Olympiad problems

2014 Taiwan TST Round 2, 6

Let $P$ be a point inside triangle $ABC$, and suppose lines $AP$, $BP$, $CP$ meet the circumcircle again at $T$, $S$, $R$ (here $T \neq A$, $S \neq B$, $R \neq C$). Let $U$ be any point in the interior of $PT$. A line through $U$ parallel to $AB$ meets $CR$ at $W$, and the line through $U$ parallel to $AC$ meets $BS$ again at $V$. Finally, the line through $B$ parallel to $CP$ and the line through $C$ parallel to $BP$ intersect at point $Q$. Given that $RS$ and $VW$ are parallel, prove that $\angle CAP = \angle BAQ$.

2022 LMT Spring, 10

Tags: combinatorics , number theory

In a room, there are $100$ light switches, labeled with the positive integers ${1,2, . . . ,100}$. They’re all initially turned off. On the $i$ th day for $1 \le i \le 100$, Bob flips every light switch with label number $k$ divisible by $i$ a total of $\frac{k}{i}$ times. Find the sum of the labels of the light switches that are turned on at the end of the $100$th day.

1962 AMC 12/AHSME, 1

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The expression $ \frac{1^{4y\minus{}1}}{5^{\minus{}1}\plus{}3^{\minus{}1}}$ is equal to: $ \textbf{(A)}\ \frac{4y\minus{}1}{8} \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ \frac{15}{2} \qquad \textbf{(D)}\ \frac{15}{8} \qquad \textbf{(E)}\ \frac{1}{8}$

2022 Israel Olympic Revenge, 4

Tags: geometry , 3d geometry , tetrahedron

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

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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

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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

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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 $?