Olympiad problems

2014 Postal Coaching, 1

Let $ABC$ be a triangle in which $\angle B$ is obtuse.Let $\Gamma$ be its circumcircle and $O$ be the centre of $\Gamma$.Let the tangent to $\Gamma$ at $C$ intersect the line $AB$ in $B_1$.Let $O_1$ be the circumcentre of the circumcircle $\Gamma_1$ of $\triangle AB_1 C$.Take any point $B_2$ on the segment $BB_1$ different from $B,B_1$.Let $C_1$ be the point of contact of the tangent from $B_2$ to $\Gamma$ which is closer to $C$.Let $O_2$ be the circumcentre of $\triangle AB_2 C_1$.Prove that $O,O_2,O_1,C_1,C$ are concyclic if $OO_2\perp AO_1$.

2018 VJIMC, 1

Tags: equation , power , real number

Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

1993 Austrian-Polish Competition, 1

Tags: diophantine equation , diophantine , number theory

Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.

VMEO III 2006 Shortlist, G3

Tags: geometry , 3d geometry , tetrahedron , geometric inequality

The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$

2001 IMO Shortlist, 8

Tags: matrix , combinatorics , ramsey theory , extremal combinatorics

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2008 Sharygin Geometry Olympiad, 6

Tags: geometry

(B.Frenkin) Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?

2019 PUMaC Geometry B, 6

Tags: geometry

Let $\Gamma$ be a circle with center $A$, radius $1$ and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$ and diameter $DY $, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.

1990 Tournament Of Towns, (266) 4

Tags: combinatorics , combinatorial geometry , tiling

A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board. (S Fomin, Leningrad)

2003 Alexandru Myller, 4

Tags: trigonometry , geometry

Let $\displaystyle ABCD$ be a a convex quadrilateral and $\displaystyle O$ be a point in its interior. Let $\displaystyle a,b,c,d,e,f$ be the areas of the triangles $\displaystyle OAB,OBC,OCD,ODA,OAC,OBD$. Prove that \[ \displaystyle \left| ac - bd \right| = ef . \]

Gheorghe Țițeica 2024, P2

Tags: geometry

$ABCD$ is a tetrahedron such that $BA\perp AC$, $DB\perp (ABC)$ and $AC\neq BD$. Denote by $O$ the midpoint of $AB$ and $K$ the foot of the perpendicular from $O$ to $DC$. Prove that $$\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$$ if and only if $2AC\cdot BD=AB^2$. [i]Vietnam Olympiad[/i]

2021 CHMMC Winter (2021-22), 5

Tags: algebra

How many cubics in the form $x^3 -ax^2 + (a+d)x -(a+2d)$ for integers $a,d$ have roots that are all non-negative integers?

2023 MOAA, 1

Tags:

Compute $\sqrt{202 \times 3 - 20 \times 23 + 2 \times 23 - 23}$. [i]Proposed by Andy Xu[/i]

2015 IFYM, Sozopol, 5

Tags: number theory

If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.

2024 Switzerland Team Selection Test, 3

Tags: algebra , polynomial

Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that \[P(a)P(b)=P(c)\]

2021 Purple Comet Problems, 30

Tags: relatively prime , number theory

For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2013 BMT Spring, 8

Tags: geometry , parabola , conic , area

A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.

2008 AMC 10, 4

Tags:

A semipro baseball league has teams with $ 21$ players each. League rules state that a player must be paid at least $ \$15,000$, and that the total of all players' salaries for each team cannot exceed $ \$700,000$. What is the maximum possiblle salary, in dollars, for a single player? $ \textbf{(A)}\ 270,000 \qquad \textbf{(B)}\ 385,000 \qquad \textbf{(C)}\ 400,000 \qquad \textbf{(D)}\ 430,000 \qquad \textbf{(E)}\ 700,000$

Kyiv City MO Juniors 2003+ geometry, 2021.8.41

Tags: geometry , equal segments

On the sides $AB$ and $BC$ of the triangle $ABC$, the points $K$ and $M$ are chosen so that $KM \parallel AC$. The segments $AM$ and $KC$ intersect at the point $O$. It is known that $AK =AO$ and $KM =MC$. Prove that $AM=KB$.

2004 Denmark MO - Mohr Contest, 4

Tags: algebra , system of equations , system

Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$

2019 India PRMO, 9

Tags: geometry , circumcircle

The centre of the circle passing through the midpoints of the sides of am isosceles triangle $ABC$ lies on the circumcircle of triangle $ABC$. If the larger angle of triangle $ABC$ is $\alpha^{\circ}$ and the smaller one $\beta^{\circ}$ then what is the value of $\alpha-\beta$?

Novosibirsk Oral Geo Oly VII, 2022.7

Tags: combinatorial geometry , combinatorics , geometry

Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?

1954 AMC 12/AHSME, 25

Tags: vieta

The two roots of the equation $ a(b\minus{}c)x^2\plus{}b(c\minus{}a)x\plus{}c(a\minus{}b)\equal{}0$ are $ 1$ and: $ \textbf{(A)}\ \frac{b(c\minus{}a)}{a(b\minus{}c)} \qquad \textbf{(B)}\ \frac{a(b\minus{}c)}{c(a\minus{}b)} \qquad \textbf{(C)}\ \frac{a(b\minus{}c)}{b(c\minus{}a)} \qquad \textbf{(D)}\ \frac{c(a\minus{}b)}{a(b\minus{}c)} \qquad \textbf{(E)}\ \frac{c(a\minus{}b)}{b(c\minus{}a)}$

2004 Poland - Second Round, 2

Tags: ratio , parallelogram , angle bisector , similar triangles , geometry

Points $D$ and $E$ are taken on sides $BC$ and $CA$ of a triangle $ BD\equal{}AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of $\angle ACB$ intersects $AD$ and $BE$ at $Q$ and $R$ respectively. Prove that $ \frac{PQ}{PR}\equal{}\frac{AD}{BE}$.

2017 Serbia Team Selection Test, 6

Tags: number theory

Let $k$ be a positive integer and let $n$ be the smallest number with exactly $k$ divisors. Given $n$ is a cube, is it possible that $k$ is divisible by a prime factor of the form $3j+2$?

1989 China National Olympiad, 2

Tags: inequalities

Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]