Olympiad problems

2019 239 Open Mathematical Olympiad, 1

Tags:

The following fractions are written on the board $\frac{1}{n}, \frac{2}{n-1}, \frac{3}{n-2}, \ldots , \frac{n}{1}$ where $n$ is a natural number. Vasya calculated the differences of the neighboring fractions in this row and found among them $10000$ fractions of type $\frac{1}{k}$ (with natural $k$). Prove that he can find even $5000$ more of such these differences.

1967 IMO Longlists, 34

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

2021 BMT, 2

Tags: algebra

Let $f$ and $g$ be linear functions such that $f(g(2021))-g(f(2021)) = 20$. Compute $f(g(2022))- g(f(2022))$. (Note: A function h is linear if $h(x) = ax + b$ for all real numbers $x$.)

2011 USA Team Selection Test, 4

Tags: algebra

Find a real number $t$ such that for any set of 120 points $P_1, \ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$.

2024-25 IOQM India, 5

Tags:

Let $a = \frac{x}{y} +\frac{y}{z} +\frac{z}{x}$, let $b = \frac{x}{z} +\frac{y}{x} +\frac{z}{y}$ and let $c = \left(\frac{x}{y} +\frac{y}{z} \right)\left(\frac{y}{z} +\frac{z}{x} \right)\left(\frac{z}{x} +\frac{x}{y} \right)$. The value of $|ab-c|$ is:

Kvant 2020, M2633

Tags: combinatorics

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? [i]Mikhail Svyatlovskiy[/i]

1998 Moldova Team Selection Test, 9

Tags: geometry , circumcircle , hexagon

A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$ and two have length $3$. Show that $r$ satisfies the equation $2r^3 - 7r - 3 = 0$.

2018 IMO, 2

Tags: sequence , system of equations , algebra

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

2007 Today's Calculation Of Integral, 199

Tags: calculus , calculus computations , integration

Let $m,\ n$ be non negative integers. Calculate \[\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}. \] where $_{i}C_{j}$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

2010 ELMO Shortlist, 1

Tags: geometry , angle bisector , asymptote , geometric transformation , circumcircle , incenter , homothety

Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent. [i]Carl Lian.[/i]

1996 AMC 12/AHSME, 22

Tags: probability

Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$? $\text{(A)}\ \frac 14 \qquad \text{(B)}\ \frac 13 \qquad \text{(C)}\ \frac 12 \qquad \text{(D)}\ \frac 23\qquad \text{(E)}\ \frac 34$

2005 Iran Team Selection Test, 1

Tags: inequalities

Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let \[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1. \] Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that: \[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2 \]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3

Tags: abstract algebra , superior algebra , group theory

Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.

PEN K Problems, 5

Tags: functional equation , function , arithmetic sequence

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

1992 IMO Longlists, 18

Tags: fibonacci , fibonacci sequence , combinatorics of words , sequence , combinatorics

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

2021 HMNT, 9

Tags: combinatorics

Let $n$ be the answer to this problem. Find the minimum number of colors needed to color the divisors of $(n - 24)!$ such that no two distinct divisors $s, t$ of the same color satisfy $s | t$.

2021 Romania National Olympiad, 2

Tags: geometry , vector , trigonometry

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] [i]Mihai Iancu[/i]

1993 All-Russian Olympiad, 2

Tags: geometry

Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Prove that the angle formed by one of the legs of one of the triangles and one of the legs of the other triangle is half the measure of the angle formed by the hypotenuses.

1991 Mexico National Olympiad, 1

Tags: fraction , irreducible , number theory

Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$.

1999 All-Russian Olympiad Regional Round, 11.4

Tags: 3d geometry , sphere , geometry

A polyhedron is circumscribed around a sphere. Let's call its face [i]large [/i] if the projection of the sphere onto the plane of the face falls entirely within the face. Prove that there are no more than 6 large faces.

2007 AMC 8, 17

Tags: percent

A mixture of 30 liters of paint is $25\%$ red tint, $30\%$ yellow tint, and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint that is the mixture? $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50$

The Golden Digits 2024, P1

Tags: functional equation , algebra

Determine all functions $f:\mathbb{R}_+\to\mathbb{R}_+$ which satisfy \[f\left(\frac{y}{f(x)}\right)+x=f(xy)+f(f(x)),\]for any positive real numbers $x$ and $y$. [i]Proposed by Pavel Ciurea[/i]

2020 Azerbaijan Senior NMO, 3

Tags: circumcircle , incenter , geometry

Let $ABC$ be a scalene triangle, and let $I$ be its incenter. A point $D$ is chosen on line $BC$, such that the circumcircle of triangle $BID$ intersects $AB$ at $E\neq B$, and the circumcircle of triangle $CID$ intersects $AC$ at $F\neq C$. Circumcircle of triangle $EDF$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. Lines $FD$ and $IC$ intersect at $Q$, and lines $ED$ and $BI$ intersect at $P$. Prove that $EN\parallel MF\parallel PQ$.

1999 Harvard-MIT Mathematics Tournament, 12

Tags: combinatorics

A fair coin is flipped every second and the results are recorded with $1$ meaning heads and $0$ meaning tails. What is the probability that the sequence $10101$ occurs before the first occurance of the sequence $010101$?

2017 Polish Junior Math Olympiad Finals, 5.

Tags: combinatorics

There are $n$ matches lying on a table, forming $n$ one-match piles. Adam wants to combine them into a single pile of $n$ matches. He will do this using $n-1$ operations, each of which consists of combining two piles into one. Adam has made a deal with Bartek that every time he combines a pile of $a$ matches with a pile of $b$ matches, he will receive $a\cdot b$ candies from Bartek. What is the greatest number of candies that Adam can receive after performing $n-1$ operations? Justify your answer.