i) Simplify $\left(a-\frac{4ab}{a+b}+b\right): \left(\frac{a}{a+b}-\frac{b}{b-a}-\frac{2ab}{a^2-b^2}\right)$ ii) Simplify $\frac{2x^2-(3a+b)x+a^2+ab}{2x^2-(a+3b)x+ab+b^2}$

All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. [b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions. [b](b)[/b]Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.

A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a sphere as shown. The volume of the sphere is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [center][img]https://snag.gy/92ikv3.jpg[/img][/center]

Prove that for a triangle $ ABC $ with $ \angle BAC \ge 90^{\circ } , $ having circumradius $ R $ and inradius $ r, $ the following inequality holds: $$ R\sin A>2r $$ [i]Romeo Ilie[/i]

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

Let $n$ be a five digit number (whose first digit is non-zero) and let $m$ be the four digit number formed from $n$ by removing its middle digit. Determine all $n$ such that $n/m$ is an integer.

Let $x$ and $y$ be two real numbers. Prove that the equations \[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\] Holds if and only if at least one of $x$ or $y$ be integer.

Give a geometric proof of the statement that the fold line on a sheet of paper is straight.

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Let $\vartriangle ABC$ be an acute triangle, and let $P$ be the foot of altitude from $C$ to $AB$. Let $\omega$ be the circle with diameter $BC$. The tangents from $A$ to $\omega$ are drawn touching $\omega$ at $D$ and $E$. Lines $AD$ and $AE$ intersect line $BC$ at $M$ and $N$ respectively, so that $B$ lies between $M$ and $C$. Let $CP$ intersect $DE$ at $Q, ME$ intersect $ND$ at $R$, and let $QR$ intersect $BC$ at $S$. Show that $QS$ bisects $\angle DSE$

In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called [i]good[/i] if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and boundary of some unit square contains exactly $n$ of the points in the set on its interior or its boundary.

Let $f:\mathbb{R} \to \mathbb{R}$, such that i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$, exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$ ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$, exists $x,y \in \mathbb{R}$ with $ b - \epsilon < x < b < y < b + \epsilon$, such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$ Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$.

Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.

Let $\mathcal{S}$ be a set of $10$ points in a plane that lie within a disk of radius $1$ billion. Define a $move$ as picking a point $P \in \mathcal{S}$ and reflecting it across $\mathcal{S}$'s centroid. Does there always exist a sequence of at most $1500$ moves after which all points of $\mathcal{S}$ are contained in a disk of radius $10$? [i]Advaith Avadhanam[/i]

Let $AC$ be the longer of the two diagonals of the parallelogram $ABCD$. Drop perpendiculars from $C$ to $AB$ and $AD$ extended. If $E$ and $F$ are the feet of these perpendiculars, prove that $$AB \cdot AE + AD \cdot AF = (AC)^2.$$

Given that $x_{n+2}=\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\sum_{n=0}^\infty\dfrac{x_{3n}}{2^n}=\dfrac pq$ for some positive integers $p$, $q$ with $GCD(p,q)=1$. Find $p+q$.

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that $$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$

A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following two conditions: \[ \left\{ \begin{array}{ll} \mbox{If } f(0) = 0, \mbox{ then } f(x) \neq 0 \mbox{ for any non-zero } x. \\ \\ f(x + y)f(y + z)f(z + x) = f(x + y + z)f(xy + yz + zx) - f(x)f(y)f(z) \quad \forall x, y, z \in \mathbb{R}. \end{array} \right. \]

The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if: $ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$

a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected. b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.

In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.

A museum has the shape of a (not necessarily convex) 3$n$-gon. Prove that $n$ custodians can be positioned so as to control all of the museum’s space.