Olympiad problems

2013 ELMO Shortlist, 4

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]

2009 Princeton University Math Competition, 7

Tags: trigonometry , inequalities , calculus , derivative

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

2008 AMC 12/AHSME, 5

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Suppose that \[ \frac {2x}{3} \minus{} \frac {x}{6} \]is an integer. Which of the following statements must be true about $ x$? $ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$ $ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$ $ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$ $ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$

2013 Dutch IMO TST, 4

Tags: modular arithmetic , number theory

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

2001 CentroAmerican, 1

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Two players $ A$, $ B$ and another 2001 people form a circle, such that $ A$ and $ B$ are not in consecutive positions. $ A$ and $ B$ play in alternating turns, starting with $ A$. A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. The winner is the last man standing. Show that one of the two players has a winning strategy, and give such strategy. Note: A player has a winning strategy if he/she is able to win no matter what the opponent does.

2016 ASDAN Math Tournament, 3

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Julia adds up the numbers from $1$ to $2016$ in a calculator. However, every time she inputs a $2$, the calculator malfunctions and inputs a $3$ instead (for example, when Julia inputs $202$, the calculator inputs $303$ instead). How much larger is the total sum returned by the broken calculator? (No $2$s are replaced by $3$s in the output, and the calculator only malfunctions while Julia is inputting numbers.)

1977 Czech and Slovak Olympiad III A, 4

Tags: algebra , system of equations , symmetry

Determine all real solutions of the system \begin{align*} x+y+z &=3, \\ \frac1x+\frac1y+\frac1z &= \frac{5}{12}, \\ x^3+y^3+z^3 &=45. \end{align*}

2012 Harvard-MIT Mathematics Tournament, 3

Tags: geometry

Let $ABC$ be a triangle with incenter $I$. Let the circle centered at $B$ and passing through $I$ intersect side $AB$ at $D$ and let the circle centered at $C$ passing through $I$ intersect side $AC$ at $E$. Suppose $DE$ is the perpendicular bisector of $AI$. What are all possible measures of angle $BAC $ in degrees?

2000 Junior Balkan Team Selection Tests - Moldova, 8

Tags: number theory

Show that the numbers $18^n$ and $2^n + 18^n$ are having the same number of digits (as written in base 10), for every natural number $n$.

2018 ASDAN Math Tournament, 5

Tags: algebra test

In the expansion of $(x + b)^{2018}$, the coefficients of $x^2$ and $x^3$ are equal. Compute $b$.

1960 AMC 12/AHSME, 21

Tags: geometry , perimeter

The diagonal of square I is $a+b$. The perimeter of square II with [i]twice[/i] the area of I is: $ \textbf{(A)}\ (a+b)^2\qquad\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad\textbf{(C)}\ 2(a+b)\qquad\textbf{(D)}\ \sqrt{8}(a+b) \qquad$ $\textbf{(E)}\ 4(a+b) $

2022 Thailand Mathematical Olympiad, 6

Tags: combinatorics

In an examination, there are $3600$ students sitting in a $60 \times 60$ grid, where everyone is facing toward the top of the grid. After the exam, it is discovered that there are $901$ students who got infected by COVID-19. Each infected student has a spreading region, which consists of students to the left, to the right, or in the front of them. Student in spreading region of at least two students are considered a close contact. Given that no infected student sat in the spreading region of other infected student, prove that there is at least one close contact.

2021 Math Prize for Girls Problems, 6

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The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?

2017 Hong Kong TST, 5

Tags: number theory

Find the first digit after the decimal point of the number $\displaystyle \frac1{1009}+\frac1{1010}+\cdots + \frac1{2016}$

2007 Indonesia MO, 2

Tags: relatively prime , diophantine equation , number theory

For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$. (a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$. (b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.

2017 Grand Duchy of Lithuania, 1

Tags: sequence , recurrence relation , algebra , inequalities

The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and $$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$

1993 AMC 12/AHSME, 12

Tags: function

If $f(2x)=\frac{2}{2+x}$ for all $x>0$, then $2f(x)=$ $ \textbf{(A)}\ \frac{2}{1+x} \qquad\textbf{(B)}\ \frac{2}{2+x} \qquad\textbf{(C)}\ \frac{4}{1+x} \qquad\textbf{(D)}\ \frac{4}{2+x} \qquad\textbf{(E)}\ \frac{8}{4+x} $

2011 LMT, 19

Tags: number theory

A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?

2001 National High School Mathematics League, 2

Tags: inequalities

If $x_i\geq0(i=1,2,\cdots,n)$, and $$\sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1.$$ Find the maximum and minumum value of $\sum_{i=1}^n x_i$.

1987 IMO Longlists, 68

Tags: trigonometry , geometric inequality , trigonometric inequality , geometry

Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma < \pi$, $\alpha+\beta > \gamma$,$ \beta+\gamma > \alpha$, $\gamma + \alpha > \beta.$ Prove that with the segments of lengths $\sin \alpha, \sin \beta, \sin \gamma $ we can construct a triangle and that its area is not greater than \[A=\dfrac 18\left( \sin 2\alpha+\sin 2\beta+ \sin 2\gamma \right).\] [i]Proposed by Soviet Union[/i]

2014 Saudi Arabia BMO TST, 2

Tags: pigeonhole principle , number theory

Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.

1960 Miklós Schweitzer, 10

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[b]10.[/b] A car is used by $n$ drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable $\mu (n)$ as the least positive integer such that each driver drives at least one day during the first $\mu (n)$ days. Find the limit distribution of the random variable $\frac {\mu (n) -n \log n}{n}$ as $n \to \infty$. [b](P. 9)[/b]

2011 Bogdan Stan, 2

Tags: linear algebra , matrix

Let be a natural number $ n\ge 2. $ Prove that there exist exactly two subsets of the set $ \left\{ \left.\left(\begin{matrix} a& b\\-b& a \end{matrix}\right)\right| a,b\in\mathbb{R} \right\} $ that are closed under multiplication and their cardinal is $ n. $ [i]Marcel Tena[/i]

2015 Thailand TSTST, 1

Tags: geometry , ratio , angle

Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.

2018 CMIMC CS, 6

Tags: computer science

For integer $n\geq 2$ and real $0\leq p\leq 1$, define $\mathcal{W}_{n,p}$ to be the complete weighted undirected random graph with vertex set $\{1,2,\ldots,n\}$: the edge $(i,j)$ will have weight $\min(i,j)$ with probability $p$ and weight $\max(i,j)$ otherwise. Let $\mathcal{L}_{n,p}$ denote the total weight of the minimum spanning tree of $\mathcal{W}_{n,p}$. Find the largest integer less than the expected value of $\mathcal{L}_{2018,1/2}$.