If this path is to continue in the same pattern: [asy] unitsize(24); draw((0,0)--(1,0)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,0)--(5,0)--(5,1)--(6,1)); draw((2/3,1/5)--(1,0)--(2/3,-1/5)); draw((4/5,2/3)--(1,1)--(6/5,2/3)); draw((5/3,6/5)--(2,1)--(5/3,4/5)); draw((9/5,1/3)--(2,0)--(11/5,1/3)); draw((8/3,1/5)--(3,0)--(8/3,-1/5)); draw((14/5,2/3)--(3,1)--(16/5,2/3)); draw((11/3,6/5)--(4,1)--(11/3,4/5)); draw((19/5,1/3)--(4,0)--(21/5,1/3)); draw((14/3,1/5)--(5,0)--(14/3,-1/5)); draw((24/5,2/3)--(5,1)--(26/5,2/3)); draw((17/3,6/5)--(6,1)--(17/3,4/5)); dot((0,0)); dot((1,0)); dot((1,1)); dot((2,1)); dot((2,0)); dot((3,0)); dot((3,1)); dot((4,1)); dot((4,0)); dot((5,0)); dot((5,1)); label("$0$",(0,0),S); label("$1$",(1,0),S); label("$2$",(1,1),N); label("$3$",(2,1),N); label("$4$",(2,0),S); label("$5$",(3,0),S); label("$6$",(3,1),N); label("$7$",(4,1),N); label("$8$",(4,0),S); label("$9$",(5,0),S); label("$10$",(5,1),N); label("$\vdots$",(5.85,0.5),E); label("$\cdots$",(6.5,0.15),S); [/asy] then which sequence of arrows goes from point $425$ to point $427$? [asy] unitsize(24); dot((0,0)); dot((0,1)); dot((1,1)); draw((0,0)--(0,1)--(1,1)); draw((-1/5,2/3)--(0,1)--(1/5,2/3)); draw((2/3,6/5)--(1,1)--(2/3,4/5)); label("(A)",(-1/3,1/3),W); dot((4,0)); dot((5,0)); dot((5,1)); draw((4,0)--(5,0)--(5,1)); draw((14/3,1/5)--(5,0)--(14/3,-1/5)); draw((24/5,2/3)--(5,1)--(26/5,2/3)); label("(B)",(11/3,1/3),W); dot((8,1)); dot((8,0)); dot((9,0)); draw((8,1)--(8,0)--(9,0)); draw((39/5,1/3)--(8,0)--(41/5,1/3)); draw((26/3,1/5)--(9,0)--(26/3,-1/5)); label("(C)",(23/3,1/3),W); dot((12,1)); dot((13,1)); dot((13,0)); draw((12,1)--(13,1)--(13,0)); draw((38/3,6/5)--(13,1)--(38/3,4/5)); draw((64/5,1/3)--(13,0)--(66/5,1/3)); label("(D)",(35/3,1/3),W); dot((17,1)); dot((17,0)); dot((16,0)); draw((17,1)--(17,0)--(16,0)); draw((84/5,1/3)--(17,0)--(86/5,1/3)); draw((49/3,1/5)--(16,0)--(49/3,-1/5)); label("(E)",(47/3,1/3),W); [/asy]

In rectangular coordinate system, give two points $M(-1,2),N(1,4)$, $P$ is a moving point on $x$-axis, when $\angle MPN$ takes its maximum value, the $x$-axis of $P$ is________.

Define the sequence $(w_n)_{n\ge0}$ by the recurrence relation $$w_{n+2}=2w_{n+1}+3w_n,\enspace\enspace w_0=1,w_1=i,\enspace n=0,1,\ldots$$ (1) Find the general formula for $w_n$ and compute the first $9$ terms. (2) Show that $|\Re w_n-\Im w_n|=1$ for all $n\ge1$. [i]Proposed by Ovidiu Bagdasar[/i]

Determine the smallest value of $x^2 + y^2 + z^2$, where $x, y, z$ are real numbers, so that $x^3 + y^3 + z^3 -3xyz = 1.$

Let $f(x)$ be a polynomial of finite degree satisfying $$(x+9)f(x+1)=(x+3)f(x+3)$$ for all real $x$. If $f(0)=1$, find the value of $f(1)$.

Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a [i]grade[/i] is a nonnegative decimal number with finitely many digits after the decimal point. Given a grade with digits $a_1a_2 \dots a_m.b_1b_2 \dots b_n$, Joe first rounds the number to the nearest $10^{-n+1}$th place. He then repeats the procedure on the new number, rounding to the nearest $10^{-n+2}$th, then rounding the result to the nearest $10^{-n+3}$th, and so on, until he obtains an integer. For example, he rounds the number $2014.456$ via $2014.456 \to 2014.46 \to 2014.5 \to 2015$. There exists a rational number $M$ such that a grade $x$ gets rounded to at least $90$ if and only if $x \ge M$. If $M = \tfrac pq$ for relatively prime integers $p$ and $q$, compute $p+q$. [i]Proposed by Yang Liu[/i]

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$. [i]Proposed by Tobi Moektijono, Indonesia[/i]

Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.

Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation: Samuel Tsui: Do our numbers have any common factors greater than $1$? Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$. Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal. What is the sumof Samuel Tsui’s and Jason Yang’s numbers? [i]Proposed by Samuel Tsui[/i]

Let $k,n,r$ be positive integers and $r<n$. Quirin owns $kn+r$ black and $kn+r$ white socks. He want to clean his cloths closet such there does not exist $2n$ consecutive socks $n$ of which black and the other $n$ white. Prove that he can clean his closet in the desired manner if and only if $r\geq k$ and $n>k+r$.

The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50\%$ probability and scissors with $50\%$ probability. If you play optimally against the computer, find the probability that after $8$ games you have won at least $4$. [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

Let $ \Gamma$ be the circumcircle of a triangle $ ABC$. A circle passing through points $ A$ and $ C$ meets the sides $ BC$ and $ BA$ at $ D$ and $ E$, respectively. The lines $ AD$ and $ CE$ meet $ \Gamma$ again at $ G$ and $ H$, respectively. The tangent lines of $ \Gamma$ at $ A$ and $ C$ meet the line $ DE$ at $ L$ and $ M$, respectively. Prove that the lines $ LH$ and $ MG$ meet at $ \Gamma$.

Let $\Phi$ denote the Euler totient function. Prove that for infinitely many $k$ we have $\Phi (2^k+1) < 2^{k-1}$ and that for infinitely many $m$ one has $\Phi (2^m+1) > 2^{m-1}$

Find all real $x$ that satisfy the equation $$\frac{1}{x+1}+\frac{1}{x+2}=\frac{1}{x}$$ [i]2015 CCA Math Bonanza Lightning Round #2.2[/i]

Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.

Compute the smallest positive integer $n$ such that there do not exist integers $x$ and $y$ satisfying $n=x^3+3y^3$. [i]Proposed by Luke Robitaille[/i]

Compute the number of integer ordered pairs $(a, b)$ such that $10!$ is a multiple of $a^2 + b^2.$

Let $ABC$ be an acute triangle with an altitude $AD$. Let $H$ be the orthocenter of triangle $ABC$. The circle $\Omega$ passes through the points $A$ and $B$, and touches the line $AC$. Let $BE$ be the diameter of $\Omega$. The lines~$BH$ and $AH$ intersect $\Omega$ for the second time at points $K$ and $L$, respectively. The lines $EK$ and $AB$ intersect at the point~$T$. Prove that $\angle BDK=\angle BLT$.

Show that for some $c > 0$, we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$.

Given is a square board measuring $1 995 \times 1 995$. These cells are painted with black and white paints in a checkerboard pattern, so that the corner cells are black. A spider sitting on one of the black cells can crawl to the cell on the same side as the one it occupies in one step. Prove that a spider can always reach a fly sitting motionless in another black cell by visiting all the cells of the board once.

Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?

Given two fractions $a/b$ and $c/d$ we define their [i]pirate sum[/i] as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$. Given an integer $n \ge 3$, initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$. At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction. Determine, in function of $n$, the maximum and the minimum possible value for the last fraction.

Find all positive integers $N$ such that there exists a triangle which can be dissected into $N$ similar quadrilaterals. [i]Proposed by Nikolai Beluhov (Bulgaria) and Morteza Saghafian[/i]