Olympiad problems

2020 Harvard-MIT Mathematics Tournament, 6

Tags:

Let $ABC$ be a triangle with $AB=5$, $BC=6$, $CA=7$. Let $D$ be a point on ray $AB$ beyond $B$ such that $BD=7$, $E$ be a point on ray $BC$ beyond $C$ such that $CE=5$, and $F$ be a point on ray $CA$ beyond $A$ such that $AF=6$. Compute the area of the circumcircle of $DEF$. [i]Proposed by James Lin.[/i]

2018 Hong Kong TST, 6

Tags: geometry

A triangle $ABC$ has its orthocentre $H$ distinct from its vertices and from the circumcenter $O$ of $\triangle ABC$. Denote by $M, N$ and $P$ respectively the circumcenters of triangles $HBC, HCA$ and $HAB$. Show that the lines $AM, BN, CP$ and $OH$ are concurrent.

2014 National Olympiad First Round, 5

Tags: geometry

Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=|AC|$, $|BD|=6$ and $|DC|=10$. If the incircles of $\triangle ABD$ and $\triangle ADC$ touch side $[AD]$ at $E$ and $F$, respectively, what is$|EF|$? $ \textbf{(A)}\ \dfrac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \dfrac{2}{\sqrt{3}} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \dfrac{9}{8} \qquad\textbf{(E)}\ 2 $

2011 National Olympiad First Round, 14

Tags: modular arithmetic , function

What is the remainder when $2011^{(2011^{(2011^{(2011^{2011})})})}$ is divided by $19$ ? $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$

2010 CHMMC Fall, 1

Tags: probability , complementary counting

Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once.

2009 Spain Mathematical Olympiad, 4

Tags: prime factorization , number theory

Find all the integer pairs $ (x,y)$ such that: \[ x^2\minus{}y^4\equal{}2009\]

2025 Canada Junior National Olympiad, 2

Tags: geometry

Let $ABCD$ be a trapezoid with parallel sides $AB$ and $CD$, where $BC\neq DA$. A circle passing through $C$ and $D$ intersects $AC, AD, BC, BD$ again at $W, X, Y, Z$ respectively. Prove that $WZ, XY, AB$ are concurrent.

1975 Miklós Schweitzer, 1

Tags: advanced fields , set theory , real analysis

Show that there exists a tournament $ (T,\rightarrow)$ of cardinality $ \aleph_1$ containing no transitive subtournament of size $ \aleph_1$. ( A structure $ (T,\rightarrow)$ is a $ \textit{tournament}$ if $ \rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $ (T,\rightarrow)$ is transitive if $ \rightarrow$ is transitive, that is, if it orders $ T$.) [i]A. Hajnal[/i]

2004 Nicolae Coculescu, 2

Tags: limit , real analysis , sequence

Let bet a sequence $\left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined as $ a_n=\sqrt[n]{1+na_{n-1}} . $ Show that $ \left( a_n \right)_{n\ge 1} $ is convergent and determine its limit. [i]Florian Dumitrel[/i]

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 $\arg z=$________.