Olympiad problems

1982 IMO Longlists, 17

Tags: sequence , rearrangement , minimization , maximization , algebra

[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes \[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\] [b](b)[/b] Find the rearrangement that minimizes $Q.$

In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.

2012 Tournament of Towns, 2

Tags: table , rectangle table , prime , combinatorics

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

Kvant 2022, M2713

Tags: graph theory , combinatorics , algebra

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

2003 Czech And Slovak Olympiad III A, 6

Tags: inequalities

a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.

MathLinks Contest 7th, 5.1

Tags: product of roots , inequalities , vieta , algebra , absolute value , polynomial

Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

2017-2018 SDML (Middle School), 10

Tags:

Mrs. Krabappel gives a five-question pop quiz one Monday. Nobody is ready, so everyone guesses and gets exactly three questions correct. The students later discover that they each answered a different set of three questions correctly. What is the largest possible number of students in the class? $\mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 10 \qquad \mathrm {(C) \ } 11 \qquad \mathrm{(D) \ } 12 \qquad \mathrm{(E) \ } 13$

2021 Thailand Online MO, P5

Tags: polynomial , algebra

Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true $\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$ $\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$

2022 BMT, 1

Tags: geometry

To fold a paper airplane, Austin starts with a square paper $F OLD$ with side length $2$. First, he folds corners $L$ and $D$ to the square’s center. Then, he folds corner $F$ to corner $O$. What is the longest distance between two corners of the resulting figure?

2014 Middle European Mathematical Olympiad, 2

Tags: algebra , inequalities , function

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ xf(xy) + xyf(x) \ge f(x^2)f(y) + x^2y \] holds for all $x,y \in \mathbb{R}$.

PEN A Problems, 27

Tags: divisibility theory , modular arithmetic , binomial coefficient

Show that the coefficients of a binomial expansion $(a+b)^n$ where $n$ is a positive integer, are all odd, if and only if $n$ is of the form $2^{k}-1$ for some positive integer $k$.

2010 South africa National Olympiad, 1

Tags: number theory

For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

2004 Junior Tuymaada Olympiad, 7

Tags: circumcircle , geometry

The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$ [i]Proposed by A. Smirnov[/i]

2023 Yasinsky Geometry Olympiad, 4

Tags: angle bisector , inradius , geometry

The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$. (Oleksii Karliuchenko)

2025 India STEMS Category A, 4

Tags: graph theory , combinatorics

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]

1986 AMC 12/AHSME, 23

Tags: polynomial , algebra

Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $

2016 BAMO, 3

Tags: prime factorization , prime factor , factor , number theory

The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$. Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$). Show that, for every integer $k$ greater than or equal to $2$, [list=i] [*] $A$ and $B$ have the same set of distinct prime factors. [*] $A+1$ and $B+1$ have the same set of distinct prime factors. [/list]

2015 NIMO Problems, 3

Tags:

How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.) [i]Based on a proposal by Yannick Yao[/i]

PEN O Problems, 16

Tags: modular arithmetic

Is it possible to find $100$ positive integers not exceeding $25000$ such that all pairwise sums of them are different?

1996 French Mathematical Olympiad, Problem 5

Tags: france , combinatorics , number theory

Let $n$ be a positive integer. We say that a natural number $k$ has the property $C_n$ if there exist $2k$ distinct positive integers $a_1,b_1,\ldots,a_k,b_k$ such that the sums $a_1+b_1,\ldots,a_k+b_k$ are distinct and strictly smaller than $n$. (a) Prove that if $k$ has the property $C_n$ then $k\le \frac{2n-3}{5}$. (b) Prove that $5$ has the property $C_{14}$. (c) If $(2n-3)/5$ is an integer, prove that it has the property $C_n$.

2024 Cono Sur Olympiad, 6

Tags: combinatorics

On a board of $8 \times 8$ exists $64$ kings, all initially placed in different squares. Alnardo and Bernaldo play alternately, with Arnaldo starting. On each move, one of the two players chooses a king and can move it one square to the right, one square up, or one square up to the right. In the event that a king is moved to an occupied square, both kings are removed from the game. The player who can remove two of the last kings or leave one last king in the upper right corner wins the game. Which of the two players can ensure victory?

2008 Spain Mathematical Olympiad, 2

Tags: reflection , geometry , geometric transformation

Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.