Olympiad problems

2014 Contests, 2

Tags: function , trigonometry

Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.

2009 National Olympiad First Round, 26

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For every $ 0 \le i \le 17$, $ a_i \equal{} \{ \minus{} 1, 0, 1\}$. How many $ (a_0,a_1, \dots , a_{17})$ $ 18 \minus{}$tuples are there satisfying : $ a_0 \plus{} 2a_1 \plus{} 2^2a_2 \plus{} \cdots \plus{} 2^{17}a_{17} \equal{} 2^{10}$ $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 1$

1988 Brazil National Olympiad, 4

Tags: geometry , porism , plane geometry , circles , incircle , circumcircle

Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too.

2021 USAMO, 4

Tags: number theory , greatest common divisor

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

2024 UMD Math Competition Part I, #20

Tags: combinatorics

There are eight seats at a round table. Six adults $A_1, \ldots, A_6$ and two children sit around the table. The two children are not allowed to six next to each other. All the seating configurations where the children are not seated next to each other are equally likely. What is the probability that the adults $A_1$ and $A_2$ end up sitting next to each other?\[ \mathrm a. ~4/15\qquad \mathrm b. ~2/7 \qquad \mathrm c. ~2/9 \qquad\mathrm d. ~1/3\qquad\mathrm e. ~1/5\qquad\]

1993 India National Olympiad, 2

Tags: quadratic , polynomial , algebra

Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.

1988 IMO Longlists, 21

Tags: geometry

Let "AB" and $CD$ be two perpendicular chords of a circle with centre $O$ and radius $r$ and let $X,Y,Z,W$ denote the cyclical order of the four parts into which the disc is thus divided. Find the maximum and minimum of the quantity \[ \frac{A(X) + A(Z)}{A(Y) + A(W)}, \] where $A(U)$ denotes the area of $U.$

2008 Spain Mathematical Olympiad, 3

Tags: modular arithmetic , combinatorics

Let $p\ge 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.

2025 Thailand Mathematical Olympiad, 10

Tags: polynomial , number theory , algebra

Let $n$ be a positive integer. Show that there exist a polynomial $P(x)$ with integer coefficient that satisfy the following [list] [*]Degree of $P(x)$ is at most $2^n - n -1$ [*]$|P(k)| = (k-1)!(2^n-k)!$ for each $k \in \{1,2,3,\dots,2^n\}$ [/list]

2009 Ukraine Team Selection Test, 3

Tags: subset , combinatorics , set

Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection. a) Show that the intersection of all subsets of $F$ is not empty. b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?

2006 China Western Mathematical Olympiad, 3

Tags: trigonometry , polynomial , algebra

Let $k$ be a positive integer not less than 3 and $x$ a real number. Prove that if $\cos (k-1)x$ and $\cos kx$ are rational, then there exists a positive integer $n>k$, such that both $\cos (n-1)x$ and $\cos nx$ are rational.

1992 French Mathematical Olympiad, Problem 4

Tags: sequence , recurrence relation , algebra

Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula $$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit. (b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.

2010 Sharygin Geometry Olympiad, 2

Tags: geometry , trigonometry , circles , circumcircle , trigonometric

Each of two equal circles $\omega_1$ and $\omega_2$ passes through the center of the other one. Triangle $ABC$ is inscribed into $\omega_1$, and lines $AC, BC$ touch $\omega_2$ . Prove that $cosA + cosB = 1$.

1997 VJIMC, Problem 4-M

Tags: limit , real analysis , number theory , digit

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

1987 AIME Problems, 10

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Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)

1999 National High School Mathematics League, 8

Tags: complex number

If $\theta=\arctan \frac{5}{12}$, $z=\frac{\cos 2\theta+\text{i}\sin2\theta}{239+\text{i}}$, then $\arg z=$________.

1999 South africa National Olympiad, 2

Tags:

$A,\ B,\ C$ and $D$ are points on a given straight line, in that order. Show how to construct a square $PQRS$, with all of $P,\ Q,\ R$ and $S$ on the same side of $AD$, such that $A,\ B,\ C$ and $D$ lie on $PQ,\ SR,\ QR$ and $PS$ produced respectively.

2022 Stanford Mathematics Tournament, 4

Tags:

Evaluate the integral: \[\int_{\frac{\pi^2}{4}}^{4\pi^2}\sin(\sqrt{x})dx.\]

2023 JBMO Shortlist, C4

Tags: combinatorics

Anna and Bob are playing the following game: The number $2$ is initially written on the blackboard. With Anna playing first, they alternately double the number currently written on the blackboard or square it. The person who first writes on the blackboard a number greater than $2023^{10}$ is the winner. Determine which player has a winning strategy.

2007 IMO Shortlist, 5

Tags: inequalities , sequence , bounded , recurrence relation , bodan last post

Let $ c > 2,$ and let $ a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that \[ a(m \plus{} n) \leq 2 \cdot a(m) \plus{} 2 \cdot a(n) \text{ for all } m,n \geq 1, \] and $ a\left(2^k \right) \leq \frac {1}{(k \plus{} 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $ a(n)$ is bounded. [i]Author: Vjekoslav Kovač, Croatia[/i]

2015 Sharygin Geometry Olympiad, P7

Tags: geometry , symmetry , orthocenter

The altitudes $AA_1$ and $CC_1$ of a triangle $ABC$ meet at point $H$. Point $H_A$ is symmetric to $H$ about $A$. Line $H_AC_1$ meets $BC$ at point $C' $, point $A' $ is defined similarly. Prove that $A' C' // AC$.

2014 China Team Selection Test, 3

Tags: combinatorics

$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$. Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.

2000 IMO Shortlist, 3

Tags: geometry , combinatorics , counting , combinatorial geometry

Let $ n \geq 4$ be a fixed positive integer. Given a set $ S \equal{} \{P_1, P_2, \ldots, P_n\}$ of $ n$ points in the plane such that no three are collinear and no four concyclic, let $ a_t,$ $ 1 \leq t \leq n,$ be the number of circles $ P_iP_jP_k$ that contain $ P_t$ in their interior, and let \[m(S)=a_1+a_2+\cdots + a_n.\] Prove that there exists a positive integer $ f(n),$ depending only on $ n,$ such that the points of $ S$ are the vertices of a convex polygon if and only if $ m(S) = f(n).$

2023 CIIM, 3

Tags: linear algebra , matrix , symmetric matrix , eigenvalue and eigenvector , sequence

Given a $3 \times 3$ symmetric real matrix $A$, we define $f(A)$ as a $3 \times 3$ matrix with the same eigenvectors of $A$ such that if $A$ has eigenvalues $a$, $b$, $c$, then $f(A)$ has eigenvalues $b+c$, $c+a$, $a+b$ (in that order). We define a sequence of symmetric real $3\times3$ matrices $A_0, A_1, A_2, \ldots$ such that $A_{n+1} = f(A_n)$ for $n \geq 0$. If the matrix $A_0$ has no zero entries, determine the maximum number of indices $j \geq 0$ for which the matrix $A_j$ has any null entries.

2021 JHMT HS, 5

Tags: algebra , polynomial

A function $f$ with domain $A$ and range $B$ is called [i]injective[/i] if every input in $A$ maps to a unique output in $B$ (equivalently, if $x, y \in A$ and $x \neq y$, then $f(x) \neq f(y)$). With $\mathbb{C}$ denoting the set of complex numbers, let $P$ be an injective polynomial with domain and range $\mathbb{C}$. Suppose further that $P(0) = 2021$ and that when $P$ is written in standard form, all coefficients of $P$ are integers. Compute the smallest possible positive integer value of $P(10)/P(1)$.