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 Live Round, Calculus 1

Tags: calculus , Live Round

Freddy the king of flavortext has an infinite chest of coins. For each number $p$ in the interval $[0, 1]$, Freddy has a coin that has probability $p$ of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is $\frac{p}{q}$ for two integers $p, q$, find $p + q$. Note: flavortext is made up

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 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 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 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 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 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 Live Round, Calculus 2

Tags: Live Round , calculus , PuMAC , probability

Three friends are trying to meet for lunch at a cafe. Each friend will arrive independently at random between $1\!:\!00$ pm and $2\!:\!00$ pm. Each friend will only wait for $5$ minutes by themselves before leaving. However, if another friend arrives within those $5$ minutes, the pair will wait $15$ minutes from the time the second friend arrives. If the probability that the three friends meet for lunch can be expressed in simplest form as $\tfrac{m}{n}$, what is $m+n$?

2018 PUMaC Live Round, 4.2

Tags: PuMAC , Live Round

Some number of regular polygons meet at a point on the plane such that the polygons' interiors do not overlap, but the polygons fully surround the point (i.e. a sufficiently small circle centered at the point would be contained in the union of the polygons). What is the largest possible number of sides in any of the polygons?

2018 PUMaC Live Round, 4.1

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

The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.

2018 PUMaC Live Round, Estimation 1

Tags: PuMAC , Live Round , rotation

A $2$-by-$2018$ grid is completely covered by non-overlapping L-tiles (see diagram below) and $1$-by-$1$ squares. If the L-tiles can be rotated and flipped, there are a total of $M$ such tilings. [asy] size(1cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0)); [/asy] What is $\ln M?$ Give your answer as an integer or decimal. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor7.5-\tfrac{|A-C|^{1.5}}{20}\rfloor,0\}.$

2018 PUMaC Live Round, Misc. 1

Tags: PuMAC , Live Round , algebra , polynomial

Consider all cubic polynomials $f(x)$ such that $f(2018)=2018$, the graph of $f$ intersects the $y$-axis at height $2018$, the coefficients of $f$ sum to $2018$, and $f(2019)>(2018)$. We define the infinimum of a set $S$ as follows. Let $L$ be the set of lower bounds of $S$. That is, $\ell\in L$ if and only if for all $s\in S$, $\ell\leq s$. Then the infinimum of $S$ is $\max(L)$. Of all such $f(x)$, what is the infinimum of the leading coefficient (the coefficient of the $x^3$ term)?

2018 PUMaC Live Round, 1.1

Tags: PuMAC , Live Round

Find the number of pairs of real numbers $(x,y)$ such that $x^4+y^4=4xy-2$.

2018 PUMaC Live Round, 2.2

Tags: PuMAC , Live Round , geometry

Let $ABC$ be a triangle with side lengths $13,14,15$. The points on the interior of $ABC$ with distance at least $1$ from each side are shaded. The area of the shaded region can be written in simplest form as $\tfrac{m}{n}$. Find $m+n$.

2018 PUMaC Live Round, 4.3

Tags: PuMAC , Live Round

Let $0\leq a,b,c,d\leq 10$. For how many ordered quadruples $(a,b,c,d)$ is $ad-bc$ a multiple of $11?$

2018 PUMaC Live Round, 2.1

Tags: PuMAC , Live Round

Compute the period (i.e. length of the repeating part) of the decimal expansion of $\tfrac{1}{729}$.

2018 PUMaC Live Round, 5.2

Tags: PuMAC , Live Round

Find $x^2$ given that $\tan^{-1}(x)+\tan^{-1}(3x)=\frac{\pi}{6}$ and $0<x<\frac{\pi}{6}$.

2018 PUMaC Live Round, 7.3

Tags: PuMAC , Live Round

Kite $ABCD$ has right angles at $B$ and $D$, and $AB<BC$. Points $E\in AB$ and $F\in AD$ satisfy $AE=4$, $EF=7$, and $FA=5$. If $AB=8$ and points $X$ lies outside $ABCD$ while satisfying $XE-XF=1$ and $XE+XF+2XA=23$, then $XA$ may be written as $\tfrac{a-b\sqrt{c}}{d}$ for $a,b,c,d$ positive integers with $\gcd(a^2,b^2,c,d^2)=1$ and $c$ squarefree. Find $a+b+c+d$.

2018 PUMaC Live Round, Misc. 2

Tags: PuMAC , Live Round , complex numbers

What is the sum of the possible values for the complex number $a$ such that the coefficient of the $x^5$ term in the power series expansion of $\tfrac{x^3+ax^2+3x-4}{2x^2+ax+2}$ is $1?$

2018 PUMaC Live Round, 1.3

Tags: PuMAC , Live Round , probability , expected value

Let a sequence be defined as follows: $a_0=1$, and for $n>0$, $a_n$ is $\tfrac{1}{3}a_{n-1}$ and is $\tfrac{1}{9}a_{n-1}$ with probability $\tfrac{1}{2}$. If the expected value of $\textstyle\sum_{n=0}^{\infty}a_n$ can be expressed in simplest form as $\tfrac{p}{q}$, what is $p+q$?

2018 PUMaC Live Round, 8.2

Tags: PuMAC , Live Round , geometry , circumcircle

The triangle $ABC$ satisfies $AB=10$ and has angles $\angle{A}=75^{\circ}$, $\angle{B}=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\tfrac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.