Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

Let $ABC$ be a triangle with $BC=9$, $CA=8$, and $AB=10$. Let the incenter and incircle of $ABC$ be $I$ and $\gamma$, respectively, and let $N$ be the midpoint of major arc $BC$ of the cirucmcircle of $ABC$. Line $NI$ meets the circumcircle of $ABC$ a second time at $P$. Let the line through $I$ perpendicular to $AI$ meet segments $AB$, $AC$, and $AP$ at $C_1$, $B_1$, and $Q$, respectively. Let $B_2$ lie on segment $CQ$ such that line $B_1B_2$ is tangent to $\gamma$, and let $C_2$ lie on segment $BQ$ such that line $C_1C_2$ tangent to $\gamma$. The length of $B_2C_2$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $100m+n$. [i]Proposed by Vincent Huang[/i]

Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

Estimate the number of times a one-digit answer (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) has been submitted as an answer for any question by any team in the first 4 sets of this competition's lightning round. An estimate $E$ earns $\frac{2}{1+|log_2(A)-log_2(E)|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.3[/i]

Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)? (A. Blinkov)

$ABC$ is an acute triangle. Three semicircles are constructed outwardly on the sides $BC$, $CA$ and $AB$ respectively. Construct points $A'$ , $B'$ and $C' $ on these semicìrcles respectively so that $AB' = AC'$, $BC' = BA'$ and $CA'= CB'$.

The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.

Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? $\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 $

(a) Find two real numebrs $a,b$ such that $|ax+b-\sqrt{x}| \le \frac{1}{24}$ for $1 \le x \le 4$. (b) Prove that the constant $\frac{1}{24}$ cannot be replaced by a smaller one.

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles. We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments. (25 points) [i]Proposed by Amirali Moinfar[/i]

Consider the isosceles triangle $ABC$ with $AB = AC$, and $M$ the midpoint of $BC$. Find the locus of the points $P$ interior to the triangle, for which $\angle BPM+\angle CPA = \pi$.

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Let $a$, $b$, and $c$ be real angles such that \newline \[3\sin a + 4\sin b + 5\sin c = 0\] \[3\cos a + 4\cos b + 5\cos c = 0.\] \newline The maximum value of the expression $\frac{\sin b \sin c}{\sin^2 a}$ can be expressed as $\frac{p}{q}$ for relatively prime $p,q$. Compute $p+q$.

A boy at point $A$ wants to get water at a circular lake and carry it to point $B$. Find the point $C$ on the lake such that the distance walked by the boy is the shortest possible given that the line $AB$ and the lake are exterior to each other.

Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$ is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it.

Let $x_1$ and $x_2$ be such that $x_1 \neq x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals $ \textbf{(A)}\ -\frac{h}{3} \qquad\textbf{(B)}\ \frac{h}{3} \qquad\textbf{(C)}\ \frac{b}{3} \qquad\textbf{(D)}\ 2b \qquad\textbf{(E)}\ -\frac{b}{3} $

Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$. [i]Proposed by Kevin Sun[/i]

In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.

An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Prove that the circumcircles of $\triangle{BFK}, \triangle{CEL}$ and $\triangle{DEF}$ concur.

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.