Found problems: 5923
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$.
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_1=1$. For each $n \geq 1$, $a_{n+1}$ is the smallest positive integer, distinct from $a_1,a_2,...,a_n$, such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. Prove that every positive integer appears in $\{a_n\}_{n=1}^{\infty}$.
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power.
Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.
Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$
For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$.
Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.
Define sequence $(a_n)$: $a_0=1,a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2},n\in\mathbb{N}$.
[b](a)[/b] Prove that for all $n\in\mathbb{N}$, $a_n$ is a positive integer.
[b](b)[/b] Prove that for all $n\in\mathbb{N}$, $a_na_{n+1}-1$ is a perfect square.
Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$
Let $(a_n)$ be a sequence with $a_1=0$ and $a_{n+1}=2+a_n$ for odd $n$ and $a_{n+1}=2a_n$ for even $n$. Prove that for each prime $p>3$, the number
\begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*}
for infinitely many values of $n$
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.
Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds:
\[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\]
Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$
\[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]
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 $n$.
a) Prove that: For all $m_1,m_2\in \mathbb{Z^+}$, we have:$$k([m_1,m_2])=[k(m_1),k(m_2)].$$(Here $[a,b]$ is the least common multiple of $a,b.$)
b) Determine $k(2),k(4),k(5),k(10).$
$ 90\plus{}91\plus{}92\plus{}93\plus{}94\plus{}95\plus{}96\plus{}97\plus{}98\plus{}99\equal{}$
\[ \textbf{(A)}\ 845 \qquad
\textbf{(B)}\ 945 \qquad
\textbf{(C)}\ 1005 \qquad
\textbf{(D)}\ 1025 \qquad
\textbf{(E)}\ 1045
\]
Let $ \left( x_n \right)_{n\ge 0} $ be a sequence of positive real numbers with $ x_0=1 $ and defined recursively:
$$ x_{n+1}=x_n+\frac{x_0}{x_1+x_2+\cdots +x_n} $$
[b]a)[/b] Show that $ \lim_{n\to\infty } x_n=\infty . $
[b]b)[/b] Calculate $ \lim_{n\to\infty }\frac{x_n}{\sqrt{\ln n}} . $
[i]Ovidiu Furdui[/i]
How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below?
1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$.
2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.
[i]20 problems for 20 minutes.[/i]
[b]p1.[/b] Evaluate $=\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}$.
[b]p2.[/b] A regular hexagon and a regular $n$-sided polygon have the same perimeter. If the ratio of the side length of the hexagon to the side length of the $n$-sided polygon is $2 : 1$, what is $n$?
[b]p3.[/b] How many nonzero digits are there in the decimal representation of $2 \cdot 10\cdot 500 \cdot 2500$?
[b]p4.[/b] When the numerator of a certain fraction is increased by $2012$, the value of the fraction increases by $2$. What is the denominator of the fraction?
[b]p5.[/b] Sam did the computation $1 - 10 \cdot a + 22$, where $a$ is some real number, except he messed up his order of operations and computed the multiplication last; that is, he found the value of $(1 - 10) \cdot (a + 22)$ instead. Luckily, he still ended up with the right answer. What is $a$?
[b]p6.[/b] Let $n! = n \cdot(n-1) \cdot\cdot\cdot 2 \cdot 1$. For how many integers $n$ between $1$ and $100$ inclusive is $n!$ divisible by $36$?
[b]p7.[/b] Simplify the expression $\sqrt{\frac{3 \cdot 27^3}{27 \cdot 3^3}}$
[b]p8.[/b] Four points $A,B,C,D$ lie on a line in that order such that $\frac{AB}{CB}=\frac{AD}{CD}$ . Let $M$ be the midpoint of segment $AC$. If $AB = 6$, $BC = 2$, compute $MB \cdot MD$.
[b]p9.[/b] Allan has a deck with $8$ cards, numbered $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$. He pulls out cards without replacement, until he pulls out an even numbered card, and then he stops. What is the probability that he pulls out exactly $2$ cards?
[b]p10.[/b] Starting from the sequence $(3, 4, 5, 6, 7, 8, ... )$, one applies the following operation repeatedly. In each operation, we change the sequence $$(a_1, a_2, a_3, ... , a_{a_1-1}, a_{a_1} , a_{a_1+1},...)$$ to the sequence $$(a_2, a_3, ... , a_{a_1} , a_1, a_{a_1+1}, ...) .$$ (In other words, for a sequence starting with$ x$, we shift each of the next $x-1$ term to the left by one, and put x immediately to the right of these numbers, and keep the rest of the terms unchanged. For example, after one operation, the sequence is $(4, 5, 3, 6, 7, 8, ... )$, and after two operations, the sequence becomes $(5, 3, 6, 4, 7, 8,... )$. How many operations will it take to obtain a sequence of the form $(7, ... )$ (that is, a sequence starting with $7$)?
[b]p11.[/b] How many ways are there to place $4$ balls into a $4\times 6$ grid such that no column or row has more than one ball in it? (Rotations and reflections are considered distinct.)
[b]p12.[/b] Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^o$ and $\angle PAC = 20^o$. If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees.
[b]p13.[/b] What is the largest prime factor of $9^3 - 4^3$?
[b]p14.[/b] Joey writes down the numbers $1$ through $10$ and crosses one number out. He then adds the remaining numbers. What is the probability that the sum is less than or equal to $47$?
[b]p15.[/b] In the coordinate plane, a lattice point is a point whose coordinates are integers. There is a pile of grass at every lattice point in the coordinate plane. A certain cow can only eat piles of grass that are at most $3$ units away from the origin. How many piles of grass can she eat?
[b]p16.[/b] A book has 1000 pages numbered $1$, $2$, $...$ , $1000$. The pages are numbered so that pages $1$ and $2$ are back to back on a single sheet, pages $3$ and $4$ are back to back on the next sheet, and so on, with pages $999$ and $1000$ being back to back on the last sheet. How many pairs of pages that are back to back (on a single sheet) share no digits in the same position? (For example, pages $9$ and $10$, and pages $89$ and $90$.)
[b]p17.[/b] Find a pair of integers $(a, b)$ for which $\frac{10^a}{a!}=\frac{10^b}{b!}$ and $a < b$.
[b]p18.[/b] Find all ordered pairs $(x, y)$ of real numbers satisfying
$$\begin{cases}
-x^2 + 3y^2 - 5x + 7y + 4 = 0 \\
2x^2 - 2y^2 - x + y + 21 = 0 \end{cases}$$
[b]p19.[/b] There are six blank fish drawn in a line on a piece of paper. Lucy wants to color them either red or blue, but will not color two adjacent fish red. In how many ways can Lucy color the fish?
[b]p20.[/b] There are sixteen $100$-gram balls and sixteen $99$-gram balls on a table (the balls are visibly indistinguishable). You are given a balance scale with two sides that reports which side is heavier or that the two sides have equal weights. A weighing is defined as reading the result of the balance scale: For example, if you place three balls on each side, look at the result, then add two more balls to each side, and look at the result again, then two weighings have been performed. You wish to pick out two different sets of balls (from the $32$ balls) with equal numbers of balls in them but different total weights. What is the minimal number of weighings needed to ensure this?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[b]p1.[/b] I have $6$ envelopes full of money. The amounts (in dollars) in the $6$ envelopes are six consecutive integers. I give you one of the envelopes. The total amount in the remaining $5$ envelopes is $\$2018$. How much money did I give you?
[b]p2. [/b]Two tangents $AB$ and $AC$ are drawn to a circle from an exterior point $A$. Let $D$ and $E$ be the midpoints of the line segments $AB$ and $AC$. Prove that the line DE does not intersect the circle.
[b]p3.[/b] Let $n \ge 2$ be an integer. A subset $S$ of {0, 1, . . . , n − 2} is said to be closed whenever it satisfies all of the following properties:
• $0 \in S$
• If $x \in S$ then $n - 2 - x \in S$
• If $x \in S$, $y \ge 0$, and $y + 1$ divides $x + 1$ then $y \in S$.
Prove that $\{0, 1, . . . , n - 2\}$ is the only closed subset if and only if $n$ is prime.
(Note: “$\in$” means “belongs to”.)
[b]p4.[/b] Consider the $3 \times 3$ grid shown below
$\begin{tabular}{|l|l|l|l|}
\hline
A & B & C \\ \hline
D & E & F \\ \hline
G & H & I \\ \hline
\end{tabular}$
A knight move is a pair of elements $(s, t)$ from $\{A, B, C, D, E, F, G, H, I\}$ such that $s$ can be reached from $t$ by moving either two spaces horizontally and one space vertically, or by moving one space horizontally and two spaces vertically. (For example, $(B, I)$ is a knight move, but $(G, E)$ is not.) A knight path of length $n$ is a sequence $s_0$, $s_1$, $s_2$, $. . . $, $s_n$ drawn from the set $\{A, B, C, D, E, F, G, H, I\}$ (with repetitions allowed) such that each pair $(s_i , s_{i+1})$ is a knight move.
Let $N$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $A$. Let $M$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $I$. Compute the value $(N- M)$, with proof. (Your answer must be in simplified form and may not involve any summations.)
[b]p5.[/b] A strip is defined to be the region of the plane lying on or between two parallel lines. The width of the strip is the distance between the two lines. Consider a finite number of strips whose widths sum to a number $d < 1$, and let $D$ be a circular closed disk of diameter $1$. Prove or disprove: no matter how the strips are placed in the plane, they cannot entirely cover the disk $D$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
Given two sequences of positive integers: the arithmetic sequence with difference $r > 0$ and the geometric sequence with ratio $q > 1$; $r$ and $q$ are coprime. Prove that if these sequences have one term in common, then they have them infinitely many.
On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over.
How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit?
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
The sequence $a_1,a_2,\dots,a_{13}$ is a geometric sequence with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that
$$\log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015,$$
find the number of possible ordered pairs $(a,r)$.