Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$

Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\] then compute $\lfloor a^r b^p c^q \rfloor$. [i]Proposed by Michael Tang[/i]

Suppose that $ k,l $ are natural numbers such that $ \gcd (11m-1,k)=\gcd (11m-1, l) , $ for any natural number $ m. $ Prove that there exists an integer $ n $ such that $ k=11^nl. $

Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not contain any of the strings Dedalo bought. Dedalo’s aim is to trap the Minotaur. For instance, if Dedalo buys the strings $00$ and $11$ for a total of half a drachma, the Minotaur is able to escape using the infinite string $01010101 \ldots$. On the other hand, Dedalo can trap the Minotaur by spending $75$ cents of a drachma: he could for example buy the strings $0$ and $11$, or the strings $00, 11, 01$. Determine all positive integers $c$ such that Dedalo can trap the Minotaur with an expense of at most $c$ cents of a drachma.

A regular polygon of $ n$ sides ($ n\geq3$) has its vertex numbered from 1 to $ n$. One draws all the diagonals of the polygon. Show that if $ n$ is odd, it is possible to assign to each side and to each diagonal an integer number between 1 and $ n$, such that the next two conditions are simultaneously satisfied: (a) The number assigned to each side or diagonal is different to the number assigned to any of the vertices that is endpoint of it. (b) For each vertex, all the sides and diagonals that have it as an endpoint, have different number assigned.

There are three video game systems: the Paystation, the WHAT, and the ZBoz2$\pi$, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2$\pi$, and Dirac owns a ZBoz2$\pi$ and a Paystation. A store sells $4$ different games for the Paystation, $6$ different games for the WHAT, and $10$ different games for the ZBoz2$\pi$. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys $3$ random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?

Let $ABCD$ be a rectangle. Points $E$ and $F$ are on diagonal $AC$ such that $F$ lies between $A$ and $E$; and $E$ lies between $C$ and $F$. The circumcircle of triangle $BEF$ intersects $AB$ and $BC$ at $G$ and $H$ respectively, and the circumcircle of triangle $DEF$ intersects $AD$ and $CD$ at $I$ and $J$ respectively. Prove that the lines $GJ, IH$ and $AC$ concur at a point.

This test and the matching AMC 12P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.

Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.

Let $\Gamma_B$ and $\Gamma_C$ be excircles of an acute-angled triangle $ABC$ opposite to its vertices $B$ and $C$, respectively. Let $C_1$ and $L$ be the tangent points of $\Gamma_C$ and the side $AB$ and the line $BC$ respectively. Let $B_1$ and $M$ be the tangent points of $\Gamma_B$ and the side $AC$ and the line $BC$, respectively. Let $X$ be the point of intersection of the lines $LC_1$ and $MB_1$. Prove that $AX$ is equal to the inradius of the triangle $ABC$. (A. Voidelevich)

Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$ [i](-InnoXenT-)[/i]

Rectangle $ABCD$ has an area of 30. Four circles of radius $r_1 = 2$, $r_2 = 3$, $r_3 = 5$, and $r_4 = 4$ are centered on the four vertices $A$, $B$, $C$, and $D$ respectively. Two pairs of external tangents are drawn for the circles at A and $C$ and for the circles at $B$ and $D$. These four tangents intersect to form a quadrilateral $W XY Z$ where $\overline{W X}$ and $\overline{Y Z}$ lie on the tangents through the circles on $A$ and $C$. If $\overline{W X} + \overline{Y Z} = 20$, find the area of quadrilateral $W XY Z$. [img]https://cdn.artofproblemsolving.com/attachments/5/a/cb3b3457f588a15ffb4c875b1646ef2aec8d11.png[/img]

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Let $ABC$ be a triangle with incentre $I$. The incircle of the triangle $ABC$ touches the sides $AC$ and $AB$ at points $E$ and $F$ respectively. Let $\ell_B$ and $\ell_C$ be the tangents to the circumcircle of $BIC$ at $B$ and $C$ respectively. Show that there is a circle tangent to $EF, \ell_B$ and $\ell_C$ with centre on the line $BC$. [i]Proposed by Navneel Singhal, India[/i]

Evaluate $$lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} ([\frac{2n}{k}] -2[\frac{n}{k}])$$ and express your answer in the form $\log a-b,$ with $a$ and $b$ positive integers. Here $[x]$ is defined to be the integer such that $[x] \leq x <[x]+1$ and $\log x$ is the logarithm of $x$ to base $e.$

Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$. [i](Karl Czakler)[/i]

Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Let $H_a, H_b, H_c$ be the orthocenter of triangles $TBC, TCA, TAB$, respectively. a) Prove that: $T$ is the centroid of the $\triangle H_aH_bH_c$. b) Denote $D, E, F$ respectively by the intersections of $H_cH_b$ and the segment $BC$, $H_cH_a$ and the segment $CA$, $H_aH_b$ and the segment $AB$. Prove that: the triangle $DEF$ is equilateral. c) Prove that: the lines passing through $D, E, F$ and are respectively perpendicular to $BC, CA, AB$ are concurrent at a point. Let that point be $S$. d) Prove that: $TS$ is parallel to the [i]Euler[/i] line of the triangle $ABC$.

Let $x,y,z$ be positive numbers such that \[ {1\over x}+{1\over y}+{1\over z}=1. \] Show that \[ \sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z} \]

Let $ABCDEF$ be a regular hexagon with side length $2$. Point $M$ is the midpoint of diagonal $AE$. The pentagon $ABCDE$ is folded along segments $BD$, $BM$, and $DM$ in such a way that points $A$, $C$, and $E$ coincide. As a result of this operation, a tetrahedron is obtained. Determine its volume.

Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

Solve the system \[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\] in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.

Find the value of $x$ such that $2^{x+3} - 2^{x-3} = 2016$.

Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

$A$ and $B$ are two points in the plane $\alpha$, and line $r$ passes through points $A, B$. There are $n$ distinct points $P_1, P_2, \ldots, P_n$ in one of the half-plane divided by line $r$. Prove that there are at least $\sqrt n$ distinct values among the distances $AP_1, AP_2, \ldots, AP_n, BP_1, BP_2, \ldots, BP_n.$