Olympiad problems

2015 Purple Comet Problems, 8

Tags: pigeonhole principle

Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. Gwendoline repeatedly rolls the two dice and records the product of the two values until one of the values she records appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?

2017 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle chasing , diameter , angle

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

Today's calculation of integrals, 890

Tags: calculus , integration , function , algorithm , calculus computations

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

Novosibirsk Oral Geo Oly VIII, 2020.4

Tags: geometry , perpendicular , equal segments

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

2007 Pre-Preparation Course Examination, 4

Tags: geometry

Let $(C)$ and $(L)$ be a circle and a line. $P_{1},\dots,P_{2n+1}$ are odd number of points on $(L)$. $A_{1}$ is an arbitrary point on $(C)$. $A_{k+1}$ is the intersection point of $A_{k}P_{k}$ and $(C)$ ($1\leq k\leq 2n+1$). Prove that $A_{1}A_{2n+2}$ passes through a constant point while $A_{1}$ varies on $(C)$.

2020-2021 OMMC, 15

Tags:

A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + b}{c \pi}$ for some positive integers $a, b, c$ with $a$ and $c$ relatively prime. Find $a+b+c$.

2015 Princeton University Math Competition, B2

Tags: algebra

Let $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we could have $f(0) = 1, f(1) = 1, f(2) = 2$. How many such functions are there which satisfy \[f(2) + f(f(0)) + f(f(f(1))) = 5?\]

2017 South East Mathematical Olympiad, 4

Tags: algebra , maximum value

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.