Olympiad problems

2016 BAMO, 1

Tags: geometry , rectangle

The diagram below is an example of a ${\textit{rectangle tiled by squares}}$: [center][img]http://i.imgur.com/XCPQJgk.png[/img][/center] Each square has been labeled with its side length. The squares fill the rectangle without overlapping. In a similar way, a rectangle can be tiled by nine squares whose side lengths are $2,5,7,9,16,25,28,33$, and $36$. Sketch one such possible arrangement of those squares. They must fill the rectangle without overlapping. Label each square in your sketch by its side length as in the picture above.

2020 Azerbaijan IMO TST, 1

Tags: algebra , inequalities

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

1993 Irish Math Olympiad, 4

Tags: polynomial , algebra

Let $ f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|\equal{}f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.

2007 AMC 12/AHSME, 13

Tags: analytic geometry

A piece of cheese is located at $ (12,10)$ in a coordinate plane. A mouse is at $ (4, \minus{} 2)$ and is running up the line $ y \equal{} \minus{} 5x \plus{} 18.$ At the point $ (a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $ a \plus{} b?$ $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 22$

2022 Korea National Olympiad, 7

Tags: inequalities , algebra , sequence

Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions: [list] [*]$a_i \leq a_j$ for every positive integers $i <j$. [*]For any positive integer $k \geq 3$, the following inequality holds: $$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$ [/list] Prove that $\{a_n\}$ is constant.

2006 Harvard-MIT Mathematics Tournament, 3

Tags: geometry

Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB=11$ and $CD=19$. Point $P$ is on segment $AB$ with $AP=6$, and $Q$ is on segment $CD$ with $CQ=7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ=27$, find $XY$.

2003 JHMMC 8, 22

Tags: absolute value

Given that $|3-a| = 2$, compute the sum of all possible values of $a$.

2011 N.N. Mihăileanu Individual, 2

Tags: equation , inequalities , algebra

Determine the real numbers $ x,y,z $ from the interval $ (0,1) $ that satisfies $ x+y+z=1, $ and $$ \sqrt{\frac{x(1-y^2)}{2}} +\sqrt{\frac{y(1-z^2)}{2}} +\sqrt{\frac{z(1-x^2)}{2}} =\sqrt{1+xy+yz+zx} . $$ [i]Gabriela Constantinescu[/i]

1967 AMC 12/AHSME, 31

Tags:

Let $D=a^2+b^2+c^2$, where $a$, $b$, are consecutive integers and $c=ab$. Then $\sqrt{D}$ is: $\textbf{(A)}\ \text{always an even integer}\qquad \textbf{(B)}\ \text{sometimes an odd integer, sometimes not}\\ \textbf{(C)}\ \text{always an odd integer}\qquad \textbf{(D)}\ \text{sometimes rational, sometimes not}\\ \textbf{(E)}\ \text{always irrational}$

2023 BMT, 6

Tags: algebra

Define a sequence $a_0$, $a_1$, $a_2$, $,...$ recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = a_{n+1} + xa_n$ for each $n \ge 0$ and some real number $x$. The infinite series $$ \sum^{\infty}_{n=0}\frac{a_n}{10^n} = 1.$$ Compute $x$.

2011 Belarus Team Selection Test, 3

Tags: algebra , functional equation , functional

Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$. I. Gorodnin

2000 Polish MO Finals, 3

Tags: polynomial , algebra

Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$

1949 Moscow Mathematical Olympiad, 156

Tags: number theory , divisible

Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.

2011 Math Prize For Girls Problems, 12

Tags: floor function , logarithm

If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by \[ S(n) = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor + 10 \left( n - 10^{\lfloor \log n \rfloor} \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor \right) \, . \] (All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?

2020 Durer Math Competition Finals, 12

Tags: combinatorics , coloring

We have a white table with $2$ rows and $5$ columns , and would like to colour all cells of the table according to the following rules: $\bullet$ We must colour the cell in the bottom left corner first. $\bullet$ After that, we can only colour a cell if some adjacent cell has already been coloured. (Two cells are adjacent if they share an edge.) How many different orders are there for colouring all $10$ squares (following these rules)?

ICMC 5, 3

Tags: determinant , linear algebra

Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$. [i]Proposed by Ethan Tan[/i]

2022 Stanford Mathematics Tournament, 1

Tags:

How many $4$ element subsets of $\{0,1,2,\dots,20\}$ contain their sum modulo $21$?

2014 Iran MO (3rd Round), 3

Tags: geometry , rectangle , combinatorics

We have a $10 \times 10$ table. $T$ is a set of rectangles with vertices from the table and sides parallel to the sides of the table such that no rectangle from the set is a subrectangle of another rectangle from the set. $t$ is the maximum number of elements of $T$. (a) Prove that $t>300$. (b) Prove that $t<600$. [i]Proposed by Mir Omid Haji Mirsadeghi and Kasra Alishahi[/i]

2022/2023 Tournament of Towns, P4

Tags: combinatorics , number theory

Is it possible to colour all integers greater than $1{}$ in three colours (each integer in one colour, all three colours must be used) so that the colour of the product of any two differently coloured numbers is different from the colour of each of the factors?

2016 BMT Spring, 9

Tags: number theory

$(\sqrt6 + \sqrt7)^{1000}$ in base ten has a tens digit of $a$ and a ones digit of $b$. Determine $10a + b$.

2022 Assara - South Russian Girl's MO, 4

Tags: greatest common divisor , number theory

Nadya has $2022$ cards, each with a number one or seven written on it. It is known that there are both cards.Nadya looked at all possible $2022$-digit numbers that can be composed from all these cards. What is the largest value that can take the greatest common divisor of all these numbers?

1980 IMO, 2

Tags: polynomial , algebra

Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies \[x \in \mathbb R \iff p(x) \in \mathbb R.\] Show that $n=1$

2009 Belarus Team Selection Test, 1

Tags: perfect square , number theory

Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares. S. Kuzmich

1992 IMO Longlists, 41

Tags: algebra , function , permutation , functional equation

Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$, where $f$ is a permutation of $\{1, 2, . . . , 6\}$. Let \[\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \} \] Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$.

2012 Online Math Open Problems, 25

Tags: probability , integration , calculus , symmetry , analytic geometry , modular arithmetic

Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000. [i]Victor Wang.[/i]