Olympiad problems

2022 Stanford Mathematics Tournament, 3

Tags:

Compute $\left\lfloor\frac{1}{\frac{1}{2022}+\frac{1}{2023}+\dots+\frac{1}{2064}}\right\rfloor$.

2008 Brazil Team Selection Test, 4

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

1983 Tournament Of Towns, (035) O4

Tags: digit , permutation , combinatorics , number theory , sum of digits

The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that (a) The sum of the digits of $2M$ equals the sum of the digits of $2K$. (b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even). (c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$. (AD Lisitskiy)

2010 Silk Road, 2

Tags: number theory

Let $N = 2010!+1$. Prove that a) $N$ is not divisible by $4021$; b) $N$ is not divisible by $2027,2029,2039$; c)$ N$ has a prime divisor greater than $2050$.

2011 Romania Team Selection Test, 2

Tags: radical axis , power of a point , geometry

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P5

Tags: double counting , graph theory , triplets , combinatorics

There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a [i]friendly triplet[/i] if any two students in the group are friends. Determine the maximal possible number of friendly triplets. [i]Proposed by Nikola Velov[/i]

1992 Tournament Of Towns, (321) 2

Tags: isosceles trapezoid , trapezoid , geometry

In trapezoid $ABCD$ the sides $BC$ and $AD$ are parallel, $AC = BC + AD$, and the angle between the diagonals is equal to $ 60^o$. Prove that $AB = CD$. (Stanislav Smirnov, St Petersburg)

2010 Hanoi Open Mathematics Competitions, 6

Tags: floor function , algebra

Find the greatest integer less than $(2 +\sqrt3)^5$ . (A): $721$ (B): $722$ (C): $723$ (D): $724$ (E) None of the above.

2019 Durer Math Competition Finals, 3

Tags: perimeter , geometric inequalities , geometry

Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?

1985 Balkan MO, 2

Tags: quadratic , algebra , function , trigonometry , inequalities

Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that $\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$. Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$

2001 Moldova National Olympiad, Problem 2

Tags: number theory

Prove that there are no $2003$ odd positive integers whose product equals their sum. Is the previous proposition true for $2001$ odd positive integers?

2024 Argentina Iberoamerican TST, 5

Tags: algebra , functional equation

Let $ \mathbb R $ be the set of real numbers. Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that, for all real numbers $ x $ and $ y $, the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$

1999 USAMTS Problems, 1

Tags:

The number $N$ consists of $1999$ digits such that if each pair of consecutive digits in $N$ were viewed as a two-digit number, then that number would either be a multiple of $17$ or a multiple of $23$. THe sum of the digits of $N$ is $9599$. Determine the rightmost ten digits of $N$.

2003 National Olympiad First Round, 15

Tags: probability

Galatasaray and Fenerbahce have qualified last $16$ in the Europen Champions League. Aftar a random draw, eight matches are regulated in that knock-out phase. The winners of the eight matches will qualify for the next round - round of $8$. Knock-out phase continues until one team remains. If each team has equal chance to win, what is the propability of having a Galatasaray-Fenerbahce match? $ \textbf{(A)}\ \dfrac {1}{32} \qquad\textbf{(B)}\ \dfrac {1}{16} \qquad\textbf{(C)}\ \dfrac {1}{8} \qquad\textbf{(D)}\ \dfrac {1}{4} \qquad\textbf{(E)}\ \text{None of the preceding} $

1994 AIME Problems, 1

Tags: modular arithmetic , arithmetic sequence , number theory

The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?

2007 Oral Moscow Geometry Olympiad, 3

Tags: geometry , trapezoid , isosceles

In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.

2024 Kyiv City MO Round 2, Problem 3

Tags: combinatorics

For a given positive integer $n$, we consider the set $M$ of all intervals of the form $[l, r]$, where the integers $l$ and $r$ satisfy the condition $0 \leq l < r \leq n$. What largest number of elements of $M$ can be chosen so that each chosen interval completely contains at most one other selected interval? [i]Proposed by Anton Trygub[/i]

2019 PUMaC Geometry A, 3

Tags: number theory , relatively prime , geometry , probability

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

2001 Croatia National Olympiad, Problem 3

Tags: number theory , combinatorics

Let there be given triples of integers $(r_j,s_j,t_j),~j=1,2,\ldots,N$, such that for each $j$, $r_j,t_j,s_j$ are not all even. Show that one can find integers $a,b,c$ such that $ar_j+bs_j+ct_j$ is odd for at least $\frac{4N}7$ of the indices $j$.

2010 AIME Problems, 12

Tags:

Let $ M \ge 3$ be an integer and let $ S \equal{} \{3,4,5,\ldots,m\}$. Find the smallest value of $ m$ such that for every partition of $ S$ into two subsets, at least one of the subsets contains integers $ a$, $ b$, and $ c$ (not necessarily distinct) such that $ ab \equal{} c$. [b]Note[/b]: a partition of $ S$ is a pair of sets $ A$, $ B$ such that $ A \cap B \equal{} \emptyset$, $ A \cup B \equal{} S$.

2017 CMIMC Team, 4

Tags: team

Say an odd positive integer $n > 1$ is $\textit{twinning}$ if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.

2023 Taiwan TST Round 1, 4

Tags: function , permutation cycles , combinatorics

Let $k$ be a positive integer, and set $n=2^k$, $N=\{1, 2, \cdots, n\}$. For any bijective function $f:N\rightarrow N$, if a set $A\subset N$ contains an element $a\in A$ such that $\{a, f(a), f(f(a)), \cdots\} = A$, then we call $A$ as a cycle of $f$. Prove that: among all bijective functions $f:N\rightarrow N$, at least $\frac{n!}{2}$ of them have number of cycles less than or equal to $2k-1$. [i]Note: A function is bijective if and only if it is injective and surjective; in other words, it is 1-1 and onto.[/i] [i]Proposed by CSJL[/i]

2013 Stanford Mathematics Tournament, 15

Tags: 3d geometry , perimeter , geometry

Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.

2004 Greece Junior Math Olympiad, 4

Tags: inequalities

Determine the rational number $\frac{a}{b}$, where $a,b$ are positive integers, with minimal denominator, which is such that $ \frac{52}{303} < \frac{a}{b}< \frac{16}{91}$

2009 All-Russian Olympiad Regional Round, 9.3

Tags: right angle , perpendicular , geometry

In an acute triangle $ABC$ the altitudes $AA_1$, $BB_1$, $CC_1$ are drawn. A line perpendicular to side $AC$ and passing through a point $A_1$, intersects the line $B_1C_1$ at point $D$. Prove that angle $ADC$ is right.