How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 3 \qquad\textbf{d)}\ 7 \qquad\textbf{e)}\ \text{Infinitely many} $

Solve in non-negative integers the equation $125.2^n-3^m=271$

If $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ are real numbers satisfying $a_1^2+\cdots+a_n^2 \le 1$ and $b_1^2+\cdots+b_n^2 \le 1$ , show that: $$(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2$$

Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M); label("$A$", A, dir(90)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$M$", M, dir(90)*dir(B--A)); label("$N$", N, dir(90)*dir(A--C)); label("$O$", O, dir(90));[/asy] $\textbf {(A) } 30 \qquad \textbf {(B) } 33 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 42$

Let $ A\equal{}A_1A_2A_3A_4$ be a tetrahedron, and suppose that for each $ j \not\equal{} k, [A_j,A_{jk}]$ is a segment of length $ \rho$ extending from $ A_j$ in the direction of $ A_k$. Let $ p_j$ be the intersection line of the planes $ [A_{jk}A_{jl}A_{jm}]$ and $ [A_kA_lA_m]$. Show that there are infinitely many straight lines that intersect the straight lines $ p_1,p_2,p_3,p_4$ simultaneously.

For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?

If on $ \triangle ABC$, trinagles $ AEB$ and $ AFC$ are constructed externally such that $ \angle AEB\equal{}2 \alpha$, $ \angle AFB\equal{} 2 \beta$. $ AE\equal{}EB$, $ AF\equal{}FC$. COnstructed externally on $ BC$ is triangle $ BDC$ with $ \angle DBC\equal{} \beta$ , $ \angle BCD\equal{} \alpha$. Prove that 1. $ DA$ is perpendicular to $ EF$. 2. If $ T$ is the projection of $ D$ on $ BC$, then prove that $ \frac{DA}{EF}\equal{} 2 \frac{DT}{BC}$.

Let a triangle $ABC$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_1$. Points $B_1$ and $C_1$ are defined similarly. a) Prove that lines $AA_1, BB_1$ and $CC1$ concur. b) Let $A_2$ be the touching point of the incircle with $BC$. Prove that lines $AA_1$ and $AA_2$ are symmetric about the bisector of angle $\angle A$.

Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$? [asy] defaultpen(linewidth(0.8)); pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D); draw(C--D--A--B--C--E); draw(Arc((0.5,0), 0.5, 0, 180)); pair point=(0.5,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E));[/asy] $ \textbf{(A)}\ \frac {2 \plus{} \sqrt5}{2} \qquad \textbf{(B)}\ \sqrt 5 \qquad \textbf{(C)}\ \sqrt 6 \qquad \textbf{(D)}\ \frac52 \qquad \textbf{(E)}\ 5 \minus{} \sqrt5$

Circle $\Omega(O,R)$ and its chord $AB$ is given. Suppose $C$ is midpoint of arc $AB$. $X$ is an arbitrary point on the cirlce. Perpendicular from $B$ to $CX$ intersects circle again in $D$. Perpendicular from $C$ to $DX$ intersects circle again in $E$. We draw three lines $\ell_{1},\ell_{2},\ell_{3}$ from $A,B,E$ parralell to $OX,OD,OC$. Prove that these lines are concurrent and find locus of concurrncy point.

\begin{quote} Ted quite likes haikus, \\ poems with five-seven-five, \\ but Ted knows few words. He knows $2n$ words \\ that contain $n$ syllables \\ for every int $n$. Ted can only write \\ $N$ distinct haikus. Find $N$. \\ Take mod one hundred. \end{quote} Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$. [i]Proposed by Lewis Chen[/i]

Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

The CCA Math Banana$^{\text{TM}}$ costs \$$100$. The cost rises $10$% then drops $10$%. Now what is the cost of the CCA Math Banana$^{\text{TM}}$? [i]2018 CCA Math Bonanza Lightning Round #1.2[/i]

The following four statements, and only these are found on a card: [asy] pair A,B,C,D,E,F,G; A=(0,1); B=(0,5); C=(11,5); D=(11,1); E=(0,4); F=(0,3); G=(0,2); draw(A--B--C--D--cycle); label("On this card exactly one statement is false.", B, SE); label("On this card exactly two statements are false.", E, SE); label("On this card exactly three statements are false.", F, SE); label("On this card exactly four statements are false.", G, SE); [/asy] (Assume each statement is either true or false.) Among them the number of false statements is exactly $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Let m be a positive integer, and let $S=\{a_1=1, a_2, ..., a_m\}$ be a set of positive integers. Prove that there exists a positive integer $n$ with $n\le m$ and a set $T={b_1, b_2, ..., b_n}$ of positive integers such that (a) all the subsets of $T$ have distinct sums of elements; (b) each of the numbers $a_1$, $a_2$, ..., $a_m$ is the sum of the elements of a subset of $T$.

Quadrilateral $ABCD$ is such that $AB \perp CD$ and $AD \perp BC$. Prove that there exist a point such that the distances from it to the sidelines are proportional to the lengths of the corresponding sides.

[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.

A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $. Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].

Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

Find $PB$, given that $PA = 15$, $PC = 20$, $PD = 7$, and $ABCD$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/7/a/cc5bf99986fea1cd75e57fe1117a4d04d3eae3.png[/img]

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

Find with proof all non–zero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.