Let $S$ be a set of $n$ points in the coordinate plane. Say that a pair of points is [i]aligned[/i] if the two points have the same $x$-coordinate or $y$-coordinate. Prove that $S$ can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most $n^{3/2}$ unordered pairs of distinct points in $S$ are aligned but not in the same subset.

Josh is older than Fred. Josh notices that if he switches the two digits of his age (an integer), he gets Fred’s age. Moreover, the difference between the squares of their ages is a square of an integer. How old are Josh and Fred?

Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove: \[ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} \] And determine when equality holds.

Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. [i]Proposed by Ivan Chan Kai Chin[/i]

Joey and his five brothers are ages $3,5,7,9,11,$ and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 13 $

An integer number $n \geq 2$ is called [i]feirense[/i] if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is. Find all the feirense numbers.

In a team of guards, each is assigned a different positive integer. For any two guards, the ratio of the two numbers assigned to them is at least $3:1$. A guard assigned the number $n$ is on duty for $n$ days in a row, off duty for $n$ days in a row, back on duty for $n$ days in a row, and so on. The guards need not start their duties on the same day. Is it possible that on any day, at least one in such a team of guards is on duty?

[b]a)[/b] Let $ AA',BB',CC' $ be the altitudes of a triangle $ ABC. $ Prove that $$ \frac{BC}{AA'}\cdot \overrightarrow{AA'} +\frac{AC}{BB'}\cdot \overrightarrow{BB'} +\frac{AB}{CC'}\cdot \overrightarrow{CC'} =0. $$ [b]b)[/b] The sum of the vectors that are perpendicular to the sides of a convex polygon and have equal lengths as those sides, respectively, is $ 0. $

In isosceles right triangle $ ABC$ a circle is inscribed. Let $ CD$ be the hypotenuse height ($ D\in AB$), and let $ P$ be the intersection of inscribed circle and height $ CD$. In which ratio does the circle divide segment $ AP$?

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day? $ \textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{9}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{4}{9}\qquad \textbf{(E)}\ \frac{5}{9}$

Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles were displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a+b+c \le 7$. At the end of the trip, where the odometer showed $cba$ miles. What is $a^2+b^2+c^2$? $ \textbf{(A) } 26 \qquad\textbf{(B) }27\qquad\textbf{(C) }36\qquad\textbf{(D) }37\qquad\textbf{(E) }41\qquad $

Let $x_1,x_2,\cdots,x_n$ be postive real numbers such that $x_1x_2\cdots x_n=1$ ,$S=x^3_1+x^3_2+\cdots+x^3_n$.Find the maximum of $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$

Positive integers $a, b, c$ satisfy $\frac{ab}{a - b} = c .$ what is the largest possible value of $a+ b+ c$ not exceeding $99$?

Let $a,b,c$ be nonnegative real numbers such that $(a+b)(b+c)(c+a) \neq0$. Find the minimum of $(a+b+c)^{2016}(\frac{1}{a^{2016}+b^{2016}}+\frac{1}{b^{2016}+c^{2016}}+\frac{1}{c^{2016}+a^{2016}})$.

Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called [i]beautiful[/i] if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$. Let $M$ be the number of beautiful labelings, and let N be the number of ordered pairs $(x, y)$ of positive integers such that $x + y \le n$ and $\gcd(x, y) = 1$. Prove that $$M = N + 1.$$

In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$? [asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,NE); label("$E$",E,NW); label("$F$",F,1.5*N); [/asy] $\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2$

In triangle $ABC$ let $A'$, $B'$, $C'$ respectively be the midpoints of the sides $BC$, $CA$, $AB$. Furthermore let $L$, $M$, $N$ be the projections of the orthocenter on the three sides $BC$, $CA$, $AB$, and let $k$ denote the nine-point circle. The lines $AA'$, $BB'$, $CC'$ intersect $k$ in the points $D$, $E$, $F$. The tangent lines on $k$ in $D$, $E$, $F$ intersect the lines $MN$, $LN$ and $LM$ in the points $P$, $Q$, $R$. Prove that $P$, $Q$ and $R$ are collinear.

Three real numbers $ a,b,c$ satisfy the equations $ a\plus{}b\plus{}c\equal{}3, a^2\plus{}b^2\plus{}c^2\equal{}9, a^3\plus{}b^3\plus{}c^3\equal{}24.$ Find $ a^4\plus{}b^4\plus{}c^4$.

A factorization of a positive integers is a way of writing it as a product of positive integers greater than $1$. Two factorizations are considered the same if they only differ in the order of terms in the product. For instance, $18$ has $4$ different factorizations: $18, 2\cdot 9, 3\cdot 6$ and $ 2\cdot 3\cdot 3$. For a positive integer $n$ we denote by $f(n)$ the number of distinct factorizations of $n$. By convention $f(1)=1$. Prove that $f(n)\leq n$ for all positive integers $n$.

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$ [b](a)[/b] Prove that $ABCD$ and $A''B''C''D''$ are similar. [b](b) [/b]The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

Let $ x_1,...,x_n$ be positive real numbers. Prove that: $ \displaystyle\sum_{k\equal{}1}^{n}x_k\plus{}\sqrt{\displaystyle\sum_{k\equal{}1}^{n}x_k^2} \le \frac{n\plus{}\sqrt{n}}{n^2} \left( \displaystyle\sum_{k\equal{}1}^{n} \frac{1}{x_k} \right) \left( \displaystyle\sum_{k\equal{}1}^{n} x_k^2 \right).$

Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees? $ \text{(A)}\ 90\qquad\text{(B)}\ 100\qquad\text{(C)}\ 105\qquad\text{(D)}\ 120\qquad\text{(E)}\ 140 $

Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is: $\textbf{(A) }4:3\qquad \textbf{(B) }3:2\qquad \textbf{(C) }7:4\qquad \textbf{(D) }78:71\qquad \textbf{(E) }\text{undetermined}$