Given a positive integer $n$. A triangular array $(a_{i,j})$ of zeros and ones, where $i$ and $j$ run through the positive integers such that $i+j\leqslant n+1$ is called a [i]binary anti-Pascal $n$-triangle[/i] if $a_{i,j}+a_{i,j+1}+a_{i+1,j}\equiv 1\pmod{2}$ for all possible values $i$ and $j$ may take on. Determine the minimum number of ones a binary anti-Pascal $n$-triangle may contain.

Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

Consider the triangle $ ABC$ with $ m(\angle BAC \equal{} 90^\circ)$ and $ AC \equal{} 2AB$. Let $ P$ and $ Q$ be the midpoints of $ AB$ and $ AC$,respectively. Let $ M$ and $ N$ be two points found on the side $ BC$ such that $ CM \equal{} BN \equal{} x$. It is also known that $ 2S[MNPQ] \equal{} S[ABC]$. Determine $ x$ in function of $ AB$.

The given sequences are $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for every real $ t $ the sequence $$ (x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })$$ has the following property: there is a number $ k $ such that $ 1 \leq k \leq n $ and all non-zero terms of the sequence with indices less than $ k $ are of the same sign and all non-zero terms of the sequence with indices not less than $ k $ are the same sign.

A regular hexagon $ABCDEF$ has perimeter $12$. $AB$, $CD$, and $EF$ are all extended, and the intersections of the line segments form an equilateral triangle. Compute the perimeter of the triangle.

We are given five watches which can be winded forward. What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?

Let us consider an infinite grid plane as shown below. We start with 4 points $A$, $B$, $C$, $D$, that form a square. We perform the following operation: We pick two points $X$ and $Y$ from the currant points. $X$ is reflected about $Y$ to get $X'$. We remove $X$ and add $X'$ to get a new set of 4 points and treat it as our currant points. For example in the figure suppose we choose $A$ and $B$ (we can choose any other pair too). Then reflect $A$ about $B$ to get $A'$. We remove $A$ and add $A'$. Thus $A'$, $B$, $C$, $D$ is our new 4 points. We may again choose $D$ and $A'$ from the currant points. Reflect $D$ about $A'$ to obtain $D'$ and hence $A'$, $B$, $C$, $D'$ are now new set of points. Then similar operation is performed on this new 4 points and so on. Starting with $A$, $B$, $C$, $D$ can you get a bigger square by some sequence of such operations?

There is an $n \times n$ table with $n -1$ cells containing ones and the remaining cells containing zeros. You can do this with the table the following operation: select the tap hole, subtract from the number in this cell, one, and to all other numbers on the same line or in the same column as the selected cell, add one. Is it possible from of this table, using the specified operations, obtain a table in which all numbers are equal?

In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]

We say that each positive number $x$ has two sons: $x+1$ and $\frac{x}{x+1}$. Characterize all the descendants of number $1$.

Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$. [i]2010 Hirosaki University entrance exam[/i]

$A, B$ are $n \times n$ matrices such that $\text{rank}(AB-BA+I) = 1.$ Prove that $\text{tr}(ABAB)-\text{tr}(A^2 B^2) = \frac{1}{2}n(n-1).$

Each of $2n + 1$ students chooses a finite, nonempty set of consecutive integers . Two students are friends if they have chosen a common number. Everyone student is friends with at least $n$ other students. Show that there is a student who is friends with everyone else.

Let $x,y,z$ be positive real numbers . Prove: $\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}$

A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? $ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$

Find all positive integers $n$ such that $9^{n}-1$ is divisible by $7^n$.

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

Can you form six squares with nine matches? How about fourteen squares with eight matches? (It is assumed that all matches have equal length, and you cannot break any of them.)

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point. (b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

Given triangle $ ABC$ with $ C$ a right angle, show that the diameter of the incenter is $ a\plus{}b\minus{}c$, where $ a\equal{}BC$, $ b\equal{}CA$, and $ c\equal{}AB$.

Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

Suppose that the sides $AB$ and $DC$ of a convex quadrilateral $ABCD$ are not parallel. On the sides $BC$ and $AD$, pairs of points $(M,N)$ and $(K,L)$ are chosen such that $BM=MN=NC$ and $AK=KL=LD$. Prove that the areas of triangles $OKM$ and $OLN$ are different, where $O$ is the intersection point of $AB$ and $CD$.

How many two digit numbers have exactly $4$ positive factors? $($Here $1$ and the number $n$ are also considered as factors of $n. )$