PUMaC

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2484   
On the number line, consider the point $x$ that corresponds to the value 10. Consider 24 distinct integer points $y_1, y_2 \cdots y_{24}$ on the number line such that for all $k$ such that $1\le k\le 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of $$\sum_{n=1}^{24}(\mid y_n-1 \mid + \mid y_n+1\mid)$$

2485   
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price p. $A$ day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has 12 plushies, Bob has 40, and Charlie has 52 but they all spent the same amount of money: \$42. How many plushies did Alice buy on the first day?

2486   
A function $f$ has its domain equal to the set of integers $\{0, 1, ..., 11\}$, and $f(n)\ge 0$ for all such $n$, and $f$ satisfies: $f(0) = 0$, $f(6) = 1$. If $x \ge 0$, $y\ge 0$, and $x + y\le 11$, then $f(x + y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$. Find $f(2)^2 + f(10)^2$.

2487   
There is a sequence with $a(2) = 0$, $a(3) = 1$ and $a(n) = a(\lfloor{\frac{n}{2}}\rfloor)+a(\lceil{\frac{n}{2}}\rceil)$ for $n\ge 4$. Find $a(2014)$.

2488   
Real numbers $x, y, z$ satisfy the following equality: $$4(x + y + z) = x^2 + y^2 + z^2$$ Let $M$ be the maximum of $xy + yz + zx$, and let $m$ be the minimum of $xy + yz + zx$. Find $M + 10m$.

2489   
Given that $x_{n+2} =\frac{20x_{n+1}}{14x_n}$, $x_0 = 25$, $x_1 = 11$, it follows that $$\sum_{n=0}^{\infty}\frac{x_{3n}}{2^n}=\frac{p}{q}$$ for some positive integers $p, q$ with $GCD(p, q) = 1$. Find $p + q$.

2490   
$x, y, z$ are positive real numbers that satisfy $x^3+2y^3+6z^3 = 1$. Let $k$ be the maximum possible value of $2x + y + 3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n + n$.

2491   
For nonnegative integer $n$, the following are true: $f(0) = 0$ $f(1) = 1$ $f(n) = f(n-\frac{m(m-1)}{2})-f(\frac{m(m+1)}{2} -n)$ for integer $m$ satisfying $m \ge 2$ and $\frac{m(m-1)}{2} < n \le \frac{m(m+1)}{2}$. Find the smallest $n$ such that $f(n) = 4$.

2492   
What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?

2493   
Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut in the shape of an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of finite pieces (the infinite pieces have infinite mass, so you can't lift them up). How many finite pieces of pizza do you have?

2494   
You have three colors {red; blue; green} with which you can color the faces of a regular octahedron ($8$ triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)

2496   
What is the size of the largest subset $S'$ of $S=\{ 2^x 3^y 5^z : 0\le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p\mid q$.

2497   
Let $f(n)$ be the number of points of intersections of diagonals of a $n$-dimensional hypercube that is not the vertice of the cube. For example, $f(3) = 7$ because the intersection points of a cube's diagonals are at the centers of each face and the center of the cube. Find $f(5)$

2498   
Tom and Jerry are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2,\cdots, 2014$ are to be written. (Each number will be written exactly once). Tom fills in a piece of paper first. How many pieces of paper must Jerry fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Tom's in at least one position?

2538   
There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends' homes.

2539   
Let $f(x) = x^3+ax^2+bx+c$ have solutions that are distinct negative integers. If $a+b+c =2014$, \ffind $c$.

2540   
What is the last digit of $17^{17^{17^{17}}}$?

2541   
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.

2542   
Find the sum of all positive integer $x$ such that $3\times 2^x = n^2-1$ for some positive integer $n$.

2543   
Find the number of pairs of integer solution $(x, y)$ that satisfies the equation $$(x-y + 2)(x-y-2) =-(x-2)(y-2)$$

2544   
Given $S = \{2, 5, 8, 11, 14, 17, 20,\cdots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1, A2,\cdots A_n\}$, s.t. $\displaystyle\sum_{i=1}^{n}\frac{1}{A_i}=1$ Find the minimum possible value of $n$.

2545   
Find the number of positive integers $n\le 2014$ such that there exists integer $x$ that satisfies the condition that $\displaystyle\frac{x + n}{x-n}$ is an odd perfect square.

2546   
Find all number sets $(a, b, c, d)$ s.t. $1 < a \le b \le c \le d, a,b,c,d \in\mathbb{N}$, and $a^2 + b + c + d, a + b^2 + c + d, a + b + c^2 + d$ and $a + b + c + d^2$ are all square numbers. Sum the value of $d$ across all solution $set(s)$.

2547   
Evaluate $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{1368}+\sqrt{1369}}$.

2548   
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n)) + f(n) = 2n + 3$ for all nonnegative integers $n$. Find $f(2014)$.

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