Olympiad problems

2022 Israel Olympic Revenge, 4

A (not necessarily regular) tetrahedron $A_1A_2A_3A_4$ is given in space. For each pair of indices $1\leq i<j\leq 4$, an ellipsoid with foci $A_i,A_j$ and string length $\ell_{ij}$, for positive numbers $\ell_{ij}$, is given (in all 6 ellipsoids were built). For each $i=1,2$, a pair of points $X_i\neq X'_i$ was chosen so that $X_i, X'_i$ both belong to all three ellipsoids with $A_i$ as one of their foci. Prove that the lines $X_1X'_1, X_2X'_2$ share a point in space if and only if \[\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23}\] [i]Remark: An [u]ellipsoid[/u] with foci $P,Q$ and string length $\ell>|PQ|$ is defined here as the set of points $X$ in space for which $|XQ|+|XP|=\ell$.[/i]

1979 Spain Mathematical Olympiad, 4

Tags: complex number , algebra

If $z_1$ , $z_2$ are the roots of the equation with real coefficients $z^2+az+b = 0$, prove that $ z^n_1 + z^n_2$ is a real number for any natural value of $n$. If particular of the equation $z^2 - 2z + 2 = 0$, express, as a function of $n$, the said sum.

2024 Centroamerican and Caribbean Math Olympiad, 3

Tags: geometry , collinearity , circumcircle

Let $ABC$ be a triangle, $H$ its orthocenter, and $\Gamma$ its circumcircle. Let $J$ be the point diametrically opposite to $A$ on $\Gamma$. The points $D$, $E$ and $F$ are the feet of the altitudes from $A$, $B$ and $C$, respectively. The line $AD$ intersects $\Gamma$ again at $P$. The circumcircle of $EFP$ intersects $\Gamma$ again at $Q$. Let $K$ be the second point of intersection of $JH$ with $\Gamma$. Prove that $K$, $D$ and $Q$ are collinear.

1991 Chile National Olympiad, 6

Tags: angle bisector , angle chasing , geometry , angle

Given a triangle with $ \triangle ABC $, with: $ \angle C = 36^o$ and $ \angle A = \angle B $. Consider the points $ D $ on $ BC $, $ E $ on $ AD $, $ F $ on $ BE $, $ G $ on $ DF $ and $ H $ on $ EG $, so that the rays $ AD, BE, DF, EG, FH $ bisect the angles $ A, B, D, E, F $ respectively. It is known that $ FH = 1 $. Calculate $ AC$.

2023 Bulgaria EGMO TST, 4

Tags: coloring , pigeonhole principle , digit , combinatorics

Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?

2010 ELMO Shortlist, 8

Tags: graph theory , probability , expected value , combinatorics

A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$? [i]David Yang.[/i]

2006 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , analytic geometry

At time $0$, an ant is at $(1,0)$ and a spider is at $(-1,0)$. The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the $x$-axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be?

2024 Baltic Way, 12

Tags: geometry

Let $ABC$ be an acute triangle with circumcircle $\omega$ such that $AB<AC$. Let $M$ be the midpoint of the arc $BC$ of~$\omega$ containing the point~$A$, and let $X\neq M$ be the other point on $\omega$ such that $AX=AM$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$ of the triangle $ABC$ such that $EX=EC$ and $FX=FB$. Prove that $AE=AF$.

2014 Singapore Senior Math Olympiad, 17

Tags:

Let $n$ be a positive integer such that $12n^2+12n+11$ is a $4$-digit number with all $4$ digits equal. Determine the value of $n$.

2000 Manhattan Mathematical Olympiad, 1

Tags: polynomial , algebra

Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.

2025 SEEMOUS, P4

Tags: real analysis , series , sequence

Let $(a_n)_{n\geq 1}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $\sum_{n=1}^{\infty}\frac{a_n}{n}$ is convergent if and only if the sequence $(a_n\ln n)_{n\geq 1}$ is bounded and $\sum_{n=1}^{\infty} (a_n-a_{n+1})\ln n$ is convergent.

2012 Mathcenter Contest + Longlist, 2

Tags: number theory , prime

Let $p=2^n+1$ and $3^{(p-1)/2}+1\equiv 0 \pmod p$. Show that $p$ is a prime. [i](Zhuge Liang) [/i]

1980 Kurschak Competition, 2

Tags: number theory

Let $n > 1$ be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers $x$ and $y$ satisfying $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}$$ is that $n$ has a prime divisor of the form $4k - 1$.

1951 AMC 12/AHSME, 39

Tags:

A stone is dropped into a well and the report of the stone striking the bottom is heard $ 7.7$ seconds after it is dropped. Assume that the stone falls $ 16t^2$ feet in $ t$ seconds and that the velocity of sound is $ 1120$ feet per second. The depth of the well is: $ \textbf{(A)}\ 784 \text{ ft.} \qquad\textbf{(B)}\ 342 \text{ ft.} \qquad\textbf{(C)}\ 1568 \text{ ft.} \qquad\textbf{(D)}\ 156.8 \text{ ft.} \qquad\textbf{(E)}\ \text{none of these}$

1986 Traian Lălescu, 1.1

Tags: equation , system of equations , algebra

Solve: $$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$

2006 Stanford Mathematics Tournament, 6

Tags: function , ratio , geometric series

The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$.

2019 India IMO Training Camp, P3

Tags: algebra

Let $n\ge 2$ be an integer. Solve in reals: \[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]

2021-2022 OMMC, 10

Tags:

A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find \[\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3.\] [i]Proposed by Evan Chang[/i]

2003 India IMO Training Camp, 10

Tags: modular arithmetic , number theory

Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

1999 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Evaluate $$\sum^{17}_{n=2} \frac{n^2+n+1}{n^4+2n^3-n^2-2n}.$$

2023 Taiwan TST Round 1, 1

Tags: algebra

Let $\mathbb{Q}_{>1}$ be the set of rational numbers greater than $1$. Let $f:\mathbb{Q}_{>1}\to \mathbb{Z}$ be a function that satisfies \[f(q)=\begin{cases} q-3&\textup{ if }q\textup{ is an integer,}\\ \lceil q\rceil-3+f\left(\frac{1}{\lceil q\rceil-q}\right)&\textup{ otherwise.} \end{cases}\] Show that for any $a,b\in\mathbb{Q}_{>1}$ with $\frac{1}{a}+\frac{1}{b}=1$, we have $f(a)+f(b)=-2$. [i]Proposed by usjl[/i]

2017 Hanoi Open Mathematics Competitions, 13

Tags: inequalities , distance , geometric inequalities , geometry

Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

2020 Bangladesh Mathematical Olympiad National, Problem 7

Tags: coordinate geometry , algebra , geometry

$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$?

2000 IMO Shortlist, 2

Tags: number theory , divisor

For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.

IMSC 2023, 5

Tags: combinatorics

In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points? [i]Proposed by Art Waeterschoot, Belgium[/i]