Olympiad problems

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.

2019 Saudi Arabia JBMO TST, 2

Tags: geometry

Two circles, having their centers in A and B, intersect at points M and N. The radii AP and BQ are parallel and are in different semi-planes determined of the line AB. If the external common tangent intersect AB in D, and PQ intersects AB at C, prove that the <CND is right.

1998 Romania National Olympiad, 2

Tags: complex numbers , inequalities , algebra

Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.

2009 Today's Calculation Of Integral, 434

Tags: calculus , integration , logarithms , function , calculus computations

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

2017 Abels Math Contest (Norwegian MO) Final, 1a

Tags: functional equation , function , algebra

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.

2022 Caucasus Mathematical Olympiad, 8

Tags: combinatorics , graph theory

There are $n > 2022$ cities in the country. Some pairs of cities are connected with straight two-ways airlines. Call the set of the cities {\it unlucky}, if it is impossible to color the airlines between them in two colors without monochromatic triangle (i.e. three cities $A$, $B$, $C$ with the airlines $AB$, $AC$ and $BC$ of the same color). The set containing all the cities is unlucky. Is there always an unlucky set containing exactly 2022 cities?

1999 AMC 8, 17

Tags: AMC , AMC 8

Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.) $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 5\qquad\text{(D)}\ 7\qquad\text{(E)}\ 15 $

1990 IMO Longlists, 68

Tags: analytic geometry , geometry

In coordinate plane, a variable point $M$, starting from the origin $O(0, 0)$, moves on the line $l$ with slope $k$, where $k$ is an irrational number. [b](i)[/b] Prove that point $O(0, 0)$ is the only rational point (namely, the coordinates of which are both rationals) on the line $l.$ [b](ii)[/b] Prove that for any number $\varepsilon > 0$, there exist integers $m, n$ such that the distance between $l$ and the point $(m, n)$ is less than $\varepsilon.$

2015 BMT Spring, 14

Tags: algebra

Determine $$ \left|\prod^{10}_{k=1}(e^{\frac{i \pi}{2^k}}+ 1) \right|$$

2020 Jozsef Wildt International Math Competition, W35

Tags: inequalities , geometry , Geometric Inequalities

In all triangles $ABC$ does it hold: $$(b^n+c^p)\tan^{n+p}\frac A2+(c^n+a^p)\tan^{n+p}\frac B2+(a^n+b^p)\tan^{n+p}\frac C2\ge6\sqrt{\left(\frac{4r^2}{R\sqrt3}\right)^{n+p}}$$ where $n,p\in(0,\infty)$. [i]Proposed by Nicolae Papacu[/i]

2014 Contests, 1

Tags: GMO-Gulf Mathmatical Olympiad , algebra

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct

2024 Yasinsky Geometry Olympiad, 5

Tags: geometry

Let $ ABCDEF $ be a cyclic hexagon such that $ AD \parallel EF $. Points $ X $ and $ Y $ are marked on diagonals $ AE $ and $ DF $, respectively, such that $ CX = EX $ and $ BY = FY $. Let $ O $ be the intersection point of $ AE $ and $ FD $, $ P $ the intersection point of $ CX $ and $ BY $, and $ Q $ the intersection point of $ BF $ and $ CE $. Prove that points $ O, P, $ and $ Q $ are collinear. [i]Proposed by Matthew Kurskyi[/i]

2013 Peru IMO TST, 1

Tags: invariant , monovariant , combinatorics , IMO Shortlist

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]

2015 ASDAN Math Tournament, 1

Tags: 2015 , Advanced Topics Tiebreaker

Rachel has $3$ children, all of which are at least $2$ years old. The ages of the children are all pairwise relatively prime, but Rachel’s age is a multiple of each of her children’s ages. What is Rachel’s minimum possible age?

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: Diophantine equation , diophantine , number theory

Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$

2009 Mathcenter Contest, 3

Tags: combinatorics , combinatorial geometry , points

Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear. [i](Anonymous314)[/i] PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]

2002 Romania National Olympiad, 4

Tags: geometry , 3D geometry , prism , geometry proposed

The right prism $[A_1A_2A_3\ldots A_nA_1'A_2'A_3'\ldots A_n'],n\in\mathbb{N},n\ge 3$, has a convex polygon as its base. It is known that $A_1A_2'\perp A_2A_3',A_2A_3'\perp A_3A_4',$$\ldots A_{n-1}A_n'\perp A_nA_1', A_nA_1'\perp A_1A_2'$. Show that: $a)$ $n=3$; $b)$ the prism is regular.

1951 Moscow Mathematical Olympiad, 204

Tags: least common multiple , reciprocal , Sum , number theory

* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.

2021 Brazil Team Selection Test, 3

Tags: algebra , IMO Shortlist , IMO Shortlist 2020 , polynomial , algorithm , lagrange s interpolation , Gerhard Woeginger

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?