Compute the sum of all positive real numbers $x \le 5$ satisfying $$x =\frac{ \lceil x^2 \rceil + \lceil x \rceil \cdot \lfloor x \rfloor}{ \lceil x\rceil + \lfloor x \rfloor}$$

Let $a$ be any positive integer. Prove that there exists a unique pair of positive integers $x$ and $y$ such that $$x +\frac12 (x + y - 1)(x + y- 2) = a.$$

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

In each square of a $6\times6$ grid there is a lightning bug on or off. One move is to choose three consecutive squares, either horizontal or vertical, and change the lightning bugs in those $3$ squares from off to on or from on to off. Show if at the beginning there is one lighting bug on and the rest of them off, it is not possible to make some moves so that at the end they are all turned off.

The numbers $1,2,...,1970$ are written on a board. One is allowed to remove $2$ numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.

Prove that for any positive integer $n$ there exists a positive integer $k$ such that $3^k+4^k-1 \vdots 12^n$

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $. a) Prove that $ I$ lies between $ M $ and $ H $. b) Calculate $ AB ^ 2-AC ^ 2 $. c) Determine $ \dfrac {AB} {AC} $. d) Find the measure of all the sides and angles of the triangle.

For each integer $i=0,1,2, \dots$, there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$? $ \textbf{a)}\ 8 \qquad\textbf{b)}\ 10 \qquad\textbf{c)}\ 12 \qquad\textbf{d)}\ 15 \qquad\textbf{e)}\ 16 $

Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome? [i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i]

In the coordinate plane are finitely many [i]walls[/i]; which are disjoint line segments, none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues moving. (Thus the bulldozer always moves parallel to the axes.) Prove that it is impossible for the bulldozer to hit both sides of every wall. [i]Proposed by Linus Hamilton and David Stoner[/i]

What is the sum of real roots of the equation $x^4-4x^3+5x^2-4x+1 = 0$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

Let $\omega$ be a circle with centre $O$, and let $\ell$ be a line that does not intersect $\omega$. Let $P$ be an arbitrary point on $\ell$. Let $A,B$ denote the tangent points of the tangent lines from $P$. Prove that $AB$ passes through a point being independent of choosing $P$.

Let $P_1P_2P_3\dots P_{12}$ be a regular dodecagon. Show that \[\left|P_1P_2\right|^2 + \left|P_1P_4\right|^2 + \left|P_1P_6\right|^2 + \left|P_1P_8\right|^2 + \left|P_1P_{10}\right|^2 + \left|P_1P_{12}\right|^2\] is equal to \[\left|P_1P_3\right|^2 + \left|P_1P_5\right|^2 + \left|P_1P_7\right|^2 + \left|P_1P_9\right|^2 + \left|P_1P_{11}\right|^2.\]

Does there exist a polynomial \( P(x,y) \) in two variables with real coefficients, such that the following two conditions hold: 1) \( P(x,y) = P(x, x-y) = P(y-x, y) \) for any real numbers \( x \) and \( y \); 2) There does not exist a polynomial \( Q(z) \) in one variable with real coefficients such that \( P(x,y) = Q(x^2 - xy + y^2) \) for any real numbers \( x \) and \( y \)?

The circles $ C_1,C_2 $ meet at the points $ A,B. $ A line thru $ A $ intersects $ C_1,C_2 $ at $ C,D, $ respectively. Point $ A $ is not on the arc $ BC $ of $ C_1, $ neither on the arc $ BD $ of $ C_2. $ On the segments $ CD,BC,BD $ there are the points $ M,N,K $ such that $ MN $ is parallel to $ BD $ and $ MK $ is parallel with $ BC. $ Upon the arc $ BC $ let $ E $ be a point having the property that $ EN $ is perpendicular to $ BC, $ and upon the arc $ BD $ let $ F $ be a point chosen so that $ FK $ is perpendicular to $ BD. $ Show that the angle $ \angle EMF $ is right.

Prove that for any non-zero real numbers $a, b, c,$ \[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]

Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$.

Let $p$ be a prime, and $n > p$ be an integer. Prove that \[ \binom{n+p-1}{p} - \binom{n}{p} \] is divisible by $n$.

The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers? $\textbf{(A) }2\text{ and }19\qquad\textbf{(B) }20\text{ and }39\qquad\textbf{(C) }40\text{ and }59\qquad\textbf{(D) }60\text{ and }79\qquad\textbf{(E) }80\text{ and }124$

Consider systems of three linear equations with unknowns $x,$ $y,$ and $z,$ \begin{align*} a_1 x + b_1 y + c_1 z = 0 \\ a_2 x + b_2 y + c_2 z = 0 \\ a_3 x + b_3 y + c_3 z = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x = y = z = 0.$ For example, one such system is $\{1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0\}$ with a nonzero solution of $\{x, y, z\} = \{1, -1, 1\}.$ How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.) $\textbf{(A) } 302 \qquad \textbf{(B) } 338 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 343 \qquad \textbf{(E) } 344$

Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$? [i]2021 CCA Math Bonanza Team Round #9[/i]

Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?

Given a sphere of radius $ R $ and a plane $ \alpha $ having no common points with this sphere. A point $ S $ moves in the plane $ \alpha $, which is the vertex of a cone tangent to the sphere along a circle with center $ C $. Find the locus of point $ C $. [hide=another is Polish MO 1967 p6] [url=https://artofproblemsolving.com/community/c6h3388032p31769739]here[/url][/hide]

Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$

Positive numbers $ a$, $ b$, $ c$ are such that $ \left(a \plus{} b\right)\left(b \plus{} c\right)\left(c \plus{} a\right) \equal{} 8$. Prove the inequality \[ \frac {a \plus{} b \plus{} c}{3}\ge\sqrt [27]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3}} \]