An increasing function $f : N \to R$ satisfies $$f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N.$$ Show that there is a real number $p > 1$ such that $f(n) =\ log_pn$ for all $n$.

Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

Kevin is in first grade, so his teacher asks him to calculate $20+1\cdot 6+k$, where $k$ is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates $(20+1)\cdot (6+k)$. Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his answer? [i]Proposed by James Lin[/i]

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.

The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find $$\max\{a+b-c,a-b+c,-a+b+c\}.$$

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 $(0,+\infty)$

Suppose that $ y \equal{} \frac34x$ and $ x^y \equal{} y^x$. The quantity $ x \plus{} y$ can be expressed as a rational number $ \frac{r}{s}$, where $ r$ and $ s$ are relatively prime positive integers. Find $ r \plus{} s$.

Find the integer solutions of the equation \[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]

Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$

This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url]. Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.

Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9.

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]

If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ and $a_1+a_2+\cdots+a_n=1$, then prove: \[a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1\]

Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.

Let $n$ be a positive integer, and let $a_1,a_2,..,a_n$ be pairwise distinct positive integers. Show that $$\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4,$$ where $[a_1,a_2,…,a_k]$ is the least common multiple of the integers $a_1,a_2,…,a_k$.

Find all positive integers $k$, so that there are only finitely many positive odd numbers $n$ satisfying $n~|~k^n+1$.

Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?

Let $n\geq 2$ be a positive integer. Prove that there exists a positive integer $m$, such that $n\mid m, \ m<n^4$ and at most four distinct digits are used in the decimal representation of $m$.

Let $(G,\cdot)$ be a group, and let $H_1,H_2$ be proper subgroups s.t. $H_1\cap H_2=\{e\}$, where $e$ is the identity element of $G$. They also have the following properties: [b]i)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_1\setminus\{e\}\Rightarrow xy\in H_2$ [b]ii)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_2\setminus\{e\}\Rightarrow xy\in H_1$ Prove that: [b]a)[/b] $|H_1|=|H_2|$ [b]b)[/b] $|G|=|H_1|\cdot |H_2|$

The graphs of $ 2x\plus{}3y\minus{}6\equal{}0$, $ 4x\minus{}3y\minus{}6\equal{}0$, $ x\equal{}2$, and $ y\equal{}\frac{2}{3}$ intersect in: $ \textbf{(A)}\ \text{6 points} \qquad \textbf{(B)}\ \text{1 point} \qquad \textbf{(C)}\ \text{2 points} \qquad \textbf{(D)}\ \text{no points} \\ \textbf{(E)}\ \text{an unlimited number of points}$

$f(x)=x^2+x$ $b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$ Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$

Let $a_n=1+n!(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!})$ for any $n\in \mathbb{Z}^{+}$. Consider $a_n$ points in the plane,no $3$ of them collinear.The segments between any $2$ of them are colored in one of $n$ colors. Prove that among them there exist $3$ points forming a monochromatic triangle.

Abolf is on the second step of a stairway to heaven in every step of this stairway except the first one which is the hell there is a devil who is either a human, an elf or a demon and tempts Abolf. The devil in the second step is Satan himself as one of three forms. Whenever an elf or a demon tries to tempt Abolf he resists and walks one step up but when a human tempts Abolf he is deceived and hence he walks one step down. However if Abolf is deceived by Satan for the first time he resists and does not fall down to hell but the second time he falls down to eternal hell. Every time a devil makes a temptation it changes its form from a human, an elf, a demon to an elf, a demon, a human respectively. Prove that Abolf passes each step after some time. [i]Proposed by Yaser Ahmadi Fouladi[/i]