Let $S$ be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that: $\textbf{(A) }\text{No member of }S\text{ is divisible by }2\qquad$ $\textbf{(B) }\text{No member of }S\text{ is divisible by }3\text{ but some member is divisible by }11\qquad$ $\textbf{(C) }\text{No member of }S\text{ is divisible by }3\text{ or }5\qquad$ $\textbf{(D) }\text{No member of }S\text{ is divisible by }3\text{ or }7\qquad$ $\textbf{(E) }\text{None of these}$

Given is a triangle $ABC$ with $\angle BAC=45$; $AD, BE, CF$ are altitudes and $EF \cap BC=X$. If $AX \parallel DE$, find the angles of the triangle.

There are three positive integers written on a blackboard every minute. You can pick two written numbers $a$ and $b$ and replace them with $a \cdot b$ and $|a-b|$. Prove that it is always possible to make two of the numbers zero.

Let $x$ be a real number. Suppose that there exist integers $a_0,a_1,\dots,a_n$, not all zero, such that \[\sum_{k=0}^n a_k\cos(kx)=\sum_{k=0}^na_k\sin(kx)=0.\] Characterize all possible values of $\cos x$. [i]Grisham Paimagam[/i]

Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.

Once upon a time there are $n$ pairs of princes and princesses who are in love with each other. One day a witch comes along and turns all the princes into frogs; the frogs can be distinguished by sight but the princesses cannot tell which frog corresponds to which prince. The witch tells the princesses that if any of them kisses the frog that corresponds to the prince very that she loves then that frog will immediately transform back into a prince. If each princess can stand kissing at most $k$ frogs, what is the maximum number of princes they can be sure to save? (The princesses may take turns kissing in any order, communicate with each other and vary their strategy for future kisses depending on information gained from past kisses.)

Determine all the two-digit numbers that satisfy the following condition: if we multiply their two digits, the result is equal to half the number. For example, $24$ does not satisfy the condition, because $2 \times 4 = 8$ and $8$ is not half of $24$.

In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\ If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\ Instead, if we exchange the questions scores, scoring D points for the easy questions and 3 for the hard ones, the total punctuation of the test is multiplied by $\frac{3}{2}$.\\ Knowing that the number of easy questions is 9 times bigger the number of hard questions, find the number of questions in this test.

(a) We say that a hyperplane $H$ that is given with this equation \[H=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n=b\}\] ($a=(a_1,\dots,a_n)\in \mathbb R^n$ and $b\in \mathbb R$ constant) bisects the finite set $A\subseteq \mathbb R^n$ if each of the two halfspaces $H^+=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n>b\}$ and $H^-=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n<b\}$ have at most $\lfloor \tfrac{|A|}{2}\rfloor$ points of $A$. Suppose that $A_1,\dots,A_n$ are finite subsets of $\mathbb R^n$. Prove that there exists a hyperplane $H$ in $\mathbb R^n$ that bisects all of them at the same time. (b) Suppose that the points in $B=A_1\cup \dots \cup A_n$ are in general position. Prove that there exists a hyperplane $H$ such that $H^+\cap A_i$ and $H^-\cap A_i$ contain exactly $\lfloor \tfrac{|A_i|}{2}\rfloor$ points of $A_i$. (c) With the help of part (b), show that the following theorem is true: Two robbers want to divide an open necklace that has $d$ different kinds of stones, where the number of stones of each kind is even, such that each of the robbers receive the same number of stones of each kind. Show that the two robbers can accomplish this by cutting the necklace in at most $d$ places.

a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the conditions $$x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y$$ for every $x,y\in A$ imply commutativity of $*$? b)a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the condition$$x*x*y=y $$ for every $x,y\in A$ implies commutativity of $*$? [i]Proposed by Paulius Drungilas, Arturas Dubickas (Vilnius University). [/i]

Marvin had a birthday on Tuesday, May $ 27$ in the leap year $ 2008$. In what year will his birthday next fall on a Saturday? $ \textbf{(A)}\ 2011 \qquad \textbf{(B)}\ 2012 \qquad \textbf{(C)}\ 2013 \qquad \textbf{(D)}\ 2015 \qquad \textbf{(E)}\ 2017$

Estimate to determine which of the following numbers is closest to $\frac{401}{.205}$. $\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000$

Let $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ be functions with the property that $$ f\left( g(x) \right) =g\left( f(x) \right) =-x,\quad\forall x\in\mathbb{R} $$ [b]a)[/b] Show that $ f,g $ are odd. [b]b)[/b] Give a concrete example of such $ f,g. $

Find all pairs $ (x; y) $ of positive integers such that $$xy | x^2 + 2y -1.$$

A cylindrical glass beaker is $8$ cm high and its circumference rim is $12$ cm wide . Inside, $3$ cm from the edge, there is a tiny drop of honey. In a point on its outer surface, belonging to the plane passing through the axis of the cylinder and for the drop of honey, and located $1$ cm from the base (or bottom) of the glass, there is a fly. What is the shortest path that the fly must travel, walking on the surface from the glass, to the drop of honey, and how long is said path? [hide=original wording]Un vaso de vidrio cil´ındrico tiene 8 cm de altura y su borde 12 cm de circunferencia. En su interior, a 3 cm del borde, hay una diminuta gota de miel. En un punto de su superficie exterior, perteneciente al plano que pasa por el eje del cilindro y por la gota de miel, y situado a 1 cm de la base (o fondo) del vaso, hay una mosca. ¿Cu´al es el camino m´as corto que la mosca debe recorrer, andando sobre la superficie del vaso, hasta la gota de miel, y qu´e longitud tiene dicho camino?[/hide]

In the figure below, the centres of four squares have been connected by two line segments. Prove that these line segments are perpendicular.

The center of a circle, the radius of which is $r$, lies on the bisector of the right angle $A$ at a distance $a$ from its sides ($a > r$). A tangent to the circle intersects the sides of the angle at points $B$ and $C$. Find the smallest possible value of the area of triangle $ABC$.

Consider the intersection of an ellipsoid with a plane $ \sigma$ passing through its center $ O$. On the line through the point $ O$ perpendicular to $ \sigma$, mark the two points at a distance from $ O$ equal to the area of the intersection. Determine the loci of the marked points as $ \sigma$ runs through all such planes. [i]L. Tamassy[/i]

Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic. Proposed by [i]Benny Wang + Oron Wang[/i]

The line parallel to side $AC$ of a right triangle $ABC$ $(\angle C=90^\circ)$ intersects sides $AB$ and $BC$ at $M$ and $N$, respectively, so that the $CN / BN = AC / BC = 2$. Let $O$ be the intersection point of the segments $AN$ and $CM$ and $K$ be a point on the segment $ON$ such that $MO + OK = KN$. The perpendicular line to $AN$ at point $K$ and the bisector of triangle $ABC$ of $\angle B$ meet at point $T$. Find the angle $\angle MTB$.

Let $n$ be an integer greater than $1$ and let $S$ be a finite set containing more than $n+1$ elements.Consider the collection of all sets $A$ of subsets of $S$ satisfying the following two conditions : [b](a)[/b] Each member of $A$ contains at least $n$ elements of $S$. [b](b)[/b] Each element of $S$ is contained in at least $n$ members of $A$. Determine $\max_A \min_B |B|$ , as $B$ runs through all subsets of $A$ whose members cover $S$ , and $A$ runs through the above collection.

Initially there is a checker on every square of a $1\times n$ board. The first move consists of moving a checker to an adjacent square thus creating a stack of two checkers. Then each time when making a move, one can choose a stack and move it in either direction as many squares on the board as there are checkers in the stack. If after the move the stack lands on a non-empty square, it is placed on top of the stack which is already there. Prove that it is possible to stack all the checkers on one square in $n - 1$ moves. (A Shapovalov)

Let $P$ be a point in the interior of triangle $ABC$. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at $L$, $M$, $N$, respec­tively. Prove that $$AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.$$

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions: a)$f(m_{i})=-1\forall i=1,2,...,n$. b)$f(x)$ is irreducible.