Olympiad problems

1988 IMO Longlists, 90

Tags: algebra

Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]

Kvant 2020, M2612

Tags: combinatorial game , combinatorics

Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.

2022 Czech and Slovak Olympiad III A, 3

Tags: geometry , rectangle , concyclic

Given a scalene acute triangle $ABC$, let M be the midpoints of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D$, $E$ its intersections with the bisector of angle $\angle BAC$. Points $D'$, $E'$ lie on $\omega$ such that $DED'E' $ is a rectangle. Prove that $D'$, $E'$, $M$, $N$ lie on a single circle. [i] (Patrik Bak)[/i]

2022 Putnam, A6

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Let $n$ be a positive integer. Determine, in terms of $n,$ the largest integer $m$ with the following property: There exist real numbers $x_1,\ldots, x_{2n}$ with $-1<x_1<x_2<\ldots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $$[x_1^{2k-1},x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \ldots, [x_{2n-1}^{2k-1},x_{2n}^{2k-1}]$$ is equal to 1 for all integers $k$ with $1\leq k \leq m.$

1994 Baltic Way, 10

Tags: absolute value , number theory

How many positive integers satisfy the following three conditions: a) All digits of the number are from the set $\{1,2,3,4,5\}$; b) The absolute value of the difference between any two consecutive digits is $1$; c) The integer has $1994$ digits?

1988 IMO Longlists, 33

Tags: geometry , rectangle , combinatorics

In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any three of them there was a question which the three students answered differently. What is the maximum number of students tested?

2018 Yasinsky Geometry Olympiad, 2

Tags: geometry , area of a triangle , area

Let $P$ the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the area of triangles $ABC$, $BCD$ and $DAP$ is equal to $8 cm^2$, $9 cm^2$ and $10 cm^2$. Find the area of the quadrilateral $ABCD$.

2019 Romania Team Selection Test, 1

Tags: geometry , incenter , circumcircle

Let $ I,O $ denote the incenter, respectively, the circumcenter of a triangle $ ABC. $ The $ A\text{-excircle} $ touches the lines $ AB,AC,BC $ at $ K,L, $ respectively, $ M. $ The midpoint of $ KL $ lies on the circumcircle of $ ABC. $ Show that the points $ I,M,O $ are collinear. [i]Павел Кожевников[/i]

2018 Brazil Undergrad MO, 2

Tags: functional equation , function , algebra

Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?

2018 Iranian Geometry Olympiad, 1

Tags: geometry

As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is to minimize the total length of the straight cuts. How to achieve this goal, and what is that minimized length? Show the correct cuts and write the final answer. There is no need to prove the answer. [i]Proposed by Morteza Saghafian[/i]

2024 Princeton University Math Competition, A6 / B8

Tags: number theory

Let Pascal’s triangle be constructed where each $\tbinom{n}{i}$ is written inside its own cell in row $n.$ Colby colors the cells red for $1 \le n \le 63$ when $\tbinom{n}{i}$ is divisible by $4.$ How many cells does he color red?

1978 Poland - Second Round, 2

Tags: intersection , parallel lines , geometry

In the plane, a set of points $ M $ is given with the following properties: 1. The points of the set $ M $ do not lie on one straight line, 2. If the points $ A, B, C$, and $D$ are vertices of a parallelogram and $ A, B, C \in M $, then $ D \in M $, 3. If $ A, B \in M $, then $ AB \geq 1 $. Prove that there exist two families of parallel lines such that $ M $ is the set of all intersection points of the lines of the first family with the lines of the second family.

2016 Fall CHMMC, 11

Tags: probability , algebra

Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?

Russian TST 2021, P1

Tags: number theory , representation

Do there exist infinitely many positive integers not expressible in the form \[(a+b)+\log_2(b+c)-2^{c+a},\]where $a,b,c$ are positive integers?

2008 Denmark MO - Mohr Contest, 4

Tags: geometry , angle bisector

In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$. Determine the length of $AD$. The answer must be given as a fraction with integer numerator and denominator.

2021 Balkan MO Shortlist, C2

Tags: shortlist , combinatorics , divisibility , maximum

Let $K$ and $N > K$ be fixed positive integers. Let $n$ be a positive integer and let $a_1, a_2, ..., a_n$ be distinct integers. Suppose that whenever $m_1, m_2, ..., m_n$ are integers, not all equal to $0$, such that $\mid{m_i}\mid \le K$ for each $i$, then the sum $$\sum_{i = 1}^{n} m_ia_i$$ is not divisible by $N$. What is the largest possible value of $n$? [i]Proposed by Ilija Jovcevski, North Macedonia[/i]

2017 AMC 8, 8

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Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true." (1) It is prime. (2) It is even. (3) It is divisible by 7. (4) One of its digits is 9. This information allows Malcolm to determine Isabella's house number. What is its units digit? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

2018 Tuymaada Olympiad, 3

Tags: combinatorics

$n$ rooks and $k$ pawns are arranged on a $100 \times 100$ board. The rooks cannot leap over pawns. For which minimum $k$ is it possible that no rook can capture any other rook? Junior League: $n=2551$ ([i]Proposed by A. Kuznetsov[/i]) Senior League: $n=2550$ ([i]Proposed by N. Vlasova[/i])

2007 Stanford Mathematics Tournament, 2

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Aliens from Lumix have one head and four legs, while those from Obscra have two heads and only one leg. If 60 aliens attend a joint Lumix and Obscra interworld conference, and there are 129 legs present, how many heads are there?

2014 Indonesia MO Shortlist, A3

Tags: algebra , inequalities , three variable inequality

Prove for each positive real number $x, y, z$, $$\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}$$

2014 Math Prize For Girls Problems, 4

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Say that an integer $A$ is [i]yummy[/i] if there exist several consecutive integers (including $A$) that add up to 2014. What is the smallest yummy integer?

1955 Putnam, A7

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Consider the function $f$ defined by the differential equation \[ f'' (x) = (x^3 + ax) f(x) \] and the initial conditions $f(0) = 1, f'(0) = 0.$ Prove that the roots of $f$ are bounded above but unbounded below.

2017 Puerto Rico Team Selection Test, 1

Tags: sum , algebra , functional equation , functional

Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.

2018 Moldova Team Selection Test, 1

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Let $x,y,z \in\mathbb{Q}$,such that $(x+y+z)^3=9(x^2y+y^2z+z^2x).$ Prove that $x=y=z$

2015 Balkan MO Shortlist, N7

Tags: number theory , digit , decimal representation

Positive integer $m$ shall be called [i]anagram [/i] of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be [i]anagram [/i] of the sum of the other $3$? (Bulgaria)