Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \] $ \mathbb {R} ^ + $ denotes the set of positive real numbers.

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

Let $m$ be a positive integer. A triangulation of a polygon is [i]$m$-balanced[/i] if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$-balanced triangulation of a regular $n$-gon. [i]Note[/i]: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior. [i]Proposed by Krit Boonsiriseth[/i]

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$

Solve the system of equation for $(x,y) \in \mathbb{R}$ $$\left\{\begin{matrix} \sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\ 3x^2+4xy=24 \end{matrix}\right.$$ \\ Explain your answer

Two positive integers are called anagrams if every decimal digit occurs the same number of times in each of them (not counting the leading zeroes). Find all non-constant polynomials $P$ with non-negative integer coefficients so that whenever $a$ and $b$ are anagrams, $P(a)$ and $P(b)$ are anagrams as well. Proposed by Sutanay Bhattacharya

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.

A sequence $a_0,a_1,a_2,...,a_n,...$ is such that $a_0=1$ and, for each $n\ge 0$ , $a_{n+1}=m \cdot a_n$ , where $m$ is an integer between $2$ and $9$ inclusive. Also, every integer between $2$ and $9$ has even been used at least once to get $a_{n+1} $ from $a_n$ . Let $Sn$ the sum of the digits of $a_n$ , $n=0,1,2,...$ . Prove that $S_n \ge S_{n+1}$ for infinite values ​​of $n$.

Determine all functions $f : R \to R$ satisfying $f(f(x) - y) \le xf(x) + f(y)$ for all real numbers $x, y$.

$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets.

$WHY$ is a triangle with angle $W \ge 90$ degrees. On the side $HY$, two distinct points $M$ and $E$ are chosen such that angle $HWM$ is equivalent to angle $ MWE$ and $HM * YE = HY * ME$. Find the angle $MWY$.

Find all the real numbers $a$ and $b$ that satisfy the relation $2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)$

Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

Let $ G$ and $ H$ be countable Abelian $ p$-groups ($ p$ an arbitrary prime). Suppose that for every positive integer $ n$, \[ p^nG \not\equal{} p^{n\plus{}1}G .\] Prove that $ H$ is a homomorphic image of $ G$. [i]M. Makkai[/i]

In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the remaining 8 socks at random and on Wednesday two of the reaining 6 socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

How many trailing zeros does the value \[300\cdot305\cdot310\dots1090\cdot1095\cdot1100\] end with?

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

8. Sebuah lantai luasnya 3 meter persegi ditutupi lima buah karpet dengan ukuran masing-masing 1 meter persegi. Buktikan bahwa ada dua karpet yang tumpang tindih dengan luas tumpang tindih minimal 0,2 meter persegi. A floor of a certain room has a $ 3 \ m^2$ area. Then the floor is covered by 5 rugs, each has an area of $ 1 \ m^2$. Prove that there exists 2 overlapping rugs, with at least $ 0.2 \ m^2$ covered by both rugs.

Beter Pai wants to tell you his fastest $40$-line clear time in Tetris, but since he does not want Qep to realize she is better at Tetris than he is, he does not tell you the time directly. Instead, he gives you the following requirements, given that the correct time is t seconds: $\bullet$ $t < 100$. $\bullet$ $t$ is prime. $\bullet$ $t -1$ has 5 proper factors. $\bullet$ all prime factors of $t +1$ are single digits. $\bullet$ $t -2$ is a multiple of $3$. $\bullet$ $t +2$ has $2$ factors. Find t.

The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$. Show that \[ f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0 \]

p1. Nurbaya's rectangular courtyard will be covered by a number of paving blocks in the form of a regular hexagon or its pieces like the picture below. The length of the side of the hexagon is $ 12$ cm. [img]https://cdn.artofproblemsolving.com/attachments/6/1/281345c8ee5b1e80167cc21ad39b825c1d8f7b.png[/img] Installation of other paving blocks or pieces thereof so that all fully covered page surface. To cover the entire surface The courtyard of the house required $603$ paving blocks. How many paving blocks must be cut into models $A, B, C$, and $D$ for the purposes of closing. If $17$ pieces of model $A$ paving blocks are needed, how many the length and width of Nurbaya's yard? Count how much how many pieces of each model $B, C$, and $D$ paving blocks are used. p2. Given the square $PQRS$. If one side lies on the line $y = 2x - 17$ and its two vertices lie on the parabola $y = x^2$, find the maximum area of possible squares $PQRS$ . p3. In the triangular pyramid $T.ABC$, the points $E, F, G$, and $H$ lie at , respectively $AB$, $AC$, $TC$, and $TB$ so that $EA : EB = FA : FC = HB : HT = GC : GT = 2:1$. Determine the ratio of the volumes of the two halves of the divided triangular pyramid by the plane $EFGH$. p4. We know that $x$ is a non-negative integer and $y$ is an integer. Define all pair $(x, y)$ that satisfy $1 + 2^x + 2^{2x + 1} = y^2$. p5. The coach of the Indonesian basketball national team will select the players for become a member of the core team. The coach will judge five players $A, B, C, D$ and $E$ in one simulation (or trial) match with total time $80$ minute match. At any time there is only one in five players that is playing. There is no limit to the number of substitutions during the match. Total playing time for each player $A, B$, and $C$ are multiples of $5$ minutes, while the total playing time of each players $D$ and $E$ are multiples of $7$ minutes. How many ways each player on the field based on total playing time?

Given that \[a^2+b^2+c^2+d^2+\frac{5}{4}=e+\sqrt{a+b+c+d-e},\] the value of $a+b+c+d+e$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Individual #12[/i]