We say that a digit is [i]high[/i] if it is placed between two other digits and it is bigger than both of them. The digits $0$,$1$,$2$,$\dots$,$9$ are used exactly once to form a 10-digit number. How many numbers can be formed with the property such that they don’t have any high digits?

Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$ Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.

Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$. Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$.

There are $6$ computers(power off) and $3$ printers. Between a printer and a computer, they are connected with a wire or not. Printer can be only activated if and only if at least one of the connected computer's power is on. Your goal is to connect wires in such a way that, no matter how you choose three computers to turn on among the six, you can activate all $3$ printers. What is the minimum number of wires required to make this possible?

A point is "bouncing" inside a unit equilateral triangle with vertices $(0,0)$, $(1,0)$, and $(1/2,\sqrt{3}/2)$. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the direction of $(1,1)$. After the ball has traveled a cumulative distance of $30\sqrt{2}$, compute its distance from the origin.

Inside the equilateral triangle $ABC$ lies the point $T$. Prove that $TA$, $TB$ and $TC$ are the lengths of the sides of a triangle.

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Let the angle bisectors of $\angle BAC,$ $\angle CBA,$ and $\angle ACB$ meets the circumcircle of $\triangle ABC$ at the points $M,N,$ and $K,$ respectively. Let the segments $AB$ and $MK$ intersects at the point $P$ and the segments $AC$ and $MN$ intersects at the point $Q.$ Prove that $PQ\parallel BC$

In a triangle $ABC$ a circle $k$ is inscribed, which is tangent to $BC$,$CA$,$AB$ in points $D,E,F$ respectively. Let point $P$ be inner for $k$. If the lines $DP$,$EP$,$FP$ intersect $k$ in points $D',E',F'$ respectively, then prove that $AD'$, $BE'$, and $CF'$ are concurrent.

A number $N$ has 2009 positive factors. What is the maximum number of positive factors that $N^2$ could have?

Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.

Two players play a game with a pile of $N$ coins on a table. On a player's turn, if there are $n$ coins, the player can take at most $n/2+1$ coins, and must take at least one coin. The player who grabs the last coin wins. For how many values of $N$ between $1$ and $100$ (inclusive) does the first player have a winning strategy?

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

Prove that however we choose the majority of numbers among an even number of the first consecutive natural numbers, there will be two numbers among this choosing whose sum is a prime.

The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$. [asy] path seg1, seg2; seg1=(6,0)--(0,3); seg2=(2,0)--(4,2); dot((0,0)); dot((1,0)); fill(circle((2,0),0.1),black); dot((3,0)); dot((4,0)); dot((5,0)); fill(circle((6,0),0.1),black); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((5,1)); dot((6,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); fill(circle((4,2),0.1),black); dot((5,2)); dot((6,2)); fill(circle((0,3),0.1),black); dot((1,3)); dot((2,3)); dot((3,3)); dot((4,3)); dot((5,3)); dot((6,3)); draw(seg1); draw(seg2); pair [] x=intersectionpoints(seg1,seg2); fill(circle(x[0],0.1),black); label("$A$",(0,3),NW); label("$B$",(6,0),SE); label("$C$",(4,2),NE); label("$D$",(2,0),S); label("$E$",x[0],N);[/asy] $\text{(A)}\ \frac{4\sqrt5}{3}\qquad\text{(B)}\ \frac{5\sqrt5}{3}\qquad\text{(C)}\ \frac{12\sqrt5}{7}\qquad\text{(D)}\ 2\sqrt5 \qquad\text{(E)}\ \frac{5\sqrt{65}}{9}$

Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$, where $n$ is an integer greater than $1$. Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.)

Find all positive integers $n$ such that there exists an infinite set $A$ of positive integers with the following property: For all pairwise distinct numbers $a_1, a_2, \ldots , a_n \in A$, the numbers $$a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n$$ are coprime.

Let points $B$ and $C$ lie on the circle with diameter $AD$ and center $O$ on the same side of $AD$. The circumcircles of triangles $ABO$ and $CDO$ meet $BC$ at points $F$ and $E$ respectively. Prove that $R^2 = AF.DE$, where $R$ is the radius of the given circle. [i]Proposed by N.Moskvitin[/i]

Compute$$\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx$$

Kevin and Yang are playing a game. Yang has $2017 + \tbinom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows: [list] [*] For each $1 \le n \le 2017$, there is a blue card with $n$ written on the back, and a fraction $\tfrac{a_n}{b_n}$ written on the front, where $\gcd(a_n, b_n) = 1$ and $a_n, b_n > 0$. [*] For each $1 \le i < j \le 2017$, there is a red card with $(i, j)$ written on the back, and a fraction $\tfrac{a_i+a_j}{b_i+b_j}$ written on the front. [/list] It is given no two cards have equal fractions. In a turn Kevin can pick any two cards and Yang tells Kevin which card has the larger fraction on the front. Show that, in fewer than $10000$ turns, Kevin can determine which red card has the largest fraction out of all of the red cards.

Let $ABCD$ be a cyclic quadrilateral with $AB=3, BC=2, CD=6, DA=8,$ and circumcircle $\Gamma.$ The tangents to $\Gamma$ at $A$ and $C$ intersect at $P$ and the tangents to $\Gamma$ at $B$ and $D$ intersect at $Q.$ Suppose lines $PB$ and $PD$ intersect $\Gamma$ at points $W \neq B$ and $X \neq D,$ respectively. Similarly, suppose lines $QA$ and $QC$ intersect $\Gamma$ at points $Y \neq A$ and $Z \neq C,$ respectively. What is the value of $\frac{{WX}^2}{{YZ}^2}?$ [i]Proposed by Kyle Lee[/i]

Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$. [i]2020 CCA Math Bonanza Lightning Round #4.1[/i]

The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $ 20$, the father is $ 48$ years old, and the average age of the mother and children is $ 16$. How many children are in the family? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Let $A$, $B$, $C$, $D$, $E$, and $F$ be $6$ points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$, $BC = 3\sqrt{3}$, $CD = 6\sqrt{6}$, $DE = 4\sqrt{2}$, and $EF = 5\sqrt{2}$. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent, determine the square of $AF$.

Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that $$\sum_{n=1}^{\infty} b_n$$ converges.