Found problems: 85335
2007 Peru IMO TST, 2
Let $ABC$ be a triangle such that $CA \neq CB$,
the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$,
respectively, and $I$ the incircle.
The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$.
The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$.
Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.
2019 USAJMO, 4
Let $ABC$ be a triangle with $\angle ABC$ obtuse. The [i]$A$-excircle[/i] is a circle in the exterior of $\triangle ABC$ that is tangent to side $BC$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?
[i]Proposed by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal[/i]
2019 USMCA, 20
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.
1984 IMO Longlists, 3
The opposite sides of the reentrant hexagon $AFBDCE$ intersect at the points $K,L,M$ (as shown in the figure). It is given that $AL = AM = a, BM = BK = b$, $CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f$.
[img]http://imgur.com/LUFUh.png[/img]
$(a)$ Given length $a$ and the three angles $\alpha, \beta$ and $\gamma$ at the vertices $A, B,$ and $C,$ respectively, satisfying the condition $\alpha+\beta+\gamma<180^{\circ}$, show that all the angles and sides of the hexagon are thereby uniquely determined.
$(b)$ Prove that
\[\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}\]
Easier version of $(b)$. Prove that
\[(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)\]
2012 Indonesia TST, 3
The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$.
Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.
2011 ELMO Shortlist, 2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.
[i]David Yang.[/i]
2010 Today's Calculation Of Integral, 634
Prove that :
\[\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)\]
[i]2010 Miyazaki University entrance exam/Medicine[/i]
2019 Grand Duchy of Lithuania, 3
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.
II Soros Olympiad 1995 - 96 (Russia), 9.10
Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about triangle $ABC$.
1978 IMO Longlists, 14
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
2024 AMC 8 -, 6
Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P$, $Q$, $R$, and $S$. What is the sorted order of the four paths from shortest to longest?
[center][img]https://wiki-images.artofproblemsolving.com/9/94/2024_AMC_8_Problem_6.png[/img][/center]
$\textbf{(A) }\text{P, Q, R, S}\qquad\textbf{(B) }\text{P, R, S, Q}\qquad\textbf{(C) }\text{Q, S, P, R}\qquad\textbf{(D) }\text{R, P, S, Q}\qquad\textbf{(E) }\text{R, S, P, Q}$
2024 Princeton University Math Competition, 2
Let real number sequences $a_k$ and $x_k$ be defined for $1 \le k \le 7$ and suppose that $a_1=1$ and $a_{k+1}=a_k+x_k$ for $1 \le k \le 7.$ Let $x_k$ be chosen such that the quantity $S=\sum_{k=1}^7 (a_k^2+x_k^2)$ is minimized. Then $S=\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m+n.$
2010 Sharygin Geometry Olympiad, 3
All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?
2017 ELMO Shortlist, 1
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]
2005 Sharygin Geometry Olympiad, 11.1
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
2013 NIMO Problems, 2
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i]
2010 National Olympiad First Round, 17
Let $A,B,C,D$ be points in the space such that $|AB|=|AC|=3$, $|DB|=|DC|=5$, $|AD|=6$, and $|BC|=2$. Let $P$ be the nearest point of $BC$ to the point $D$, and $Q$ be the nearest point of the plane $ABC$ to the point $D$. What is $|PQ|$?
$ \textbf{(A)}\ \frac{1}{\sqrt 2}
\qquad\textbf{(B)}\ \frac{3\sqrt 7}{2}
\qquad\textbf{(C)}\ \frac{57}{2\sqrt{11}}
\qquad\textbf{(D)}\ \frac{9}{2\sqrt 2}
\qquad\textbf{(E)}\ 2\sqrt 2
$
1997 Irish Math Olympiad, 4
Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a\plus{}b\plus{}c\ge abc$. Prove that:
$ a^2\plus{}b^2\plus{}c^2 \ge abc.$
2022 Belarusian National Olympiad, 10.5
$n$ distinct integers are given, all of which are bigger than $-a$, where $a$ is a positive integer. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even numbers is equal to the biggest odd numbers
a) Find the least possible value of $n$ for all $a$
b) For each $a \geq 2$ find the maximum possible value of $n$
2009 Tournament Of Towns, 2
$A; B; C; D; E$ and $F$ are points in space such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, $CD$ is parallel to $FA$, but $AB \neq DE$. Prove that all six points lie in the same plane.
[i](4 points)[/i]
2021 JBMO Shortlist, C6
Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$
and $n$ where $m, n \ge 3$.
Proposed by [i]Toghrul Abbasov, Azerbaijan[/i]
2002 SNSB Admission, 6
Find a Galois extension of the field $ \mathbb{Q} $ whose Galois group is isomorphic with $ \mathbb{Z}/3\mathbb{Z} . $
2014 Contests, 1
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
2018 Bulgaria EGMO TST, 2
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?
2009 All-Russian Olympiad Regional Round, 9.5
There are $11$ phrases written on $11$ pieces of paper (one per sheet):
1) To the left of this sheet there are no sheets with false statements.
2) Exactly one sheet to the left of this one contains a false statement.
3) Exactly $2$ sheets to the left of this one contain false statements
...
11) Exactly $10$ sheets to the left of this one contain false statements.
The sheets of paper were laid out in some order in a row, going from left to right. After this, some of the written statements became true and some became false. What is the greatest possible number of true statements?