Found problems: 5923
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that:
$ \textbf{(A)}\ \text{they form an arithmetic progression} \\
\textbf{(B)}\ \text{they form a geometric progression} \\
\textbf{(C)}\ \text{they are unequal} \\
\textbf{(D)}\ \text{they are all negative numbers} \\
\textbf{(E)}\ \text{only b is negative and a and c are positive}$
For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g $P(6)=36$). For every positive integer $n$ define the sequence
$$a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n))\quad (k=1,2,\dots,2016)$$
Determine whether for every set $S\subset\{1,2,\dots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:
For every $k$ with $1\leq k\leq 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
A positive integer is called [i]uphill [/i] if the digits in its decimal representation form an increasing sequence from left to right. That is, a number $\overline{a_1a_2... a_n}$ is uphill if $a_i \le a_{i+1}$ for all $i$. For example, $123$ and $114$ are both uphill. Suppose a polynomial $P(x)$ with rational coefficients takes on an integer value for each uphill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?
$ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $
Let $ \{k_n\}_{n \equal{} 1}^{\infty}$ be a sequence of real numbers having the properties $ k_1 > 1$ and $ k_1 \plus{} k_2 \plus{} \cdots \plus{} k_n < 2k_n$ for $ n \equal{} 1,2,...$. Prove that there exists a number $ q > 1$ such that $ k_n > q^n$ for every positive integer $ n$.
A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.
Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$.
[b]a) [/b]Is it possible that the sequence contains exactly $11$ terms?
[b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?
The sequence $\{a_{n}\}_{n \ge 1}$ is defined by $a_{1}=1$ and \[a_{n+1}= \frac{a_{n}}{2}+\frac{1}{4a_{n}}\; (n \in \mathbb{N}).\] Prove that $\sqrt{\frac{2}{2a_{n}^{2}-1}}$ is a positive integer for $n>1$.
Let $a_1, . . . , a_{2020}$ be a sequence of real numbers such that $a_1 = 2^{-2019}$, and $a^2_{n-1}a_n = a_n-a_{n-1}$.
Prove that $a_{2020} <\frac{1}{2^{2019} -1}$
Does there exist an infinite sequence $\{x_{n}\}$ of reals satisfying the following conditions
i)$|x_{n}|\leq 0,666$ for all $n=1,2,...$
ii)$|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)}$ for all $m\not = n$?
How many ways are there to arrange the letters $A,A,A,H,H$ in a row so that the sequence $HA$ appears at least once?
[i]Author: Ray Li[/i]
On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
The sides of $\triangle ABC$ form an arithmetic sequence of integers. Incircle $I$ is tangent to $AB$, $BC$, and $CA$ at $D$, $E$, and $F$, respectively. Given that $DB = \tfrac32$, $FA = \tfrac12$, find the radius of $I$.
$\textbf{(A) } \dfrac12\qquad\textbf{(B) } \dfrac{\sqrt{15}}7\qquad\textbf{(C) } \dfrac{\sqrt{15}}6\qquad\textbf{(D) } \dfrac{2\sqrt{15}}{9}\qquad\textbf{(E) } \dfrac{\sqrt{15}}{4}$
Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the[i] odd part[/i] of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the [i]odd part[/i] of $3a_k+1$. Find $a_n$.
Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
Let $X$ and $Y$ be the following sums of arithmetic sequences: \begin{eqnarray*} X &=& 10 + 12 + 14 + \cdots + 100, \\ Y &=& 12 + 14 + 16 + \cdots + 102. \end{eqnarray*} What is the value of $Y - X$?
$ \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 $
[u]Round 5[/u]
[b]5.1.[/b] Julia baked a pie for herself to celebrate pi day this year. If Julia bakes anyone pie on pi day, the following year on pi day she bakes a pie for herself with $1/3$ probability, she bakes her friend a pie with $1/6$ probability, and she doesn't bake anyone a pie with $1/2$ probability. However, if Julia doesn't make pie on pi day, the following year on pi day she bakes a pie for herself with $1/2$ probability, she bakes her friend a pie with $1/3$ probability, and she doesn't bake anyone a pie with $1/6$ probability. The probability that Julia bakes at least $2$ pies on pi day in the next $5$ years can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$.
[b]5.2.[/b] Steven is flipping a coin but doesn't want to appear too lucky. If he ips the coin $8$ times, the probability he only gets sequences of consecutive heads or consecutive tails that are of length $4$ or less can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$.
[b]5.3.[/b] Let $ABCD$ be a square with side length $3$. Further, let $E$ be a point on side$ AD$, such that $AE = 2$ and $DE = 1$, and let $F$ be the point on side $AB$ such that triangle $CEF$ is right with hypotenuse $CF$. The value $CF^2$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
[u]Round 6[/u]
[b]6.1.[/b] Let $P$ be a point outside circle $\omega$ with center $O$. Let $A,B$ be points on circle $\omega$ such that $PB$ is a tangent to $\omega$ and $PA = AB$. Let $M$ be the midpoint of $AB$. Given $OM = 1$, $PB = 3$, the value of $AB^2$ can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$.
[b]6.2.[/b] Let $a_0, a_1, a_2,...$with each term defined as $a_n = 3a_{n-1} + 5a_{n-2}$ and $a_0 = 0$, $a_1 = 1$. Find the remainder when $a_{2020}$ is divided by $360$.
[b]6.3.[/b] James and Charles each randomly pick two points on distinct sides of a square, and they each connect their chosen pair of points with a line segment. The probability that the two line segments intersect can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$.
[u]Round 7[/u]
[b]7.1.[/b] For some positive integers $x, y$ let $g = gcd (x, y)$ and $\ell = lcm (2x, y)$: Given that the equation $xy+3g+7\ell = 168$ holds, find the largest possible value of $2x + y$.
[b]7.2.[/b] Marco writes the polynomials $$f(x) = nx^4 +2x^3 +3x^2 +4x+5$$ and $$g(x) = a(x-1)^4 +b(x-1)^3 +6(x-1)^2 + d(x - 1) + e,$$ where $n, a, b, d, e$ are real numbers. He notices that $g(i) = f(i) - |i|$ for each integer $i$ satisfying $-5 \le i \le -1$. Then $n^2$ can be expressed as $p/q$ for relatively prime positive integers $p, q$. Find $p + q$.
[b]7.3. [/b]Equilateral $\vartriangle ABC$ is inscribed in a circle with center $O$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $BC$, respectively. Segment $\overline{CD}$ intersects $\overline{AB}$ and $\overline{AE}$ at $Y$ and $X$, respectively. Given that $\vartriangle DXE$ and $\vartriangle AXC$ have equal area, $\vartriangle AXY$ has area $ 1$, and $\vartriangle ABC$ has area $52$, find the area of $\vartriangle BXC$.
[u]Round 8[/u]
[b]8.[/b] Let $A$ be the number of total webpage visits our website received last month. Let $B$ be the number photos in our photo collection from ABMC onsite 2017. Let $M$ be the mean speed round score. Further, let $C$ be the number of times the letter c appears in our problem bank. Estimate
$$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766251p24226451]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Dertemine integers $a_{1},a_{2},...,a_{99}=a_{0}$ satisfying $|a_{k}-a_{k-1}|\geq 1996$ for all $k=1,2,...,99$, such that $m=\max_{1\leq k\leq 99} |a_{k}-a_{k-1}|$ is minimum possible, and find the minimum value $m^{*}$ of $m$.
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that
$$k\le2\lfloor\sqrt n\rfloor.$$
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent