Consider the sequence $\{a_n\}$ defined so that $a_n$ is the leftmost digit of $2^n$. The first few terms of this sequence are $1,2,4,8,1,3,6,\dots$. For how many $0\le n\le100000$ is $a_n=1$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max\left(0,25-\tfrac{1}{6}\sqrt{|A-C|}\right)$.

Let $K_1, K_2, K_3, K_4, K_5$ be 5 distinguishable keys, and let $D_1, D_2, D_3, D_4, D_5$ be $5$ distinguishable doors. For $1 \leq i \leq 5$, key $K_i$ opens doors $D_{i}$ and $D_{i+1}$ (where $D_6 = D_1$) and can only be used once. The keys and doors are placed in some order along a hallway. Key\$ha walks into the hallway, picks a key and opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many orders can the keys and doors be placed such that Key\$ha can open all of the doors? [i]Author: Mitchell Lee[/i] [hide="Clarifications"] [list=1][*]The doors and keys are in series. In other words, the doors aren't lined up along the side of the hallway. They are blocking Key\$ha's path to the end, and the only way she can get past them is by getting the appropriate keys along the hallway. [*]The doors and keys appear consecutively along the hallway. For example, she might find $K_1 D_1 K_2 D_2 K_3 D_3 K_4 D_4 K_5 D_5$ down the hallway in that order. Also, by "she never obtains a key before all the doors in front of it are unlocked," we mean that she cannot obtain a key before all the doors appearing before the key are unlocked. In essence, it merely states that locked doors cannot be passed. [*]The doors and keys do not need to alternate down the hallway.[/list][/hide]

Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.

Let $A$ and $B$ infinite sets of positive real numbers such that: 1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$. 2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$. Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.

Let $D$ be a point inside of equilateral $\triangle ABC$, and $E$ be a point outside of equilateral $\triangle ABC$ such that $m(\widehat{BAD})=m(\widehat{ABD})=m(\widehat{CAE})=m(\widehat{ACE})=5^\circ$. What is $m(\widehat{EDC})$ ? $ \textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 25^\circ $

A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

Let $x, y, z$ are real numbers such that $x<y<z .$ Prove that $$ (x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0 $$

A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$. Restore the triangle.

If $ \log_{10} (x^2\minus{}3x\plus{}6)\equal{}1$, the value of $ x$ is: $ \textbf{(A)}\ 10\text{ or }2 \qquad\textbf{(B)}\ 4\text{ or }\minus{}2 \qquad\textbf{(C)}\ 3\text{ or }\minus{}1 \qquad\textbf{(D)}\ 4\text{ or }\minus{}1\\ \textbf{(E)}\ \text{none of these}$

The Fibonacci sequence is defined as $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$ ($n\in\mathbb{N}$). Suppose that $a$ and $b$ are positive integers such that $\frac ab$ lies between the two fractions $\frac{f_n}{f_{n-1}}$ and $\frac{f_{n+1}}{f_{n}}$. Show that $b\ge f_{n+1}$.

We wish to lay down boards on a floor with width $B$ in the direction across the boards. We have $n$ boards of width $b$, and $B/b$ is an integer, and $nb \le B$. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end. [img]https://cdn.artofproblemsolving.com/attachments/f/f/24ce8ae05d85fd522da0e18c0bb8017ca3c8e8.png[/img]

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

Find all ordered pairs of integers $(a,b)$ such that there exists a function $f\colon \mathbb{N} \to \mathbb{N}$ satisfying $$f^{f(n)}(n)=an+b$$ For all $n\in \mathbb{N}$.

Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients

Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.

Cole is trying to solve the [i]Collatz conjecture[/i]. She decides to make a model with a piece of wood with a hole for every natural number. For every even number there is a rope from $n$ to $\frac{n}{2}$ and for every odd number there is a rope from $n$ to $3n+1$. She wants to bring her model to a convention but in order to do that she needs to cut off the part containing the first $240$ holes. How many ropes did she break? [i]2017 CCA Math Bonanza Individual Round #4[/i]