Olympiad problems

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]

MathLinks Contest 6th, 4.2

Tags: number theory

Let $n$ be a positive integer. Prove that there exist an infinity of multiples of $n$ which do not contain the digit “$9$” in their decimal representation

2016 Miklós Schweitzer, 4

Tags: algebra , function , sequence

Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that \[ a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)} \] for all integers $m,n\ge 1$, and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line. [i]Remark.[/i] A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$, then $a(n)/n$ converges or tends to $(-\infty)$.

2012 IMAC Arhimede, 2

Tags: parallel , geometry , circles , tangent

Circles $k_1,k_2$ intersect at $B,C$ such that $BC$ is diameter of $k_1$.Tangent of $k_1$ at $C$ touches $k_2$ for the second time at $A$.Line $AB$ intersects $k_1$ at $E$ different from $B$, and line $CE$ intersects $k_2$ at F different from $C$. An arbitrary line through $E$ intersects segment $AF$ at $H$ and $k_1$ for the second time at $G$.If $BG$ and $AC$ intersect at $D$, prove $CH//DF$ .

2021 ISI Entrance Examination, 2

Tags: functional equation , isi entrance

Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers $x,y$. [b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$. [b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.

2005 Iran MO (3rd Round), 1

Tags: rotation , limit , geometry

An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed $v$. Suppose the radius of earth is $R$. a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole? b) Will the airplne rotate finitely many times around the north pole? If yes how many times?

1979 IMO Shortlist, 10

Tags: vector , trigonometry , geometry , geometric inequality , euclidean geometry

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

1999 Croatia National Olympiad, Problem 3

Tags: sum , summation , fibonacci , fibonacci sequence , algebra

Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.

2005 Austrian-Polish Competition, 7

Tags: system of equations , number theory , algebra

For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$: $$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$ where $$S=x_1^2+x_2^2+\dots+x_n^2.$$

2014 AMC 8, 13

Tags: modular arithmetic

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible? $\textbf{(A) }n$ and $m$ are even $\qquad\textbf{(B) }n$ and $m$ are odd $\qquad\textbf{(C) }n+m$ is even $\qquad\textbf{(D) }n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible

2017 CCA Math Bonanza, T1

Tags: arithmetic sequence

Given that $9\times10\times11\times\cdots\times15=32432400$, what is $1\times3\times5\times\cdots\times15$? [i]2017 CCA Math Bonanza Team Round #1[/i]