$ a$ is a real number. $ x_1$ and $ x_2$ are the distinct roots of $ x^2 \plus{} ax \plus{} 2 \equal{} x$. $ x_3$ and $ x_4$ are the distinct roots of $ (x \minus{} a)^2 \plus{} a(x \minus{} a) \plus{} 2 \equal{} x$. If $ x_3 \minus{} x_1 \equal{} 3(x_4 \minus{} x_2)$, then $ x_4 \minus{} x_2$ will be ? $\textbf{(A)}\ \frac {a}{2} \qquad\textbf{(B)}\ \frac {a}{3} \qquad\textbf{(C)}\ \frac {2a}{3} \qquad\textbf{(D)}\ \frac {3a}{2} \qquad\textbf{(E)}\ \text{None}$

Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Let $b\geq2$ and $w\geq2$ be fixed integers, and $n=b+w$. Given are $2b$ identical black rods and $2w$ identical white rods, each of side length 1. We assemble a regular $2n-$gon using these rods so that parallel sides are the same color. Then, a convex $2b$-gon $B$ is formed by translating the black rods, and a convex $2w$-gon $W$ is formed by translating the white rods. An example of one way of doing the assembly when $b=3$ and $w=2$ is shown below, as well as the resulting polygons $B$ and $W$. [asy]size(10cm); real w = 2*Sin(18); real h = 0.10 * w; real d = 0.33 * h; picture wht; picture blk; draw(wht, (0,0)--(w,0)--(w+d,h)--(-d,h)--cycle); fill(blk, (0,0)--(w,0)--(w+d,h)--(-d,h)--cycle, black); // draw(unitcircle, blue+dotted); // Original polygon add(shift(dir(108))*blk); add(shift(dir(72))*rotate(324)*blk); add(shift(dir(36))*rotate(288)*wht); add(shift(dir(0))*rotate(252)*blk); add(shift(dir(324))*rotate(216)*wht); add(shift(dir(288))*rotate(180)*blk); add(shift(dir(252))*rotate(144)*blk); add(shift(dir(216))*rotate(108)*wht); add(shift(dir(180))*rotate(72)*blk); add(shift(dir(144))*rotate(36)*wht); // White shifted real Wk = 1.2; pair W1 = (1.8,0.1); pair W2 = W1 + w*dir(36); pair W3 = W2 + w*dir(108); pair W4 = W3 + w*dir(216); path Wgon = W1--W2--W3--W4--cycle; draw(Wgon); pair WO = (W1+W3)/2; transform Wt = shift(WO)*scale(Wk)*shift(-WO); draw(Wt * Wgon); label("$W$", WO); /* draw(W1--Wt*W1); draw(W2--Wt*W2); draw(W3--Wt*W3); draw(W4--Wt*W4); */ // Black shifted real Bk = 1.10; pair B1 = (1.5,-0.1); pair B2 = B1 + w*dir(0); pair B3 = B2 + w*dir(324); pair B4 = B3 + w*dir(252); pair B5 = B4 + w*dir(180); pair B6 = B5 + w*dir(144); path Bgon = B1--B2--B3--B4--B5--B6--cycle; pair BO = (B1+B4)/2; transform Bt = shift(BO)*scale(Bk)*shift(-BO); fill(Bt * Bgon, black); fill(Bgon, white); label("$B$", BO);[/asy] Prove that the difference of the areas of $B$ and $W$ depends only on the numbers $b$ and $w$, and not on how the $2n$-gon was assembled. [i]Proposed by Ankan Bhattacharya[/i]

A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square. (V Proizvolov)

Consider a triangle $ABC$ with incenter $I$. Let $D$ be the intersection of $BI,AC$ and $CI$ intersects the circumcircle of $ABC$ at $M$. Point $K$ lies on the line $MD$ and $\angle KIA=90^\circ$. Let $F$ be the reflection of $B$ about $C$. Prove that $BIKF$ is cyclic.

Let $O$ be the circumcenter of $\triangle ABC$($AB<AC$), and $D$ be a point on the internal angle bisector of $\angle BAC$. Point $E$ lies on $BC$, satisfying $OE\parallel AD$, $DE\perp BC$. Point $K$ lies on $EB$ extended such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$, and meets the circumcircle of $\triangle ABC$ at $Q\neq A$. Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$.

Bob is flipping bottles. Each time he flips the bottle, he has a $0.25$ probability of landing it. After successfully flipping a bottle, he has a $0.8$ probability of landing his next flip. What is the expected value of the number of times he has to flip the bottle in order to flip it twice in a row? [i]2017 CCA Math Bonanza Lightning Round #3.2[/i]

Let be two $ 2\times 2 $ real matrices $A,B$ having the property that all their natural powers are not real multiples of the identity. Prove that if some natural power of $ A $ is equal to some natural power of $ B, $ then, $ A,B $ commute. Is the converse statement true? [i]Cosmin Nitu[/i]

In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at $(0,0)$ and the line segment from $(-1,0)$ to $(1,0)$. David stands at the point $(-1,0)$ and shines a flashlight into the room at an angle of $46^{\circ}$ above the horizontal. How many times does the light beam reflect off the walls before coming back to David at $(-1,0)$ for the first time?

For every sequence $\{a_n\}~(n\in\mathbb N)$ we define the sequences $\{\Delta a_n\}$ and $\{\Delta^2a_n\}$ by the following formulas: \begin{align*}\Delta a_n&=a_{n+1}-a_n,\\\Delta^2a_n&=\Delta a_{n+1}-\Delta a_n.\end{align*}Further, for all $n\in\mathbb N$ for which $\Delta a_n^2\ne0$, define $$a_n'=a_n-\frac{(\Delta a_n)^2}{\Delta^2a_n}.$$ (a) For which sequences $\{a_n\}$ is the sequence $\{\Delta^2a_n\}$ constant? (b) Find all sequences $\{a_n\}$, for which the numbers $a_n'$ are defined for all $n\in\mathbb N$ and for which the sequence $\{a_n'\}$ is constant. (c) Assume that the sequence $\{a_n\}$ converges to $a=0$, and $a_n\ne a$ for all $n\in\mathbb N$ and the sequence $\{\tfrac{a_{n+1}-a}{a_n-a}\}$ converges to $\lambda\ne1$. i. Prove that $\lambda\in[-1,1)$. ii. Prove that there exists $n_0\in\mathbb N$ such that for all integers $n\ge n_0$ we have $\Delta^2a_n\ne0$. iii. Let $\lambda\ne0$. For which $k\in\mathbb Z$ is the sequence $\{\tfrac{a_n'}{a_{n+k}}\}$ not convergent? iv. Let $\lambda=0$. Prove that the sequences $\{a_n'/a_n\}$ and $\{a_n'/a_{n+1}\}$ converge to $0$. Find an example of $\{a_n\}$ for which the sequence $\{a_n'/a_{n+2}\}$ has a non-zero limit. (d) What happens with part (c) if we remove the condition $a=0$?

Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$ where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

Assume that in the $\triangle ABC$ there exists a point $D$ on $BC$ and a line $l$ passing through $A$ such that $l$ is tangent to $(ADC)$ and $l$ bisects $BD.$ Prove that $a\sqrt{2}\geq b+c.$

a) Let $x$ be a positive real number. Show that $x + 1 / x\ge 2$. b) Let $n\ge 2$ be a positive integer and let $x _1,y_1,x_2,y_2,...,x_n,y_n$ be positive real numbers such that $x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n$. Show that $x _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}$

For positive real numbers $ x,y,z$ the inequality \[\frac1{x^2\plus{}1}\plus{}\frac1{y^2\plus{}1}\plus{}\frac1{z^2\plus{}1}\equal{}\frac12\] holds. Prove the inequality \[\frac1{x^3\plus{}2}\plus{}\frac1{y^3\plus{}2}\plus{}\frac1{z^3\plus{}2}<\frac13.\]

A charity sells 140 benefit tickets for a total of $ \$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? $ \textbf{(A)} \ \$782 \qquad \textbf{(B)} \ \$986 \qquad \textbf{(C)} \ \$1158 \qquad \textbf{(D)} \ \$1219 \qquad \textbf{(E)} \ \$1449$

A problem author for a math competition was looking through a tentative exam when he realized that he could not use one of his proposed problems. Frustrated, he decided to take a nap instead, and slept from $10:47\text{ AM}$ to $7:32\text{ PM}$. For how many minutes did he sleep?

Let $\Delta ABC$ be an isosceles triangle with base $AB$. Point $P\in AB$ is such that $AP=2PB$. Point $Q$ from the segment $CP$ is such that $\angle AQP=\angle ACB$. Prove that $\angle PQB=\frac{1}{2}\angle ACB$.

Let $ABC$ be a triangle with orthocenter $H$. Let $P,Q,R$ be the reflections of $H$ with respect to sides $BC,AC,AB$, respectively. Show that $H$ is incenter of $PQR$.

The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively. (Grigory Filippovsky)

[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series $\sum_{n=1}^{\infty} \epsilon_n z_n$ is convergente. [b](F. 9)[/b]

Streets of Moscow are some circles (rings) with common center $O$ and some straight lines from center $O$ to external ring. Point $A,B$ - two crossroads on external ring. Three friends want to move from $A$ to $B$. Dima goes by external ring, Kostya goes from $A$ to $O$ then to $B$. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.

The sequence of letters [b]TAGC[/b] is written in succession 55 times on a strip, as shown below. The strip is to be cut into segments between letters, leaving strings of letters on each segment, which we call words. For example, a cut after the first G, after the second T, and after the second C would yield the words [b]TAG[/b], [b]CT[/b] and [b]AGC[/b]. At most how many distinct words could be found if the entire strip were cut? Justify your answer. \[\boxed{\textbf{T A G C T A G C T A G}}\ldots\boxed{\textbf{C T A G C}}\]

Let ${ABC}$ be a triangle. Line [i]l[/i] is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on [i]l[/i]. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear. (Slovakia)

The points $P,Q,R$ and $A,B,C,D$ lie on a circle (clockwise) such that $\vartriangle PQR$ is equilateral and $ABCD$ is a square. The points $A$ and $P$ coincide. Prove that the symmetric of $B$ and $D$ wrt $PQ$ and $PR$ respectively lie on the sidelines of the symmetric square wrt $QR$.