Found problems: 341
Define a sequence by $a_0=2019$ and $a_n=a_{n-1}^{2019}$ for all positive integers $n$. Compute the remainder when \[a_0+a_1+a_2+\dots+a_{51}\] is divided by $856$.
[i]Proposed by Tristan Shin[/i]
For a natural number $n$, let $n'$ denote the number obtained by deleting zero digits, if any. (For example, if $n = 260$, $n' = 26$, if $n = 2020$, $n' = 22$.),Find the number of $3$-digit numbers $n$ for which $n'$ is a divisor of $n$, different from $n$.
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36));
draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61));
draw((34,36)--(34,45)--(25,45));
draw((36,36)--(36,38)--(34,38));
draw((36,38)--(41,38));
draw((34,45)--(41,45));[/asy]
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.
The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?
Prove that the fraction $ \dfrac{21n \plus{} 4}{14n \plus{} 3}$ is irreducible for every natural number $ n$.
For some positive integer $n,$ Elmo writes down the equation
\[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\]
Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation
\[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\]
Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$?
[i]Srinivas Arun[/i]
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.
Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that
$$\displaystyle P(x)=R(T(x))$$
a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$
b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$, what are all possible values of $a$?
[i]2019 CCA Math Bonanza Individual Round #8[/i]
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
Prove that the fraction $ \dfrac{21n \plus{} 4}{14n \plus{} 3}$ is irreducible for every natural number $ n$.
Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called [i]beautiful[/i] if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$.
Let $M$ be the number of beautiful labelings, and let N be the number of ordered pairs $(x, y)$ of positive integers such that $x + y \le n$ and $\gcd(x, y) = 1$. Prove that $$M = N + 1.$$
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the integers $x$ and $y$, where $x\leq y$ and replace them with $2x$ and $y-x.$
Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\]
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting point?
$\textbf{(A)}~405\qquad\textbf{(B)}~407\qquad\textbf{(C)}~409\qquad\textbf{(D)}~411\qquad\textbf{(E)}~413$
Find all ordered triplets $(p,q,r)$ of positive integers such that $p$ and $q$ are two (not necessarily distinct) primes, $r$ is even, and
\[p^3+q^2=4r^2+45r+103.\]