Olympiad problems

2004 Romania National Olympiad, 4

Let $p,q \in \mathbb N^{\ast}$, $p,q \geq 2$. We say that a set $X$ has the property $\left( \mathcal S \right)$ if no matter how we choose $p$ subsets $B_i \subset X$, $i = \overline{1,n}$, not necessarily distinct, each with $q$ elements, there is a subset $Y \subset X$ with $p$ elements s.t. the intersection of $Y$ with each of the $B_i$'s has an element at most, $i=\overline{1,p}$. Prove that: (a) if $p=4,q=3$ then any set composed of $9$ elements doesn't have $\left( \mathcal S \right)$; (b) any set $X$ composed of $pq-q$ elements doesn't have the property $\left( \mathcal S \right)$; (c) any set $X$ composed of $pq-q+1$ elements has the property $\left( \mathcal S \right)$. [i]Dan Schwarz[/i]

1994 Dutch Mathematical Olympiad, 5

Tags: inequalities , algebra , polynomial , algebra unsolved

Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.

2025 Philippine MO, P6

Tags: combinatorics , cartesian plane , knight , paths

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list] [*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or [*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down). [/list] Thus, for any $k$, the ant can choose to go to one of eight possible points. \\ Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

2010 Malaysia National Olympiad, 3

Tags: combinatorics unsolved , combinatorics

Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10?

2006 Hungary-Israel Binational, 3

Tags: geometry unsolved , geometry

Let $ \mathcal{H} \equal{} A_1A_2\ldots A_n$ be a convex $ n$-gon. For $ i \equal{} 1, 2, \ldots, n$, let $ A'_{i}$ be the point symmetric to $ A_i$ with respect to the midpoint of $ A_{i \minus{} 1}A_{i \plus{} 1}$ (where $ A_{n \plus{} 1} \equal{} A_1$). We say that the vertex $ A_i$ is [i]good[/i] if $ A'_{i}$ lies inside $ \mathcal{H}$. Show that at least $ n \minus{} 3$ vertices of $ \mathcal{H}$ are [i]good[/i].

1997 Estonia National Olympiad, 2

Tags: trinomial , algebra , quadratic polynomial

Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.

2014 BMT Spring, 2

Tags: combinatorics

A mathematician is walking through a library with twenty-six shelves, one for each letter of the alphabet. As he walks, the mathematician will take at most one book off each shelf. He likes symmetry, so if the letter of a shelf has at least one line of symmetry (e.g., M works, L does not), he will pick a book with probability $\frac12$. Otherwise he has a $\frac14$ probability of taking a book. What is the expected number of books that the mathematician will take?

1998 Cono Sur Olympiad, 1

Tags: combinatorics , algebra

We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$. We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!

2015 NIMO Problems, 6

Tags: geometry , geometric transformation , reflection , circumcircle , homothety , trigonometry , parallelogram

Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Sreejato Bhattacharya[/i]

1999 AMC 8, 12

Tags: ratio , percent

The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11:4$ . To the nearest whole percent, what percent of its games did the team lose? $ \text{(A)}\ 24\qquad\text{(B)}\ 27\qquad\text{(C)}\ 36\qquad\text{(D)}\ 45\qquad\text{(E)}\ 73 $

2019 Romania National Olympiad, 1

Tags: algebra , inequalities

a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$ b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$

2020 Harvard-MIT Mathematics Tournament, 5

Tags:

Let $ABCDEF$ be a regular hexagon with side length $2$. A circle with radius $3$ and center at $A$ is drawn. Find the area inside quadrilateral $BCDE$ but outside the circle. [i]Proposed by Carl Joshua Quines.[/i]

2004 All-Russian Olympiad Regional Round, 9.7

Tags: parallel , geometry , angles , parallelogram

Inside the parallelogram $ABCD$, point $M$ is chosen, and inside the triangle $AMD$, point $N$ is chosen in such a way that $$\angle MNA + \angle MCB =\angle MND + \angle MBC = 180^o.$$ Prove that lines $MN$ and $AB$ are parallel.

2023 May Olympiad, 3

Tags: geometry , areas

On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

LMT Team Rounds 2021+, 14

Tags: geometry

In a cone with height $3$ and base radius $4$, let $X$ be a point on the circumference of the base. Let $Y$ be a point on the surface of the cone such that the distance from $Y$ to the vertex of the cone is $2$, and $Y$ is diametrically opposite $X$ with respect to the base of the cone. The length of the shortest path across the surface of the cone from $X$ to $Y$ can be expressed as $\sqrt{a +\sqrt{b}}$, where a and b are positive integers. Find $a +b$.

2000 AMC 8, 10

Tags: AMC

Ara and Shea were once the same height. Since then Shea has grown $20\%$ while Ara has grow half as many inches as Shea. Shea is now $60$ inches tall. How tall, in inches, is Ara now? $\text{(A)}\ 48 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$

2018 BMT Spring, Tie 1

Tags: combinatorics

Every face of a cube is colored one of $3$ colors at random. What is the expected number of edges that lie along two faces of different colors?

2025 Kosovo National Mathematical Olympiad`, P3

Tags: geometry , similar triangles , national olympiad , Kosovo

Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively. Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if $$g_a : g_b : g_c = g_x : g_y : g_z, $$ then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.

2016 Balkan MO Shortlist, C2

Tags: combinatorics , Maximal

There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.) Find the maximal $k$ such that the following holds: There are $k$ customers such that either all of them were in the shop at a specic time instance or no two of them were both in the shop at any time instance.

2007 Alexandru Myller, 1

Tags: algebra , modular arithmetic , number theory

[b]a)[/b] Show that $ n^2+2n+2007 $ is squarefree for any natural number $ n. $ [b]b)[/b] Prove that for any natural number $ k\ge 2 $ there is a nonnegative integer $ m $ such that $ m^2+2m+2k $ is a perfect square.

2023 Romania Team Selection Test, P1

Tags: number theory , sum of digits

Let $m$ and $n$ be positive integers, where $m < 2^n.$ Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to $m\cdot(2^n- 1).$ [i]The Problem Selection Committee[/i]

2022 Canada National Olympiad, 2

Tags: number theory

I think we are allowed to discuss since its after 24 hours How do you do this Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function

1985 AMC 12/AHSME, 10

Tags: trigonometry

An arbitrary circle can intersect the graph $ y \equal{} \sin x$ in $ \textbf{(A)} \text{ at most 2 points} \qquad \textbf{(B)} \text{ at most 4 points} \qquad$ $ \textbf{(C)} \text{ at most 6 points} \qquad \textbf{(D)} \text{ at most 8 points} \qquad$ $ \textbf{(E)} \text{ more than 16 points}$

2015 Sharygin Geometry Olympiad, 2

Tags: geometry , parallelogram , construction , projection

A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram. (A. Zaslavsky)

1947 Putnam, A4

Tags: Putnam , physics

A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ($=v_0 $), determine the set $H$ of points in the plane and above the horizontal which can be hit.