This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2013 Iran MO (3rd Round), 4

In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.

2023 MIG, 3

A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle? $\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$

2006 Switzerland Team Selection Test, 2

Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.

1955 Kurschak Competition, 3

The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.

1976 IMO Longlists, 35

Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

2002 China Girls Math Olympiad, 7

An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$

MBMT Team Rounds, 2020.7

Tags:
Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product? [i]Proposed by Gabriel Wu[/i]

2016 CIIM, Problem 2

Tags: graph
A boa of size $k$ is a graph with $k+1$ vertices $\{0,1,\dots,k-1,k\}$ and edges only between the vertices $i$ and $i+1$ for $0\leq i < k.$ The boa is place in a graph $G$ through a injection of graphs. (This is an injective function form the vertices of the boa to the vertices of the graph in such a way that if there is an edge between the vertices $x$ and $y$ in the boa then there must be an edge between $f(x)$ and $f(y)$ in $G$). The Boa can move in the graph $G$ using to type of movement each time. If the boa is initially on the vertices $f(0),f(1),\dots,f(k)$ then it moves in one of the following ways: (i) It choose $v$ a neighbor of $f(k)$ such that $v\not\in\{f(0),f(1),\dots,f(k-1)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(k)=v$ and $f'(i) = f(i+1)$ for $0 \leq i < k,$ or (ii) It choose $v$ a neighbor of $f(0)$ such that $v\not\in\{f(1),f(2),\dots,f(k)\}$ and the boa now moves to $f(0),f(1),\dots,f(k)$ with $f'(0)=v$ and $f'(i) = f'(i-1)$ for $0 < i \leq k.$ Prove that if $G$ is a connected graph with diameter $d$, then it is possible to put a size $\lceil d/2 \rceil$ boa in $G$ such that the boa can reach any vertex of $G$.

Novosibirsk Oral Geo Oly VIII, 2020.3

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

2006 Junior Balkan Team Selection Tests - Romania, 1

Tags: angle , ratio , geometry , median
Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.

2002 AMC 10, 20

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Points $ A,B,C,D,E$ and $ F$ lie, in that order, on $ \overline{AF}$, dividing it into five segments, each of length 1. Point $ G$ is not on line $ AF$. Point $ H$ lies on $ \overline{GD}$, and point $ J$ lies on $ \overline{GF}$. The line segments $ \overline{HC}, \overline{JE},$ and $ \overline{AG}$ are parallel. Find $ HC/JE$. $ \text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2$

2002 AMC 12/AHSME, 17

Several sets of prime numbers, such as $ \{ 7, 83, 421, 659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? $ \textbf{(A)}\ 193\qquad\textbf{(B)}\ 207\qquad\textbf{(C)}\ 225\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 447$

2017 Junior Regional Olympiad - FBH, 5

Tags: equation
Fathers childhood lasted for one sixth part of his life, and he married one $8$th after that and he immediately left to army. When one $12$th of his life passed, father returned from the army and $5$ years after he got a son. Son who lived for one half of fathers years, died $4$ years before his father. How many years lived his father, and how many years he had when his son was born?

2011 Kosovo Team Selection Test, 1

Tags: inequalities
Let $a,b,c$ be real positive numbers. Prove that the following inequality holds: \[{ \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }\]

2014 Cezar Ivănescu, 2

[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $ [b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $

2004 China Team Selection Test, 2

Tags: inequalities
Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

2023 China Girls Math Olympiad, 5

Tags: geometry
Let $\Delta ABC$ be an acute-angled triangle with $AB < AC$, $H$ be a point on $BC$ such that $AH\ \bot BC$ and $G$ be the centroid of $\Delta ABC$. Let $P,Q$ be the point of tangency of the inscribed circle of $\Delta ABC$ with $AB,AC$, correspondingly. Define $M,N$ to be the midpoint of $PB,QC$, correspondingly. Let $D,E$ be points on the inscribed circle of $\Delta ABC$ such that $\angle BDH + \angle ABC = 180^{\circ}$, $\angle CEH + \angle ACB = 180^{\circ}$. Prove that lines $MD,NE,HG$ share a common point.

2002 HKIMO Preliminary Selection Contest, 7

Tags: geometry
In $\triangle ABC$, $X, Y$, are points on BC such that $BX=XY=YC$, $M , N$ are points on $AC$ such that $AM=MN=NC$. $BM$ and $BN$ intersect $AY$ at $S$ and $R$ and respectively. If the area of $\triangle ABC$ is $1$, find the area of $SMNR$.

1990 Swedish Mathematical Competition, 4

$ABCD$ is a quadrilateral. The bisectors of $\angle A$ and $\angle B$ meet at $E$. The line through $E$ parallel to $CD$ meets $AD$ at $L$ and $BC$ at $M$. Show that $LM = AL + BM$.

2022 BMT, 12

Tags: geometry
Let circles $C_1$ and $C_2$ be internally tangent at point $P$, with $C_1$ being the smaller circle. Consider a line passing through $P$ which intersects $C_1$ at $Q$ and $C_2$ at $R$. Let the line tangent to $C_2$ at $R$ and the line perpendicular to $\overline{PR}$ passing through $Q$ intersect at a point $S$ outside both circles. Given that $SR = 5$, $RQ = 3$, and $QP = 2$, compute the radius of $C_2$.

2005 IMAR Test, 2

Tags: geometry , incenter
Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle. Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.

PEN A Problems, 24

Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

2021 South Africa National Olympiad, 2

Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.

2009 Turkey Junior National Olympiad, 1

Tags:
Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$. Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$. If the lines $AD$ and $OE$ meet at $F$, find $|AF|/|FD|$.

2015 CCA Math Bonanza, L1.4

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How many digits are in the base $10$ representation of $3^{30}$ given $\log 3 = 0.47712$? [i]2015 CCA Math Bonanza Lightning Round #1.4[/i]