Olympiad problems

2010 ISI B.Stat Entrance Exam, 6

Tags: algebra , polynomial

Consider the equation $n^2+(n+1)^4=5(n+2)^3$ (a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation. (b) Does the equation have a solution in positive integers?

1954 AMC 12/AHSME, 19

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If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle: $ \textbf{(A)}\ \text{are always equal to }60^\circ \\ \textbf{(B)}\ \text{are always one obtuse angle and two unequal acute angles} \\ \textbf{(C)}\ \text{are always one obtuse angle and two equal acute angles} \\ \textbf{(D)}\ \text{are always acute angles} \\ \textbf{(E)}\ \text{are always unequal to each other}$

2012 Lusophon Mathematical Olympiad, 1

Tags: combinatorics , combinatorial geometry , collinear , geometry , incenter

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

2024-25 IOQM India, 18

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Let $p,q$ be two-digit number neither of which are divisible by $10$. Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p,q$ very, a computer prints $r$ on the screen if $\gcd(p,q) = 1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).

1969 IMO Shortlist, 20

Tags: polygon , lattice points , area , geometry

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

2021 MOAA, 8

Tags: accuracy

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2010 Baltic Way, 20

Tags: induction , number theory

Determine all positive integers $n$ for which there exists an infinite subset $A$ of the set $\mathbb{N}$ of positive integers such that for all pairwise distinct $a_1,\ldots , a_n \in A$ the numbers $a_1+\ldots +a_n$ and $a_1a_2\ldots a_n$ are coprime.

2003 AMC 10, 16

Tags: modular arithmetic , algorithm , number theory , greatest common divisor , euclidean algorithm

What is the units digit of $ 13^{2003}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2010 Poland - Second Round, 2

Tags: 3d geometry , tetrahedron , circumcircle , incenter , geometry

The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

2013 NIMO Problems, 4

Tags: trigonometry

Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey \begin{align*} 0.1 &= \sin \gamma \cos \theta \sin \alpha, \\ 0.2 &= \sin \gamma \sin \theta \cos \alpha, \\ 0.3 &= \cos \gamma \cos \theta \sin \beta, \\ 0.4 &= \cos \gamma \sin \theta \cos \beta, \\ 0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert. \end{align*}[i]Proposed by Evan Chen[/i]

2004 Indonesia MO, 3

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In how many ways can we change the sign $ *$ with $ \plus{}$ or $ \minus{}$, such that the following equation is true? \[ 1 *2*3*4*5*6*7*8*9*10\equal{}29\]

1993 Austrian-Polish Competition, 8

Tags: polynomial , functional , functional equation , algebra

Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.

2020 Purple Comet Problems, 15

Tags: geometry

Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece he formed the perimeter of a rectangle whose width and length are in the ratio $3 : 8$. The two rectangles that Daniel formed have the same area, and each of those areas is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

1964 AMC 12/AHSME, 10

Tags: geometry

Given a square side of length $s$. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is: ${{ \textbf{(A)}\ s\sqrt{2} \qquad\textbf{(B)}\ s/\sqrt{2} \qquad\textbf{(C)}\ 2s \qquad\textbf{(D)}\ 2\sqrt{s} }\qquad\textbf{(E)}\ 2/ \sqrt{s} } $

1989 AMC 8, 17

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The number $\text{N}$ is between $9$ and $17$. The average of $6$, $10$, and $\text{N}$ could be $\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

2022 JHMT HS, 8

Tags: algebra , complex number

Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

2024 TASIMO, 6

Tags: abstract algebra , number theory , complete residue system

We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$

1999 Harvard-MIT Mathematics Tournament, 5

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For any finite set $S$, let $f(S)$ be the sum of the elements of $S$ (if $S$ is empty then $f(S)=0$). Find the sum over all subsets $E$ of $S$ of $\dfrac{f(E)}{f(S)}$ for $S=\{1,2,\cdots,1999\}$.

2012 Pre-Preparation Course Examination, 4

Tags: linear algebra , matrix , vector

Prove that these two statements are equivalent for an $n$ dimensional vector space $V$: [b]$\cdot$[/b] For the linear transformation $T:V\longrightarrow V$ there exists a base for $V$ such that the representation of $T$ in that base is an upper triangular matrix. [b]$\cdot$[/b] There exist subspaces $\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V$ such that for all $i$, $T(V_i)\subseteq V_i$.

Brazil L2 Finals (OBM) - geometry, 2011.2

Tags: geometry

Let $ ABCD $ be a convex quadrilateral such that $ AD = DC, AC = AB $ and $ \angle ADC = \angle CAB $. If $ M $ and $ N $ are midpoints of the $ AD $ and $ AB $ sides, prove that the $ MNC $ triangle is isosceles.

2007 Stanford Mathematics Tournament, 8

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If $r+s+t=3$, $r^2+s^2+t^2=1$, and $r^3+s^3+t^3=3$, compute $rst$.

2021 Israel TST, 2

Tags: functional equation , functional equation in z , algebra

Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.

2013 Costa Rica - Final Round, F2

Tags: functional equation , functional , algebra

Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$

1972 IMO, 2

Tags: geometry , trapezoid , dissection , cyclic quadrilateral

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

2022 Princeton University Math Competition, A4 / B6

Tags: geometry

Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. (Here $[P]$ denotes the area of polygon $P$.)