Olympiad problems

2024 Malaysia IMONST 2, 5

Tags: geometry

A duck drew a square $ABCD$, then he reflected $C$ across $B$ to obtain a point $E$. He also drew the center of the square to be $F$. Then, he drew a point $G$ on ray $EF$ beyond $F$ such that $\angle AGC = 135^{\circ}$. Help the Duck prove that $\angle CGD = 135^{\circ}$ as well.

2008 Iran MO (2nd Round), 1

Tags: combinatorics

In how many ways, can we draw $n-3$ diagonals of a $n$-gon with equal sides and equal angles such that: $i)$ none of them intersect each other in the polygonal. $ii)$ each of the produced triangles has at least one common side with the polygonal.

1996 IMO Shortlist, 1

Tags: inequalities , three variable inequality , algebra

Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that \[ \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. \]

2024 Brazil EGMO TST, 4

Tags: geometry , bicentric quadrilateral , tangent , projective geometry

Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.

OMMC POTM, 2022 7

Tags: diophantine equation , number theory

Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$ [i]Proposed by vsamc[/i]

2018 Azerbaijan JBMO TST, 2

Tags: geometry

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2022 AMC 10, 18

Tags: transformation

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself? $\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$

2021 China Girls Math Olympiad, 4

Tags: number theory , infinite sequence , integer sequence

Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square. Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.

Cono Sur Shortlist - geometry, 2020.G2

Tags: geometry , incircle , circumcircle

Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.

2021 IMO Shortlist, C2

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

1972 Swedish Mathematical Competition, 2

Tags: combinatorial geometry , combinatorics

A rectangular grid of streets has $m$ north-south streets and $n$ east-west streets. For which $m, n > 1$ is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?

2017 QEDMO 15th, 4

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

2023 New Zealand MO, 6

Tags: geometry

Let triangle $ABC$ be right-angled at $A$. Let $D$ be the point on $AC$ such that $BD$ bisects angle $\angle ABC$. Prove that $BC - BD = 2AB$ if and only if $\frac{1}{BD} - \frac{1}{BC} =\frac{1}{2AB}$.

2024 Iberoamerican, 2

Tags: geometry , circumcircle

Let $\triangle ABC$ be an acute triangle and let $M, N$ be the midpoints of $AB, AC$ respectively. Given a point $D$ in the interior of segment $BC$ with $DB<DC$, let $P, Q$ the intersections of $DM, DN$ with $AC, AB$ respectively. Let $R \ne A$ be the intersection of circumcircles of triangles $\triangle PAQ$ and $\triangle AMN$. If $K$ is midpoint of $AR$, prove that $\angle MKN=2\angle BAC$

1980 AMC 12/AHSME, 2

Tags: algebra , polynomial

The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is $\text{(A)} \ 5 \qquad \text{(B)} \ 7 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 17 \qquad \text{(E)} \ 72$

1991 Kurschak Competition, 3

Tags: combinatorics

Consider $998$ red points on the plane with no three collinear. We select $k$ blue points in such a way that inside each triangle whose vertices are red points, there is a blue point as well. Find the smallest $k$ for which the described selection of blue points is possible for any configuration of $998$ red points.

2018 ISI Entrance Examination, 8

Tags: linear algebra

Let $n\geqslant 3$. Let $A=((a_{ij}))_{1\leqslant i,j\leqslant n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leqslant i,j\leqslant n$. Suppose that $$a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n$$ and $~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j$. Show that $n$ is a multiple of $4$.

2015 South East Mathematical Olympiad, 4

Tags: number theory

For any positive integer $n$, we have the set $P_n = \{ n^k \mid k=0,1,2, \ldots \}$. For positive integers $a,b,c$, we define the group of $(a,b,c)$ as lucky if there is a positive integer $m$ such that $a-1$, $ab-12$, $abc-2015$ (the three numbers need not be different from each other) belong to the set $P_m$. Find the number of lucky groups.

2000 Singapore Team Selection Test, 3

Tags: number theory , sum

Let $n$ be any integer $\ge 2$. Prove that $\sum 1/pq = 1/2$, where the summation is over all integers$ p, q$ which satisfy $0 < p < q \le n$,$ p + q > n$, $(p, q) = 1$.

2010 VJIMC, Problem 1

Tags: number theory

Let $a$ and $b$ be given positive coprime integers. Then for every integer $n$ there exist integers $x,y$ such that $$n=ax+by.$$Prove that $n=ab$ is the greatest integer for which $xy\le0$ in all such representations of $n$.

2025 Bulgarian Spring Mathematical Competition, 9.2

Tags: geometry , angle chasing , similarity , circumcircle

Let $ABC$ be an acute scalene triangle inscribed in a circle $ \Gamma $. The angle bisector of $ \angle BAC $ intersects $ BC $ at $ L $ and $ \Gamma $ at $ S $. The point $ M $ is the midpoint of $ AL $. Let $ AD $ be the altitude in $ \triangle ABC $, and the circumcircle of $ \triangle DSL $ intersects $ \Gamma $ again at $ P $. Let $ N $ be the midpoint of $ BC $, and let $ K $ be the reflection of $ D $ with respect to $ N $. Prove that the triangles $ \triangle MPS $ and $ \triangle ADK $ are similar.

2012 India Regional Mathematical Olympiad, 5

Tags: ratio , geometry , trigonometry , area of a triangle

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

2011 AMC 12/AHSME, 21

Tags: quadratic , algebra , difference of squares , special factorizations

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

1997 Turkey Junior National Olympiad, 2

Tags: symmetry , pythagorean theorem , power of a point , geometry

Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.

2020 Belarusian National Olympiad, 11.7

Tags: geometry

Line $AL$ is an angle bisector in the triangle $ABC$ ($L \in BC$), and $\omega$ is its circumcircle. Chords $X_1X_2$ and $Y_1Y_2$ pass through $L$ such that points $X_1,Y_1$ and $A$ lie in the same half-plane with respect to $BC$. Lines $X_1Y_2$ and $Y_1X_2$ intersect side $BC$ in points $Z_1$ and $Z_2$ respectively. Prove that $\angle BAZ_1=\angle CAZ_2$.