Found problems: 5923
Let $c_1, \cdots, c_n \ (n \geq 2)$ be real numbers such that $0 \leq \sum c_i \leq n$. Prove that there exist integers $k_1, \cdots , k_n$ such that $\sum k_i=0$ and $1-n \leq c_i + nk_i \leq n$ for every $i = 1, \cdots , n.$
Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.
Let $a_1 \le a_2 \le ... \le a_n$ be real numbers for which $$\sum_{i=1}^{n} a_i^{2k+1} = 0$$ holds for all integers $0 \le k < n$. Show that in this case, $a_i = -a_{n+1-i}$ holds for all $1 \le i \le n$.
Determine how many $100$-positive integer sequences satisfy the two conditions following:
- At least one term of the sequence is equal to $4$ or $5$.
- Any two adjacent terms differ as a maximum in $2$.
A positive integer $k$ is given. Initially, $N$ cells are marked on an infinite checkered plane. We say that the cross of a cell $A$ is the set of all cells lying in the same row or in the same column as $A$. By a turn, it is allowed to mark an unmarked cell $A$ if the cross of $A$ contains at least $k$ marked cells. It appears that every cell can be marked in a sequence of such turns. Determine the smallest possible value of $N$.
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
$ \textbf{(A)}\ 0.10 \qquad
\textbf{(B)}\ 0.15 \qquad
\textbf{(C)}\ 0.20 \qquad
\textbf{(D)}\ 0.25 \qquad
\textbf{(E)}\ 0.30$
Let $({{x}_{n}})$ be an integer sequence such that $0\le {{x}_{0}}<{{x}_{1}}\le 100$ and
$${{x}_{n+2}}=7{{x}_{n+1}}-{{x}_{n}}+280,\text{ }\forall n\ge 0.$$
a) Prove that if ${{x}_{0}}=2,{{x}_{1}}=3$ then for each positive integer $n,$ the sum of divisors of the following number is divisible by $24$ $${{x}_{n}}{{x}_{n+1}}+{{x}_{n+1}}{{x}_{n+2}}+{{x}_{n+2}}{{x}_{n+3}}+2018.$$
b) Find all pairs of numbers $({{x}_{0}},{{x}_{1}})$ such that ${{x}_{n}}{{x}_{n+1}}+2019$ is a perfect square for infinitely many nonnegative integer numbers $n.$
Compute the number of rearrangements $a_1, a_2, \dots, a_{2018}$ of the sequence $1, 2, \dots, 2018$ such that $a_k > k$ for $\textit{exactly}$ one value of $k$.
Given an alfabet of $n$ letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a [i]word[/i].
a) Find the maximal possible length of a [i]word[/i].
b) Find the number of the [i]words[/i] of maximal length.
A coin is flipped $20$ times. Let $p$ be the probability that each of the following sequences of flips occur exactly twice:
[list]
[*] one head, two tails, one head
[*] one head, one tails, two heads.
[/list]
Given that $p$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $\gcd (m,n)$.
[i]2021 CCA Math Bonanza Lightning Round #1.3[/i]
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]C.1 / G.1[/b] Daniel is exactly one year younger than his friend David. If David was born in the year $2008$, in what year was Daniel born?
[b]C.2 / G.3[/b] Mr. Pham flips three coins. What is the probability that no two coins show the same side?
[b]C.3 / G.2[/b] John has a sheet of white paper which is $3$ cm in height and $4$ cm in width. He wants to paint the sky blue and the ground green so the entire paper is painted. If the ground takes up a third of the page, how much space (in cm$^2$) does the sky take up?
[b]C.4 / G.5[/b] Jihang and Eric are busy fidget spinning. While Jihang spins his fidget spinner at $15$ revolutions per second, Eric only manages $10$ revolutions per second. How many total revolutions will the two have made after $5$ continuous seconds of spinning?
[b]C.5 / G.4[/b] Find the last digit of $1333337777 \cdot 209347802 \cdot 3940704 \cdot 2309476091$.
[u]Set 2[/u]
[b]C.6[/b] Evan, Chloe, Rachel, and Joe are splitting a cake. Evan takes $\frac13$ of the cake, Chloe takes $\frac14$, Rachel takes $\frac15$, and Joe takes $\frac16$. There is $\frac{1}{x}$ of the original cake left. What is $x$?
[b]C.7[/b] Pacman is a $330^o$ sector of a circle of radius $4$. Pacman has an eye of radius $1$, located entirely inside Pacman. Find the area of Pacman, not including the eye.
[b]C.8[/b] The sum of two prime numbers $a$ and $b$ is also a prime number. If $a < b$, find $a$.
[b]C.9[/b] A bus has $54$ seats for passengers. On the first stop, $36$ people get onto an empty bus. Every subsequent stop, $1$ person gets off and $3$ people get on. After the last stop, the bus is full. How many stops are there?
[b]C.10[/b] In a game, jumps are worth $1$ point, punches are worth $2$ points, and kicks are worth $3$ points. The player must perform a sequence of $1$ jump, $1$ punch, and $1$ kick. To compute the player’s score, we multiply the 1st action’s point value by $1$, the $2$nd action’s point value by $2$, the 3rd action’s point value by $3$, and then take the sum. For example, if we performed a punch, kick, jump, in that order, our score would be $1 \times 2 + 2 \times 3 + 3 \times 1 = 11$. What is the maximal score the player can get?
[u]Set 3[/u]
[b]C.11[/b] $6$ students are sitting around a circle, and each one randomly picks either the number $1$ or $2$. What is the probability that there will be two people sitting next to each other who pick the same number?
[b]C.12 / G. 8[/b] You can buy a single piece of chocolate for $60$ cents. You can also buy a packet with two pieces of chocolate for $\$1.00$. Additionally, if you buy four single pieces of chocolate, the fifth one is free. What is the lowest amount of money you have to pay for $44$ pieces of chocolate? Express your answer in dollars and cents (ex. $\$3.70$).
[b]C.13 / G.12[/b] For how many integers $k$ is there an integer solution $x$ to the linear equation $kx + 2 = 14$?
[b]C.14 / G.9[/b] Ten teams face off in a swim meet. The boys teams and girls teams are ranked independently, each team receiving some number of positive integer points, and the final results are obtained by adding the points for the boys and the points for the girls. If Blair’s boys got $7$th place while the girls got $5$th place (no ties), what is the best possible total rank for Blair?
[b]C.15 / G.11[/b] Arlene has a square of side length $1$, an equilateral triangle with side length $1$, and two circles with radius $1/6$. She wants to pack her four shapes in a rectangle without items piling on top of each other. What is the minimum possible area of the rectangle?
PS. You should use hide for answers. C16-30/G10-15, G25-30 have been posted [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here [/url] . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$
(1) Find the minimum of $f_{2020}$.
(2) Find the minimum of $f_{2020} \cdot f_{2021}$.
Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n \geq 2$ and sequence of non-integer positive real numbers $a_1, a_2, \dots, a_n$ satisfying $$\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = 1,$$ it's possible to choose positive integers $b_i$ such that
(i) for each $i = 1, 2, \dots, n$, either $b_i = \lfloor a_i \rfloor$ or $b_i = \lfloor a_i \rfloor + 1$, and
(ii) we have $$1 < \frac{1}{b_1} + \frac{1}{b_2} + \cdots + \frac{1}{b_n} \leq C.$$
(Here $\lfloor \bullet \rfloor$ denotes the floor function, as usual.)
[i]Merlijn Staps[/i]
A regular $n$-gon is inscribed in a circle of radius $1$. Let $a_1,\cdots,a_{n-1}$ be the distances of one of the vertices of the polygon to all the other vertices. Prove that
\[(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2\]
where $F_n$ is the $n^{th}$ term of the Fibonacci sequence $1,1,2,\cdots$
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
Prove that for any three infinite sequences of natural numbers $(a_n)_{n\ge 1}$, $(b_n)_{n\ge 1}$, $(c_n)_{n\ge 1}$, there exist numbers $p$ and $q$ such that $a_p\ge a_q$, $b_p\ge b_q$ and $c_p\ge c_q$.
Which positive integers are missing in the sequence $ \left\{a_n\right\}$, with $ a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right]$ for all $ n \ge 1$? ($ \left[x\right]$ denotes the largest integer less than or equal to $ x$, i.e. $ g$ with $ g \le x < g \plus{} 1$.)
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.
Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$.
Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture.
(yes, $cd$ is parallel to $a_1b_1$ there)
Show $A_k < S$ for every positive integer $k$.
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284[/img]
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.)
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that:
a) ${5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).$
b) ${F_{p - 1}} \equiv 0(\bmod p).$
c) $k(p)|p-1.$
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that
\[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection.
Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.
[i]Proposed by Warut Suksompong, Thailand[/i]