Found problems: 5923
For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.
We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex.
(a) Show that the task cannot be done if $ r$ is divisible by 2 or 3.
(b) Prove that the task is possible when $ r \equal{} 73$.
(c) Can the task be done when $ r \equal{} 97$?
Determine all sequences of real numbers $a_1$, $a_2$, $\ldots$, $a_{1995}$ which satisfy:
\[ 2\sqrt{a_n - (n - 1)} \geq a_{n+1} - (n - 1), \ \mbox{for} \ n = 1, 2, \ldots 1994, \] and \[ 2\sqrt{a_{1995} - 1994} \geq a_1 + 1. \]
Let $ \{\xi_{k \ell} \}_{k,\ell=1}^{\infty}$ be a double sequence of random variables such that
\[ \Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \\\ .\]
Prove that with probability one,
\[ \frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \; \max (m,n)\rightarrow \infty\ \\ .\]
[i]F. Moricz[/i]
Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied:
[b](i)[/b] each integer belongs to at least one of them;
[b](ii)[/b] each progression contains a number which does not belong to other progressions.
Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization.
Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\]
[i]Proposed by Dierk Schleicher, Germany[/i]
A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$
Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$
When rolling two normal dice, the set of possible outcomes of the sum of the points is $2, 3, 3, 4,4, 4,..., 11, 11,12$. Notice that this sequence can be obtained from the identity $$(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.$$ Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.
Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence:
$$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$
where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that
$$\lim_{n\to\infty}a_n=0$$
[i]Proposed by Laurențiu Modan[/i]
Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$.
[i]Mircea Lascu, Dorel Mihet[/i]
define $(a_n)_{n \in \mathbb{N}}$ such that $a_1=2$ and $$a_{n+1}=\left(1+\frac{1}{n}\right)^n \times a_{n}$$
Prove that there exists infinite number of $n$ such that $\frac{a_1a_2 \ldots a_n}{n+1}$ is a square of an integer.
Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had form a geometric progression. Alex eats 5 of his peanuts, Betty eats 9 of her peanuts, and Charlie eats 25 of his peanuts. Now the three numbers of peanuts that each person has form an arithmetic progression. Find the number of peanuts Alex had initially.
A sequence starts at some rational number $x_1>1$, and is subsequently defined using the recurrence relation \[x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }\]
Show that $k>0$ exists with $x_k=1$.
[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes
\[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\]
[b](b)[/b] Find the rearrangement that minimizes $Q.$
The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$.
(a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$.
(b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.
Sequences $(a_n), (b_n)$ are defined by $ a_1 = 1, b_1 = 2$, $a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}$, $ b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}$ for all positive integers $n$. Prove that $a_{2020} < 5$.
Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]
Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$
\[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \]
Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.
Define the sequence $a_n$ by the rule $$a_{n+1} =\left \lfloor \frac{a_n} 2 \right \rfloor + \left \lfloor \frac{a_n}3 \right \rfloor$$ for $n \in \{1, 2, 3, 4, 5, 6, 7\}$, where $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$. If $a_8=8$, how many possible values are there for $a_1$ given that it is a positive integer?
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 10)[/i]
[b]p1.[/b] Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once.
[b]p2.[/b] In triangles $\vartriangle ABC$ and $\vartriangle DEF$, $DE = 4AB$, $EF = 4BC$, and $FD = 4CA$. The area of $\vartriangle DEF$ is $360$ units more than the area of $\vartriangle ABC$. Compute the area of $\vartriangle ABC$.
[b]p3.[/b] Andy has $2010$ square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an $m\times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a $1\times 2010$ rectangle is considered to be the same as a $2010\times 1$ rectangle.
[b]p4.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ... \log_{999} 1000000.$$
Compute the greatest integer less than or equal to $\log_2 S$.
[b]p5.[/b] Let $A$ and $B$ be fixed points in the plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point $C$ could possibly be located.
[b]p6.[/b] Lisette notices that $2^{10} = 1024$ and $2^{20} = 1 048 576$. Based on these facts, she claims that every number of the form $2^{10k}$ begins with the digit $1$, where k is a positive integer. Compute the smallest $k$ such that Lisette's claim is false. You may or may not find it helpful to know that $ln 2 \approx 0.69314718$, $ln 5 \approx 1.60943791$, and $log_{10} 2 \approx 0:30103000$.
[b]p7.[/b] Let $S$ be the set of all positive integers relatively prime to $6$. Find the value of $\sum_{k\in S}\frac{1}{2^k}$ .
[b]p8.[/b] Euclid's algorithm is a way of computing the greatest common divisor of two positive integers $a$ and $b$ with $a > b$. The algorithm works by writing a sequence of pairs of integers as follows.
1. Write down $(a, b)$.
2. Look at the last pair of integers you wrote down, and call it $(c, d)$.
$\bullet$ If $d \ne 0$, let r be the remainder when c is divided by d. Write down $(d, r)$.
$\bullet$ If $d = 0$, then write down c. Once this happens, you're done, and the number you just wrote down is the greatest common divisor of a and b.
3. Repeat step 2 until you're done.
For example, with $a = 7$ and $b = 4$, Euclid's algorithm computes the greatest common divisor in $4$ steps:
$$(7, 4) \to (4, 3) \to (3, 1) \to (1, 0) \to 1$$
For $a > b > 0,$ compute the least value of a such that Euclid's algorithm takes $10$ steps to compute the greatest common divisor of $a$ and $b$.
[b]p9.[/b] Let $ABCD$ be a square of unit side length. Inscribe a circle $C_0$ tangent to all of the sides of the square. For each positive integer $n$, draw a circle Cn that is externally tangent to $C_{n-1}$ and also tangent to sides $AB$ and $AD$. Suppose $r_i$ is the radius of circle $C_i$ for every nonnegative integer $i$. Compute $\sqrt[200]{r_0/r_{100}}$.
[b]p10.[/b] Rachel and Mike are playing a game. They start at $0$ on the number line. At each positive integer on the number line, there is a carrot. At the beginning of the game, Mike picks a positive integer $n$ other than $30$. Every minute, Rachel moves to the next multiple of $30$ on the number line that has a carrot on it and eats that carrot. At the same time, every minute, Mike moves to the next multiple of $n$ on the number line that has a carrot on it and eats that carrot. Mike wants to pick an $n$ such that, as the game goes on, he is always within $1000$ units of Rachel. Compute the average (arithmetic mean) of all such $n$.
[b]p11.[/b] Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$.
[b]p12.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$.
[b]p13.[/b] Let $\theta \ne 0$ be the smallest acute angle for which $\sin \theta$, $\sin (2\theta)$, $\sin (3\theta)$, when sorted in increasing order, form an arithmetic progression. Compute $\cos (\theta/2)$.
[b]p14.[/b] A $4$-dimensional hypercube of edge length 1 is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by the intersection.
[b]p15.[/b] A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The sequence of integers $(a_{n})_{n \ge 1}$ is defined by $a_{1}= 2$, $a_{n+1}= 2a_{n}^{2}-1$.
Prove that for each positive integer n, $n$ and $a_{n}$ are coprime.
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.
An infinite sequence of digits $1$ and $2$ is determined by the following two properties:
i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$
ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained.
In which position is the hundredth digit $1$? What is the thousandth digit of the sequence?
Does there exist a non-constant infinite sequence of distinct natural numbers such that for all large enough $k$ we have that sum of any $k$-sized sub-sequence is square-free if and only if $k$ is square-free.
[i]Proposed by Alireza Haqi, Amir Rezaie Moghadam [/i] [b]Rated 3[/b]
The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.
For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer)