Olympiad problems

2012 Stanford Mathematics Tournament, 10

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

2020 CHMMC Winter (2020-21), 3

Tags: number theory

[i](6 pts)[/i] Find all positive integers $n \ge 3$ such that there exists a permutation $a_{1}, a_{2}, \dots, a_{n}$ of $1, 2, \dots, n$ such that $a_{1}, 2a_{2}, \dots, na_{n}$ can be rearranged into an arithmetic progression.

2020 Nigerian MO round 3, #2

Tags: algebra

a sequence $(a_n)$ $n$ $\geq 1$ is defined by the following equations; $a_1=1$, $a_2=2$ ,$a_3=1$, $a_{2n-1}$$a_{2n}$=$a_2$$a_{2n-3}$+$(a_2a_{2n-3}+a_4a_{2n-5}.....+a_{2n-2}a_1)$ for $n$ $\geq 2$ $na_{2n}$$a_{2n+1}$=$a_2$$a_{2n-2}$+$(a_2a_{2n-2}+a_4a_{2n-4}.....+a_{2n-2}a_2)$ for $n$ $\geq 2$ find $a_{2020}$

2019 IMC, 3

Tags: function , real analysis

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan[/i]

2008 Harvard-MIT Mathematics Tournament, 2

Tags: ratio , inradius , geometry

Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.

2012 Turkey Team Selection Test, 3

Tags: algebra , inequalities

For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that \[ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) \]

2017 Brazil National Olympiad, 2.

Tags: number theory , greatest common divisor , combinatorics

[b]2.[/b] Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements.

2021 Kosovo National Mathematical Olympiad, 2

Tags:

Dua has all the odd natural numbers less than 20. Asija has all the even numbers less than 21. They play the following game. In each round, they take a number from each other and after every round, they may fix two or more consecutive numbers so that their opponent cannot take these fixed numbers in the next round. The game is won by the player who attains 10 consecutive numbers first. Does either player have a winning strategy?

2015 Romania National Olympiad, 2

Tags: linear algebra , matrix

Let be a $ 5\times 5 $ complex matrix $ A $ whose trace is $ 0, $ and such that $ I_5-A $ is invertible. Prove that $ A^5\neq I_5. $

1995 AMC 8, 4

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A teacher tells the class, [i]"Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer."[/i] Ben thinks of $6$, and gives his answer to Sue. What should Sue's answer be? $\text{(A)}\ 18 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 30$

1998 Irish Math Olympiad, 4

Tags: algebra

A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.

2022 Harvard-MIT Mathematics Tournament, 6

Tags: geometry

Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on minor arc $AB$ of $\Gamma$. Suppose that $P A \cdot P B = 2$, $P C \cdot P D = 18$, and $P B \cdot P C = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100a + 10b + c$.

2007 India Regional Mathematical Olympiad, 4

Tags: combinatorics

How many 6-digit numbers are there such that-: a)The digits of each number are all from the set $ \{1,2,3,4,5\}$ b)any digit that appears in the number appears at least twice ? (Example: $ 225252$ is valid while $ 222133$ is not) [b][weightage 17/100][/b]

2017 Polish Junior Math Olympiad Finals, 3.

Tags: number theory

Positive integers $a$ and $b$ are given such that each of the numbers $ab$ and $(a+1)(b+1)$ is a perfect square. Prove that there exists an integer $n>1$ such that the number $(a+n)(b+n)$ is a perfect square.

2014 Putnam, 1

Tags: modular arithmetic , induction

A [i]base[/i] 10 [i]over-expansion[/i] of a positive integer $N$ is an expression of the form $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$ with $d_k\ne 0$ and $d_i\in\{0,1,2,\dots,10\}$ for all $i.$ For instance, the integer $N=10$ has two base 10 over-expansions: $10=10\cdot 10^0$ and the usual base 10 expansion $10=1\cdot 10^1+0\cdot 10^0.$ Which positive integers have a unique base 10 over-expansion?

2003 China Western Mathematical Olympiad, 3

Tags: inequalities

The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1$. Prove that $ \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1$.

CIME I 2018, 2

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An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\leq x < y \leq 26$ there is exactly one passenger that goes from the $x$-th stop to the $y$-th one. If every passenger wants to take a seat during his journey, find the minimum number of seats that must be available on the train. [i]Proposed by [b]FedeX333X[/b][/i]

2007 Grigore Moisil Intercounty, 2

Tags: algebra , polynomial , complex number

Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.

1979 IMO Longlists, 33

Tags: trigonometry , inequality , geometric inequality , trigonometric identities , inequalities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

2018-2019 Winter SDPC, 2

Tags: number theory

Call a number [i]precious[/i] if it is the sum of two distinct powers of two. Find all precious numbers $n$ such that $n^2$ is also precious.

2016 PUMaC Geometry A, 7

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.

1987 Polish MO Finals, 1

Tags: point , combinatorial geometry , combinatorics , square , distance , geometry

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

2017 Online Math Open Problems, 2

Tags:

The numbers $a,b,c,d$ are $1,2,2,3$ in some order. What is the greatest possible value of $a^{b^{c^d}}$? [i]Proposed by Yannick Yao and James Lin[/i]

2023 Iran MO (3rd Round), 2

Tags: geometry

In triangle $\triangle ABC$ , $M$ is the midpoint of arc $(BAC)$ and $N$ is the antipode of $A$ in $(ABC)$. The line through $B$ perpendicular to $AM$ , intersects $AM , (ABC)$ at $D,P$ respectively and a line through $D$ perpendicular to $AC$ , intersects $BC,AC$ at $F,E$ respectively. Prove that $PE,MF,ND$ are concurrent.

2021 BMT, 12

Tags: algebra

Let $a$, $b$, and $c$ be the solutions of the equation $$x^3 - 3 \cdot 2021^2x = 2 \cdot 20213.$$ Compute $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$