Found problems: 10
2018 PUMaC Individual Finals B, 3
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
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 $ABC$.
2015 Princeton University Math Competition, A3
Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.
2018 PUMaC Individual Finals A, 3
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
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 $n^2=d_1^3+d_2^3+d_4^3$.
2015 Princeton University Math Competition, A1/B1
Alice places down $n$ bishops on a $2015\times 2015$ chessboard such that no two bishops are attacking each other. (Bishops attack each other if they are on a diagonal.)
[list=a]
[*]Find, with proof, the maximum possible value of $n$.
[*](A1 only) For this maximal $n$, find, with proof, the number of ways she could place her bishops on the chessboard.
[/list]
2015 Princeton University Math Competition, A2/B3
For an odd prime number $p$, let $S$ denote the following sum taken modulo $p$:
\[ S \equiv \frac{1}{1 \cdot 2} + \frac{1}{3\cdot 4} + \ldots + \frac{1}{(p-2)\cdot(p-1)} \equiv \sum_{i=1}^{\frac{p-1}{2}} \frac{1}{(2i-1) \cdot 2i} \pmod p\]
Prove that $p^2 | 2^p - 2$ if and only if $S \equiv 0 \pmod p$.
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.
2015 Princeton University Math Competition, B2
On a circle $\omega_1$, four points $A$, $C$, $B$, $D$ lie in that order. Prove that $CD^2 = AC \cdot BC + AD \cdot BD$ if and only if at least one of $C$ and $D$ is the midpoint of arc $AB$.
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
$$f\left(xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)=f\left(xy\left(\frac{1}{x}+\frac{1}{y}\right)\right)+f(x)f\left(\frac{y}{x+y}\right).$$