Olympiad problems

2017 South East Mathematical Olympiad, 4

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

2019 Online Math Open Problems, 9

Tags:

Susan is presented with six boxes $B_1, \dots, B_6$, each of which is initially empty, and two identical coins of denomination $2^k$ for each $k = 0, \dots, 5$. Compute the number of ways for Susan to place the coins in the boxes such that each box $B_k$ contains coins of total value $2^k$. [i]Proposed by Ankan Bhattacharya[/i]

2008 Miklós Schweitzer, 4

Tags: group theory , abstract algebra

Let $A$ be a subgroup of the symmetric group $S_n$, and $G$ be a normal subgroup of $A$. Show that if $G$ is transitive, then $|A\colon G|\le 5^{n-1}$ (translated by Miklós Maróti)

2008 USA Team Selection Test, 4

Tags: number theory

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

1996 Tuymaada Olympiad, 1

Tags: algebra , inequalities

Prove the inequality $x_1y_1+x_2y_2+x_2y_1+2x_2y_2\le 1996$ if $x_1^2+2x_1x_2+2x_2^2\le 998$ and $y_1^2+2y_1y_2+2y_2^2\le 3992$.

2004 AMC 12/AHSME, 17

Tags: vieta , logarithm

For some real numbers $ a$ and $ b$, the equation \[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0 \]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$? $ \textbf{(A)}\minus{}\!256 \qquad \textbf{(B)}\minus{}\!64 \qquad \textbf{(C)}\minus{}\!8 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 256$

2003 Bulgaria Team Selection Test, 2

Tags: functional equation , algebra

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

2014 IMO Shortlist, C5

Tags: probability , combinatorics , geometry

A set of lines in the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note[/i]: Results with $\sqrt{n}$ replaced by $c\sqrt{n}$ will be awarded points depending on the value of the constant $c$.

2002 AMC 10, 25

Tags: algebra , system of equations

When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

1991 Austrian-Polish Competition, 7

Tags: function , max , algebra

For a given positive integer $n$ determine the maximum value of the function $f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}$ over all $x \ge 0$ and find all positive $x$ for which the maximum is attained.

2003 Greece National Olympiad, 4

Tags: analytic geometry , combinatorics

On the set $\Sigma$ of points of the plane $\Pi$ we deﬁne the operation $*$ which maps each pair $(X, Y )$ of points in $\Sigma$ to the point $Z = X * Y$ that is symmetric to $X$ with respect to $Y .$ Consider a square $ABCD$ in $\Pi$. Is it possible, using the points $A, B, C$ and applying the operation $*$ ﬁnitely many times, to construct the point $D?$

PEN R Problems, 2

Tags: 3d geometry , tetrahedron , vector , modular arithmetic , the geometry of numbers , geometry

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

Cono Sur Shortlist - geometry, 2012.G4.2

Tags: rhombus , circumcircle , projective geometry , geometry

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

MathLinks Contest 2nd, 3.2

Tags: geometry

Let $ABC$ be a triangle with altitudes $AD, BE, CF$. Choose the points $A_1, B_1, C_1$ on the lines $AD, BE, CF$ respectively, such that $$\frac{AA_1}{AD}= \frac{BB_1}{BE}= \frac{CC_1}{CF} = k.$$ Find all values of $k$ such that the triangle $A_1B_1C_1$ is similar to the triangle $ABC$ for all triangles $ABC$.

2005 Silk Road, 1

Tags: number theory

Let $n \geq 2$ be natural number. Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1 $ and $(p-1)| \frac{n}{p}-1$.

2022 JBMO Shortlist, C5

Tags: combinatorics

Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]

2009 Tuymaada Olympiad, 2

Tags: ceiling function , ratio , combinatorics

A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace? [i]Proposed by A. Golovanov[/i]

2023 Harvard-MIT Mathematics Tournament, 6

Tags:

For any odd positive integer $n$, let $r(n)$ be the odd positive integer such that the binary representation of $r(n)$ is the binary representation of $n$ written backwards. For example, $r(2023)=r(111111001112)=111001111112=1855$. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression $a_1, \ldots, a_8$ of odd positive integers such that $r(a_1), \ldots , r(a_8)$ is an arithmetic progression in that order.

1973 Spain Mathematical Olympiad, 7

Tags: analytic geometry , geometry , inequalities , geometric inequalities

The two points $P(8, 2)$ and $Q(5, 11)$ are considered in the plane. A mobile moves from $P$ to $Q$ according to a path that has to fulfill the following conditions: The moving part of $ P$ and arrives at a point on the $x$-axis, along which it travels a segment of length $1$, then it departs from this axis and goes towards a point on the $y$ axis, on which travels a segment of length $2$, separates from the $y$ axis finally and goes towards the point $Q$. Among all the possible paths, determine the one with the minimum length, thus like this same length.

2008 Argentina Iberoamerican TST, 3

Tags: inequalities , induction , graph theory , combinatorics

The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$

2020 USMCA, 18

Tags:

Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide what time. Each of the four group members sign on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?

2013 India IMO Training Camp, 1

Tags: modular arithmetic , diophantine equation , number theory

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

2018 Peru Iberoamerican Team Selection Test, P5

Tags: number theory

Find all positive integers $a, b$, and $c$ such that the numbers $$\frac{a+1}{b}, \frac{b+1}{c} \quad \text{and} \quad \frac{c+1}{a}$$ are positive integers.

2004 Thailand Mathematical Olympiad, 14

Tags: number theory , greatest common divisor , gcd

Compute gcd$(5^{2547} - 1, 5^{2004} - 1)$.

2020-IMOC, A3

Tags: quadratic , inequalities

$\definecolor{A}{RGB}{250,120,0}\color{A}\fbox{A3.}$ Assume that $a, b, c$ are positive reals such that $a + b + c = 3$. Prove that $$\definecolor{A}{RGB}{200,0,200}\color{A} \frac{1}{8a^2-18a+11}+\frac{1}{8b^2-18b+11}+\frac{1}{8c^2-18c+11}\le 3.$$ [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1734[/color]