Olympiad problems

1991 French Mathematical Olympiad, Problem 3

Tags: 3d geometry , geometry

Let $S$ be a fixed point on a sphere $\Sigma$ with center $\Omega$. Consider all tetrahedra $SABC$ inscribed in $\Sigma$ such that $SA,SB,SC$ are pairwise orthogonal. (a) Prove that all the planes $ABC$ pass through a single point. (b) In one such tetrahedron, $H$ and $O$ are the orthogonal projections of $S$ and $\Omega$ onto the plane $ABC$, respectively. Let $R$ denote the circumradius of $\triangle ABC$. Prove that $R^2=OH^2+2SH^2$.

2022 Canada National Olympiad, 1

Tags: algebra , inequalities

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

2023 ITAMO, 1

Tags: number theory

Let $a, b$ be positive integers such that $54^a=a^b$. Prove that $a$ is a power of $54$.

1998 Brazil National Olympiad, 1

Tags: quadratic , game , combinatorial game , algebra

Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.

2003 Gheorghe Vranceanu, 2

Tags: inequalities

Let be a natural number $ n $ and $ 2n $ positive real numbers $ v_1,v_2,\ldots ,v_{2n} $ such that the last $ n $ of them are greater than $ 1. $ Prove that: $$ \sum_{i=1}^n v_iv_{n+i}\le \max_{1\le k\le n}\left( \left( -1+\prod_{l=n}^{2n} v_l \right) v_k +\sum_{m=1}^n v_m \right) $$

LMT Team Rounds 2010-20, 2020.S23

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Let $\triangle ABC$ be a triangle such that $AB=AC=40$ and $BC=79.$ Let $X$ and $Y$ be the points on segments $AB$ and $AC$ such that $AX=5, AY=25.$ Given that $P$ is the intersection of lines $XY$ and $BC,$ compute $PX\cdot PY-PB\cdot PC.$

2022 239 Open Mathematical Olympiad, 4

Tags: combinatorics , graph theory

The degrees of all vertices of a graph are not less than 100 and not more than 200. Prove that its vertices can be divided into connected pairs and triples.

2025 China Team Selection Test, 13

Tags: number theory

Find all positive integers $ m $ for which there exists an infinite subset $ A $ of the positive integers such that: for any pairwise distinct positive integers $ a_1, a_2, \cdots, a_m \in A $, the sum $ a_1 + a_2 + \cdots + a_m $ and the product $ a_1a_2 \cdots a_m $ are both square-free.

2003 Alexandru Myller, 1

Tags: complex number , inequalities , algebra

Let be a natural number $ n, $ a positive real number $ \lambda , $ and a complex number $ z. $ Prove the following inequalities. $$ 0\le -\lambda +\frac{1}{n}\sum_{\stackrel{w\in\mathbb{C}}{w^n=1 }} \left| z-\lambda w \right|\le |z| $$ [i]Gheorghe Iurea[/i]

2012 Purple Comet Problems, 7

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A snail crawls $2\frac12$ centimeters in $4\frac14$ minutes. At this rate, how many centimeters can the snail crawl is 85 minutes?

2022 HMNT, 28

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Let $ABC$ be a triangle with $AB = 13, BC = 14,$ and $CA = 15.$ Pick points $Q$ and $R$ on $AC$ and $AB$ such that $\angle CBQ = \angle BCR = 90^\circ.$ There exist two points $P_1\neq P_2$ in the plane of $ABC$ such that $\triangle P_1 QR, \triangle P_2QR,$ and $\triangle ABC$ are similar (with vertices in order). Compute the sum of the distances from $P_1$ to $BC$ and $P_2$ to $BC.$

2014 Taiwan TST Round 2, 6

Tags: circumcircle , geometric transformation , similar triangles , geometry

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.