Found problems: 85335
Ukraine Correspondence MO - geometry, 2014.7
Let $ABC$ be an isosceles triangle ($AB = AC$). The points $D$ and $E$ were marked on the ray $AC$ so that $AC = 2AD$ and $AE = 2AC$. Prove that $BC$ is the bisector of the angle $\angle DBE$.
2023 Kazakhstan National Olympiad, 1
A triangle $ABC$ with obtuse angle $C$ and $AC>BC$ has center $O$ of its circumcircle $\omega$. The tangent at $C$ to $\omega$ meets $AB$ at $D$. Let $\Omega$ be the circumcircle of $AOB$. Let $OD, AC$ meet $\Omega$ at $E, F$ and let $OF \cap CE=T$, $OD \cap BC=K$. Prove that $OTBK$ is cyclic.
1999 Federal Competition For Advanced Students, Part 2, 2
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.
2010 Belarus Team Selection Test, 8.2
Prove that for positive real numbers $a, b, c$ such that $abc=1$, the following inequality holds:
$$\frac{a}{b(a+b)}+\frac{b}{c(b+c)}+\frac{c}{a(c+a)} \ge \frac32$$
(I. Voronovich)
2014 China Team Selection Test, 1
$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.
2022 Germany Team Selection Test, 2
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$.
Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.
2011 NIMO Problems, 1
A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$, with $a,b > 0$. Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$, while rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. What is the ratio of the area of $M$ to that of $N$?
[i]Proposed by Eugene Chen[/i]
2024 Bulgarian Winter Tournament, 10.4
Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$, satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors.
2023 Czech-Polish-Slovak Junior Match, 4
In triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. The bisectors of interior angles $\angle ABC$ and $\angle BCA$ intersect the line $MN$ at points $P$ and $Q$, respectively. Let $p$ be the tangent to the circumscribed circle of the triangle $AMP$ passing through point $P$, and $q$ be the tangent to the circumscribed circle of the triangle $ANQ$ passing through point $Q$. Prove that the lines $p$ and $q$ intersect on line $BC$.
2003 Romania Team Selection Test, 13
A parliament has $n$ senators. The senators form 10 parties and 10 committees, such that any senator belongs to exactly one party and one committee. Find the least possible $n$ for which it is possible to label the parties and the committees with numbers from 1 to 10, such that there are at least 11 senators for which the numbers of the corresponding party and committee are equal.
2022 Argentina National Olympiad Level 2, 5
Determine all positive integers that cannot be written as $\dfrac{a}{b}+\dfrac{a+1}{b+1}$, where $a$ and $b$ are positive integers.
2024 Singapore Junior Maths Olympiad, Q4
Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit?
Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]
2004 May Olympiad, 1
Julián writes five positive integers, not necessarily different, such that their product is equal to their sum. What could be the numbers that Julian writes?
2011 Benelux, 1
An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a [i]Benelux couple[/i] if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$.
(a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$.
(b) Prove that there are infinitely many Benelux couples
2004 AMC 10, 15
Given that $ \minus{} 4\le x\le \minus{} 2$ and $ 2\le y\le 4$, what is the largest possible value of $ (x \plus{} y)/x$?
$ \textbf{(A)}\ \minus{}\!1\qquad
\textbf{(B)}\ \minus{}\!\frac {1}{2}\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ \frac {1}{2}\qquad
\textbf{(E)}\ 1$
India EGMO 2023 TST, 3
Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
[i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]
2016 Japan Mathematical Olympiad Preliminary, 10
Boy A and $2016$ flags are on a circumference whose length is $1$ of a circle. He wants to get all flags by moving on the circumference. He can get all flags by moving distance $l$ regardless of the positions of boy A and flags. Find the possible minimum value as $l$ like this.
Note that boy A doesn’t have to return to the starting point to leave gotten flags.
2011 All-Russian Olympiad, 2
Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extensions, points $P$ and $Q$ are chosen, such that angle $PAQ$ is right. Let $AF$ be a height of triangle $APQ$. Prove that angle $BFC$ is a right angle.
2008 Vietnam National Olympiad, 7
Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.
2019 China Team Selection Test, 5
In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .
1969 Yugoslav Team Selection Test, Problem 6
Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.
2009 IMO Shortlist, 2
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
JOM 2015 Shortlist, G2
Let $ ABC $ be a triangle, and let $M$ be midpoint of $BC$. Let $ I_b $ and $ I_c $ be incenters of $ AMB $ and $ AMC $. Prove that the second intersection of circumcircles of $ ABI_b $ and $ ACI_c $ distinct from $A$ lies on line $AM$.
PEN H Problems, 70
Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.
2021 All-Russian Olympiad, 3
Some language has only three letters - $A, B$ and $C$. A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all $A$. What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?