Olympiad problems

2010 Purple Comet Problems, 11

A jar contains one white marble, two blue marbles, three red marbles, and four green marbles. If you select two of these marbles without replacement, the probability that both marbles will be the same color is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

2010 Purple Comet Problems, 15

Tags: number theory , relatively prime

Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions $\star$ each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are [b]not[/b] relatively prime $\star$ all other pairs of the five integers [b]are[/b] relatively prime.

2010 Indonesia TST, 1

Tags: polynomial , relatively prime , number theory , algebra

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

2013 Online Math Open Problems, 30

Tags: algebra , polynomial , number theory , relatively prime , area of a triangle , geometry

Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[ \sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

2019 PUMaC Geometry B, 4

Tags: geometry , probability , number theory , relatively prime

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

2015 Hanoi Open Mathematics Competitions, 13

Tags: number theory , relatively prime , exponential

Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.

2010 Tournament Of Towns, 2

Tags: ratio , number theory , relatively prime , geometry

Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?

PEN J Problems, 6

Tags: divisor function , modular arithmetic , quadratic , number theory , relatively prime , prime factorization

Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.

PEN H Problems, 77

Tags: diophantine equation , ratio , relatively prime , number theory

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2010 Indonesia MO, 6

Tags: relatively prime , number theory

Find all positive integers $n>1$ such that \[\tau(n)+\phi(n)=n+1\] Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$, and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$. [i]Raja Oktovin, Pekanbaru[/i]

1996 AIME Problems, 13

Tags: geometry , ratio , trigonometry , number theory , relatively prime

In triangle $ABC, AB=\sqrt{30}, AC=\sqrt{6},$ and $BC=\sqrt{15}.$ There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ and $\angle ADB$ is a right angle. The ratio \[ \frac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)} \] can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2007 ITest, 48

Tags: trigonometry , function , number theory , relatively prime , complex analysis

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

1977 IMO Longlists, 12

Tags: modular arithmetic , relatively prime , number theory

Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$

2008 Argentina Iberoamerican TST, 2

Tags: number theory , prime number , relatively prime

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

1985 IberoAmerican, 2

Tags: function , pigeonhole principle , relatively prime , number theory

To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied: (1) $ f(rs) \equal{} f(r)\plus{}f(s)$ (2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3. (3) $ f(10) \equal{} 0$. Find $f(1985)$. Justify your answer.

2008 ITest, 37

Tags: geometry , number theory , relatively prime

A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.

2019 USAMO, 5

Tags: number theory , relatively prime

Two rational numbers $\tfrac{m}{n}$ and $\tfrac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\tfrac{x+y}{2}$ or their harmonic mean $\tfrac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write 1 on the board in finitely many steps. [i]Proposed by Yannick Yao[/i]

2012 USA TSTST, 3

Tags: function , modular arithmetic , induction , number theory , greatest common divisor , relatively prime , arithmetic sequence

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

1995 AIME Problems, 15

Tags: probability , ratio , number theory , relatively prime

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2014 NIMO Problems, 6

Tags: modular arithmetic , number theory , relatively prime , arithmetic sequence , logarithm

Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]

2013 AIME Problems, 12

Tags: geometry , trigonometry , number theory , relatively prime , area of a triangle

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

1993 Greece National Olympiad, 11

Tags: probability , number theory , relatively prime

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?

1999 AIME Problems, 10

Tags: probability , number theory , relatively prime

Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$