Olympiad problems

2000 Turkey MO (2nd round), 2

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

1983 AMC 12/AHSME, 13

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If $xy = a, xz =b,$ and $yz = c$, and none of these quantities is zero, then $x^2+y^2+z^2$ equals: $ \textbf{(A)}\ \frac{ab+ac+bc}{abc}\qquad\textbf{(B)}\ \frac{a^2+b^2+c^2}{abc}\qquad\textbf{(C)}\ \frac{(a+b+c)^2}{abc}\qquad\textbf{(D)}\ \frac{(ab+ac+bc)^2}{abc}\qquad\textbf{(E)}\ \frac{(ab)^2+(ac)^2+(bc)^2}{abc} $

2018 PUMaC Team Round, 4

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For how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13?$

2019 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry

Let $A_1$, $B_1$, $C_1$ be the midpoints of sides $BC$, $AC$ and $AB$ of triangle $ABC$, $AK$ be the altitude from $A$, and $L$ be the tangency point of the incircle $\gamma$ with $BC$. Let the circumcircles of triangles $LKB_1$ and $A_1LC_1$ meet $B_1C_1$ for the second time at points $X$ and $Y$ respectively, and $\gamma$ meet this line at points $Z$ and $T$. Prove that $XZ = YT$.

2017 Puerto Rico Team Selection Test, 3

Tags: parallelogram , geometry , collinear

In triangle $ABC$, the altitude through $B$ intersects $AC$ at $E$ and the altitude through $C$ intersects $AB$ at $F$. Point $T$ is such that $AETF$ is a parallelogram and points $ A$ ,$T$ lie on different half-planes wrt the line $EF$. Point $D$ is such that $ABDC$ is a parallelogram and points $ A$ ,$D$ lie in different half-planes wrt line $BC$. Prove that $T, D$ and the orthocenter of $ABC$ are collinear.

1981 Vietnam National Olympiad, 3

Tags: geometry , perpendicular bisector , circumcircle

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\Delta ABC$ is tangent to $O_1O_2$.

1987 IMO Shortlist, 2

Tags: combinatorial inequality , graph theory , extremal graph theory , combinatorics

At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2n$. [i]Proposed by USA.[/i]

2019 Jozsef Wildt International Math Competition, W. 28

Tags: combinatorics , puzzle

In a room, we have 2019 aligned switches, connected to 2019 light bulbs, all initially switched on. Then, 2019 people enter the room one by one, performing the operation: The first, uses all the switches; the second, every second switch; the third, every third switch, and so on. How many lightbulbs remain switched on, after all the people entered ?

LMT Speed Rounds, 18

Tags: geometry

In square $ABCD$ with side length $2$, let $M$ be the midpoint of $AB$. Let $N$ be a point on $AD$ such that $AN = 2ND$. Let point $P$ be the intersection of segment $MN$ and diagonal $AC$. Find the area of triangle $BPM$. [i]Proposed by Jacob Xu[/i]

2013 Cuba MO, 7

Tags: inequalities , algebra

Let $x, y, z$ be positive real numbers whose sum is $2013$. Find the maximum possible value of $$\frac{(x^2+y^2+z^2)(x^3+y^3+z^3)}{ (x^4+y^4+z^4)}.$$

2021 IMO Shortlist, C3

Tags: swap , bushy , jumpy , combinatorics

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

2013 German National Olympiad, 1

Tags: diophantine equation , number theory

Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

1998 AMC 8, 16

Tags: geometry

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 16. Estimate the year in which the population of Nisos will be approximately 6,000. $ \text{(A)}\ 2050\qquad\text{(B)}\ 2075\qquad\text{(C)}\ 2100\qquad\text{(D)}\ 2125\qquad\text{(E)}\ 2150 $

1950 Moscow Mathematical Olympiad, 187

Tags: combinatorics

Is it possible to draw $10$ bus routes with stops such that for any $8$ routes there is a stop that does not belong to any of the routes, but any $9$ routes pass through all the stops?

2020 USMCA, 11

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What is the largest real $x$ satisfying $(x+1)(x+2)(x+3)(x+6) = 2x+1$?

2010 Pan African, 1

Tags: pigeonhole principle , combinatorics

Seven distinct points are marked on a circle of circumference $c$. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal $\frac{c}{24}$.

1989 Tournament Of Towns, (205) 3

Tags: number theory , divide , digit , divisible

What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

2010 Lithuania National Olympiad, 2

Tags: perpendicular bisector , circumcircle , geometry , inradius , incenter

Let $I$ be the incenter of a triangle $ABC$. $D,E,F$ are the symmetric points of $I$ with respect to $BC,AC,AB$ respectively. Knowing that $D,E,F,B$ are concyclic,find all possible values of $\angle B$.

LMT Speed Rounds, 8

Tags: combinatorics

To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$. [i]Proposed by Christopher Cheng[/i]

2007 China Team Selection Test, 1

Tags: algebra , polynomial , geometry

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

2020 Durer Math Competition Finals, 13

Tags: probability , combinatorics

In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability? Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.

1976 IMO Longlists, 17

Tags: ratio , 3d geometry , sphere , geometry

Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.

2008 Philippine MO, 1

Tags: combinatorics

Prove that the set $\{1, 2, \cdots, 2007\}$ can be expressed as the union of disjoint subsets $A_i$ for $i=1,2,\cdots, 223$ such that each $A_i$ contains nine elements and the sum of all the elements in each $A_i$ is the same.

Indonesia MO Shortlist - geometry, g2.6

Tags: geometry , circumcircle

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

2014 Saudi Arabia Pre-TST, 2.4

Tags: geometry , circumcircle , perpendicular bisector

Let $ABC$ be an acute triangle with $\angle A < \angle B \le \angle C$, and $O$ its circumcenter. The perpendicular bisector of side $AB$ intersects side $AC$ at $D$. The perpendicular bisector of side $AC$ intersects side $AB$ at $E$. Express the angles of triangle $DEO$ in terms of the angles of triangle $ABC$.