Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Welcome to the [b]USAYNO[/b], a twelve-part question where each part has a yes/no answer. If you provide $C$ correct answers, your score on this problem will be $\frac{C}{6}$. Your answer should be a twelve-character string containing `Y' (for yes) and `N' (for no). For instance if you think a, c, and f are `yes' and the rest are `no', you should answer YNYNNYNNNNNN. (a) Is there a positive integer $n$ such that the sum of the digits of $2018n+1337$ in base $10$ is $2018$ more than the sum of the digits of $2018n+1337$ in base $4$? (b) Is there a fixed constant $\theta$ such that for all triangles $ABC$ with $$2018AB^2=2018CA^2+2017CA\cdot CB+2018CB^2,$$ one of the angles of $ABC$ is $\theta$? (c) Adam lists out every possible way to arrange the letters of ``CCACCACCA'' (including the given arrangement) at $1$ arrangement every $5$ seconds. Madam lists out every possible way to arrange the letters of ``CCACCAA'' (including the given arrangement) at $1$ arrangement every $12$ seconds. Does Adam finish first? (d) Do there exist real numbers $a,b,c$, none of which is the average of the other two, such that \[\frac{1}{b+c-2a}+\frac{1}{c+a-2b}+\frac{1}{a+b-2c}=0?\] (e) Let $f\left(x\right)=\frac{2^x-2}{x}-1$. Is there an integer $n$ such that $$f\left(n\right),f\left(f\left(n\right)\right),f\left(f\left(f\left(n\right)\right)\right),\ldots$$ are all integers? (f) In an elementary school with $2585$ students and $159$ classes (every student is in exactly one class), each student reports the size of their class. The principal of the school takes the average of all of these numbers and calls it $X$. The principal then computes the average size of each class and calls it $Y$. Is it necessarily true that $X>Y$? (g) Six sticks of lengths $3$, $5$, $7$, $11$, $13$, and $17$ are put together to form a hexagon. From a point inside the hexagon, a circular water balloon begins to expand and will stop expanding once it hits any stick. Is it possible that once the balloon stops expanding, it is touching each of the six sticks? (h) A coin is biased so that it flips heads and tails (and only heads or tails) each with a positive rational probability (not necessarily $\frac{1}{2}$). Is it possible that on average, it takes exactly twice as long to flip two heads in a row as it is to flip two tails in a row? (i) Does there exist a base $b$ such that $2018_b$ is prime? (j) Does there exist a sequence of $2018$ distinct real numbers such that no $45$ terms (not necessarily consecutive) can be examined, in order, and be in strictly increasing or strictly decreasing order? (k) Does there exist a scalene triangle $ABC$ such that there exist two distinct rectangles $PQRS$ inscribed in $\triangle{ABC}$ with $P\in AB$, $Q,R\in BC$, $S\in AC$ such that the angle bisectors of $\angle{PAS}$, $\angle{PQR}$, and $\angle{SRQ}$ concur? (l) For three vectors $\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3$ with $\mathbf{u}_i=\left(x_{i,1},x_{i,2},x_{i,3},x_{i,4}\right)$, define \[f\left(\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\right)=1-\displaystyle\prod_{j=1}^4\left(1+\left(x_{2,j}-x_{3,j}\right)^2+\left(x_{3,j}-x_{1,j}\right)^2+\left(x_{1,j}-x_{2,j}\right)^2\right).\] Are there any sequences $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_{18}$ of distinct vectors with four components, with all components in $\left\{1,2,3\right\}$, such that \[\displaystyle\prod_{1\leq i<j<k\leq18}f\left(\mathbf{v}_i,\mathbf{v}_j,\mathbf{v}_k\right)\equiv1\pmod3?\] [i]2018 CCA Math Bonanza Lightning Round #5.4[/i]

Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions : (a) $P(2017) = 2016$ and (b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$. [i]proposed by Walther Janous[/i]

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? [asy] defaultpen(linewidth(0.65)); real d=90-63.43494882; draw(ellipse((origin), 2, 4)); fill((0,4)--(0,-4)--(-8,-4)--(-8,4)--cycle, white); draw(ellipse((-4,0), 2, 4)); draw((0,4)--(-4,4)); draw((0,-4)--(-4,-4)); draw(shift(-2,0)*rotate(-d-5)*ellipse(origin, 1.82, 4.56), linetype("10 10")); draw((-4,4)--(-8,4), dashed); draw((-4,-4)--(-8,-4), dashed); draw((-4,4.3)--(-4,5)); draw((0,4.3)--(0,5)); draw((-7,4)--(-7,-4), Arrows(5)); draw((-4,4.7)--(0,4.7), Arrows(5)); label("$8$ cm", (-7,0), W); label("$6$ cm", (-2,4.7), N);[/asy] $\textbf{(A)} 48 \qquad \textbf{(B)} 75 \qquad \textbf{(C)}151\qquad \textbf{(D)}192 \qquad \textbf{(E)}603$

The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11:4$ . To the nearest whole percent, what percent of its games did the team lose? $ \text{(A)}\ 24\qquad\text{(B)}\ 27\qquad\text{(C)}\ 36\qquad\text{(D)}\ 45\qquad\text{(E)}\ 73 $

For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$. [i]Proposed by Gerhard Woeginger, Austria[/i]

Let $S$ be a set of real numbers which is closed under multiplication (that is $a,b\in S\implies ab\in S$). Let $T,U\subset S$ such that $T\cap U=\emptyset, T\cup U=S$. Given that for any three elements $a,b,c$ in $T$, not necessarily distinct, we have $abc\in T$ and also if $a,b,c\in U$, not necessarily distinct then $abc\in U$. Show at least one of $T$ and $U$ is closed under multiplication.

Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?

Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.

In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).

Points $A, B$ move with equal speeds along two equal circles. Prove that the perpendicular bisector of $AB$ passes through a fixed point.

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.

In trapezoid $ABCD$ with bases $AD, BC$, $AD = 2BC$ and $M$ is midpoint of $AB$. Prove that line $BD$ passes through the midpoint of segment $CM$.

Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one. The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins. Determine which player has a winning strategy and describe this strategy.

An infinite number of lilypads grow in a line, numbered $\dots$, $-2$, $-1$, $0$, $1$, $2$, $\dots$ Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad $0$ or pad $96$. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad $0$ or pad $96$, regardless of her frog's decisions?

The partition of $2n$ positive integers into $n$ pairs is called [i]square-free[/i] if the product of numbers in each pair is not a perfect square.Prove that if for $2n$ distinct positive integers, there exists one square-free partition, then there exists at least $n!$ square-free partitions.

The real-valued function $f$ is defined for $0 \le x \le 1, f(0) = 0, f(1) = 1$, and $\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2$ for all $0 \le x < y < z \le 1$ with $z - y = y -x$. Prove that $\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}$.

On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove $ AB\plus{}BC\plus{}CA\geq 8DE$

A regular hexagon $ABCDEF$ has area $36$. Find the area of the region which lies in the overlap of the triangles $ACE$ and $BDF$. $\text{(A) }3\qquad\text{(B) }9\qquad\text{(C) }12\qquad\text{(D) }18\qquad\text{(E) }24$

If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality: $\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$

A convex quadrilateral $ABCD$ has an inscribed circle touching its sides $AB$, $BC$, $CD$, $DA$ at the points $M$,$N$,$P$,$K$, respectively. Let $O$ be the center of the inscribed circle, the area of the quadrilateral $MNPK$ is equal to $8$. Prove the inequality $$2S \le OA \cdot OC+ OB \cdot OD.$$