Found problems: 85335
Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$, then $N = 7$.
Let $ a, b, c $ be the side length of a triangle, with $ ab + bc + ca = 18 $ and $ a, b, c > 1 $. Prove that $$ \sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3} $$
In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.
Suppose $N$ is an $n$ digit positive integer such that
(a) all its digits are distinct;
(b) the sum of any three consecutive digits is divisible by $5$.
Prove that $n \leq 6$. Further, show that starting with any digit, one can find a six digit number with these properties.
For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds
$$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$
Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$.
[i]Proposed by Sammy Luo[/i]
A function $g$ is [i]ever more[/i] than a function $h$ if, for all real numbers $x$, we have $g(x) \ge h(x)$. Consider all quadratic functions $f(x)$ such that $f(1) = 16$ and $f(x)$ is ever more than both $(x + 3)^2$ and $x^2 + 9$. Across all such quadratic functions $f$, compute the minimum value of $f(0)$.
Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]2021 CCA Math Bonanza Individual Round #8[/i]
Find positive integers $ a,b$ if for every $ x,y\in[a,b]$, $ \frac1x\plus{}\frac1y\in[a,b]$.
Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$
If $a_1, a_2, \cdots, a_7$ are not necessarily distinct real numbers such that $1 < a_i < 13$ for all $i$, then show that we can choose three of them such that they are the lengths of the sides of a triangle.
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.
Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?
We fill in an $n\times n$ table with real numbers such that the sum of the numbers in each row and each coloumn equals $1$. For which values of $K$ is the following statement true: if the sum of the absolute values of the negative entries in the table is at most $K$, then it's always possible to choose $n$ positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries.
[i]Proposed by Dávid Bencsik, Budapest[/i]
Let $\oplus$ denote the xor binary operation. Define $x \star y=(x+y)-(x\oplus y).$ Compute $$\sum_{k=1}^{63} (k \star 45).$$([i]Remark:[/i] The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2=1001_2=9.$)
101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers".
What is the maximum possible amount of the statements that can be true?
[i]M. Karpuk[/i]
The angles of quadrilateral $ ABCD$ satisfy $ \angle A \equal{} 2 \angle B \equal{} 3 \angle C \equal{} 4 \angle D$. What is the degree measure of $ \angle A$, rounded to the nearest whole number?
$ \textbf{(A)}\ 125 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 153 \qquad \textbf{(D)}\ 173 \qquad \textbf{(E)}\ 180$
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes.
$ (b)$ Prove that every integer can be written as a sum of five cubes.
You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.
Find all polynomials $P(x)$ of degree $1$ such that
$\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$
Let $a, b, c \in \mathbb{R}^+$ such that $a + b + c = 3$. Prove that $$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.$$
Let $n$ be a positive integer. Calculate the sum $\sum_{k=1}^n\ \ {\sum_{1\le i_1 < \ldots < i_k\le n}^{}{\frac {2^k}{(i_1 + 1)(i_2 + 1)\ldots (i_k + 1)}}}$
Let $ ABCD $ be an inscriptible quadrilateral and $ M $ be a point on its circumcircle, distinct from its vertices. Let $ H_1,H_2,H_3,H_4 $ be the orthocenters of $ MAB,MBC, MCD, $ respectively, $ MDA, $ and $ E,F, $ the midpoints of the segments $ AB, $ respectivley, $ CD. $ Prove that:
[b]a)[/b] $ H_1H_2H_3H_4 $ is a parallelogram.
[b]b)[/b] $ H_1H_3=2\cdot EF. $