Olympiad problems

2019 IMO Shortlist, G1

Tags: geometry

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

2012 Pre-Preparation Course Examination, 3

Tags: function , integration , real analysis

Consider the set $\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$. Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?

2011 Brazil Team Selection Test, 1

Tags: trinomial , polynomial , algebra

Let $P_1$, $P_2$ and $P_3$ be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial $P_1 + P_2 + P_3$ has also real roots.

2008 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: modular arithmetic

Let $ b$ be an even positive integer. Assume that there exist integer $ n > 1$ such that $ \frac {b^{n} \minus{} 1}{b \minus{} 1}$ is perfect square. Prove that $ b$ is divisible by 8.

2018 JBMO TST-Turkey, 8

Tags: algebra , triangle inequality , positive real numbers , inequalities

Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$

2006 Hanoi Open Mathematics Competitions, 7

Tags: geometry , circles

On the circle $(O)$ of radius $15$ cm are given $2$ points $A, B$. The altitude $OH$ of the triangle $OAB$ intersect $(O)$ at $C$. What is $AC$ if $AB = 16$ cm?

Novosibirsk Oral Geo Oly VIII, 2023.1

Tags: geometry , square , area

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]

1948 Moscow Mathematical Olympiad, 154

Tags: diophantine , number theory , inequalities , combinatorics

How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

2020 Czech-Austrian-Polish-Slovak Match, 4

Tags: algebra , functional equation , functional

Let $a$ be a given real number. Find all functions $f : R \to R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in R$. (Walther Janous, Austria)

2015 District Olympiad, 4

Tags: word , base 10 arithmetic , arithmetic

Determine all pairs of natural numbers, the components of which have the same number of digits and the double of their product is equal with the number formed by concatenating them.

2017 Kyiv Mathematical Festival, 5

Tags: geometry , number theory , analytic geometry

A triangle $ABC$ is given on the plane, such that all its vertices have integer coordinates. Does there necessarily exist a straight line which intersects the straight lines $AB,$ $BC,$ and $AC$ at three distinct points with integer coordinates?

1995 Grosman Memorial Mathematical Olympiad, 1

Tags: number theory , divisor

Positive integers $d_1,d_2,...,d_n$ are divisors of $1995$. Prove that there exist $d_i$ and $d_j$ among them, such the denominator of the reduced fraction $d_i/d_j$ is at least $n$

2009 Danube Mathematical Competition, 2

Tags: sum , consecutive , number theory

Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .

2018 Moldova Team Selection Test, 9

Tags: number theory , number of divisors

The positive integers $a $ and $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where $ a_1 <a_2 <\dots $ and $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ . Find $a$ and $b $ .

2012 Indonesia TST, 3

Tags: combinatorics

Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.

2023 Bulgaria EGMO TST, 1

Tags: geometry , symmedian , angle chasing , similar triangles , circumcircle

Let $ABC$ be a triangle with circumcircle $k$. The tangents at $A$ and $C$ intersect at $T$. The circumcircle of triangle $ABT$ intersects the line $CT$ at $X$ and $Y$ is the midpoint of $CX$. Prove that the lines $AX$ and $BY$ intersect on $k$.

2004 National Chemistry Olympiad, 59

Tags:

What substance is formed when $\ce{CF2=CF2}$ is polymerized? $ \textbf{(A) } \text{Polyethylene} \qquad\textbf{(B) } \text{Polyurethane}\qquad\textbf{(C) } \text{PVC}\qquad\textbf{(D) } \text{Teflon}\qquad$

2018 Lusophon Mathematical Olympiad, 2

Tags: right triangle , isosceles , square , geometry

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

2019 Macedonia National Olympiad, 3

Tags: symmetry , angle chasing , miquel point , geometry

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

1953 Moscow Mathematical Olympiad, 254

Tags: game , combinatorics , reflection , combinatorial geometry

Given a $101\times 200$ sheet of graph paper, we start moving from a corner square in the direction of the square’s diagonal (not the sheet’s diagonal) to the border of the sheet, then change direction obeying the laws of light’s reflection. Will we ever reach a corner square? [img]https://cdn.artofproblemsolving.com/attachments/b/8/4ec2f4583f406feda004c7fb4f11a424c9b9ae.png[/img]

2024 CCA Math Bonanza, I3

Tags:

Find the units digit of $2^{2^{\iddots^{2}}}$, where there are $2024$ $2$s. [i]Individual #3[/i]

2023 Princeton University Math Competition, 4

Tags: number theory

Find the largest integer $x<1000$ such that $\left(\begin{array}{c}1515 \\ x\end{array}\right)$ and $\left(\begin{array}{c}1975 \\ x\end{array}\right)$ are both odd.

MIPT student olimpiad spring 2022, 1

Tags: real analysis , function

Sequence of uniformly continuous functions $f_n:R \to R$ uniformly converges to a function $f:R\to R$. Can we say that $f$ is uniformly continuous?

1966 IMO Shortlist, 23

Tags: geometry , 3d geometry , tetrahedron

Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. [i](a) [/i]Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. [i](b)[/i] Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.

2019 Hanoi Open Mathematics Competitions, 3

Tags: algebra , polynomial

Let $a$ and $b$ be real numbers, and the polynomial $P(x) =ax + b$ such that $P(2)- P(1)= 3$: Compute the value of $P(5)- P(0)$. [b]A.[/b] $11$ [b]B.[/b] $13$ [b]C.[/b] $15$ [b]D.[/b] $17$ [b]E.[/b] $19$