Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.

Estimate the number of primes among the first thousand primes divide some term of the sequence \[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\] An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer. [i]2021 CCA Math Bonanza Lightning Round #5.4[/i]

Definition: A natural number is [i]abundant [/i] if the sum of its positive divisors is greater than its double. Find an odd abundant number and prove that there are infinitely many odd abundant numbers.

3. Let $ABC$ be a triangle with $AB=30$, $BC=14$, and $CA=26$. Let $N$ be the center of the equilateral triangle constructed externally on side $AB$. Let $M$ be the center of the square constructed externally on side $BC$. Given that the area of quadrilateral $ACMN$ can be expressed as $a+b\sqrt{c}$ for positive integers $a$, $b$ and $c$ such that $c$ is not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by winnertakeover[/i]

Sequence of real numbers $a_1, a_2, . . . , a_{2000}$ is such that for any natural number $n$, $1\le n \le 2000$, the equality $$a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.$$ Prove that all terms of this sequence are integers.

Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

Denote in the triangle $ABC$ by $h$ the length of the height drawn from vertex $A$, and by $\alpha = \angle BAC$. Prove that the inequality $AB + AC \ge BC \cdot \cos \alpha + 2h \cdot \sin \alpha$ . Are there triangles for which this inequality turns into equality?

Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ onto sides $AC$ and $AB$, respectively. Lines $BD$ and $CE$ intersect $\omega$ again at points $P \neq B$ and $Q \neq C$. Suppose that $PD=3$, $QE=2$, and $AP \parallel BC$. Compute $DE$. [i]Proposed by Kyle Lee[/i]

A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

Let $\Gamma_A$, $\Gamma_B$, $\Gamma_C$ be circles such that $\Gamma_A$ is tangent to $\Gamma_B$ and $\Gamma_B$ is tangent to $\Gamma_C$. All three circles are tangent to lines $L$ and $M$, with $A$, $B$, $C$ being the tangency points of $M$ with $\Gamma_A$, $\Gamma_B$, $\Gamma_C$, respectively. Given that $12=r_A<r_B<r_C=75$, calculate: a) the length of $r_B$. b) the distance between point $A$ and the point of intersection of lines $L$ and $M$.

Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$? $ \textbf{(A)}\ 840 \qquad\textbf{(B)}\ 816 \qquad\textbf{(C)}\ 969 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$

Determine all four-digit numbers $\overline{abcd} $ such that $(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd} $:

Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years. (a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end. (b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor . \] [i]Proposed by Sammy Luo[/i]

Let $ABC$ be a triangle and $A'$ be the reflection of $A$ about $BC$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, such that $PA'=PC$ and $QA'=QB$. Prove that the perpendicular from $A'$ to $PQ$ passes through the circumcenter of $\triangle ABC$. [i]Fedir Yudin[/i]

Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has a) $2012$, b) $2013$ symmetry planes? c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?

Let be three real numbers $ a,b,c, $ all in the interval $ (0,\infty ) $ or all in the interval $ (0,1). $ Prove the following inequality: $$ \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c . $$

Let $p(n) = n^4-6n^2-160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)

Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.

Joana divided $365$ by all integers from $1$ to $365$ and added all the remainders. Then she divided $366$ by all the integers from $1$ to $366$ and also added all the remainders. Which of the two sums is greater and what is the difference between them?

Let $ABC$ be a triangle and $n$ a positive integer. Consider on the side $BC$ the points $A_1, A_2, ..., A_{2^n-1}$ that divide the side into $2^n$ equal parts, that is, $BA_1=A_1A_2=...=A_{2^n-2}A_{2^n-1}=A_{2^n-1}C$. Set the points $B_1, B_2, ..., B_{2^n-1}$ and $C_1, C_2, ..., C_{2^n-1}$ on the sides $CA$ and $AB$, respectively, analogously. Draw the line segments $AA_1, AA_2, ..., AA_{2^n-1}$, $BB_1, BB_2, ..., BB_{2^n-1}$ and $CC_1, CC_2, ..., CC_{2^n-1}$. Find, in terms of $n$, the number of regions into which the triangle is divided.

How many integer values of $k$, with $1 \leq k \leq 70$, are such that $x^{k}-1 \equiv 0 \pmod{71}$ has at least $\sqrt{k}$ solutions?

In triangle $\triangle ABC$ , $M, N$ are midpoints of $AC,AB$ respectively. Assume that $BM,CN$ cuts $(ABC)$ at $M',N'$ respectively. Let $X$ be on the extention of $BC$ from $B$ st $\angle N'XB=\angle ACN$. And define $Y$ similarly on the extention of $BC$ from $C$. Prove that $AX=AY$.

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$. [i]Proposed by Serbia[/i]

Determine all real numbers $x\in\mathbb{R}$ for which \[ \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor=x-2022. \] The notation $\lfloor z \rfloor$, for $z\in\mathbb{R}$, denotes the largest integer which is less than or equal to $z$. For example: \[\lfloor 3.98 \rfloor =3 \quad \text{and} \quad \lfloor 0.14 \rfloor =0.\]