Solve in integers the equation : $x^2y+y^2=x^3$

A ladder is leaning against a house with its lower end $15$ feet from the house. When the lower end is pulled $9$ feet farther from the house, the upper end slides $13$ feet down. How long is the ladder (in feet)?

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?

There are $n$ people with hats present at a party. Each two of them greeted each other exactly once and each greeting consisted of exchanging the hats that the two persons had at the moment. Find all $n \ge 2$ for which the order of greetings can be arranged in such a way that after all of them, each person has their own hat back.

$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$. [hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]

Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?

Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?

Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.

If $a679b$ is the decimal expansion of a number in base $10$, such that it is divisible by $72$, determine $a,b$.

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$ ${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?

Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?

The polynomial $x^{2n}+1+(x+1)^{2n}$ is not divisible by $x^2+x+1$ if $n$ equals $\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

If $A$ is the area of a triangle with perimeter $ 1$, what is the largest possible value of $A^2$?

Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_1, \ldots , a_n$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_1, \ldots , a_n$. [i]Proposed by James Lin.[/i]

How many neutrons are in $0.025$ mol of the isotope ${ }_{24}^{54}\text{Cr}$? $ \textbf{(A)}\hspace{.05in}1.5\times10^{22} \qquad\textbf{(B)}\hspace{.05in}3.6\times10^{23} \qquad\textbf{(C)}\hspace{.05in}4.5\times10^{23} \qquad\textbf{(D)}\hspace{.05in}8.1\times10^{23} \qquad $

In the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\frac{p}{q}$. Determine $p+q$. [i]2022 CCA Math Bonanza Lightning Round 3.2[/i]

Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps. While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana. How many points in the plane, apart from $S$, are marked with a banana by the end of the $6$ minutes.

Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

Alice and Bob now play a magic show. There are $101 $ different hats lie on the table and they form a circle. Firstly, Bob choose a positive integer $n$(Alice doesn’t know it). Then Bob puts a rabbit under one of the hats and Alice doesn’t know which hat contains the rabbit. Each time, she can choose a hat and see whether the rabbit is under the hat. If not, then Bob will move the rabbit from the current hat to the $n$th hat in a clockwise direction. They will repeat these steps until Alice find the rabbit. Prove that Alice can find the rabbit in $201$ steps.

A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12% $ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?

Prove that if a quadrilateral $ABCD$ can be cut into a finite number of parallelograms, then $ABCD$ is a parallelogram.

Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.