Found problems: 85335
Let $P (x)$ be a polynomial with integer coefficients. Given that for some integer $a$ and some positive integer $n$, where
\[\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a,\]
is it true that $P (P (a)) = a$?
Let $a>0$. Show that
$$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$
lies between $e^a$ and $e^{a+1}.$
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum
$$p_1^4+p_2^4+\dots +p_{2021}^4$$
is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$.
$\textit{Stefan Bălăucă}$
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$. An estimate $E$ earns $\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer.
[i]2022 CCA Math Bonanza Lightning Round 5.2[/i]
Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds
$$f(xf(x) + f(y)) = y + f(x)^2$$
The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.
For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.
Tom, Dick and Harry started out on a $ 100$-mile journey. Tom and Harry went by automobile at the rate of $ 25$ mph, while Dick walked at the rate of $ 5$ mph. After a certain distance, Harry got off and walked on at $ 5$ mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{none of these answers}$
Let $\triangle ABC$ such that $AB=AC$, $D$ and $E$ points on $AB$ and $BC$, respectively, with $DE\parallel AC$. Let $F$ on line $DE$ such that $CADF$ it´s a parallelogram. If $O$ is the circumcenter of $\triangle BDE$, prove that $O,F,A$ and $D$ lie on a circle.
If a $5 \times n$ rectangle can be tiled using $n$ pieces like those shown in the diagram, prove that $n$ is even. Show that there are more than $2 \cdot 3^{k-1}$ ways to file a fixed $5 \times 2k$ rectangle $(k \geq 3)$ with $2k$ pieces. (symmetric constructions are supposed to be different.)
Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial
\[d_9x^9+d_8x^8+\cdots+d_1x+d_0\]
is irreducible in $\Bbb Q$. Prove that $N$ is even.
(A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)
What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?
Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M,$ or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order.
Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS| = \sqrt2 |SQ|$.
Determine the natural numbers $m,n$ such as $85^m-n^4=4$
$ \lim_{n\to\infty } \left( (1+1/n)^{-n}\sum_{i=0}^n\frac{1}{i!} \right)^{2n} $
Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.
Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f(x^2 + f(y)) = (f(x) + y^2)^ 2 \] , for all $x,y\in \Bbb{R}.$
[u]Round 5[/u]
[b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle.
[b]5.2.[/b] A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus?
[b]5.3.[/b] A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is $\frac23$ , the volume of the combined cup and lid is $296\pi$, and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container?
[u]Round 6[/u]
[i]Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.[/i]
[b]6.1.[/b] Let the answer to problem $2$ be $b$. There are b people in a room, each of which is either a truth-teller or a liar. Person $1$ claims “Person $2$ is a liar,” Person $2$ claims “Person $3$ is a liar,” and so on until Person $b$ claims “Person $1$ is a liar.” How many people are truth-tellers?
[b]6.2.[/b] Let the answer to problem $3$ be $c$. What is twice the area of a triangle with coordinates $(0, 0)$, $(c, 3)$ and $(7, c)$ ?
[b]6.3.[/b] Let the answer to problem $ 1$ be $a$. Compute the smaller zero to the polynomial $x^2 - ax + 189$ which has $2$ integer roots.
[u]Round 7[/u]
[b]7.1. [/b]Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states:
[i]Every minute, either $\left\lfloor \frac{1000}{d} \right\rfloor$ or $\left\lceil \frac{1000}{d} \right\rceil$ kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.[/i]
Over the next minute, $n$ kiwis fall on Neeton’s head. Let $S$ be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that $m < x < M$ for all elements $x$ in $S$. If the least possible value of $M - m$ is $\frac{2000}{16899}$ centimeters, what is the value of $n$?
Note that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the least integer greater than or equal to $x$.
[b]7.2.[/b] Nithin is playing chess. If one queen is randomly placed on an $ 8 \times 8$ chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.)
[b]7.3.[/b] Nithin is writing binary strings, where each character is either a $0$ or a $1$. How many binary strings of length $12$ can he write down such that $0000$ and $1111$ do not appear?
[u]Round 8[/u]
[b]8.[/b] What is the period of the fraction $1/2018$? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2765571p24215461]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$
Triangle $ABC$ is such that $\angle BAC>\angle ABC>60^o$. The perpendicular bisector of $\overline{AB}$ intersects the segment $\overline {BC}$ at $O$. Suppose there exists a point $D$ on the segment $\overline{AC}$ such that $OD=AB$ and $\angle ODA=30^o$. Find $\angle BAC$.
[i](Vlad Robu)[/i]
What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed).