Functions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2n $ and $g(n+1) = 2g(n) + 3 f(n) + 5$. Find $f(2005) - g(2005)$.

Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.

A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$? [asy]unitsize(2mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B=(0,0), A=(-10,0), C=20*dir(50); draw(A--B--C); draw(A--C,linetype("4 4")); dot(A); dot(B); dot(C); label("$10$",midpoint(A--B),S); label("$20$",midpoint(B--C),SE); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE);[/asy]$ \textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]$ $ \textbf{(E)}\ [800,900]$

Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds: \[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\] where $S=\sum_{i=1}^n x_i$.

Let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$ which does not lie on either diagonal. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $ meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.

Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that: \[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\] [i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]

Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality \[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$ and let $(c)$ be its circumcircle with center $O$. Let $M$ be the midpoint of $BC$. The line $AM$ meets the circle $(c)$ again at the point $D$. The circumcircle $(c_1)$ of triangle $\triangle MDC$ intersects the line $AC$ at the points $C$ and $I$, and the circumcircle $(c_2)$ of $\triangle AMI$ intersects the line $AB$ at the points $A$ and $Z$. If $N$ is the foot of the perpendicular from $B$ on $AC$, and $P$ is the second point of intersection of $ZN$ with $(c_2)$, prove that the quadrilateral with vertices the points $N, P, I$ and $M$ is a parallelogram.

Let $ABC$ be an isosceles triangle with $AB=AC$. Circle $\Gamma$ lies outside $ABC$ and touches line $AC$ at point $C$. The point $D$ is chosen on circle $\Gamma$ such that the circumscribed circle of the triangle $ABD$ touches externally circle $\Gamma$. The segment $AD$ intersects circle $\Gamma$ at a point $E$ other than $D$. Prove that $BE$ is tangent to circle $\Gamma$ .

If $(2^x - 4^x) + (2^{-x} - 4^{-x}) = 3$, find the numerical value of the expression $$(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).$$

Find all relatively distinct integers $m, n, p\in \mathbb{Z}_{\ne 0}$ such that the polynomial $$F(x) = x(x - m)(x - n)(x - p) + 1$$is reducible in $\mathbb{Z}[x].$

Determine all real numbers $x>1$ such that \[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \] for any positive integer $n$.

A chessboard is covered with $32$ dominos. Each domino covers two adjacent squares. Show that the number of horizontal dominos with a white square on the left equals the number with a white square on the right.

A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]

Let $n\geq 3$ be an integer. Find the number of functions $f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that \[ f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 . \]

Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawig the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies on between the parallel lines (excluding the intersection points on the lines) ? $ \textbf{(A)}\ 5600 \qquad\textbf{(B)}\ 5650 \qquad\textbf{(C)}\ 6500 \qquad\textbf{(D)}\ 6560 \qquad\textbf{(E)}\ 6600 $

Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.

Source: 1976 Euclid Part A Problem 8 ----- Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is $\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E) } \text{none of these}$

Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a,\, H_b,\, H_c$ be the projections of $A,\, B,\, C$ on respective opposite sides and $D,\, E,\, F$ be the midpoints of $BC,\, CA,\, AB$ in that order. $\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}$ intersect $(O)$ at $X,\,Y,\,Z$ respectively. a. Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$ b. Let $M,\, N,\, P$ be the midpoints of $AX,\, BY,\, CZ$ respectively. Prove that $\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.

Compute the remainder when $98!$ is divided by $101$.

Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.

A league consists of $2024$ players. A [i]round[/i] involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called [i]balanced[/i] if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of $N$ rounds. Prove that the committee can organise a tournament this year with $N$ balanced rounds. [i]Proposed by Anant Mudgal and Navilarekallu Tejaswi[/i]