Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.

Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$. [i](Proposed by Oleksii Masalitin)[/i]

Let $a_1,\ldots,a_{n+1}$ be positive real numbers satisfying $1/(a_1+1)+\cdots+1/(a_{n+1}+1)=n$. Prove that \[\sum_{i=1}^{n+1}\prod_{j\neq i}\sqrt[n]{a_j}\leqslant\frac{n+1}{n}.\]

Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x\plus{}y}$ is a prime number

Consider the nonconvex quadrilateral $ABCD$ with $\angle C>180$ degrees. Let the side $DC$ extended to meet $AB$ at $F$ and the side $BC$ extended to meet $AD$ at $E$. A line intersects the interiors of the sides $AB,AD,BC,CD$ at points $K,L,J,I$ respectively. Prove that if $DI=CF$ and $BJ=CE$, then $KJ=IL$

An unfair coin has probability $p$ of coming up heads on a single toss. Let $w$ be the probability that, in $5$ independent toss of this coin, heads come up exactly $3$ times. If $w = 144 / 625$, then $ \textbf{(A)}\ p\text{ must be }2/5$ $ \textbf{(B)}\ p\text{ must be }3/5$ $ \textbf{(C)}\ p\text{ must be greater than }3/5$ $ \textbf{(D)}\ p\text{ is not uniquely determined}$ $ \textbf{(E)}\ \text{there is no value of }p\text{ for which }w = 144/625$

In the beginnig, each square of a strip formed by $n$ adjacent squares contains $0$ or $1$. At each step, we are writing $1$ to the squares containing $0$ and to the squares having exactly one neighbour containing $1$, and we are writing $0$s into the other squares. Determine all possible values of $n$ such that whatever the initial arrangement of $0$ and $1$ is, after finite number of steps, all squares can turn into $0$.

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

In the plane are fixed two internally tangent circles $\omega$ and $\Omega$, so that $\omega$ is inside $\Omega$. Denote their common point by $T$. The point $A \neq T$ moves on $\Omega$ and point $B$ on $\Omega$ is such that $AB$ is tangent to $\omega$. The line through $B$, perpendicular to $AB$, meets the external angle bisector of $\angle ATB$ at $P$. Prove that, as $A$ varies on $\Omega$, the line $AP$ passes through a fixed point.

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]