Olympiad problems

2016 Puerto Rico Team Selection Test, 1

Tags: algebra

The integers $1, 2, 3,. . . , 2016$ are written in a board. You can choose any pair of numbers in the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After such replacements, the board will have only one number. (a) Prove that there is a sequence of substitutions that will make let the final number be $2$. (b) Prove that there is a sequence of substitutions that will make let the final number be $1000$.

2015 Balkan MO Shortlist, A4

Tags: functional equation , algebra , function

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$ (x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$ (Albania)

2019 Puerto Rico Team Selection Test, 5

Tags: combinatorics , coloring

The wizard Gandalf has a necklace that is shaped like a row of magic pearls. The necklace has $2019$ pearls, $2018$ are black and the last one is white. Everytime that the magician Gandalf touches the necklace, the following occurs: the pearl in position $i$ is move to position $i-1$, for $1 <i <2020$, furthermore the pearl in position $1$ moves to position $2019$. But something else happens, if the pearl in position $1$ now is white, then the last pearl turns white without the need for Gandalf to touch the necklace again. How many times does Gandalf have to touch the necklace to be sure that all pearls are white?

2018 Purple Comet Problems, 2

Tags:

The following figure is made up of many $2$ × $4$ tiles such that adjacent tiles always share an edge of length $2$. Find the perimeter of this figure.

2021 JHMT HS, 3

Tags: function , calculus

There is a unique ordered triple of real numbers $(a, b, c)$ that makes the piecewise function \begin{align*} f(x) = \begin{cases} (x - a)^2 + b & \text{if } x \geq c \\ x^3 - x & \text{if } x < c \end{cases} \end{align*} twice continuously differentiable for all real $x.$ The value of $a + b + c$ can be expressed as a common fraction $p/q.$ Compute $p + q.$

2002 AMC 12/AHSME, 2

Tags:

Cindy was asked by her teacher to subtract $ 3$ from a certain number and then divide the result by $ 9$. Instead, she subtracted $ 9$ and then divided the result by $ 3$, giving an answer of $ 43$. What would her answer have been had she worked the problem correctly? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 51 \qquad \textbf{(E)}\ 138$

2023 UMD Math Competition Part I, #3

Tags: geometry

Adam is walking in the city. In order to get around a large building, he walks $12$ miles east and then $5$ miles north, then stop. His friend Neutrino, who can go through buildings, starts in the same place as Adam but walks in a straight line to where Adam stops. How much farther than Neutrino does Adam walk? $$ \mathrm a. ~ 1~\mathrm{mile}\qquad \mathrm b.~2 ~\mathrm{miles}\qquad \mathrm c. ~3~\mathrm{miles} \qquad \mathrm d. ~4~\mathrm{miles} \qquad \mathrm e. ~5~\mathrm{miles} $$

2018 Brazil National Olympiad, 5

Tags: combinatorics

One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum value of $x$.

2006 IMO Shortlist, 6

Tags: tiling , rhombus , combinatorics , hall s marriage theorem

A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus. Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$. [i]Proposed by Federico Ardila, Colombia [/i]

1980 Bundeswettbewerb Mathematik, 4

Tags: sequence , sum , number theory

Consider the sequence $a_1, a_2, a_3, \ldots$ with $$ a_n = \frac{1}{n(n+1)}.$$ In how many ways can the number $\frac{1}{1980}$ be represented as the sum of finitely many consecutive terms of this sequence?

2010 May Olympiad, 3

Tags: coloring , combinatorics

Is it possible to color positive integers with three colors so that whenever two numbers with different colors are added, the result of their addition is the third color? (All three colors must be used.) If the answer is yes, indicate a possible coloration; if not, explain why.

1963 Putnam, B5

Tags: inequalities , limit , series

Let $(a_n )$ be a sequence of real numbers satisfying the inequalities $$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$ and such that the series $$\sum_{n=0}^{\infty} a_n $$ converges. Prove that $$\lim_{n\to \infty} n a_n = 0.$$

2023 India IMO Training Camp, 3

Tags: algebra , polynomial

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

2022 VIASM Summer Challenge, Problem 4

Tags: geometry

Given a triangle $ABC$ inscribed in $(O)$. Choose points $M,N,P$ on the sides $AB,BC,CA$ such that $AMNP$ is a parallelogram. The segment $CM$ intersects $NP$ at $E$; the segment $BP$ intersects $NM$ at $F$; and the segment $BE$ intersects $CF$ at $D.$ a) Prove that: $A,D,N$ are collinear. b) Let $I,J$ be the circumcenters of $\triangle MBF, \triangle PCE,$ respectively. Prove that: $OD$ passes through the midpoint of $IJ.$

LMT Accuracy Rounds, 2023 S5

Tags: number theory

Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?

1998 Tournament Of Towns, 4

Tags: 3-digit , sum , product , digit , number theory

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

2016 AMC 12/AHSME, 2

Tags:

For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

2019 Middle European Mathematical Olympiad, 1

Tags: algebra , functional equation , injective function

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds $$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ [i]Proposed by Patrik Bak, Slovakia[/i]

2012 Albania National Olympiad, 2

Tags: analytic geometry , polynomial , quadratic , algebra

The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.

1997 Miklós Schweitzer, 2

Tags: number theory , real analysis

Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.

2008 Postal Coaching, 6

Tags: point , combinatorial geometry , line

Suppose $n$ straight lines are in the plane so that there exist seven points such that any of these line passes through at least three of these points. Find the largest possible value of $n$.

2012 Czech-Polish-Slovak Match, 3

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$. Let $I, J$ and $K$ be the incentres of the triangles $ABC, ACD$ and $ABD$ respectively. Let $E$ be the midpoint of the arc $DB$ of circle $\omega$ containing the point $A$. The line $EK$ intersects again the circle $\omega$ at point $F$ $(F \neq E)$. Prove that the points $C, F, I, J$ lie on a circle.

2014 Contests, 4

Tags: inequalities , calculus , three variable inequality

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]

2008 ISI B.Math Entrance Exam, 1

Tags: calculus , integration , function , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

1996 German National Olympiad, 2

Tags: inequalities , algebra

Let $a$ and $b$ be positive real numbers smaller than $1$. Prove that the following two statements are equivalent: (i) $a+b = 1$, (ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$, the following inequlity holds: $$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$$