Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle? (I. Bogdanov)

Let $ABC$ be a triangle and let $N$ and $M$ be the midpoints of $AB$ and $CA$, respectively. Let $H$ be the foot of altitude from $A$. The circumcircle of $ABH$ intersects $MN$ at $P$, with $P$ and $M$ on the same side relative to $N$, and the circumcircle of $ACH$ intersects $MN$ at $Q$, with $Q$ and $N$ on the same side relative to $M$. $BP$ and $CQ$ intersect at $X$. Prove that $AX$ is the angle bisector of $\angle CAB$.

In triangle $ABC$ the orthocenter is $H$ and the foot of the altitude from $A$ is $D$. Point $P$ satisfies $AP=HP$, and the line $PA$ is tangent to $(ABC)$. Line $PD$ intersects lines $AB, AC$ at points $X,Y$ respectively. Prove that $\angle YHX = \angle BAC$ or $\angle YHX+\angle BAC= 180^\circ$.

Find the units digit of $2^{2^{\iddots^{2}}}$, where there are $2024$ $2$s. [i]Individual #3[/i]

An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible? $\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\textbf{(E)}\mbox{ }9999$

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

If an angle $0^\circ<\theta<30^\circ$ satisfies $\sin\left(90^\circ-\theta\right)\sin\left(60^\circ-\theta\right)\sin\left(30^\circ-\theta\right)=\sin^3\left(\theta\right)$, compute $\sin\left(\theta\right)$. [i]2019 CCA Math Bonanza Lightning Round #4.4[/i]

Eight consecutive positive integers are divided into 2 sets, such that the sum of the squares of the elements in the first set is equal to the sum of the squares of the elements in the second set. Prove that the sum of the lements in the first set is equal to the sum of the elements in the second one.

Prove that there exists $N\in\mathbb{N}$ so that for all integer $n > N$, one may find $2019$ pairwise co-prime positive integers with \[n=a_1+a_2+\cdots+a_{2019}\] and \[2019<a_1<a_2<\cdots<a_{2019}\]

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

Point $D$ lies on the side $AB$ of triangle $ABC$. The circle inscribed in angle $ADC$ touches internally the circumcircle of triangle $ACD$. Another circle inscribed in angle $BDC$ touches internally the circumcircle of triangle $BCD$. These two circles touch segment $CD$ in the same point $X$. Prove that the perpendicular from $X$ to $AB$ passes through the incenter of triangle $ABC$

Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$

Consider two concentric circles $\Omega$ and $\omega$. Chord $AD$ of the circle $\Omega$ is tangent to $\omega$. Inside the minor disk segment $AD$ of $\Omega$, an arbitrary point $P{}$ is selected. The tangent lines drawn from the point $P{}$ to the circle $\omega$ intersect the major arc $AD$ of the circle $\Omega$ at points $B{}$ and $C{}$. The line segments $BD$ and $AC$ intersect at the point $Q{}$. Prove that the line segment $PQ$ passes through the midpoint of line segment $AD$. [i]Note.[/i] A circle together with its interior is called a disk, and a chord $XY$ of the circle divides the disk into disk segments, a minor disk segment $XY$ (the one of smaller area) and a major disk segment $XY$.

Compute the remainder when $2^{3^5}+ 3^{5^2}+ 5^{2^3}$ is divided by $30$.