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]

In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of $2: 1, 2, 4, 8, 16, ...$ Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form an arithmetic progression with common difference $r$ and whose first term is the favourite number of the mayor. For which positive integers $r$ is it always possible that, no matter the favourite number of the mayor, given any $m$ stations, there is a bus going by all of them? Proposed by [i]Savinien Kreczman and Martin Rakovsky, France[/i]

The cyclic octagon $ABCDEFGH$ has sides $a,a,a,a,b,b,b,b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH.$

Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered $70\%$ of the arithmetic, $40\%$ of the algebra, and $60\%$ of the geometry problems correctly, she did not pass the test because she got less than $60\%$ of the problems right. How many more problems would she have needed to answer correctly to earn a $60\%$ passing grade? $ \text{(A)}\ 1\qquad\text{(B)}\ 5\qquad\text{(C)}\ 7\qquad\text{(D)}\ 9\qquad\text{(E)}\ 11 $

Show that if natural numbers $a,b, p,q,r,s$ satisfy the conditions $$qr- ps = 1 \,\,\,\,\, and \,\,\,\,\, \frac{p}{q}<\frac{a}{b}<\frac{r}{s},$$ then $b \ge q+s.$

$\vartriangle ABC$ is a triangle with $AB < BC, \Gamma$ its circumcircle, $K$ the midpoint of the minor arc $CA$ of the circle $C$ and $T$ a point on $\Gamma$ such that $KT$ is perpendicular to $BC$. If $A',B'$ are the intouch points of the incircle of $\vartriangle ABC$ with the sides $BC,AC$, prove that the lines $AT,BK,A'B'$ are concurrent.

Two circles $\Gamma_1,\Gamma_2$ intersect at $A,B$. Through $B$ a straight line $\ell$ is drawn and $\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B)$. We are given $\ell_1\parallel AM$ is tangent to $\Gamma_1$ at $Q$. $QA\cap \Gamma_2=R\;(\neq A)$ and further $\ell_2$ is tangent to $\Gamma_2$ at $R$. Prove that: [list] [*]$\ell_2\parallel AK$ [*]$\ell,\ell_1,\ell_2$ have a common point.[/list]

Consider a circle centered at $O$ with radius $r$ and a line $\ell$ not passing through $O$. A grasshopper is jumping to and fro between the points of the circle and the line, the length of each jump being $r$. Prove that there are at most $8$ points for the grasshopper to reach.

Let $T$ be the answer to question $18$. Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$. Point $B$ lies on segment $ZO$, $O'$ lies on segment $OM$, and $E$ lies on segment $RM$ such that $BR = BE = EO'$, and $\angle BEO' = 90^o$. Compute $2(ZO + O'M + ER)$. PS. You had better calculate it in terms of $T$.

Let $n \neq 0$. For every sequence of integers \[ A = a_0,a_1,a_2,\dots, a_n \] satisfying $0 \le a_i \le i$, for $i=0,\dots,n$, define another sequence \[ t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n) \] by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

For $k=1,2 \dots, n$ let $z_k=x_k+iy_k,$ where the $x_k$ and $y_k$ are real and $i=\sqrt{-1}$. Let $r$ bet the absolute value of the real part of $$\pm \sqrt{z_1^2+z_2^2+\dots z_n^2}.$$ Prove that $r\leq |x_1|+|x_2|+ \dots +|x_n|.$

A positive integer $N$ is [i]interoceanic[/i] if its prime factorization $$N=p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}$$ satisfies $$x_1+x_2+\dots +x_k=p_1+p_2+\cdots +p_k.$$ Find all interoceanic numbers less than 2020.

Rectangle $ ABCD$ has area 2006. An ellipse with area $ 2006\pi$ passes through $ A$ and $ C$ and has foci at $ B$ and $ D$. What is the perimeter of the rectangle? (The area of an ellipse is $ \pi ab$, where $ 2a$ and $ 2b$ are the lengths of its axes.) $ \textbf{(A) } \frac {16\sqrt {2006}}{\pi} \qquad \textbf{(B) } \frac {1003}4 \qquad \textbf{(C) } 8\sqrt {1003} \qquad \textbf{(D) } 6\sqrt {2006} \qquad \textbf{(E) } \frac {32\sqrt {1003}}\pi$