Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $ h$ hours and $ m$ minutes, with $ 0<m<60$, what is $ h\plus{}m$? $ \textbf{(A)}\ 46 \qquad \textbf{(B)}\ 47 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 53 \qquad \textbf{(E)}\ 54$

p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$. Find the number of subsets of $H$. p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$. a. Give an example of your own from the above process. b. Explain mathematically the general form of the process. p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn? p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made. p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?

Given a circle with center $O$, such that $A$ is not on the circumcircle. Let $B$ be the reflection of $A$ with respect to $O$. Now let $P$ be a point on the circumcircle. The line perpendicular to $AP$ through $P$ intersects the circle at $Q$. Prove that $AP \times BQ$ remains constant as $P$ varies.

How many positive integers less than 4000 are not divisible by 2, not divisible by 3, not divisible by 5, and not divisible by 7?

Let $ABC$ be a triangle with $AB < AC$. The incircle of triangle $ABC$ is tangent to side $BC$ at $D$ and intersects the perpendicular bisector of segment $BC$ at distinct points $X$ and $Y$. Lines $AX$ and $AY$ intersect line $BC$ at $P$ and $Q$, respectively. Prove that, if $DP \cdot DQ = (AC-AB)^2$ then $AB + AC = 3BC.$

In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.

Let $ n\ge 3$ distinct non-collinear points be given on a plane. Show that there is a closed simple polygonal line passing through each point.

In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the diagonals $AC$ and $BD$.

Compute $$\sum_{n=1}^{\infty} \frac{2^{n+1}}{8 \cdot 4^n - 6 \cdot 2^n +1}$$

Suppose that $ \omega, a,b,c$ are distinct real numbers for which there exist real numbers $ x,y,z$ that satisfy the following equations: $ x\plus{}y\plus{}z\equal{}1,$ $ a^2 x\plus{}b^2 y \plus{}c^2 z\equal{}\omega ^2,$ $ a^3 x\plus{}b^3 y \plus{}c^3 z\equal{}\omega ^3,$ $ a^4 x\plus{}b^4 y \plus{}c^4 z\equal{}\omega ^4.$ Express $ \omega$ in terms of $ a,b,c$.

A binoku is a $9 \times 9$ grid that is divided into nine $3 \times 3$ subgrids with the following properties: - each cell contains either a $0$ or a $1$, - each row contains at least one $0$ and at least one $1$, - each column contains at least one $0$ and at least one $1$, and - each of the nine subgrids contains at least one $0$ and at least one $1$. An incomplete binoku is obtained from a binoku by removing the numbers from some of the cells. What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way? [i]Proposed by Stijn Cambie, South Korea[/i]

Fernando has six coins, one of which is fake. We do not know what the weight of a fake coin is or the weight of a real coin, we only know that real coins all have the same weight and that the weight of the fake coin is different. Using a two-pan scale, show that it is possible to discover the fake coin using just $3$ weighings.

If $x$ and $y$ are positive real numbers such that $\sqrt(2x)+\sqrt(y)=13$ and $\sqrt(8x)+\sqrt(9y)=35$, calculate $20x+23y$.

Quadrilateral $ABCD$ is inscribed into a circle with center $O$. The bisectors of its angles form a cyclic quadrilateral with circumcenter $I$, and its external bisectors form a cyclic quadrilateral with circumcenter $J$. Prove that $O$ is the midpoint of $IJ$.

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.

In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$. $\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$ $\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$ $\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$ $\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$

If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x)$ is divisible by $ p(x)\minus{}x$

Suppose there are $n$ points on the plane, no three of which are collinear. Draw $n-1$ non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a [i]network[/i]. Given a network, we may assign labels from $1$ to $n-1$ to each segment such that each segment gets a different label. Define a [i]spin[/i] as the following operation: $\bullet$ Choose a point $v$ and rotate the labels of its adjacent segments clockwise. Formally, let $e_1,e_2,\cdots,e_k$ be the segments which contain $v$ as an endpoint, sorted in clockwise order (it does not matter which segment we choose as $e_1$). Then, the label of $e_{i+1}$ is replaced with the label of $e_{i}$ simultaneously for all $1 \le i \le k$. (where $e_{k+1}=e_{1}$) A network is [i]nontrivial[/i] if there exists at least $2$ points with at least $2$ adjacent segments each. A network is [i]versatile[/i] if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers $n \ge 5$ such that any nontrivial network with $n$ points is versatile. [i]Proposed by Yeoh Zi Song[/i]

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

How many real numbers $x$ are solutions to the following equation? \[ |x-1| = |x-2| + |x-3| \]

Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then \[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]

A mushroom is called [i]bad [/i] if it contains at least $10$ worms. A basket contains $90$ bad and $10$ good mushrooms. Can all mushrooms become good after some worms crawl from bad mushrooms to good ones? [hide=original wording]Гриб называется плохим, если в нем не менее 10 червей. В лукошке 90 плохих и 10 хороших грибов. Могут ли все грибы стать хорошими после того, как некоторые черви переползут из плохих грибов в хорошие?[/hide]

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]