Olympiad problems

2018 PUMaC Live Round, 7.1

Find the number of nonzero terms of the polynomial $P(x)$ if $$x^{2018}+x^{2017}+x^{2016}+x^{999}+1=(x^4+x^3+x^2+x+1)P(x).$$

2018 PUMaC Algebra B, 4

Tags: PuMAC , algebra

If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$, $$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$ find the smallest $n$ such that $a_n < \frac{1}{2018}$.

2018 PUMaC Individual Finals B, 3

Tags: PuMAC , Individual Finals , geometry

Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.

2018 PUMaC Individual Finals A, 1

Tags: PuMAC , Individual Finals , geometry

Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.

2018 PUMaC Team Round, 16

Tags: PuMAC , Team Round , abstract algebra

Let $N$ be the number of subsets $B$ of the set $\{1,2,\dots,2018\}$ such that the sum of the elements of $B$ is congruent to $2018$ modulo $2048$. Find the remainder when $N$ is divided by $1000$.

2018 PUMaC Team Round, 6

Tags: PuMAC , Team Round

Let $\tau(n)$ be the number of distinct positive divisors of $n$ (including $1$ and itself). Find the sum of all positive integers $n$ satisfying $n=\tau(n)^3.$

2018 PUMaC Live Round, 5.3

Tags: PuMAC , Live Round

Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.

2018 PUMaC Live Round, 2.3

Tags: PuMAC , Live Round , counting , distinguishability , probability , expected value

Sophie has $20$ indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out $30$ socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as $\tfrac{m}{n}$. Find $m+n$.

2018 PUMaC Combinatorics B, 6

Tags: PuMAC , combinatorics , probability , expected value

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.

2018 PUMaC Live Round, Estimation 2

Tags: Live Round , PuMAC

How many perfect squares have the digits $1$ through $9$ each exactly once when written in base $10$? You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, your score will be $\lfloor12.5\cdot\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor.$

2018 PUMaC Algebra A, 7

Tags: PuMAC , algebra

Let the sequence $\left \{ a_n \right \}_{n = -2}^\infty$ satisfy $a_{-1} = a_{-2} = 0, a_0 = 1$, and for all non-negative integers $n$, $$n^2 = \sum_{k = 0}^n a_{n - k}a_{k - 1} + \sum_{k = 0}^n a_{n - k}a_{k - 2}$$ Given $a_{2018}$ is rational, find the maximum integer $m$ such that $2^m$ divides the denominator of the reduced form of $a_{2018}$.

2018 PUMaC Team Round, 11

Tags: PuMAC , Team Round

Let $\tfrac{a}{b}$ be a fraction such that $a$ and $b$ are positive integers and the first three digits of its decimal expansion are $527$. What is the smallest possible value of $a+b?$

2018 PUMaC Team Round, 9

Tags: PuMAC , Team Round

There are numerous sets of $17$ distinct positive integers that sum to $2018$, such that each integer has the same sum of digits in base $10$. Let $M$ be the maximum possible integer that could exist in any such set. Find the sum of $M$ and the number of such sets that contain $M$.

2018 PUMaC Live Round, 1.2

Tags: function , PuMAC , Live Round

Define a function given the following $2$ rules: $\qquad$ 1) for prime $p$, $f(p)=p+1$. $\qquad$ 2) for positive integers $a$ and $b$, $f(ab)=f(a)\cdot f(b)$. For how many positive integers $n\leq 100$ is $f(n)$ divisible by $3$?

2018 PUMaC Combinatorics A, 4

Tags: PuMAC , combinatorics , probability , expected value

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.

2018 PUMaC Team Round, 7

Tags: PuMAC , Team Round

Let triangle $\triangle{MNP}$ have side lengths $MN=13$, $NP=89$, and $PM=100$. Define points $S$, $R$, and $B$ as the midpoints of $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ respectively. A line $\ell$ cuts lines $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ at points $I$, $J$, and $A$ respectively. Find the minimum value of $(SI+RJ+BA)^2.$

2018 PUMaC Live Round, Misc. 3

Tags: PuMAC , Live Round

Suppose $x,y\in\mathbb{Z}$ satisfy $$y^4+4y^3+28y+8x^3+6y^2+32x+1=(x^2-y^2)(x^2+y^2+24).$$ Find the sum of all possible values of $|xy|$.

2018 PUMaC Team Round, 0

Tags: PuMAC

For each problem, you will be asked to submit two integers. The first value that you submit represents what you think the correct answer to the problem is. The second value that you submit represents [b]how many teams[/b] you think will submit the correct answer. For example, consider 0. What is $32\div 2 \times 4 +3?$ The correct answer would be $67$. If you think every team will get it right, you should submit the number of teams competing at PUMaC. Therefore, a viable submission for the first entry could be $67$ and $n$ for the second, where $n$ is the number of teams taking the team round. There are $\mathbf{72}$ [b]teams[/b] signed up to take this round at PUMaC: 45 in A division and 27 in B division. You will receive $\left(\min\left(\tfrac{a}{b},\tfrac{b}{a}\right)\right)^2$ points for your guess, where $a$ is the number of teams that correctly answered the question and $b$ is the number of teams you guessed would get it correct (Note that in the case that no teams answer correctly or you guess $0$, you will receive $0$ points).

2018 PUMaC Live Round, 8.1

Tags: PuMAC , Live Round

Let $a$, $b$, and $c$ be such that the coefficient of the $x^ay^bz^c$ term in the expansion of $(x+2y+3z)^{100}$ is maximal (no other term has a strictly larger coefficient). Find the sum of all possible values of $1,000,000a+1,000b+c$.

2018 PUMaC Live Round, 8.3

Tags: PuMAC , Live Round

If $a$ and $b$ are positive integers such that $3\sqrt{2+\sqrt{2+\sqrt{3}}}=a\cos\frac{\pi}{b}$, find $a+b$.

2018 PUMaC Individual Finals A, 3

Tags: PuMAC , Individual Finals , number theory , prime numbers

We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?

2018 PUMaC Individual Finals B, 1

Tags: PuMAC , Individual Finals

Let a positive integer $n$ have at least four positive divisors. Let the least four positive divisors be $1=d_1<d_2<d_3<d_4$. Find, with proof, all solutions to $n^2=d_1^3+d_2^3+d_4^3$.

2018 PUMaC Live Round, 7.2

Tags: PuMAC , Live Round , number theory , relatively prime

Compute the smallest positive integer $n$ that is a multiple of $29$ with the property that for every positive integer that is relatively prime to $n$, $k^{n}\equiv 1\pmod{n}.$

2018 PUMaC Live Round, Estimation 3

Tags: PuMAC , Live Round , probability , expected value

Andrew starts with the $2018$-tuple of binary digits $(0,0,\dots,0)$. On each turn, he randomly chooses one index (between $1$ and $2018$) and flips the digit at that index (makes it $1$ if it was a $0$ and vice versa). What is the smallest $k$ such that, after $k$ steps, the expected number of ones in the sequence is greater than $1008?$ You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.$

2018 PUMaC Team Round, 2

Tags: PuMAC , Team Round

Let triangle $\triangle{ABC}$ have $AB=90$ and $AC=66$. Suppose that the line $IG$ is perpendicular to side $BC$, where $I$ and $G$ are the incenter and centroid, respectively. Find the length of $BC$.