In a perpendicular triangle the perimeter is 60 and the altitude on the hypotenuse is 12. Then, the length of the hypotenuse is $ \text{(A)}\ 24 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 28$

Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively. Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles $(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$. [i]Proposed by Ivan Chai, Malaysia.[/i]

The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one. If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!! Show that, we can repeat this operation until we have all $1996$ numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!!

Let $d_i(k)$ the number of divisors of $k$ greater than $i$. Let $f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)$. Find all $n \in N$ such that $f(n)$ is a perfect square.

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

Given $0 \le x_0 <1$, let \[ x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} \] for all integers $n>0$. For how many $x_0$ is it true that $x_0=x_5$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{infinitely many} $

Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below:

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

Let $AB\varGamma\varDelta$ be a rhombus. (a) Prove that you can construct a circle $(c)$ which is inscribed in the rhombus and is tangent to its sides. (b) The points $\varTheta,H,K,I$ are on the sides $\varDelta\varGamma,B\varGamma,AB,A\varDelta$ of the rhombus respectively, such that the line segments $KH$ and $I\varTheta$ are tangent on the circle $(c)$. Prove that the quadrilateral defined from the points $\varTheta,H,K,I$ is a trapezium.

How many real solutions does the equation $ x^{3} 3^{1/x^{3} } \plus{}\frac{1}{x^{3} } 3^{x^{3} } \equal{}6$ have? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None}$

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

Let $f(x)$ be a polynomial, such that $f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}$. Velly and Polly are taking turns, starting from Velly changing the coefficients $a_i$ with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?

For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]

In a school there are $ 10$ rooms. Each student from a room knows exactly one student from each one of the other $ 9$ rooms. Prove that the rooms have the same number of students (we suppose that if $ A$ knows $ B$ then $ B$ knows $ A$).

Given a triangle $ABC$ where $AC > BC$, $D$ is located on the circumcircle of $ABC$ such that $D$ is the midpoint of the arc $AB$ that contains $C$. $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$. Prove that $AE = EC + CB$.

Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$

Let $ABCD$ be a cyclic quadrilateral. Let $E$ be a point on side $BC,$ $F$ be a point on side $AE,$ $G$ be a point on the exterior angle bisector of $\angle BCD,$ such that $EG=FG,$ $\angle EAG=\dfrac12\angle BAD.$ Prove that $AB\cdot AF=AD\cdot AE.$

A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up $36\%$ of the area of the flag, what percent of the area of the flag is blue? [asy] draw((0,0)--(5,0)--(5,5)--(0,5)--cycle); draw((1,0)--(5,4)); draw((0,1)--(4,5)); draw((0,4)--(4,0)); draw((1,5)--(5,1)); label("RED", (1.2,3.7)); label("RED", (3.8,3.7)); label("RED", (1.2,1.3)); label("RED", (3.8,1.3)); label("BLUE", (2.5,2.5)); [/asy] $ \textbf{(A)}\ 0.5 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6 $

The first four terms of an arithmetic sequence are $ p,9,3p\minus{}q,$ and $ 3p\plus{}q$. What is the $ 2010^{\text{th}}$ term of the sequence? $ \textbf{(A)}\ 8041\qquad \textbf{(B)}\ 8043\qquad \textbf{(C)}\ 8045\qquad \textbf{(D)}\ 8047\qquad \textbf{(E)}\ 8049$

An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.

If $10\%$ of $\left(x+10\right)$ is $\left(x-10\right)$, what is $10\%$ of $x$? $\text{(A) }\frac{11}{90}\qquad\text{(B) }\frac{9}{11}\qquad\text{(C) }1\qquad\text{(D) }\frac{11}{9}\qquad\text{(E) }\frac{110}{9}$

Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make? [i]Proposed by Anthony Yang[/i]

An $n\times n$ table whose cells are numerated with numbers $1, 2,\cdots, n^2$ in some order is called [i]Naissus[/i] if all products of $n$ numbers written in $n$ [i]scattered[/i] cells give the same residue when divided by $n^2+1$. Does there exist a Naissus table for $(a) n = 8;$ $(b) n = 10?$ ($n$ cells are [i]scattered[/i] if no two are in the same row or column.) [i]Proposed by Marko Djikic[/i]

A square and a line intersect at a $45^o$ angle. The line bisects the square into two unequal pieces such that the area of one piece is twice that of the other. If the square has side length $6$, compute the length of the cut due to the line. [img]https://cdn.artofproblemsolving.com/attachments/6/4/2eb33fb9766497d25d342001cdbae9a7ffd4b4.png[/img]