Olympiad problems

2008 Princeton University Math Competition, A4/B7

Tags: geometry

How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and \[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\] is an integer?

1957 AMC 12/AHSME, 11

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The angle formed by the hands of a clock at $ 2: 15$ is: $ \textbf{(A)}\ 30^\circ \qquad \textbf{(B)}\ 27\frac{1}{2}^\circ\qquad \textbf{(C)}\ 157\frac{1}{2}^\circ\qquad \textbf{(D)}\ 172\frac{1}{2}^\circ\qquad \textbf{(E)}\ \text{none of these}$

2011 IMAC Arhimede, 2

Tags: angle bisector , geometry

Let $ABCD$ be a cyclic quadrilatetral inscribed in a circle $k$. Let $M$ and $N$ be the midpoints of the arcs $AB$ and $CD$ which do not contain $C$ and $A$ respectively. If $MN$ meets side $AB$ at $P$, then show that $\frac{AP}{BP}=\frac{AC+AD}{BC+BD}$

2013 Iran MO (3rd Round), 8

Tags: analytic geometry , geometry

Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$. ($A_i=A_{i+5}$) Prove that at most 3 lines from the lines $A_iB_i$ ($1\leq i \leq 5$) are concurrent. Time allowed for this problem was 75 minutes.

2024 Azerbaijan IZhO TST, 4

Tags: number theory

Take a sequence $(a_n)_{n=1}^\infty$ such that $a_1=3$ $a_n=a_1a_2a_3...a_{n-1}-1$ [b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence. [b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.

1973 Bundeswettbewerb Mathematik, 4

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Prove: for every positive integer there exists a positive integer having n digits, all of them being $1$'s and $2$'s only, such that this number is divisible by $2^{n}$. Is this still true in base $4$ or $6$¿

2020 CCA Math Bonanza, T7

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Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$ (note that the first one starts at $99$, and the second one ends at $98$). [i]2020 CCA Math Bonanza Team Round #7[/i]

2023 Singapore Junior Math Olympiad, 1

Tags: geometry

In a convex quadrilateral $ABCD$, the diagonals intersect at $O$, and $M$ and $N$ are points on the segments $OA$ and $OD$ respectively. Suppose $MN$ is parallel to $AD$ and $NC$ is parallel to $AB$. Prove that $\angle ABM=\angle NCD$.

2010 Today's Calculation Of Integral, 567

Tags: calculus , integration , analytic geometry , geometry , trigonometry , calculus computations

Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.

2020 Iran MO (2nd Round), P3

Tags: geometry

let $\omega_1$ be a circle with $O_1$ as its center , let $\omega_2$ be a circle passing through $O_1$ with center $O_2$ let $A$ be one of the intersection of $\omega_1$ and $\omega_2$ let $x$ be a line tangent line to $\omega_1$ passing from $A$ let $\omega_3$ be a circle passing through $O_1,O_2$ with its center on the line $x$ and intersect $\omega_2$ at $P$ (not $O_1$) prove that the reflection of $P$ through $x$ is on $\omega_1$

2011 ELMO Problems, 5

Tags: modular arithmetic , quadratic , number theory

Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

1999 Harvard-MIT Mathematics Tournament, 3

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Alex is stuck on a platform floating over an abyss at $1$ ft/s. An evil physicist has arranged for the platform to fall in (taking Alex with it) after traveling $100$ft. One minute after the platform was launched, Edward arrives with a second platform capable of floating all the way across the abyss. He calculates for 5 seconds, then launches the second platform in such a way as to maximize the time that one end of Alex's platform is between the two ends of the new platform, thus giving Alex as much time as possible to switch. If both platforms are $5$ ft long and move with constant velocity once launched, what is the speed of the second platform (in ft/s)?

KoMaL A Problems 2022/2023, A. 855

Tags: geometry , circumcircle

In scalene triangle $ABC$ the shortest side is $BC$. Let points $M$ and $N$ be chosen on sides $AB$ and $AC$, respectively, such that $BM=CN=BC$. Let $I$ and $O$ denote the incenter and circumcentre of triangle $ABC$, and let $D$ and $E$ denote the incenter and circumcenter of triangle $AMN$. Prove that lines $IO$ and $DE$ intersect each other on the circumcircle of triangle $ABC$. [i]Submitted by Luu Dong, Vietnam[/i]

LMT Team Rounds 2021+, A29 B30

Tags: combinatorics

In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. [i]Proposed by Sammy Charney[/i] [b]Due to the problem having infinitely many solutions, all teams who inputted answers received points.[/b]

1969 IMO Shortlist, 40

Tags: combinatorics , decimal representation , counting , digit

$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

2023 Belarus - Iran Friendly Competition, 1

Tags: factorial , number theory

Find all positive integers n such that the product $1! \cdot 2! \cdot \cdot \cdot \cdot n!$ is a perfect square

1975 IMO Shortlist, 8

Tags: geometry , trigonometry , triangle , angle

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

2011 Bogdan Stan, 1

Tags: group theory , abstract algebra , algebra , polynomial , linear algebra , matrix

Consider the multiplicative group $ \left\{ \left.A_k:=\left(\begin{matrix} 2^k& 2^k\\2^k& 2^k\end{matrix}\right)\right| k\in\mathbb{Z} \right\} . $ [b]a)[/b] Prove that $A_xA_y=A_{x+y+1} , $ for all integers $ x,y. $ [b]b)[/b] Show that, for all integers $ t, $ the multiplicative group $ \left\{ A_{jt-1}|j\in\mathbb{Z} \right\} $ is a subgroup of $ G. $ [b]c)[/b] Determine the linear integer polynomials $ P $ for which it exists an isomorphism $ \left( G,\cdot \right)\stackrel{\eta}{\cong}\left( \mathbb{Z} ,+ \right) $ such that $ \eta\left( A_k \right) =P(k). $

2021 Korea Winter Program Practice Test, 4

Tags: geometry , combinatorial geometry , combinatorics

A positive integer $m(\ge 2$) is given. From circle $C_1$ with a radius 1, construct $C_2, C_3, C_4, ... $ through following acts: In the $i$th act, select a circle $P_i$ inside $C_i$ with a area $\frac{1}{m}$ of $C_i$. If such circle dosen't exist, the act ends. If not, let $C_{i+1}$ a difference of sets $C_i -P_i$. Prove that this act ends within a finite number of times.

2014 IFYM, Sozopol, 4

Tags: polynomial , algebra

Find all polynomials $P,Q\in \mathbb{R}[x]$, such that $P(2)=2$ , $Q(x)$ has no negative roots, and $(x-2)P(x^2-1)Q(x+1)=P(x)Q(x^2 )+Q(x+1)$.

2006 France Team Selection Test, 2

Tags: algebra , inequalities

Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?

2020 Caucasus Mathematical Olympiad, 6

Tags: combinatorics

All vertices of a regular 100-gon are colored in 10 colors. Prove that there exist 4 vertices of the given 100-gon which are the vertices of a rectangle and which are colored in at most 2 colors.

2023 Durer Math Competition Finals, 12

Tags: number theory

Marvin really likes pancakes, so he asked his grandma to make pancakes for him. Every time Grandma sends pancakes, she sends a package of $32$. When Marvin is in the mood for pancakes, he eats half of the pancakes he has. Marvin ate $157$ pancakes for lunch today. At least how many times has Grandma sent pancakes to Marvin so far? Marvin does not necessarily eat an integer number of pancakes at once, and he is in the mood for pancakes at most once a day.

2007 India IMO Training Camp, 1

Tags: geometry , trapezoid , circumcircle , ratio , homothety

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

2017 CMIMC Algebra, 7

Tags: algebra , system of equations

Let $a$, $b$, and $c$ be complex numbers satisfying the system of equations \begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*} Find $abc$.