Olympiad problems

2011 ELMO Shortlist, 4

Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal. [i]David Yang.[/i]

2022 Swedish Mathematical Competition, 1

Tags: combinatorics , tiling , tiles , combinatorial geometry

What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? [center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]

1994 Poland - First Round, 12

Tags: geometric series , inequalities

The sequence $(x_n)$ is given by $x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$ Prove that for all natural numbers $n \geq 1$ the following inequality holds $x_1+x_2+...+x_n < 1$.

2008 Postal Coaching, 1

Tags: combination , lcm , least common multiple , number theory

Prove that for any $n \ge 1$, $LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$

2013 Greece Team Selection Test, 2

Tags: parametric equation , analytic geometry , parameterization

For the several values of the parameter $m\in \mathbb{N^{*}}$,find the pairs of integers $(a,b)$ that satisfy the relation $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11}$, and,moreover,on the Cartesian plane $Oxy$ the lie in the square $D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}$. [i][u]Note:[/u]$[k,l]$ denotes the least common multiple of the positive integers $k,l$.[/i]

1957 AMC 12/AHSME, 26

Tags: geometry , circumcircle , ratio

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be: $ \textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad \\ \textbf{(C)}\ \text{such that the three angles fromed at the point each be }{120^\circ}\qquad \\ \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad \\ \textbf{(E)}\ \text{the intersection of the medians of the triangle}$

2016-2017 SDML (Middle School), 12

Tags:

What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x-3$? $\text{(A) }6\pi\qquad\text{(B) }7\pi\qquad\text{(C) }8\pi\qquad\text{(D) }9\pi\qquad\text{(E) }10\pi$

1985 All Soviet Union Mathematical Olympiad, 408

Tags: circumcircle , geometry , arc , circumradius

The $[A_0A_5]$ diameter divides a circumference with the $O$ centre onto two hemicircumferences. One of them is divided onto five equal arcs $A_0A_1, A_1A_2, A_2A_3, A_3A_4, A_4A_5$. The $(A_1A_4)$ line crosses $(OA_2)$ and $(OA_3)$ lines in $M$ and $N$ points. Prove that $(|A_2A_3| + |MN|)$ equals to the circumference radius.

1978 Vietnam National Olympiad, 2

Tags: parameterization , algebra , system of equations

Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.

2001 Taiwan National Olympiad, 3

Tags: search , combinatorics

Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint. what i have is [hide="this"]we may assume that the union of the $A_{i}$s is $S$. [/hide]

LMT Speed Rounds, 11

Tags: geometry

Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees. [i]Proposed by Edwin Zhao[/i]

2018 Regional Competition For Advanced Students, 2

Tags: geometry , equilateral triangle , chord , perpendicular bisector

Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. [i]Proposed by Stefan Leopoldseder[/i]

2019 IMO Shortlist, A4

Tags: algebra

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

1996 USAMO, 3

Tags: ratio , geometric transformation , trapezoid , geometry , angle bisector , inequalities , reflection

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

1993 AMC 12/AHSME, 5

Tags: percent , percent increase

Last year a bicycle cost $\$160$ and a cycling helmet cost $ \$ 40$. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 7\% \qquad\textbf{(C)}\ 7.5\% \qquad\textbf{(D)}\ 8\% \qquad\textbf{(E)}\ 15\% $

1994 IMO Shortlist, 3

Tags: invariant , algorithm , game , calculus , combinatorics

Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?

2012 Balkan MO Shortlist, A3

Tags:

Determine the maximum possible number of distinct real roots of a polynomial $P(x)$ of degree $2012$ with real coefficients satisfying the condition \begin{align*} P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*} for all real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=0$

1972 Miklós Schweitzer, 2

Tags: advanced fields

Let $ \leq$ be a reflexive, antisymmetric relation on a finite set $ A$. Show that this relation can be extended to an appropriate finite superset $ B$ of $ A$ such that $ \leq$ on $ B$ remains reflexive, antisymmetric, and any two elements of $ B$ have a least upper bound as well as a greatest lower bound. (The relation $ \leq$ is extended to $ B$ if for $ x,y \in A , x \leq y$ holds in $ A$ if and only if it holds in $ B$.) [i]E. Freid[/i]

2013 Purple Comet Problems, 4

Tags:

One of the two Purple Comet! question writers is an adult whose age is the same as the last two digits of the year he was born. His birthday is in August. What is his age today?

1951 AMC 12/AHSME, 18

Tags:

The expression $ 21x^2 \plus{} ax \plus{} 21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be one if $ a$ is: $ \textbf{(A)}\ \text{any odd number} \qquad\textbf{(B)}\ \text{some odd number} \qquad\textbf{(C)}\ \text{any even number}$ $ \textbf{(D)}\ \text{some even number} \qquad\textbf{(E)}\ \text{zero}$

2001 Switzerland Team Selection Test, 9

Tags: combinatorics

In Geneva there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

2010 Peru IMO TST, 6

Tags: trigonometry , symmetry , circumcircle , geometry

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

2007 VJIMC, Problem 4

Tags: inequalities , function

Let $f:[0,1]\to[0,\infty)$ be an arbitrary function satisfying $$\frac{f(x)+f(y)}2\le f\left(\frac{x+y}2\right)+1$$ for all pairs $x,y\in[0,1]$. Prove that for all $0\le u<v<w\le1$, $$\frac{w-v}{w-u}f(u)+\frac{v-u}{w-u}f(w)\le f(v)+2.$$

2015 IFYM, Sozopol, 3

Tags: game strategy , combinatorics

A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?

2008 ITest, 88

Tags: 3d geometry , geometry , analytic geometry

A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$ This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share 2-D square faces with neighbors ($\textit{one}$ 2-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many 2-D squares are faces of one or more of the unit $6$-cubes?