Found problems: 83
2018 PUMaC Individual Finals B, 2
Aumann, Bill, and Charlie each roll a fair $6$-sided die with sides labeled $1$ through $6$ and look at their individual rolls. Each flips a fair coin and, depending on the outcome, looks at the roll of either the player to his right or the player to his left, without anyone else knowing which die he observed. Then, at the same time, each of the three players states the expected value of the sum of the rolls based on the information he has. After hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Then, for the third time, after hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Prove that Aumann, Bill, and Charlie say the same number the third time.
2018 PUMaC Live Round, 1.1
Find the number of pairs of real numbers $(x,y)$ such that $x^4+y^4=4xy-2$.
2018 PUMaC Live Round, 2.2
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 Individual Finals A, 2
Find all functions $f:\mathbb{R^{+}}\to\mathbb{R^+}$ such that for all $x,y\in\mathbb{R^+}$ it holds that
$$f\left(xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)=f\left(xy\left(\frac{1}{x}+\frac{1}{y}\right)\right)+f(x)f\left(\frac{y}{x+y}\right).$$
2018 PUMaC Combinatorics A, 3
Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.
2018 PUMaC Combinatorics B, 7
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
2018 PUMaC Combinatorics B, 4
Let $N$ be the number of sequences of natural numbers $d_1,d_2,\dots,d_{10}$ such that the following conditions hold: $d_1|d_2$, $\dots$, $d_9|d_{10}$ and $d_{10}|6^{2018}$. Evaluate the remainder when $N$ is divided by $2017$.
2018 PUMaC Live Round, 4.3
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
Compute the period (i.e. length of the repeating part) of the decimal expansion of $\tfrac{1}{729}$.
2018 PUMaC Combinatorics A, 1
There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?
2018 PUMaC Algebra B, 5
Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$.
Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.
2018 PUMaC Team Round, 1
Let $T=\{a_1,a_2,\dots,a_{1000}\}$, where $a_1<a_2<\dots<a_{1000}$, be a uniformly randomly selected subset of $\{1,2,\dots,2018\}$ with cardinality $1000$. The expected value of $a_7$ can be written in reduced form as $\tfrac{m}{n}$. Find $m+n$.
2018 PUMaC Team Round, 15
Aaron the Ant is somewhere on the exterior of a hollow cube of side length $2$ inches, and Fred the Flea is on the inside, at one of the vertices. At some instant, Fred flies in a straight line towards the opposite vertex, and simultaneously Aaron begins crawling on the exterior of the cube towards that same vertex. Fred moves at $\sqrt{3}$ inches per second and Aaron moves at $\sqrt{2}$ inches per second. If Aaron arrives before Fred, the area of the surface on the cube from which Aaron could have started can be written as $a\pi+\sqrt{b}+c$ where $a$, $b$, and $c$ are integers. Find $a+b+c.$
2018 PUMaC Live Round, 5.2
Find $x^2$ given that $\tan^{-1}(x)+\tan^{-1}(3x)=\frac{\pi}{6}$ and $0<x<\frac{\pi}{6}$.
2012 Princeton University Math Competition, Team Round
[hide=instructions]Time limit: 20 minutes.
Fill in the crossword above with answers to the problems below.
Notice that there are three directions instead of two. You are probably used to "down" and "across," but this crossword has "1," $e^{4\pi i/3}$, and $e^{5\pi i/3}$. You can think of these labels as complex numbers pointing in the direction to fill in the spaces. In other words "1" means "across", $e^{4\pi i/3}$ means "down and to the left," and $e^{5\pi i/3}$ means "down and to the right."
To fill in the answer to, for example, $12$ across, start at the hexagon labeled $12$, and write the digits, proceeding to the right along the gray line. (Note: $12$ across has space for exactly $5$ digits.)
Each hexagon is worth one point, and must be filled by something from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Note that $\pi$ is not in the set, and neither is $i$, nor $\sqrt2$, nor $\heartsuit$,etc.
None of the answers will begin with a $0$.
"Concatenate $a$ and $b$" means to write the digits of $a$, followed by the digits of $b$. For example, concatenating $10$ and $3$ gives $103$. (It's not the same as concatenating $3$ and $10$.)
Calculators are allowed!
THIS SHEET IS PROVIDED FOR YOUR REFERENCE ONLY. DO NOT TURN IN THIS SHEET. TURN IN THE OFFICIAL ANSWER SHEET PROVIDED TO THE TEAM. OTHERWISE YOU WILL GET A SCORE OF ZERO! ZERO! ZERO! AND WHILE SOMETIMES "!" MEANS FACTORIAL, IN THIS CASE IT DOES NOT.
Good luck, and have fun![/hide]
[img]https://cdn.artofproblemsolving.com/attachments/b/f/f7445136e40bf4889a328da640f0935b2b8b82.png[/img]
[u][b][i]Across[/i][/b][/u] (1)
[b]A 3.[/b] (3 digits) Suppose you draw $5$ vertices of a convex pentagon (but not the sides!). Let $N$ be the number of ways you can draw at least $0$ straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is $N - 64$? (Note what the question is asking for! You have been warned!)
[b]A 5.[/b] (3 digits) Among integers $\{1, 2,..., 10^{2012}\}$, let $n$ be the number of numbers for which the sum of the digits is divisible by $5$. What are the first three digits (from the left) of $n$?
[b]A 6.[/b] (3 digits) Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly $2012$ digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If $n^2$ is the largest perfect square he writes, what is $n$?
[b]A 8. [/b](3 digits) How many positive integers $n$ are there such that $n \le 2012$, and the greatest common divisor of $n$ and $2012$ is a prime number?
[b]A 9.[/b] (4 digits) I have a random number machine generator that is very good at generating integers between $1$ and $256$, inclusive, with equal probability. However, right now, I want to produce a random number between $1$ and $n$, inclusive, so I do the following:
$\bullet$ I use my machine to generate a number between $1$ and $256$. Call this $a$.
$\bullet$ I take a and divide it by $n$ to get remainder $r$. If $r \ne 0$, then I record $r$ as the randomly generated number. If $r = 0$, then I record $n$ instead.
Note that this process does not necessarily produce all numbers with equal probability, but that is okay. I apply this process twice to generate two numbers randomly between $1$ and $10$. Let $p$ be the probability that the two numbers are equal. What is $p \cdot 2^{16}$?
[b]A 12.[/b] (5 digits) You and your friend play the following dangerous game. You two start off at some point $(x, y)$ on the plane, where $x$ and $y$ are nonnegative integers.
When it is player $A$'s turn, A tells his opponent $B$ to move to another point on the plane. Then $A$ waits for a while. If $B$ is not eaten by a tiger, then $A$ moves to that point as well.
From a point $(x, y)$ there are three places $A$ can tell $B$ to walk to: leftwards to $(x - 1, y)$, downwards to $(x, y-1)$, and simultaneously downwards and leftwards to $(x-1, y-1)$. However, you cannot move to a point with a negative coordinate.
Now, what was this about being eaten by a tiger? There is a tiger at the origin, which will eat the first person that goes there! Needless to say, you lose if you are eaten.
Consider all possible starting points $(x, y)$ with $0 \le x \le 346$ and $0 \le y \le 346$, and $x$ and $y$ are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win?
[b][u][i]Down and to the left [/i][/u][/b] $e^{4\pi i/3}$
[b]DL 2.[/b] (2 digits) ABCDE is a pentagon with $AB = BC = CD = \sqrt2$, $\angle ABC = \angle BCD = 120$ degrees, and $\angle BAE = \angle CDE = 105$ degrees. Find the area of triangle $\vartriangle BDE$. Your answer in its simplest form can be written as $\frac{a+\sqrt{b}}{c}$ , where where $a, b, c$ are integers and $b$ is square-free. Find $abc$.
[b]DL 3.[/b] (3 digits) Suppose $x$ and $y$ are integers which satisfy $$\frac{4x^2}{y^2} + \frac{25y^2}{x^2} =
\frac{10055}{x^2} +\frac{4022}{y^2} +\frac{2012}{x^2y^2}- 20. $$ What is the maximum possible value of $xy -1$?
[b]DL 5.[/b] (3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties:
$\bullet$ The square has side length $7\sqrt2$.
$\bullet$ The boundary of the square intersects the graph of $xy = 0$ at at least $3$ points.
[b]DL 8.[/b] (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred:
$\bullet$ Mom: Tell me the number of zeros at the end of $2012!$
$\bullet$ PT: Huh? $2012$ ends in $2$, so there aren't any zeros.
$\bullet$ Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about $2012$, I was asking about $2012!$.
What is the correct answer?
[b]DL 9.[/b] (4 digits) Define the following:
$\bullet$ $A = \sum^{\infty}_{n=1}\frac{1}{n^6}$
$\bullet$ $B = \sum^{\infty}_{n=1}\frac{1}{n^6+1}$
$\bullet$ $C = \sum^{\infty}_{n=1}\frac{1}{(n+1)^6}$
$\bullet$ $D = \sum^{\infty}_{n=1}\frac{1}{(2n-1)^6}$
$\bullet$ $E = \sum^{\infty}_{n=1}\frac{1}{(2n+1)^6}$
Consider the ratios $\frac{B}{A}, \frac{C}{A}, \frac{D}{A} , \frac{E}{A}$. Exactly one of the four is a rational number. Let that number be $r/s$, where $r$ and $s$ are nonnegative integers and $gcd \,(r, s) = 1$. Concatenate $r, s$.
(It might be helpful to know that $A = \frac{\pi^6}{945}$ .)
[b]DL 10.[/b] (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at $(-1, 0)$, $(1, 0)$, $(1, 100)$, $(-1, 100)$. You remove a section of the bottom of the paper by cutting along the function $y = f(x)$, where $f$ satisfies $f(1) = f(-1) = 0$. (In other words, you keep the bottom two vertices.)
You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an $L$-shaped elbow tube.
We can write $f\left( \frac13 \right)+f\left( \frac16 \right) = \frac{a+\sqrt{b}}{\pi c}$ , where $a, b, c$ are integers and $b$ is square-free. $Find a+b+c$.
[b]DL 11.[/b] (3 digits) Let
$$\Xi (x) = 2012(x - 2)^2 + 278(x - 2)\sqrt{2012 + e^{x^2-4x+4}} + 1392 + (x^2 - 4x + 4)e^{x^2-4x+4}$$
find the area of the region in the $xy$-plane satisfying:
$$\{x \ge 0 \,\,\, and x \le 4 \,\,\, and \,\,\, y \ge 0 \,\,\, and \,\,\, y \le \sqrt{\Xi(x)}\}$$
[b]DL 13.[/b] (3 digits) Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height $6$. Find $V^2$.
[b][u][i]Down and to the right[/i][/u][/b] $e^{5\pi i/3}$
[b]DR 1.[/b] (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking "What? That's not right!" but I did not make a mistake.
When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done.
When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as $r/s$, where r and s are integers and $gcd \,(r, s) = 1$. What is $r + s$?
[b]DR 4.[/b] (3 digits) Let $a_1 = 2 +\sqrt2$ and $b_1 =\sqrt2$, and for $n \ge 1$, $a_{n+1} = |a_n - b_n|$ and $b_{n+1} = a_n + b_n$. The minimum value of $\frac{a^2_n+a_nb_n-6b^2_n}{6b^2_n-a^2_n}$ can be written in the form $a\sqrt{b} - c$, where $a, b, c$ are integers and $b$ is square-free. Concatenate $c, b, a$ (in that order!).
[b]DR 7.[/b] (3 digits) How many solutions are there to $a^{503} + b^{1006} = c^{2012}$, where $a, b, c$ are integers and $|a|$,$|b|$, $|c|$ are all less than $2012$?
PS. You should use hide for answers.
2018 PUMaC Combinatorics A, 6
Michael is trying to drive a bus from his home, $(0,0)$, to school, located at $(6,6)$. There are horizontal and vertical roads at every line $x=0,1,\ldots,6$ and $y=0,1,\ldots,6$. The city has placed $6$ roadblocks on lattice point intersections $(x,y)$ with $0\leq x,y \leq 6$. Michael notices that the only path he can take that only goes up and to the right is directly up from $(0,0)$ to $(0,6)$, and then right to $(6,6)$. How many sets of $6$ locations could the city have blocked?
2018 PUMaC Team Round, 5
There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have
$$\sqrt{n}=\sum_{i=0}^{n-1}\sqrt[5]{\sqrt{ai^5+bi^4+ci^3+di^2+ei+1}-\sqrt{ai^5+bi^4+ci^3+di^2+ei}}.$$
Find $a+b+c+d$.
2018 PUMaC Live Round, 7.3
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
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 Team Round, 3
The value of
$$\frac{\log_35\log_25}{\log_35+\log_25}$$
can be expressed as $a\log_bc$, where $a$, $b$, and $c$ are positive integers, and $a+b$ is as small as possible. Find $a+2b+3c$.
2018 PUMaC Live Round, 1.3
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
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 $a+b+c$.
2012 Princeton University Math Competition, B2
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors.
Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers.
Find $\Sigma_{k=1}^{n}(p_k + a_k)$
2018 PUMaC Combinatorics A, 2
In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.
2018 PUMaC Combinatorics B, 8
Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of
$$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$
then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?