Olympiad problems

2003 Moldova National Olympiad, 12.5

Tags: polynomial , algebra

Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where $n\in{N^*}$. Find the remainder of the division of polynomial $P(x^{2n+1})$ by $P(x)$.

2011 Princeton University Math Competition, A6 / B8

Tags: geometry

Let $\omega_1$ be a circle of radius 6, and let $\omega_2$ be a circle of radius 5 that passes through the center $O$ of $\omega_1$. Let $A$ and $B$ be the points of intersection of the two circles, and let $P$ be a point on major arc $AB$ of $\omega_2$. Let $M$ and $N$ be the second intersections of $PA$ and $PB$ with $\omega_1$, respectively. Let $S$ be the midpoint of $MN$. As $P$ ranges over major arc $AB$ of $\omega_2$, the minimum length of segment $SA$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$. Find $a+b$.

2024 Korea Junior Math Olympiad (First Round), 6.

Tags: gauss , algebra

Find the number of $ x $ which follows the following : $ x-\frac{1}{x}=[x]-[\frac{1}{x}] $ $ ( \frac{1}{100} \le x \le {100} ) $

2003 All-Russian Olympiad, 4

Tags: symmetry , ceiling function , geometry , rectangle , combinatorics

Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$

2013 Romania National Olympiad, 4

Tags: number theory

Let $n$ be a positive integer and $M = {1, 2, . . . , 2n + 1}$. Find out in how many ways we can split the set $M$ into three mutually disjoint nonempty sets $A,B,C$ so that both the following are true: (i) for each $a \in A$ and $b \in B$, the remainder of the division of $a$ by $b$ belongs to $C$ (ii) for each $c \in C$ there exists $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.

2017 IOM, 3

Tags: polynomial , algebra

Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.

1974 IMO Longlists, 48

Tags: 3d geometry , sphere , geometry

We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.

2014 Contests, 3

Tags: trigonometry , power of a point , radical axis , geometry

Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.

2023 Sharygin Geometry Olympiad, 9.1

Tags: median , concyclic , geometry

The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.

1986 Putnam, A1

Tags:

Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36\leq 13x^2$.

2017 ELMO Problems, 2

Tags: orthocenter , circumcircle , geometry

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

2008 Tournament Of Towns, 5

Tags: combinatorics

Standing in a circle are $99$ girls, each with a candy. In each move, each girl gives her candy to either neighbour. If a girl receives two candies in the same move, she eats one of them. What is the minimum number of moves after which only one candy remains?

Russian TST 2020, P3

Tags: combinatorics , combinatorial geometry , angle

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

2018 Regional Olympiad of Mexico Northeast, 6

Tags: geometry

Let $ABC$ be a triangle with $AB < AC$ and $M$ the midpoint of the arc $BC$ containing $A$, plus $T$ the foot of the perpendicular from $M$ on side $AC$. Prove that $AB + AT = TC$. [img]https://cdn.artofproblemsolving.com/attachments/0/a/5c90d7001f73c2f8ff2b0e69078f9a2a5cd606.png[/img]

2008 Princeton University Math Competition, B5

Tags: number theory

How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$)?

2018 Moscow Mathematical Olympiad, 4

Tags: combinatorics

We call the arrangement of $n$ ones and $m$ zeros around the circle as good, if we can swap neighboring zero and one in such a way that we get an arrangement, that differs from the original by rotation. For what natural $m$ and $n$ does a good arrangement exist?

1958 AMC 12/AHSME, 19

Tags: ratio

The sides of a right triangle are $ a$ and $ b$ and the hypotenuse is $ c$. A perpendicular from the vertex divides $ c$ into segments $ r$ and $ s$, adjacent respectively to $ a$ and $ b$. If $ a : b \equal{} 1 : 3$, then the ratio of $ r$ to $ s$ is: $ \textbf{(A)}\ 1 : 3\qquad \textbf{(B)}\ 1 : 9\qquad \textbf{(C)}\ 1 : 10\qquad \textbf{(D)}\ 3 : 10\qquad \textbf{(E)}\ 1 : \sqrt{10}$

1998 IMO Shortlist, 5

Tags: number theory , prime number , algebra , functional equation

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

2018 CMIMC Algebra, 6

Tags: algebra

We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \[k = \sum_{i=2}^n a_i F_i,\] where $a_i\in\{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i\ge2$). This representation is said to be $\textit{minimal}$ if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.

2018 Iran Team Selection Test, 3

Tags: geometry , pole and polar , pappus

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

2017 NIMO Summer Contest, 13

Tags: quadratic , algebra , polynomial

We say that $1\leq a\leq101$ is a quadratic polynomial residue modulo $101$ with respect to a quadratic polynomial $f(x)$ with integer coefficients if there exists an integer $b$ such that $101 \mid a-f(b)$. For a quadratic polynomial $f$, we define its quadratic residue set as the set of quadratic residues modulo $101$ with respect to $f(x)$. Compute the number of quadratic residue sets. [i]Proposed by Michael Ren[/i]

2014 IFYM, Sozopol, 4

Tags: algebra , inequality , inequalities

Prove that for $\forall$ $x,y,z\in \mathbb{R}^+$ the following inequality is true: $\frac{x}{y+z}+\frac{25y}{z+x}+\frac{4z}{x+y}>2$.

2006 Princeton University Math Competition, 10

Tags: graph theory , combinatorics

What is the largest possible number of vertices one can have in a graph that satisfies the following conditions: each vertex is connected to exactly $3$ other vertices, and there always exists a path of length less than or equal to $2$ between any two vertices?

1988 AMC 12/AHSME, 30

Tags:

Let $f(x) = 4x - x^{2}$. Give $x_{0}$, consider the sequence defined by $x_{n} = f(x_{n-1})$ for all $n \ge 1$. For how many real numbers $x_{0}$ will the sequence $x_{0}, x_{1}, x_{2}, \ldots$ take on only a finite number of different values? $ \textbf{(A)}\ \text{0}\qquad\textbf{(B)}\ \text{1 or 2}\qquad\textbf{(C)}\ \text{3, 4, 5 or 6}\qquad\textbf{(D)}\ \text{more than 6 but finitely many}\qquad\textbf{(E)}\ \text{infinitely many} $

2014 Iran MO (3rd Round), 5

Tags: geometry , geometric transformation , rotation , combinatorics

An $n$-mino is a connected figure made by connecting $n$ $1 \times 1 $ squares. Two polyminos are the same if moving the first we can reach the second. For a polymino $P$ ,let $|P|$ be the number of $1 \times 1$ squares in it and $\partial P$ be number of squares out of $P$ such that each of the squares have at least on edge in common with a square from $P$. (a) Prove that for every $x \in (0,1)$:\[\sum_P x^{|P|}(1-x)^{\partial P}=1\] The sum is on all different polyminos. (b) Prove that for every polymino $P$, $\partial P \leq 2|P|+2$ (c) Prove that the number of $n$-minos is less than $6.75^n$. [i]Proposed by Kasra Alishahi[/i]