Olympiad problems

1989 IMO Longlists, 9

Tags: algebra

Do there exist two sequences of real numbers $ \{a_i\}, \{b_i\},$ $ i \in \mathbb{N},$ satisfying the following conditions: \[ \frac{3 \cdot \pi}{2} \leq a_i \leq b_i\] and \[ \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i}\] $ \forall i \in \mathbb{N}$ and all $ x,$ with $ 0 < x < 1?$

2007 China Northern MO, 2

Tags: function , algebra , logarithm

Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$. a) Solve the equation $ f(x) = 0$. b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]

2010 Iran Team Selection Test, 2

Tags: function , inequalities , algebra

Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$ \[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]

2005 AMC 10, 2

Tags:

For each pair of real numbers $ a\not\equal{} b$, define the operation $ \star$ as \[(a \star b) \equal{} \frac{a \plus{} b}{a \minus{} b}.\] What is the value of $ ((1 \star 2) \star 3)$? $ \textbf{(A)}\ \minus{}\frac{2}{3}\qquad \textbf{(B)}\ \minus{}\frac{1}{5}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac{1}{2}\qquad \textbf{(E)}\ \text{This value is not defined.}$

2016 VJIMC, 4

Tags: function , integral , bounded variation , differentiable function , real analysis

Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$ for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

2001 All-Russian Olympiad, 3

Tags: graph theory , combinatorics

The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set $D$ of towns [i]dominant[/i] if every town not in $D$ is connected by a road to a town in $D$. Suppose that each dominant set consists of at least $k$ towns. Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road.

2020 Jozsef Wildt International Math Competition, W49

Tags: inequalities

Let $a,b,c>0$ so that $a+b+c=1$. Then prove that $$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$ [i]Proposed by Marius Drăgan[/i]

2001 Estonia National Olympiad, 3

Tags: square , geometry , circles

A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.

2012 Kosovo National Mathematical Olympiad, 1

Tags: number theory

Find the two last digits of $2012^{2012}$.

2016 NIMO Problems, 3

Tags: calculus

Let $f$ be the quadratic function with leading coefficient $1$ whose graph is tangent to that of the lines $y=-5x+6$ and $y=x-1$. The sum of the coefficients of $f$ is $\tfrac pq$, where $p$ and $q$ are positive relatively prime integers. Find $100p + q$. [i]Proposed by David Altizio[/i]

1981 Canada National Olympiad, 3

Tags: combinatorics

Given a finite collection of lines in a plane $P$, show that it is possible to draw an arbitrarily large circle in $P$ which does not meet any of them. On the other hand, show that it is possible to arrange a countable infinite sequence of lines (first line, second line, third line, etc.) in $P$ so that every circle in $P$ meets at least one of the lines. (A point is not considered to be a circle.)

2005 Bulgaria National Olympiad, 1

Tags: number theory

Determine all triples $\left( x,y,z\right)$ of positive integers for which the number \[ \sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}} \] is an integer .

2013 Iran Team Selection Test, 5

Tags: modular arithmetic , algebra , polynomial , quadratic , vieta , number theory

Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

2019 Teodor Topan, 4

Tags: group theory

Let $ S $ be a finite [url=https://en.wikipedia.org/wiki/Cancellation_property]cancellative semigroup.[/url] [b]a)[/b] Prove that $ S $ contains an idempotent element. [b]b)[/b] Prove that $ S $ is a group. [b]c)[/b] Disprove subpoint [b]b)[/b] in the case that $ S $ would not be finite. [i]Vlad Mihaly[/i]

2021 Princeton University Math Competition, 5

Tags: analytic geometry , geometry

Given a real number $t$ with $0 < t < 1$, define the real-valued function $f(t, \theta) = \sum^{\infty}_{n=-\infty} t^{|n|}\omega^n$, where $\omega = e^{i \theta} = \cos \theta + i\sin \theta$. For $\theta \in [0, 2\pi)$, the polar curve $r(\theta) = f(t, \theta)$ traces out an ellipse $E_t$ with a horizontal major axis whose left focus is at the origin. Let $A(t)$ be the area of the ellipse $E_t$. Let $A\left( \frac12 \right) = \frac{a\pi}{b}$ , where $a, b$ are relatively prime positive integers. Find $100a +b$ .

2013 IMC, 5

Tags: function , complex number

Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime? [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

2021 Oral Moscow Geometry Olympiad, 1

Tags: geometry , trigonometry , grid

Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$. [img]https://cdn.artofproblemsolving.com/attachments/6/1/eeb98ccdee801361f9f66b8f6b2da4714e659f.png[/img]

2010 Serbia National Math Olympiad, 1

Tags: circumcircle , geometric transformation , reflection , trigonometry , incenter , geometry

Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$. [i]Proposed by Dusan Djukic[/i]

2015 İberoAmerican, 4

Tags: geometry

Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $P$ be a point on segment $AD$. Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $J$ and $K$ be the feet of the peroendiculars from $E$ and $F$ to $AD$, respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$.

Russian TST 2016, P1

Tags: number theory , diophantine equation

Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]

2016 CMIMC, 3

Tags: geometry

Let $ABC$ be a triangle. The angle bisector of $\angle B$ intersects $AC$ at point $P$, while the angle bisector of $\angle C$ intersects $AB$ at a point $Q$. Suppose the area of $\triangle ABP$ is 27, the area of $\triangle ACQ$ is 32, and the area of $\triangle ABC$ is $72$. The length of $\overline{BC}$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers with $n$ as small as possible. What is $m+n$?

2012 Online Math Open Problems, 13

Tags:

A number is called [i]6-composite[/i] if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is [i]composite[/i] if it has a factor not equal to 1 or itself. In particular, 1 is not composite.) [i]Ray Li.[/i]

1976 Putnam, 3

Tags:

Find all integral solutions of the equation $$|p^r-q^s|=1,$$ where $p$ and $q$ are prime numbers and $r$ and $s$ are positive integers larger than unity. Prove that there are no other solutions.

2024 JHMT HS, 11

Tags: number theory

Let $N_{10}$ be the answer to problem 10. Compute the number of ordered pairs of integers $(m,n)$ that satisfy the equation \[ m^2+n^2=mn+N_{10}. \]

2020 Harvest Math Invitational Team Round Problems, HMI Team #5

Tags: geometry , euclidean geometry

5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by winnertakeover and Monkey_king1[/i]