Olympiad problems

2024 Ukraine National Mathematical Olympiad, Problem 3

Tags: algebra , inequalities

$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum? [i]Proposed by Oleksiy Masalitin[/i]

1985 IMO Longlists, 81

Tags: geometry

Given the side $a$ and the corresponding altitude $h_a$ of a triangle $ABC$, find a relation between $a$ and $h_a$ such that it is possible to construct, with straightedge and compass, triangle $ABC$ such that the altitudes of $ABC$ form a right triangle admitting $h_a$ as hypotenuse.

Kvant 2024, M2808

Tags: combinatorics , algebra

Some participants of the tournament are friends with each other, and everyone has at least one friend. Each participant of the tournament was given a T-shirt with the number of his friends at the tournament written on it. Prove that at least one participant has the arithmetic mean of the numbers written on his friends' T-shirts, not less than the arithmetic mean of the numbers on all T-shirts. [i] From Czech-Slovak Olympiad 1991 [/i]

2007 Croatia Team Selection Test, 6

Tags: combinatorics

$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this: - there is only one winner; - there are $\displaystyle 3$ students on the second place; - no student lost all $\displaystyle 4$ matches. How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)

1966 IMO Longlists, 54

Tags: number theory , decimal representation , digit , modular arithmetic , last digit

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

2017 Taiwan TST Round 2, 6

Tags: geometry

Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$. [list=a] [*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$. [*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$. [/list]

2014 Stanford Mathematics Tournament, 1

Tags: geometry

The coordinates of three vertices of a parallelogram are $A(1, 1)$, $B(2, 4)$, and $C(-5, 1)$. Compute the area of the parallelogram.

1995 AMC 12/AHSME, 23

Tags: inequalities , calculus , integration , trigonometry , triangle inequality , pythagorean theorem , geometry

The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

2005 China Girls Math Olympiad, 2

Tags: trigonometry , geometry , trig identities , law of sines , algebra

Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

2023 Assam Mathematics Olympiad, 16

Tags:

$n$ is a positive integer such that the product of all its positive divisors is $n^3$. Find all such $n$ less than $100$.

2001 Bosnia and Herzegovina Team Selection Test, 1

Tags: ratio , geometry , arc , angle , circles

On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$

2003 IMAR Test, 2

Tags: geometry , angle bisector , geometric inequality , semiperimeter

Prove that in a triangle the following inequality holds: $$s\sqrt3 \ge \ell_a + \ell_b + \ell_c$$ where $\ell_a$ is the length of the angle bisector from angle $A$, and $s$ is the semiperimeter of the triangle

2017 SG Originals, N6

Tags: number theory , vieta jumping

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2019 Polish Junior MO Second Round, 1.

Tags: algebra , inequalities

Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.

2019 Math Prize for Girls Problems, 4

Tags:

A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?

1997 Bulgaria National Olympiad, 1

Tags: function , inequalities

Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$. Prove that $ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.

2012 Danube Mathematical Competition, 3

Tags: number theory

Let $p$ and $q, p < q,$ be two primes such that $1 + p + p^2+...+p^m$ is a power of $q$ for some positive integer $m$, and $1 + q + q^2+...+q^n$ is a power of $p$ for some positive integer $n$. Show that $p = 2$ and $q = 2^t-1$ where $t$ is prime.

2021 HMIC, 2

Tags:

Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.

2000 Harvard-MIT Mathematics Tournament, 47

Tags:

Find an $n<100$ such that $n\cdot 2^n-1$ is prime. Score will be $n-5$ for correct $n$, $5-n$ for incorrect $n$ ($0$ points for answer $<5$)

2012 ELMO Shortlist, 4

Tags: factorial , number theory

Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

2017 Pan-African Shortlist, A?

Tags: algebra

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

2023 Auckland Mathematical Olympiad, 6

Tags: combinatorics , number theory

Suppose there is an infinite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work: $\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on. $\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off. Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?

1994 Miklós Schweitzer, 1

Tags: ordered set

An ordered set of numbers is mean-free if for all $x < y < z$ , $y \neq \frac{x + z}{2}$. Is it possible to order the real numbers so it becomes mean-free? related: [url]https://www.youtube.com/watch?v=ppaXUxsEjMQ[/url]

2011-2012 SDML (High School), 12

Tags: factorial

Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate? $\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$

2014 Kyiv Mathematical Festival, 4a

Tags: least common multiple , modular arithmetic , number theory

a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$