Olympiad problems

2019 India IMO Training Camp, P1

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

2020 Macedonia Additional BMO TST, 2

Tags: angle chasing , geometry , parallel

Let $ABCD$ be a convex quadrilateral. On the sides $AB$ and $CD$ there are interior points $K$ and $L$, respectively, such that $\angle BAL = \angle CDK$. Prove that the following statements are equivalent: $i) \angle BLA= \angle CKD$ $ii) AD \parallel BC $

1968 AMC 12/AHSME, 12

Tags: ratio , heron's formula , circumcircle , geometry , area of a triangle

A circle passes through the vertices of a triangle with side-lengths of $7\tfrac{1}{2},10,12\tfrac{1}{2}$. The radius of the circle is: $\textbf{(A)}\ \dfrac{15}{4} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ \dfrac{25}{4} \qquad \textbf{(D)}\ \dfrac{35}{4} \qquad \textbf{(E)}\ \dfrac{15\sqrt2}{2} $

2011 Polish MO Finals, 2

Tags: dilation , circumcircle , power of a point , geometry , perpendicular bisector , geometric transformation , incenter

The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$, the line joining the midpoints of $BF,BD$, and the line joining the midpoints of $CD,CE$. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$.

1991 Tournament Of Towns, (292) 2

Tags: triangle construction , geometric construction , geometry

Two points $K$ and $L$ are given on a circle. Construct a triangle $ABC$ so that its vertex $C$ and the intersection points of its medians $AK$ and $BL$ both lie on the circle, $K$ and $L$ being the midpoints of its sides $BC$ and $AC$.

1991 Czech And Slovak Olympiad IIIA, 4

Tags: geometry , ratio , distance

Prove that in all triangles $ABC$ with $\angle A = 2\angle B$ the distance from $C$ to $A$ and to the perpendicular bisector of $AB$ are in the same ratio.

2003 Baltic Way, 7

Tags: combinatorics

A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?

2005 South East Mathematical Olympiad, 1

Tags: derivative , parabola , conic , calculus , function , algebra

Let $a \in \mathbb{R}$ be a parameter. (1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola. (2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

2006 Sharygin Geometry Olympiad, 26

Tags: cone , 3d geometry , concyclic , geometry , tangent circle

Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

1969 IMO Longlists, 47

Tags: tangent , reflection , geometry , circumcircle

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

1981 Romania Team Selection Tests, 3.

Tags: inequalities , binomial coefficient , algebra

Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\] for $t=1,2,\ldots,n-1$ [i]Vasile Brânzănescu[/i]

2023 MOAA, 10

Tags:

A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1? [i]Proposed by Andy Xu[/i]

2012 Iran Team Selection Test, 2

Tags: combinatorial geometry , inequalities , combinatorics , rotation

Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$. Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$-gon (are in [i]convex position[/i]), but the points of $A$ are not, prove that $T(B)<T(A)$. [i]Proposed by Ali Khezeli[/i]

2017 Harvard-MIT Mathematics Tournament, 10

Tags: tangential quadrilateral , geometry , ratio

Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$. Let $I$ be the center of $\omega$, and let $IA=12,$ $IB=16,$ $IC=14,$ and $ID=11$. Let $M$ be the midpoint of segment $AC$. Compute the ratio $\frac{IM}{IN}$, where $N$ is the midpoint of segment $BD$.

V Soros Olympiad 1998 - 99 (Russia), 11.1

Tags: algebra , logarithm , trigonometry

Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.

1998 Vietnam Team Selection Test, 1

Tags: polynomial , algebra , rational root theorem , calculus , integration , number theory

Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: geometry , circumcircle

Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$

2023 Kyiv City MO Round 1, Problem 4

Tags: combinatorics

For $n \ge 2$ consider $n \times n$ board and mark all $n^2$ centres of all unit squares. What is the maximal possible number of marked points that we can take such that there don't exist three taken points which form right triangle? [i]Proposed by Mykhailo Shtandenko[/i]

2003 Flanders Math Olympiad, 1

Tags:

Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc. Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal. Who made it?

2012 CentroAmerican, 1

Tags: digit sum , sum of digits , number theory

Find all positive integers that are equal to $700$ times the sum of its digits.

2017 Turkey Team Selection Test, 7

Tags: functional equation , algebra

Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.

2013 Tournament of Towns, 2

Tags: polynomial , algebra

Find all positive integers $n$ for which the following statement holds: For any two polynomials $P(x)$ and $Q(x)$ of degree $n$ there exist monomials $ax^k$ and $bx^{ell}, 0 \le k,\ ell \le n$, such that the graphs of $P(x) + ax^k$ and $Q(x) + bx^{ell}$ have no common points.

DMM Individual Rounds, 2021 Tie

Tags: combinatorics

You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ the probability that you land on the starting face after $n$ turns. Supposed that $F(8) = \frac{43}{256}$ , and F(10) can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.

May Olympiad L2 - geometry, 2023.4

Tags: paper , rectangle , folding , geometry

Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

1988 IMO Longlists, 50

Tags: rectangle , geometry , combinatorics

Prove that the numbers $A,B$ and $C$ are equal, where: - $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles. - $B=$ number of sequences of ones and twos that add up to $n$ - $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and - $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$