Olympiad problems

1939 Eotvos Mathematical Competition, 3

$ABC$ is an acute triangle. Three semicircles are constructed outwardly on the sides $BC$, $CA$ and $AB$ respectively. Construct points $A'$ , $B'$ and $C' $ on these semicìrcles respectively so that $AB' = AC'$, $BC' = BA'$ and $CA'= CB'$.

1998 Hong kong National Olympiad, 2

Tags: geometry , 3d geometry , pyramid , combinatorics

The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )

2007 Italy TST, 3

Tags: inequalities , geometry , geometric transformation , reflection , prime number , number theory

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

2001 China Second Round Olympiad, 3

Tags: combinatorial geometry , combinatorics

An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.

2014 IFYM, Sozopol, 5

Tags: geometry , isogonal lines

Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

2018 AMC 10, 6

Tags:

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? $\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 $

1998 Israel National Olympiad, 5

Tags: inequalities , algebra

(a) Find two real numebrs $a,b$ such that $|ax+b-\sqrt{x}| \le \frac{1}{24}$ for $1 \le x \le 4$. (b) Prove that the constant $\frac{1}{24}$ cannot be replaced by a smaller one.

2004 IMO Shortlist, 6

Tags: linear algebra , matrix , combinatorics , extremal combinatorics

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

2013 Iran MO (3rd Round), 4

Tags: geometry , rhombus , combinatorics

We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles. We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments. (25 points) [i]Proposed by Amirali Moinfar[/i]

2006 IMAR Test, 3

Tags: trigonometry , circumcircle , incenter , geometry

Consider the isosceles triangle $ABC$ with $AB = AC$, and $M$ the midpoint of $BC$. Find the locus of the points $P$ interior to the triangle, for which $\angle BPM+\angle CPA = \pi$.

2005 AMC 12/AHSME, 10

Tags: geometry , 3d geometry

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

LMT Team Rounds 2010-20, 2020.S24

Tags:

Let $a$, $b$, and $c$ be real angles such that \newline \[3\sin a + 4\sin b + 5\sin c = 0\] \[3\cos a + 4\cos b + 5\cos c = 0.\] \newline The maximum value of the expression $\frac{\sin b \sin c}{\sin^2 a}$ can be expressed as $\frac{p}{q}$ for relatively prime $p,q$. Compute $p+q$.

1983 IMO Longlists, 11

Tags: geometry

A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible given that the line $AB$ and the lake are exterior to each other.

1990 Spain Mathematical Olympiad, 4

Tags: algebra , radical

Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$ is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it.

1974 AMC 12/AHSME, 2

Tags:

Let $x_1$ and $x_2$ be such that $x_1 \neq x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals $ \textbf{(A)}\ -\frac{h}{3} \qquad\textbf{(B)}\ \frac{h}{3} \qquad\textbf{(C)}\ \frac{b}{3} \qquad\textbf{(D)}\ 2b \qquad\textbf{(E)}\ -\frac{b}{3} $

2013 NIMO Problems, 3

Tags: floor function

Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$. [i]Proposed by Kevin Sun[/i]

2016 China Team Selection Test, 3

Tags: geometry , cyclic quadrilateral , incenter

In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.

STEMS 2021 Math Cat A, Q3

Tags: geometry , angle chasing

An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Prove that the circumcircles of $\triangle{BFK}, \triangle{CEL}$ and $\triangle{DEF}$ concur.

2010 Contests, 2

Tags: rectangle , geometry

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

2010 AMC 10, 8

Tags:

Tony works $ 2$ hours a day and is paid $ \$0.50$ per hour for each full year of his age. During a six month period Tony worked $ 50$ days and earned $ \$630$. How old was Tony at the end of the six month period? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

1995 Tuymaada Olympiad, 6

Tags: geometry , number theory , right triangle , pythagorean triple , circles

Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.

2022-IMOC, G6

Tags: geometry , concurrency

Let $D$ be a point on the circumcircle of some triangle $ABC$. Let $E, F$ be points on $AC$, $AB$, respectively, such that $A,D,E,F$ are concyclic. Let $M$ be the midpoint of $BC$. Show that if $DM$, $BE$, $CF$ are concurrent, then either $BE \cap CF$ is on the circle $ADEF$, or $EF$ is parallel to $BC$. [i]proposed by USJL[/i]

1939 Eotvos Mathematical Competition, 3

1998 Hong kong National Olympiad, 2

2007 Italy TST, 3

2001 China Second Round Olympiad, 3

2014 IFYM, Sozopol, 5

2018 AMC 10, 6

1998 Israel National Olympiad, 5

2004 IMO Shortlist, 6

2013 Iran MO (3rd Round), 4

2006 IMAR Test, 3

2005 AMC 12/AHSME, 10

LMT Team Rounds 2010-20, 2020.S24

1983 IMO Longlists, 11

1990 Spain Mathematical Olympiad, 4

1974 AMC 12/AHSME, 2

2013 NIMO Problems, 3

2016 China Team Selection Test, 3

STEMS 2021 Math Cat A, Q3

2010 Contests, 2

2010 AMC 10, 8

1995 Tuymaada Olympiad, 6

2022-IMOC, G6

2007 Harvard-MIT Mathematics Tournament, 3

1995 Turkey Team Selection Test, 1

2020 Novosibirsk Oral Olympiad in Geometry, 2