by at least one student. Each student answered an even number of problems. Any two students answered an even number of problems in common. What is the number of values that $n$ cannot take?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of above}
$
2021 China Team Selection Test, 5
Given a triangle $ABC$, a circle $\Omega$ is tangent to $AB,AC$ at $B,C,$ respectively. Point $D$ is the midpoint of $AC$, $O$ is the circumcenter of triangle $ABC$. A circle $\Gamma$ passing through $A,C$ intersects the minor arc $BC$ on $\Omega$ at $P$, and intersects $AB$ at $Q$. It is known that the midpoint $R$ of minor arc $PQ$ satisfies that $CR \perp AB$. Ray $PQ$ intersects line $AC$ at $L$, $M$ is the midpoint of $AL$, $N$ is the midpoint of $DR$, and $X$ is the projection of $M$ onto $ON$. Prove that the circumcircle of triangle $DNX$ passes through the center of $\Gamma$.
2007 Nicolae Păun, 1
Consider a finite group $ G $ and the sequence of functions $ \left( A_n \right)_{n\ge 1} :G\longrightarrow \mathcal{P} (G) $ defined as $ A_n(g) = \left\{ x\in G|x^n=g \right\} , $ where $ \mathcal{P} (G) $ is the power of $ G. $
[b]a)[/b] Prove that if $ G $ is commutative, then for any natural numbers $ n, $ either $ A_n(g) =\emptyset , $ or $ \left| A_n(g) \right| =\left| A_n(1) \right| . $
[b]b)[/b] Provide an example of what $ G $ could be in the case that there exists an element $ g_0 $ of $ G $ and a natural number $ n_0 $ such that $ \left| A_{n_0}\left( g_0 \right) \right| >\left| A_{n_0}(1) \right| . $
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]