Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number is now greater than Paula's by $5070$. What was the sum of their original numbers? $\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

Let $O$ and $P$ be fixed points on a plane, and let $ABCD$ be any parallelogram with center $O$. Let $M$ and $N$ be the midpoints of $AP$ and $BP$ respectively. Lines $MC$ and $ND$ meet at $Q$. Prove that the point $Q$ lies on the lines $OP$, and show that it is independent of the choice of the parallelogram $ABCD$.

Given triangle $ABC$. $D$ is a point on $BC$. $AC$ meets $(ABD)$ again at $E$,and $AB$ meets $(ACD)$ again at $F$. $M$ is the midpoint of $EF$. $BC$ meets $(DEF)$ again at $P$. Prove that $\angle BAP = \angle MAC$.

Let $ABCD$ be a tetrahedron inscribed in a sphere with center $O$, and $G$ and $I$ be its barycenter and incenter respectively. Prove that the following are equivalent: (i) Points $O$ and $G$ coincide. (ii) The four faces of the tetrahedron are congruent. (iii) Points $O$ and $I$ coincide.

A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column. (a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table. (b) Show that there is such a table not all of whose entries are $0$.

Penniless Pete’s piggy bank has no pennies in it, but it has $ 100$ coins, all nickels, dimes, and quarters, whose total value is $ \$8.35$. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 83$

We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that: $i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$ $ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$ $iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$ Find all [i]interesting [/i]$n$. [i]Proposed by Mojtaba Zare Bidaki[/i]

Let $n$ be a positive integer. In the $\mathit{philand}$ language, words are all finite sequences formed by the letters "$P$", "$H$" and "$I$". Philipe, who speaks only the $\mathit{philand}$ language, writes the word $PHIPHI\ldots PHI$ on a piece of paper, where $PHI$ is repeated $n$ times. He can do the following operations: • Erase two identical letters and write in their place two different letters from the original and from each other; (Ex: $PP\rightarrow HI$) • Erase two distinct letters and rewrite them changing the order in which they appear; (Ex: $PI\rightarrow IP$) • Erase two distinct letters and write the letter distinct from the two he erased. (Ex: $PH\rightarrow I$) Find the largest integer $C$ such that any Philandese word of up to $C$ letters can be written by Philip through the above operations. Note: Operations are taken on adjacent letters.

Suppose $a$ is a non-zero real number such that $a +\frac{1}{a}$ is a whole number. (a) Prove that $a^2 +\frac{1}{a^2}$ is also an integer. (b) Prove that $a^n+\frac{1}{a^n}$ is also an integer, for any integer value positive of $n$.

a) Prove that \[ \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} ... \plus{} \dfrac{1}{2^{2n}} > n, \] for all positive integers $ n$. b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} \cdots \plus{} \dfrac{1}{k}>n \right\} > 2^n$.

$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$

Determine all pairs $(a,b) \in \mathbb{C} \times \mathbb{C}$ of complex numbers satisfying $|a|=|b|=1$ and $a+b+a\overline{b} \in \mathbb{R}$.

Determine, with proof, whether $22!6! + 1$ is prime.

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

$x,y,z$ are nonnegative real numbers, and $4^{\sqrt{5x+9y+4z}}-68\times2^{\sqrt{5x+9y+4z}}+256=0$. Then, the product of the maximum and minimum value of $x+y+z$ is________.

How many three-digit positive integers contain both even and odd digits?

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

For a sequence with some ones and zeros, we count the number of continuous runs of equal digits in it. (For example the sequence $011001010$ has $7$ continuous runs: $0,11,00,1,0,1,0$.) Find the sum of the number of all continuous runs for all possible sequences with $2019$ ones and $2019$ zeros.

Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

Unit square $ABCD$ is drawn on a plane. Point $O$ is drawn outside of $ABCD$ such that lines $AO$ and $BO$ are perpendicular. Square $F ROG$ is drawn with $F$ on $AB$ such that $AF =\frac23$, $R$ is on $\overline{BO}$, and $G$ is on $\overline{AO}$. Extend segment $\overline{OF}$ past $\overline{AB}$ to intersect side $\overline{CD}$ at $E$. Compute $DE$.

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$? $\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$

Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .

Let $A$ and $B$ different points of a circle $k$ centered at $O$ in such a way such that $AB$ is not a diagonal of $k$. Furthermore, let $X$ be an arbitrary inner point of the segment $AB$. Let $k_1$ be the circle that passes through the points $A$ and $X$, and $A$ is the only common point of $k$ and $k_1$. Similarly, let $k_2$ be the circle that passes through the points $B$ and $X$, and $B$ is the only common point of $k$ and $k_2$. Let $M$ be the second intersection point of $k_1$ and $k_2$. Let $Q$ denote the center of circumscribed circle of the triangle $AOB$. Let $O_1$ and $O_2$ be the centers of $k_1$ and $k_2$. Show that the points $M,O,O_1,O_2,Q$ are on a circle.

Let $A_1A_2A_3A_4A_5A_6$ be a convex hexagon such that $A_iA_{i+2} \parallel A_{i+3}A_{i+5}$ for $i = 1, 2, 3$ (we take $A_{i+6} = A_i$ for each $i$). Segment $A_iA_{i+2}$ intersects segment $A_{i+1}A_{i+3}$ at $B_i$, for $1 \le i \le 6$, as shown. Furthermore, suppose that $\vartriangle A_1A_3A_5 \cong \vartriangle A_4A_6A_2$. Given that $[A_1B_5B_6] = 1$, $[A_2B_6B_1] = 4$, and $[A_3B_1B_2] = 9$ (by $[XY Z]$ we mean the area of $ \vartriangle XY Z$), determine the area of hexagon $B_1B_2B_3B_4B_5B_6$. [img]https://cdn.artofproblemsolving.com/attachments/d/0/1a8997c9eb7dea5223b6805dacd79c10a2cd33.png[/img]