Found problems: 5923
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$?
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
A sequence of positive integers $a_0, a_1, a_2, \dots$ satisfies $a_0 = 83$ and $a_n = (a_{n-1})^{a_{n-1}}$ for all positive integers $n$. Compute the remainder when $a_{2021}$ is divided by $60$.
Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.
Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?
$ a_{1} \equal{}1$, $ a_{n\plus{}1} \equal{}\frac{a_{n} }{\sqrt{1\plus{}4a_{n}^{2} } }$ for $ n\ge 1$. What is the least $ k$ such that $ a_{k} <10^{\minus{}2}$ ?
$\textbf{(A)}\ 2501 \qquad\textbf{(B)}\ 251 \qquad\textbf{(C)}\ 2499 \qquad\textbf{(D)}\ 249 \qquad\textbf{(E)}\ \text{None}$
Decide if it is possible to generate an infinite sequence of positive integers $a_n$ such that in the sequence there are no three terms that are in arithmetic progression and that for all $n$ $\left |a_n-n^2\right | <\frac{n}{2}$.
Clarification: Three numbers $a$, $b$, $c$ are in arithmetic progression if and only if $2b=a+c$.
The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term?
$\textbf{(A) }1\qquad
\textbf{(B) }\sqrt[7]3\qquad
\textbf{(C) }\sqrt[8]3\qquad
\textbf{(D) }\sqrt[9]3\qquad
\textbf{(E) }\sqrt[10]3\qquad$
Sergio thinks of a positive integer $S$, less than or equal to $100$. Iván must guess the number that Sergio thought of, using the following procedure: in each step, he chooses two positive integers $A$ and $B$ less than $100$, and asks Sergio what is the greatest common factor between $A+ S$ and $B$. Give a sequence of seven steps that ensures Iván guesses the number $S$ that Sergio thought of.
Clarification:In each step, Sergio correctly answers Iván's question.
[b]p1.[/b] What is $\sqrt[2015]{2^01^5}$?
[b]p2.[/b] What is the ratio of the area of square $ABCD$ to the area of square $ACEF$?
[b]p3.[/b] $2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property?
[b]p4.[/b] What is the next number in the following geometric series: $1020100$, $10303010$, $104060401$?
[b]p5.[/b] A circle has radius $A$ and area $r$. If $A = r^2\pi$, then what is the diameter, $C$, of the circle?
[b]p6.[/b] If
$$O + N + E = 1$$
$$T + H + R + E + E = 3$$
$$N + I + N + E = 9$$
$$T + E + N = 10$$
$$T + H + I + R + T + E + E + N = 13$$
Then what is the value of $O$?
[b]p7.[/b] By shifting the initial digit, which is $6$, of the positive integer $N$ to the end (for example, $65$ becomes $56$), we obtain a number equal to $\frac{N}{4}$ . What is the smallest such $N$?
[b]p8.[/b] What is $\sqrt[3]{\frac{2015!(2013!)+2014!(2012!)}{2013!(2012!)}}$ ?
[b]p9.[/b] How many permutations of the digits of $1234$ are divisible by $11$?
[b]p10.[/b] If you choose $4$ cards from a normal $52$ card deck (with replacement), what is the probability that you will get exactly one of each suit (there are $4$ suits)?
[b]p11.[/b] If $LMT$ is an equilateral triangle, and $MATH$ is a square, such that point $A$ is in the triangle, then what is $HL/AL$?
[b]p12.[/b] If
$$\begin{tabular}{cccccccc}
& & & & & L & H & S\\
+ & & & & H & I & G & H \\
+ & & S & C & H & O & O & L \\
\hline
= & & S & O & C & O & O & L \\
\end{tabular}$$ and $\{M, A, T,H, S, L,O, G, I,C\} = \{0, 1, 2, 3,4, 5, 6, 7, 8, 9\} $, then what is the ordered pair $(M + A +T + H, [T + e + A +M])$ where $e$ is $2.718...$and $[n]$ is the greatest integer less than or equal to $n$ ?
[b]p13.[/b] There are $5$ marbles in a bag. One is red, one is blue, one is green, one is yellow, and the last is white. There are $4$ people who take turns reaching into the bag and drawing out a marble without replacement. If the marble they draw out is green, they get to draw another marble out of the bag. What is the probability that the $3$rd person to draw a marble gets the white marble?
[b]p14.[/b] Let a "palindromic product" be a product of numbers which is written the same when written back to front, including the multiplication signs. For example, $234 * 545 * 432$, $2 * 2 *2 *2$, and $14 * 41$ are palindromic products whereas $2 *14 * 4 * 12$, $567 * 567$, and $2* 2 * 3* 3 *2$ are not. 2015 can be written as a "palindromic product" in two ways, namely $13 * 5 * 31$ and $31 * 5 * 13$. How many ways can you write $2016$ as a palindromic product without using 1 as a factor?
[b]p15.[/b] Let a sequence be defined as $S_n = S_{n-1} + 2S_{n-2}$, and $S_1 = 3$ and $S_2 = 4$. What is $\sum_{n=1}^{\infty}\frac{S_n}{3^n}$ ?
[b]p16.[/b] Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$.
[b]p17.[/b] Evaluate
$\dfrac{8+ \dfrac{8+ \dfrac{8+...}{3+...}}{3+ \dfrac{8+...}{3+...}}}{3+\dfrac{8+ \dfrac{8+...}{3+...}}{
3+ \dfrac{8+...}{3+...}}}$, assuming that it is a positive real number.
[b]p18.[/b] $4$ non-overlapping triangles, each of area $A$, are placed in a unit circle. What is the maximum value of $A$?
[b]p19.[/b] What is the sum of the reciprocals of all the (positive integer) factors of $120$ (including $1$ and $120$ itself).
[b]p20.[/b] How many ways can you choose $3$ distinct elements of $\{1, 2, 3,...,4000\}$ to make an increasing arithmetic series?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]
[i]Proposed by Denmark[/i]
Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
Let $f_1,f_2,\ldots $ be a sequence of non-increasing functions from the naturals to the naturals. Show there exists $i < j$ such that
$$f_i(n) \leq f_j(n) \text{ for all } n \in \mathbb{N}.$$
We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
The sequence $(\chi_n) _{n=1}^{\infty}$ is defined as follows
\begin{align*} \chi_{n+1}=\chi_n + \chi _{\lceil \frac{n}{2} \rceil} ~, \chi_1 =1 \end{align*}
Prove that none of the terms of this sequence are divisible by $4$
Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$?
Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$
(a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$).
(b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$
Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most
\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]
Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that:
\[
45 < x_{1000} < 45.1.
\]
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$,
$a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$.
Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
We consider the sequence $t_1,t_2,t_3,...$ By $P_n$ we mean the product of the first $n$ terms of the sequence. Given that $t_{n+1} = t_n \cdot t_{n+2}$ for each $n$, and that $P_{40} = P_{80} = 8$. Calculate $t_1$ and $t_2$.
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 $x$. )
[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$.
[b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive?
[b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$.
[b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$.
[b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ .
[b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list?
[b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$.
[b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits?
[b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces
$$2\_3\_4\_5$$
[b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ .
[b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes?
p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence?
[b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits.
[b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$?
[b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$.
[b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$?
[b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$
[b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$.
[b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$.
[b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number.
[b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$?
[b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers.
[b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$.
[b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute
$$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$
[b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].