Olympiad problems

2020 Estonia Team Selection Test, 2

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

2006 CentroAmerican, 1

Tags: modular arithmetic , geometry , 3D geometry

For $0 \leq d \leq 9$, we define the numbers \[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]Find the last digit of the number \[S_{0}+S_{1}+\cdots+S_{9}.\]

2015 AIME Problems, 5

Tags: AMC , AIME , AIME I

In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the remaining 8 socks at random and on Wednesday two of the reaining 6 socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1998 Belarusian National Olympiad, 8

Tags: algebra , functional equation , functional

a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$ b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.

2025 Portugal MO, 5

Tags: number theory

An integer number $n \geq 2$ is called [i]feirense[/i] if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is. Find all the feirense numbers.

Russian TST 2020, P3

Tags: geometry , IMO Shortlist

Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

2003 China Team Selection Test, 2

Tags: geometry , circumcircle , symmetry , projective geometry , power of a point , radical axis , geometry proposed

Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$. Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$. Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$). Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$). Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$. Find the length of $AP$.

2017 Math Prize for Girls Problems, 16

Tags: Math Prize for Girls

Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form SWEET, in which the two E's represent the same digit but otherwise the digits are distinct. (The leading digit S can't be 0.) How many such integers are divisible by 16?

2013 Tournament of Towns, 5

Tags: algebra , trinomial , integer roots

A quadratic trinomial with integer coefficients is called [i]admissible [/i] if its leading coefficient is $1$, its roots are integers and the absolute values of coefficients do not exceed $2013$. Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.

2011 Thailand Mathematical Olympiad, 9

Tags:

Prove that, for all $n \in \mathbb{N}$ \begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}

2004 Gheorghe Vranceanu, 4

Tags: function , algebra , bijectivity , combinatorics

Let be three finite and nonempty sets $ A,B,C $ such that $ |A|=|C|\le |B| , $ and a bijection $ A\stackrel{\beta }{\longrightarrow } C. $ How many pairs of functions $ A\stackrel{f_2 }{\longrightarrow } B\stackrel{f_1 }{\longrightarrow } C $ that satisfy $ f_1\circ f_2=\beta $ are there?

2018 Singapore MO Open, 3

Tags: number theory

Let $n$ be a positive integer. Show that there exists an integer $m$ such that \[ 2018m^2+20182017m+2017 \] is divisible by $2^n$.

2012 India PRMO, 6

Tags: combinatorics

A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?

1956 AMC 12/AHSME, 17

Tags:

The fraction $ \frac {5x \minus{} 11}{2x^2 \plus{} x \minus{} 6}$ was obtained by adding the two fractions $ \frac {A}{x \plus{} 2}$ and $ \frac {B}{2x \minus{} 3}$. The values of $ A$ and $ B$ must be, respectively: $ \textbf{(A)}\ 5x, \minus{} 11 \qquad\textbf{(B)}\ \minus{} 11,5x \qquad\textbf{(C)}\ \minus{} 1,3 \qquad\textbf{(D)}\ 3, \minus{} 1 \qquad\textbf{(E)}\ 5, \minus{} 11$

2014 Contests, 2

Tags: function , algebra , functional equation , algebra proposed

Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions: (i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and (ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ? [i]Proposed by Igor I. Voronovich, Belarus[/i]

2017 VTRMC, 4

Tags: geometry

Let $P$ be an interior point of a triangle of area $T$. Through the point $P$, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let $a$, $b$ and $c$ be the areas of the three triangles. Prove that $ \sqrt { T } = \sqrt { a } + \sqrt { b } + \sqrt { c } $.

2023 AIME, 5

Tags: 2023 aime , P5 , geometry

Let $P$ be a point on the circumcircle of square $ABCD$ such that $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ What is the area of square $ABCD?$

2025 Nepal National Olympiad, 4

Tags: number theory , factorial

Find all pairs of positive integers $ n $ and $ x $ such that \[ 1^n + 2^n + 3^n + \cdots + n^n = x! \] [i](Petko Lazarov, Bulgaria)[/i]

2019 Serbia National Math Olympiad, 6

Tags: algebra , Sequence

Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations : $$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and $$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$

1994 Argentina National Olympiad, 5

Tags: combinatorics , combinatorial geometry , geometry , analytic geometry

Let $A$ be an infinite set of points in the plane such that inside each circle there are only a finite number of points of $A$, with the following properties: $\bullet$ $(0, 0)$ belongs to $A$. $\bullet$ If $(a, b)$ and $(c, d)$ belong to $A$, then $(a-c, b-d)$ belongs to $A$. $\bullet$ There is a value of $\alpha$ such that by rotating the set $A$ with center at $(0, 0)$ and angle $\alpha$, the set $A$ is obtained again. Prove that $\alpha$ must be equal to $\pm 60^{\circ}$ or $\pm 90^{\circ}$ or $\pm 120^{\circ}$ or $\pm 180^{\circ}$.

2011 Indonesia MO, 3

Tags: geometry , circumcircle , trigonometry , inequalities , trig identities , Law of Sines , Law of Cosines

Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.

2021 Romania Team Selection Test, 2

Tags: combinatorics , Sets , romania , Romanian TST , TST

Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$

2005 Sharygin Geometry Olympiad, 7

Tags: Equilateral , Equilateral Triangle , concentric circles , geometry , angles

Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?

1992 Chile National Olympiad, 2

Tags: number theory , algebra , Sum

For a finite set of naturals $(C)$, the product of its elements is going to be noted $P(C)$. We are going to define $P (\phi) = 1$. Calculate the value of the expression $$\sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)}$$

2013 Federal Competition For Advanced Students, Part 2, 6

Tags: geometry , 3D geometry , octahedron , sphere , geometry proposed

Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$. Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.