Olympiad problems

1993 Cono Sur Olympiad, 3

Tags: combinatorics

Prove that, given a positive integrer $n$, there exists a positive integrer $k_n$ with the following property: Given any $k_n$ points in the space, $4$ by $4$ non-coplanar, and associated integrer numbers between $1$ and $n$ to each sharp edge that meets $2$ of this points, there's necessairly a triangle determined by $3$ of them, whose sharp edges have associated the same number.

2009 Sharygin Geometry Olympiad, 3

Tags: trapezoid , parallelogram , rectangle , incenter , geometry

The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.

1996 Czech and Slovak Match, 5

Tags: set , interval , combinatorics

Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .

2012 Miklós Schweitzer, 3

Tags: combinatorics

There is a simple graph which chromatic number is equal to $k$. We painted all of the edges of graph using two colors. Prove that there exist a monochromatic tree with $k$ vertices

1978 USAMO, 1

Tags: inequalities , vector , marvio

Given that $a,b,c,d,e$ are real numbers such that $a+b+c+d+e=8$, $a^2+b^2+c^2+d^2+e^2=16$. Determine the maximum value of $e$.

2019 Moldova EGMO TST, 7

Tags: combinatorics

Let $A{}$ be a subset formed of $16$ elements of the set $B=\{1, 2, 3, \ldots, 105, 106\}$ such that the difference between every two elements from $A$ is different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements in $A{}$ whose difference is $3$.

2014 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , algebra , polynomial , vieta , sum of roots

Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.

2014 China Girls Math Olympiad, 5

Tags: algebra , polynomial , number theory

Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $ r+\sqrt{a}$ is an irrational number.

2013 AMC 12/AHSME, 7

Tags:

The sequence $S_1, S_2, S_3, \cdots, S_{10}$ has the property that every term beginning with the third is the sum of the previous two. That is, \[ S_n = S_{n-2} + S_{n-1} \text{ for } n \ge 3. \] Suppose that $S_9 = 110$ and $S_7 = 42$. What is $S_4$? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16\qquad $

2025 Iran MO (2nd Round), 1

Tags: number theory

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

2014 PUMaC Number Theory B, 4

Tags:

Find the number of fractions in the following list that is in its lowest form. (ie. for $\tfrac pq$, $\gcd(p,q) = 1$.) \[\frac{1}{2014}, \frac{2}{2013}, \dots, \frac{1007}{1008}\]

1997 All-Russian Olympiad Regional Round, 10.1

Tags: algebra , combinatorics

The microcalculator ''MK-97'' can work out the numbers entered in memory, perform only three operations: a) check whether the selected two numbers are equal; b) add the selected numbers; c) using the selected numbers $a$ and $b$, find the equation $x^2 +ax+b = 0$, and if there are no roots, display a message about this. The results of all actions are stored in memory. Initially, one number $x$ is stored in memory. How to use ''MK-97'' to find out whether is this number one?

2022 Romania Team Selection Test, 4

Tags: perfect square , number theory

Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?

2021 Regional Competition For Advanced Students, 2

Tags: geometry , tangent , isosceles

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

2011 Middle European Mathematical Olympiad, 3

Tags: combinatorics

For an integer $n \geq 3$, let $\mathcal M$ be the set $\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane. What is the maximum possible number of points in a subset $S \subseteq \mathcal M$ which does not contain three distinct points being the vertices of a right triangle?

2011 IMAC Arhimede, 3

Tags: combinatorics

Place $n$ points on a circle and draw all possible chord joining these points. If no three chord are concurent, find the number of disjoint regions created. [color=#008000]Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=260926&hilit=circle+points+segments+regions[/color]

2018 Latvia Baltic Way TST, P4

Tags: function , inequalities , algebra

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that satisfies $$\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2$$ for all real $x$. Prove for all real $x$: [i](a)[/i] $f(x)\ge 4$; [i](b)[/i] $f(x)\ge 7.$

2021 China Team Selection Test, 4

Tags: algebra , polynomial , number theory , modular arithmetic

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

1959 Putnam, A3

Tags: function , functional equation , complex

Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$ for all $z\in \mathbb{C}$.

2011 Cono Sur Olympiad, 5

Tags: similar triangles , geometry

Let $ABC$ be a triangle and $D$ a point in $AC$. If $\angle{CBD} - \angle{ABD} = 60^{\circ}, \hat{BDC} = 30^{\circ}$ and also $AB \cdot BC = BD^{2}$, determine the measure of all the angles of triangle $ABC$.

1975 AMC 12/AHSME, 19

Tags: logarithm

Which positive numbers $ x$ satisfy the equation $ (\log_3x)(\log_x5)\equal{}\log_35$? $ \textbf{(A)}\ 3 \text{ and } 5 \text{ only} \qquad \textbf{(B)}\ 3, 5, \text{ and } 15 \text{ only} \qquad$ $ \textbf{(C)}\ \text{only numbers of the form } 5^n \cdot 3^m, \text{ where } n \text{ and } m \text{ are }$ $ \text{positive integers} \qquad$ $ \textbf{(D)}\ \text{all positive } x \neq 1 \qquad \textbf{(E)}\ \text{none of these}$

1994 Irish Math Olympiad, 3

Tags: inequalities

Prove that for every integer $ n>1$, $ n((n\plus{}1)^{\frac{2}{n}}\minus{}1)<\displaystyle\sum_{i\equal{}1}^{n}\frac{2i\plus{}1}{i^2}<n(1\minus{}n^{\minus{}\frac{2}{n\minus{}1}})\plus{}4$.

2009 Jozsef Wildt International Math Competition, W. 25

Tags: geometry , cyclic quadrilateral , inequalities

Let $ABCD$ be a quadrilateral in which $\widehat{A}=\widehat{C}=90^{\circ}$. Prove that $$\frac{1}{BD}(AB+BC+CD+DA)+BD^2\left (\frac{1}{AB\cdot AD}+\frac{1}{CB\cdot CD}\right )\geq 2\left (2+\sqrt{2}\right )$$

2008 AMC 12/AHSME, 20

Tags: ratio

Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

2015 Oral Moscow Geometry Olympiad, 6

Tags: altitude , geometry , angle bisector , cyclic quadrilateral

In the acute-angled non-isosceles triangle $ABC$, the height $AH$ is drawn. Points $B_1$ and $C_1$ are marked on the sides $AC$ and $AB$, respectively, so that $HA$ is the angle bisector of $B_1HC_1$ and quadrangle $BC_1B_1C$ is cyclic. Prove that $B_1$ and $C_1$ are feet of the altitudes of triangle $ABC$.