Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

Let the points $A,B,C$ lie on a line in this order. $AB$ is the diameter of semicircle $\omega_1$, $AC$ is the diameter of semicircle $\omega_2$. Assume both $\omega_1$ and $\omega_2$ lie on the same side of $AC$. $D$ is a point on $\omega_2$ such that $BD\perp AC$. A circle centered at $B$ with radius $BD$ intersects $\omega_1$ at $E$. $F$ is on $AC$ such that $EF\perp AC$. Prove that $BC=BF$.

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

$2023$ balls are divided into several buckets such that no bucket contains more than $99$ balls. We can remove balls from any bucket or remove an entire bucket, as many times as we want. Prove that we can remove them in such a way that each of the remaining buckets will have an equal number of balls and the total number of remaining balls will be at least $100$.

Let $m$, $n$ and $p$ are real numbers such that $\left(m+n+p\right)\left(\frac 1m + \frac 1n + \frac1p\right) =1$. Find all possible values of $$\frac 1{(m+n+p)^{2023}} -\frac 1{m^{2023}} -\frac 1{n^{2023}} -\frac 1{p^{2023}}.$$

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.

Prove that every positive integer can be represented in the form $$3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}$$where $m_1 > m_2 > \dots > m_k \geq 0$ and $0 \leq n_1 < n_2 < \dots < n_k$ are integers.

We are given $n$ intervals $[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]$ in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most $n-1$ pairs of overlapping intervals.

Let $A_1A_2\dots A_{2n}$ be a regular $2n$-gon inscribed in circle $\omega$. Let $P$ be any point on the circle $\omega$. Let $H_1,H_2,\dots, H_n$ be orthocenters of triangles $PA_1A_2, PA_3A_4,\dots, PA_{2n-1}A_{2n}$ respectively. Prove that $H_1H_2\dots H_n$ is a regular $n$-gon.

Joy has a square board of size $n \times n$. At every step, he colours a cell of the board. He cannot colour any cell more than once. He also counts points while colouring the cells. At first, he has $0$ points. Every step, after colouring a cell $c$, he takes the largest possible set $S$ that creates a "$+$" sign where all cells are coloured and $c$ lies in the centre. Then, he gets the size of set $S$ as points. After colouring the whole $n \times n$ board, what is the maximum possible amount of points he can get?

Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

Let {$a_1, a_2,\cdots,a_n$} be a set of $n$ real numbers whos sym equals S. It is known that each number in the set is less than $\frac{S}{n-1}$. Prove that for any three numbers $a_i$, $a_j$ and $a_k$ in the set, $a_i+a_j>a_k$.

For any positive integer $n$, define $f(n)$ to be the smallest positive integer that does not divide $n$. For example, $f(1)=2$, $f(6)=4$. Prove that for any positive integer $n$, either $f(f(n))$ or $f(f(f(n)))$ must be equal to $2$.

Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$, such that $AB||CD$. Let $P$ be a point on the circle $\omega$. Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle.

Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration: $$ \int_{0}^{\infty} f(x) \,dx $$

Let $\Delta ABC$ be an acute triangle and $\omega$ be its circumcircle. Perpendicular from $A$ to $BC$ intersects $BC$ at $D$ and $\omega$ at $K$. Circle through $A$, $D$ and tangent to $BC$ at $D$ intersect $\omega$ at $E$. $AE$ intersects $BC$ at $T$. $TK$ intersects $\omega$ at $S$. Assume, $SD$ intersects $\omega$ at $X$. Prove that $X$ is the reflection of $A$ with respect to the perpendicular bisector of $BC$.

Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.

Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$, where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$