Olympiad problems

1989 IMO Longlists, 48

Tags: geometry , circumcircle , quadrilateral , collinearity , IMO Shortlist

A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.

2007 JBMO Shortlist, 2

Tags: algebra , Inequality

$\boxed{\text{A2}}$ Prove that for all Positive reals $a,b,c$ $\frac{a^2-bc}{2a^2+bc}+\frac{b^2-ca}{2b^2+ca}+\frac{c^2-ab}{2c^2+ab}\leq 0$

V Soros Olympiad 1998 - 99 (Russia), 11.7

Tags: inequalities , algebra , trigonometry

Prove that for all positive and admissible values of $x$ the following inequality holds: $$\sin x + arc \sin x>2x$$

1990 Baltic Way, 12

Tags: algebra

Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.

2024 Middle European Mathematical Olympiad, 1

Tags: algebra , Sequences , functions , monotone , construction , Functional inequality , function

Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.

2017 Baltic Way, 16

Tags: number theory , graph theory

Is it possible for any finite group of people to choose a positive integer $N$ and assign a positive integer to each person in the group such that the product of two persons' number is divisible by $N$ if and only if they are friends?

2002 Manhattan Mathematical Olympiad, 4

Tags:

Find six points $A_1, A_2, \ldots , A_6$ in the plane, such that for each point $A_i, i = 1, 2, \ldots , 6$ there are exactly three of the remaining five points exactly $1$ cm from $A_i$.

2021 Iranian Geometry Olympiad, 3

Tags: geometry , igo p3 , Inversion , Angle Chasing

Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.

1983 IMO Longlists, 38

Tags: algebra , polynomial , algebra unsolved

Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula \[u_{n+2 }= u_n - u_{n+1}.\] [b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined. [b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.

2019 Thailand Mathematical Olympiad, 1

Tags: geometry , circumcircle , right triangle , pentagon , Concyclic

Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$. (a) Show that $P$ is the circumcenter of $BDE$. (b) Show that $A, C, D, E$ are concyclic.

2013 Tournament of Towns, 2

Tags: geometry , similar triangles , parallelogram

On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram

2015 IMO Shortlist, G5

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2014 Contests, 2

Tags: geometry , geometric transformation , reflection , circumcircle , rotation , trapezoid , trigonometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

2017 Iran Team Selection Test, 2

Tags: combinatorics , Iran , Iranian TST

In the country of [i]Sugarland[/i], there are $13$ students in the IMO team selection camp. $6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation. Is it possible that all $13$ students have a chance of being a team member? [i]Proposed by Morteza Saghafian[/i]

1984 Balkan MO, 4

Tags: algebra proposed , algebra

Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system: \[ ax+by=(x-y)^{2} \\ by+cz=(y-z)^{2} \\ cz+ax=(z-x)^{2}\]

2016 Postal Coaching, 5

Tags: set theory , group theory , Binary operation , number theory

Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds; [*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]

2024 Taiwan TST Round 1, C

Tags: combinatorics , Taiwan

A $k$-set is a set with exactly $k$ elements. For a $6$-set $A$ and any collection $\mathcal{F}$ of $4$-sets, we say that $A$ is [i]$\mathcal{F}$-good[/i] if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy $$(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.$$ Find all $n \geq 6$ so that there exists a collection $\mathcal{F}$ of $4$-subsets of $\{1, 2, \ldots , n\}$ such that every $6$-set $A \subseteq \{1, 2, \ldots , n\}$ is $\mathcal{F}$-good. [i] Proposed by usjl[/i]

2023 AMC 10, 23

Tags: AMC , AMC 10 , 2023 AMC , 2023 amc 10A , digit problems

Positive integer divisors $a$ and $b$ of $n$ are called [i]complementary[/i] if $ab=n$. Given that $N$ has a pair of complementary divisors that differ by $20$ and a pair of complementary divisors that differ by $23$, find the sum of the digits of $N$. $\textbf{(A) } 11 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

2004 AMC 10, 14

Tags:

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $ 20$ cents. If she had one more quarter, the average value would be $ 21$ cents. How many dimes does she have in her purse? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

1992 Chile National Olympiad, 4

Tags: Equilateral , parallel

Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.

2004 AIME Problems, 8

Tags: AMC , AIME , AIME I

How many positive integer divisors of $2004^{2004}$ are divisible by exactly $2004$ positive integers?

2014 IFYM, Sozopol, 5

Tags: geometry

Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.

1961 Miklós Schweitzer, 5

Tags: college contests

[b]5.[/b] Determine the functions $G$ defined on the set of all non-zero real numbers the values of which are regular matrices of order $2$, and the functions $f$ mapping the two-dimensional real vector space $E_2$ into itself, such that for any vector $y \in E_2$ and for any regular matrix $X$ of order $2$, $f(X_y)= G(det X)Xf(y)$ ($det X $ denotes the determinant of $X$).[b](A. 5)[/b]

2020 Sharygin Geometry Olympiad, 7

Tags: geometry

Prove that the medial lines of triangle $ABC$ meets the sides of triangle formed by its excenters at six concyclic points.

2003 Tuymaada Olympiad, 3

Tags: geometry , circumcircle , angle bisector , geometry unsolved

In a convex quadrilateral $ABCD$ we have $AB\cdot CD=BC\cdot DA$ and $2\angle A+\angle C=180^\circ$. Point $P$ lies on the circumcircle of triangle $ABD$ and is the midpoint of the arc $BD$ not containing $A$. It is known that the point $P$ lies inside the quadrilateral $ABCD$. Prove that $\angle BCA=\angle DCP$ [i]Proposed by S. Berlov[/i]