Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

What is the sum of distinct real numbers $x$ such that $(2x^2+5x+9)^2=56(x^3+1)$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ \dfrac{7}{4} \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \dfrac{9}{2} \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ABC$ be an acute triangle with orthocenter $H$ and let $M$ be the midpoint of $\overline{BC}$. (The [i]orthocenter[/i] is the point at the intersection of the three altitudes.) Denote by $\omega_B$ the circle passing through $B$, $H$, and $M$, and denote by $\omega_C$ the circle passing through $C$, $H$, and $M$. Lines $AB$ and $AC$ meet $\omega_B$ and $\omega_C$ again at $P$ and $Q$, respectively. Rays $PH$ and $QH$ meet $\omega_C$ and $\omega_B$ again at $R$ and $S$, respectively. Show that $\triangle BRS$ and $\triangle CRS$ have the same area. [i]Proposed by Aaron Lin[/i]

Find the median value of $m$ over all integers $m$ where $|m^2 + 8m - 65|$ is a perfect power. A perfect power is any integer at least $2$ which can be written as $a^b$, where $a$, $b$ are integers and $b \ge 2$. [i]Individual #9[/i]

There are $36$ gangsters bands.And there are war between some bands. Every gangster can belongs to several bands and every 2 gangsters belongs to different set of bands. Gangster can not be in feuding bands. Also for every gangster is true, that every band, where this gangster is not in, is in war with some band, where this gangster is in. What is maximum number of gangsters in city?

Assume that real numbers $x$, $y$ and $z$ satisfy $x + y + z = 3$. Prove that $xy + xz + yz \leq 3$.

Let $I$ be a point on the bisector of angle $BAC$ of a triangle $ABC$. Points $M,N$ are taken on the respective sides $AB$ and $AC$ so that $\angle ABI=\angle NIC$ and $\angle ACI=\angle MIB$. Show that $I$ is the incenter of triangle $ABC$ if and only if points $M,N$ and $I$ are collinear.

Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that: \[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\] [hide=Edit.] The $77777^{\text{th}}$ topic in College Math :coolspeak: [/hide]

Does there exist an infinite set $ M$ of straight lines on the coordinate plane such that (i) no two lines are parallel, and (ii) for any integer point there is a line from $ M$ containing it?

$F_{n}$ denotes the Fibonacci Sequence where $F_{1}=0, F_{2}=1, F_{n}=F_{n-1}+F_{n-2},\ \forall \ n \geq 3$ Find$$\sum\limits_{n=3}^{\infty}\frac{18+999F_n}{F_{n-1}\times F_{n+1}}$$ [list=1] [*] 2016 [*] 2017 [*] 2018 [*] None of these [/list]

Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

Sets $A,B$ satisfy that $A\cup B=\{a_1,a_2,a_3\}$. If $A\neq B$, then $(A,B)$ is different from $(B,A)$. The number of such sets $(A,B)$ is $\text{(A)}8\qquad\text{(B)}9\qquad\text{(C)}26\qquad\text{(D)}27$

Find the value of $ x$ that satisfies the equation \[ 25^{\minus{}2}\equal{}\frac{5^{48/x}}{5^{26/x}\cdot25^{17/x}}. \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair one number is divisible by another.

Let $a$ be a positive real number. Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$

Does a convex heptagon exist which can be divided into 2011 equal triangles?

Let be some points on a plane, no three collinear. We associate a positive or a negative value to every segment formed by these. Prove that the number of points, the number of segments with negative associated value, and the number of triangles that has a negative product of the values of its sides, share the same parity.

In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$. [i]proposed by Mr.Etesami[/i]

Let $n$ be an integer $> 1$. In a circular arrangement of $n$ lamps $L_0, \cdots, L_{n-1}$, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \cdots$. If $L_{j-1}$ ($j$ is taken mod n) is ON, then $Step_j$ changes the status of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If $L_{j-1}$ is OFF, then $Step_j$ does not change anything at all. Show that: [i](a)[/i] There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again. [i](b)[/i] If $n$ has the form $2^k$, then all lamps are ON after $n^2 - 1$ steps. [i](c) [/i]If $n$ has the form $2^k +1$, then all lamps are ON after $n^2 -n+1$ steps.

Let $I$ be the incenter of triangle $ABC$, and let $A_1$, $B_1$, and $C_1$ be arbitrary points lying on segments $AI$,$BI$, and $CI$, respectively. The perpendicular bisectors of segments $AA_1$, $BB_1$, and $CC_1$ form triangles $A_2B_2C_2$. Prove that the circumcenter of triangle $A_2B_2C_2$ coincides with the circumcenter of triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$.

$ c$ is a positive integer. Consider the following recursive sequence: $ a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}$, for all $ n \in N$. Prove that all the terms of the sequence are positive integers.

In an acute-angled triangle $ ABC$, $ D$ is the foot of the altitude from $ A$, and $ P$ a point on segment $ AD$. The lines $ BP$ and $ CP$ meet $ AC$ and $ AB$ at $ E$ and $ F$ respectively. Prove that $ AD$ bisects the angle $ EDF$.

The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says: "Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers.'' How many possibilities are there for the set of truth-tellers on the U.S. Senate? [i]Proposed by James Lin[/i]

Let $p$ be a prime number, and $X$ be the set of cubes modulo $p$, including $0$. Denote by $C_2(k)$ the number of ordered pairs $(x, y) \in X \times X$ such that $x + y \equiv k \pmod p$. Likewise, denote by $C_3(k)$ the number of ordered pairs $(x, y, z) \in X \times X \times X$ such that $x + y + z \equiv k \pmod p$. Prove that there are integers $a, b$ such that for all $k$ not in $X$, we have \[ C_3(k) = a\cdot C_2(k) + b. \] [i]Proposed by Murilo Corato, Brazil.[/i]

Compute $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}.$