Olympiad problems

1996 Estonia National Olympiad, 5

Suppose that $n$ teterahedra are given in space such that any two of them have at least two common vertices, but any three have at most one common vertex. Find the greatest possible $n$.

1993 Mexico National Olympiad, 1

Tags: right triangle , geometry , area of a triangle

$ABC$ is a triangle with $\angle A = 90^o$. Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$. Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$. $O$ is the midpoint of $BC$. Let the lines $OD$ and $EC$ meet at $D'$, and the lines $OE$ and $BD$ meet at $E'$. Find area $DED'E'$ in terms of the sides of $ABC$.

1971 AMC 12/AHSME, 12

Tags: modular arithmetic

For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }8$

Brazil L2 Finals (OBM) - geometry, 2004.2

Tags: geometry , isosceles , angle

In the figure, $ABC$ and $DAE$ are isosceles triangles ($AB = AC = AD = DE$) and the angles $BAC$ and $ADE$ have measures $36^o$. a) Using geometric properties, calculate the measure of angle $\angle EDC$. b) Knowing that $BC = 2$, calculate the length of segment $DC$. c) Calculate the length of segment $AC$ . [img]https://1.bp.blogspot.com/-mv43_pSjBxE/XqBMTfNlRKI/AAAAAAAAL2c/5ILlM0n7A2IQleu9T4yNmIY_1DtrxzsJgCK4BGAYYCw/s400/2004%2Bobm%2Bl2.png[/img]

2014 ASDAN Math Tournament, 6

Tags: algebra test

Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.

2018 May Olympiad, 1

Tags: algebra

Juan makes a list of $2018$ numbers. The first is $ 1$. Then each number is obtained by adding to the previous number, one of the numbers $ 1$, $2$, $3$, $4$, $5$, $6$, $7$, $ 8$ or $9$. Knowing that none of the numbers in the list ends in $0$, what is the largest value you can have the last number on the list?

2005 District Olympiad, 1

Tags: function , algebra

Let $a,b>1$ be two real numbers. Prove that $a>b$ if and only if there exists a function $f: (0,\infty)\to\mathbb{R}$ such that i) the function $g:\mathbb{R}\to\mathbb{R}$, $g(x)=f(a^x)-x$ is increasing; ii) the function $h:\mathbb{R}\to\mathbb{R}$, $h(x)=f(b^x)-x$ is decreasing.

1958 Czech and Slovak Olympiad III A, 1

Tags: equation

Find all real solutions of equation $x + \sqrt{2p - x^2} = 8$ with real parameter $p$.

1998 Croatia National Olympiad, Problem 3

Tags: rates , algebra

Ivan and Krešo started to travel from Crikvenica to Kraljevica, whose distance is $15$ km, and at the same time Marko started from Kraljevica to Crikvenica. Each of them can go either walking at a speed of $5$ km/h, or by bicycle with the speed of $15$ km/h. Ivan started walking, and Krešo was driving a bicycle until meeting Marko. Then Krešo gave the bicycle to Marko and continued walking to Kraljevica, while Marko continued to Crikvenica by bicycle. When Marko met Ivan, he gave him the bicycle and continued on foot, so Ivan arrived at Kraljevica by bicycle. Find, for each of them, the time he spent in travel as well as the time spent in walking.

2017 Greece Team Selection Test, 1

Tags: geometry

Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$, and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at $F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals $AEFA', BDFB', CDEC'$ are inscribable. (1) Prove that $DEA'B'$ is inscribable. (2) Prove that $DA', EB', FC'$ are concurrent.

2003 Junior Tuymaada Olympiad, 8

Tags: combinatorics

A few people came to the party. Prove that they can be placed in two rooms so that each of them has in their own room an even number of acquaintances. (One of the rooms can be left empty.)

CIME II 2018, 1

Tags:

Find the largest positive integer value of $n<1000$ such that $\phi(36n)=\phi(25n)+\phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$. [i]Proposed by [b]AOPS12142015[/b][/i]

2009 Belarus Team Selection Test, 4

Tags: perfect square , diophantine , diophantine equation , number theory

Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$ a) Prove that number $xy$ is a perfect square. b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality. I.Voronovich

1988 IMO Longlists, 46

Tags: combinatorics

$A_1, A_2, \ldots, A_{29}$ are $29$ different sequences of positive integers. For $1 \leq i < j \leq 29$ and any natural number $x,$ we define $N_i(x) =$ number of elements of the sequence $A_i$ which are less or equal to $x,$ and $N_{ij}(x) =$ number of elements of the intersection $A_i \cap A_j$ which are less than or equal to $x.$ It is given for all $1 \leq i \leq 29$ and every natural number $x,$ \[ N_i(x) \geq \frac{x}{e}, \] where $e = 2.71828 \ldots$ Prove that there exist at least one pair $i,j$ ($1 \leq i < j \leq 29$) such that \[ N_{ij}(1988) > 200. \]

2017 Latvia Baltic Way TST, 9

Tags: parallel , perpendicular , geometry , isosceles

In an isosceles triangle $ABC$ in which $AC = BC$ and $\angle ABC < 60^o$, $I$ and $O$ are the centers of the inscribed and circumscribed circles, respectively. For the triangle $BIO$, the circumscribed circle intersects the side $BC$ again at $D$. Prove that: i) lines $AC$ and $DI$ are parallel, ii) lines $OD$ and $IB$ are perpendicular.

India EGMO 2025 TST, 1

Tags: combinatorics , set

Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing? Proposed by Shantanu Nene

2011 Pre - Vietnam Mathematical Olympiad, 4

Tags: combinatorics

For a table $n \times 9$ ($n$ rows and $9$ columns), determine the maximum of $n$ that we can write one number in the set $\left\{ {1,2,...,9} \right\}$ in each cell such that these conditions are satisfied: 1. Each row contains enough $9$ numbers of the set $\left\{ {1,2,...,9} \right\}$. 2. Any two rows are distinct. 3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

1988 IMO Longlists, 4

Tags: circumcircle , trigonometry , inradius , geometric inequality , geometry

The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$

2025 Harvard-MIT Mathematics Tournament, 29

Tags: guts

Points $A$ and $B$ lie on circle $\omega$ with center $O.$ Let $X$ be a point inside $\omega.$ Suppose that $XO=2\sqrt{2}, XA=1, XB=3,$ and $\angle{AXB}=90^\circ.$ Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle{AXB}$ and $\triangle{YXZ}$ are similar with the same orientation. Compute $XY.$

2007 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry

Let $ABC$ a triangle and $M,N,P$ points on $AB,BC$, respective $CA$, such that the quadrilateral $CPMN$ is a paralelogram. Denote $R \in AN \cap MP$, $S \in BP \cap MN$, and $Q \in AN \cap BP$. Prove that $[MRQS]=[NQP]$.

1965 IMO Shortlist, 1

Tags: inequalities , trigonometry

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

2022 Chile Junior Math Olympiad, 6

Tags: combinatorics , combinatorial geometry

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?

2001 China Team Selection Test, 2

Tags: circumcircle , incenter , geometry

In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively. $O_1I_1$ intersects $O_2I_2$ at $P$. Find the locus of point $P$ as $D$ moves along $BC$.

2006 Pre-Preparation Course Examination, 1

Tags: limit , vector , probability , invariant , topology , real analysis , advanced fields

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

1970 Putnam, A5

Tags: ellipsoid , circles

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$