Olympiad problems

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 $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$

2014 IMO Shortlist, N1

Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , induction

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

MathLinks Contest 3rd, 3

Tags: combinatorics , geometry

An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$. Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.

2003 Kazakhstan National Olympiad, 6

Tags: geometry , circumcircle , circles

Let the point $ B $ lie on the circle $ S_1 $ and let the point $ A $, other than the point $ B $, lie on the tangent to the circle $ S_1 $ passing through the point $ B $. Let a point $ C $ be chosen outside the circle $ S_1 $, so that the segment $ AC $ intersects $ S_1 $ at two different points. Let the circle $ S_2 $ touch the line $ AC $ at the point $ C $ and the circle $ S_1 $ at the point $ D $, on the opposite side from the point $ B $ with respect to the line $ AC $. Prove that the center of the circumcircle of triangle $ BCD $ lies on the circumcircle of triangle $ ABC $.

2005 Sharygin Geometry Olympiad, 10.3

Tags: circles , parallel , distance , chord , geometry

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$. Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.

2006 Oral Moscow Geometry Olympiad, 1

Tags: equal circles , construction , geometry , cyclic

An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles. (L. Blinkov)

2019 Junior Balkan MO, 3

Tags: perpendicular bisector , circumcircle , geometry

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , number theory , difference , divisor

Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most $n$.