Found problems: 373
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
If the sum of the real roots $x$ to each of the equations
\[2^{2x}-2^{x+1}+1-\frac{1}{k^2}=0\]
for $k=2,3,\dots,2023$ is $N$, what is $2^N$?
For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following:
\begin{align*}
001 &&001 && 000 \\
2017 &&4034 &&6051 \\
+2017 &&+2017 &&+2017\\
=4034 &&=6051 &&=8068\\
\end{align*}
Prove that
$$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$
Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$
for $n\ge2$.
Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\,
\frac{1}{F(2^n)}}$ is a rational number.
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).
(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.
(b) Find the largest possible size of a very set not contained in any line.
(Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)
[i]Proposed by Sammy Luo[/i]
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if
\[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$. If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{M}{N} \] for relatively prime positive integers $M$ and $N$, compute $100M+N$.
[i]Based on a proposal by ssilwa[/i]
How many positive integer factors does the following expression have? \[ \sum_{n=1}^{999} \log_{10} \left(\frac{n+1}{n} \right) \]
[i]2022 CCA Math Bonanza Tiebreaker Round #1[/i]
The sequences $(a_n), (b_n),$ and $(c_n)$ are de fined by $a_0 = 1, b_0 = 0, c_0 = 0,$ and
\[a_n = a_{n-1} +
\frac{c_{n-1}}{n}, b_n = b_{n-1} +\frac{a_{n-1}}{n}, c_n = c_{n-1} +\frac{b_{n-1}}{n}\]
for all $n \geq1$. Prove that
\[\left|a_n -\frac{n + 1}{3}\right|<\frac{2}{\sqrt{3n}}\]
for all $n \geq 1$.
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Given that positive integers $a,b$ satisfy \[\frac{1}{a+\sqrt{b}}=\sum_{i=0}^\infty \frac{\sin^2\left(\frac{10^\circ}{3^i}\right)}{\cos\left(\frac{30^\circ}{3^i}\right)},\] where all angles are in degrees, compute $a+b$.
[i]2021 CCA Math Bonanza Team Round #10[/i]
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Solve for $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that
$$g(m + n) = g(m) + mn(m + n) + g(n)$$
Show that $g(n)$ is of the form $\sum_{i=0}^{d} {c_i n^i}$ \\
and find necessary and sufficient conditions on $d$ and $c_0, c_1, \cdots , c_d$
Let $ f(r) \equal{} \sum_{j \equal{} 2}^{2008} \frac {1}{j^r} \equal{} \frac {1}{2^r} \plus{} \frac {1}{3^r} \plus{} \dots \plus{} \frac {1}{2008^r}$. Find $ \sum_{k \equal{} 2}^{\infty} f(k)$.
Evaluate $$\sum^{17}_{n=2} \frac{n^2+n+1}{n^4+2n^3-n^2-2n}.$$
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]
Define the sequence $a_1,a_2,a_3,\ldots$ by $a_n=\sum_{k=1}^n\sin(k)$, where $k$ represents radian measure. Find the index of the $100$th term for which $a_n<0$.
Is there exist a sequence $a_0,a_1,a_2,\cdots $ consisting of non-zero integers that satisfies the following condition?
[b]Condition[/b]: For all integers $n$ ($\ge 2020$), equation
$$a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0$$
has a real root with its absolute value larger than $2.001$.
Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of
\[
\left|1-\sum_{i \in X} a_{i}\right|
\]
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that
\[
\sum_{i \in X} b_{i}=1.
\]
[i]25 problems for 30 minutes[/i]
[b]p1.[/b] Today is Saturday, April $25$, $2020$. What is the value of $6 + 4 + 25 + 2020$?
[b]p2.[/b] The figure below consists of a $2$ by $3$ grid of squares. How many squares of any size are in the grid?
$\begin{tabular}{|l|l|l|}
\hline
& & \\ \hline
& & \\ \hline
\end{tabular}$
[b]p3.[/b] James is playing a game. He first rolls a six-sided dice which contains a different number on each side, then randomly picks one of twelve di erent colors, and finally ips a quarter. How many different possible combinations of a number, a color and a flip are there in this game?
[b]p4.[/b] What is the sum of the number of diagonals and sides in a regular hexagon?
[b]p5.[/b] Mickey Mouse and Minnie Mouse are best friends but they often fight. Each of their fights take up exactly one hour, and they always fight on prime days. For example, they fight on January $2$nd, $3$rd, but not the $4$th. Knowing this, how many total times do Mickey and Minnie fight in the months of April, May and June?
[b]p6.[/b] Apple always loved eating watermelons. Normal watermelons have around $13$ black seeds and $25$ brown seeds, whereas strange watermelons had $45$ black seeds and $2$ brown seeds. If Apple bought $14$ normal watermelons and $7$ strange watermelons, then let $a$ be the total number of black seeds and $b$ be the total number of brown seeds. What is $a - b$?
[b]p7.[/b] Jerry and Justin both roll a die once. The probability that Jerry's roll is greater than Justin's can be expressed as a fraction in the form $\frac{m}{n}$ in simplified terms. What is $m + n$?
[b]p8.[/b] Taylor wants to color the sides of an octagon. What is the minimum number of colors Taylor will need so that no adjacent sides of the octagon will be filled in with the same color?
[b]p9.[/b] The point $\frac23$ of the way from ($-6, 8$) to ($-3, 5$) can be expressed as an ordered pair $(a, b)$. What is $|a - b|$?
[b]p10.[/b] Mary Price Maddox laughs $7$ times per class. If she teaches $4$ classes a day for the $5$ weekdays every week but doesn't laugh on Wednesdays, then how many times does she laugh after $5$ weeks of teaching?
[b]p11.[/b] Let $ABCD$ be a unit square. If $E$ is the midpoint of $AB$ and $F$ lies inside $ABCD$ such that $CFD$ is an equilateral triangle, the positive difference between the area of $CED$ and $CFD$ can be expressed in the form $\frac{a-\sqrt{b}}{c}$ , where $a$, $b$, $c$ are in lowest simplified terms. What is $a + b + c$?
[b]p12.[/b] Eddie has musician's syndrome. Whenever a song is a $C$, $A$, or $F$ minor, he begins to cry and his body becomes very stiff. On the other hand, if the song is in $G$ minor, $A$ at major, or $E$ at major, his eyes open wide and he feels like the happiest human being ever alive. There are a total of $24$ keys. How many different possibilities are there in which he cries while playing one song with two distinct keys?
[b]p13.[/b] What positive integer must be added to both the numerator and denominator of $\frac{12}{40}$ to make a fraction that is equivalent to $\frac{4}{11}$ ?
[b]p14.[/b] The number $0$ is written on the board. Each minute, Gene the genie either multiplies the number on the board by $3$ or $9$, each with equal probability, and then adds either $1$,$2$, or $3$, each with equal probability. Find the expected value of the number after $3$ minutes.
[b]p15.[/b] $x$ satisfies $\dfrac{1}{x+ \dfrac{1}{1+\frac{1}{2}}}=\dfrac{1}{2+ \dfrac{1}{1- \dfrac{1}{2+\frac{1}{2}}}}$
Find $x$.
[b]p16.[/b] How many different points in a coordinate plane can a bug end up on if the bug starts at the origin and moves one unit to the right, left, up or down every minute for $8$ minutes?
[b]p17.[/b] The triplets Addie, Allie, and Annie, are racing against the triplets Bobby, Billy, and Bonnie in a relay race on a track that is $100$ feet long. The first person of each team must run around the entire track twice and tag the second person for the second person to start running. Then, the second person must run once around the entire track and tag the third person, and finally, the third person would only have to run around half the track. Addie and Bob run first, Allie and Billy second, Annie and Bonnie third. Addie, Allie, and Annie run at $50$ feet per minute (ft/m), $25$ ft/m, and $20$ ft/m, respectively. If Bob, Billy, and Bonnie run half as fast as Addie, Allie, and Annie, respectively, then how many minutes will it take Bob, Billy, and Bonnie to finish the race. Assume that everyone runs at a constant rate.
[b]p18.[/b] James likes to play with Jane and Jason. If the probability that Jason and Jane play together is $\frac13$, while the probability that James and Jason is $\frac14$ and the probability that James and Jane play together is $\frac15$, then the probability that they all play together is $\frac{\sqrt{p}}{q}$ for positive integers $p$, $q$ where $p$ is not divisible by the square of any prime. Find $p + q$.
[b]p19.[/b] Call an integer a near-prime if it is one more than a prime number. Find the sum of all near-primes less than$ 1000$ that are perfect powers. (Note: a perfect power is an integer of the form $n^k$ where $n, k \ge 2$ are integers.)
[b]p20.[/b] What is the integer solution to $\sqrt{\frac{2x-6}{x-11}} = \frac{3x-7}{x+6}$ ?
[b]p21.[/b] Consider rectangle $ABCD$ with $AB = 12$ and $BC = 4$ with $F$,$G$ trisecting $DC$ so that $F$ is closer to $D$. Then $E$ is on $AB$. We call the intersection of $EF$ and $DB$ $X$, and the intersection of $EG$ and $DB$ is $Y$. If the area of $\vartriangle XY E$ is \frac{8}{15} , then what is the length of $EB$?
[b]p22.[/b] The sum $$\sum^{\infty}_{n=2} \frac{1}{4n^2-1}$$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$.
[b]p23.[/b] In square $ABCD$, $M$, $N$, $O$, $P$ are points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, respectively. If $AB = 4$, $AM = BM$ and $DP = 3AP$, the least possible value of $MN + NO + OP$ can be expressed as $\sqrt{x}$ forsome integer x. Find x:
[b]p24.[/b] Grand-Ovich the ant is at a vertex of a regular hexagon and he moves to one of the adjacent vertices every minute with equal probability. Let the probability that after $8$ minutes he will have returned to the starting vertex at least once be the common fraction $\frac{a}{b}$ in lowest terms. What is $a + b$?
[b]p25.[/b] Find the last two non-zero digits at the end of $2020!$ written as a two digit number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$