We say that a finite set $S$ of red and green points in the plane is [i]separable[/i] if there exists a triangle $\delta$ such that all points of one colour lie strictly inside $\delta$ and all points of the other colour lie strictly outside of $\delta$. Let $A$ be a finite set of red and green points in the plane, in general position. Is it always true that if every $1000$ points in $A$ form a separable set then $A$ is also separable?

Let $P$ be a convex polygon, so have all interior angles smaller than $180^o$, and let $X$ be a point in the interior of $P$. Prove that $P$ has a side $[AB]$ such that the perpendicular from $X$ to the line $AB$ lies on the side $[AB]$.

Find the number of trailing $0$s in the base $12$ expression of $99!$ (Note: $99$ is in base $10$).

Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent. (S. Arkhipov)

Find all positive integers $k$, so that there are only finitely many positive odd numbers $n$ satisfying $n~|~k^n+1$.

Prove that $\sqrt 3$ is irrational.

Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\). [i]Proposed by Karthik Vedula[/i]

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

In a rectangle triangle, let $I$ be its incenter and $G$ its geocenter. If $IG$ is parallel to one of the catheti and measures $10 cm$, find the lengths of the two catheti of the triangle.

Three parallel lines $\ell_1, \ell_2$ and $\ell_3$ of a plane are given such that the line $\ell_2$ has the same distance $a$ from $\ell_1$ and $\ell_3$. We put $5$ points $M_1, M_2, M_3,M_4$ and $M_5$ on the lines $\ell_1, \ell_2$ and $\ell_3$ in such a way that each line contains at least one point. Detennine the maximal number of isosceles triangles that are possible to be formed with vertices three of the points $M_1, M_2, M_3, M_4$ and $M_5$ in the following cases: (i) $M_1,M_2,M_3 \in \ell_2, M_4 \in \ell_1$ and $M_5 \in \ell_3$. (ii) $M_1,M_2 \in \ell_1, M_3,M_4 \in \ell_3$ and $M_5 \in \ell_2$.

(a) Divide the line segment $[0,1]$ into smaller black and white segments so that, for any polynomial $p(x)$ of degree no greater than two, the sum of increments of $p(x)$ along all the black segments is equal to that along the white ones. (The increment of $p(x)$ along the segment $[a, n]$ is the number $p(b) - p(a)$.) (b) Can this be done for all polynomials of degree no greater than $1995$? (Burkov)

And it came to pass that Jeb owned over a thousand chickens. So Jeb counted his chickens. And Jeb reported the count to Hannah. And Hannah reported the count to Joshua. And Joshua reported the count to Caleb. And Caleb reported the count to Rachel. But as fate would have it, Jeb had over-counted his chickens by nine chickens. Then Hannah interchanged the last two digits of the count before reporting it to Joshua. And Joshua interchanged the first and the third digits of the number reported to him before reporting it to Caleb. Then Caleb doubled the number reported to him before reporting it to Rachel. Now it so happens that the count reported to Rachel was the correct number of chickens that Jeb owned. How many chickens was that?

Let P be a point inside triangle ABC such that 3<ABP = 3<ACP = <ABC + <ACB. Prove that AB/(AC + PB) = AC/(AB + PC).

Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$ be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold. $$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$ Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.

Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD=\angle DAC$. Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$. Prove that lines $DS$ and $AT$ are parallel. [i]Michael Ren[/i]

Determine all polynomials $P(x)$ with real coefficients such that $P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.

For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$

Let $n$ be a given positive integer and $$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$ Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.

Let $a,b,c,d$ to be non negative real numbers satisfying $ab+ac+ad+bc+bd+cd=6$. Prove that \[\dfrac{1}{a^2+1} + \dfrac{1}{b^2+1} + \dfrac{1}{c^2+1} + \dfrac{1}{d^2+1} \ge 2\]

On a circle there are $2n+1$ points, dividing it into equal arcs ($n\ge 2$). Two players take turns to erase one point. If after one player's turn, it turned out that all the triangles formed by the remaining points on the circle were obtuse, then the player wins and the game ends. Who has a winning strategy: the starting player or his opponent?

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

For the quadrilateral $ABCD$, let $AC$ and $BD$ intersect at $E$, $AB$ and $CD$ intersect at $F$, and $AD$ and $BC$ intersect at $G$. Additionally, let $W, X, Y$, and $Z$ be the points of symmetry to $E$ with respect to $AB, BC, CD,$ and $DA$ respectively. Prove that one of the intersection points of $\odot(FWY)$ and $\odot(GXZ)$ lies on the line $FG$. [i]Proposed by chengbilly[/i]

Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root. Proposed by N. Agakhanov

For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is $7$. The roots may be characterized as: $ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$ $\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $

The area of the region bound by the equations $y = 2\cos(4x)$, $y = \sin(2x) + 10$, $x=0$, and $x = 5\pi$ can be expressed as $x\pi$ for some value of $x$. Evaluate $x$. [i]2022 CCA Math Bonanza Lightning Round 1.3[/i]