Found problems: 5923
Suppose that the positive numbers $a_1, a_2,.. , a_n$ form an arithmetic progression; hence $a_{k+1}- a_k = d,$ for $k = 1, 2,... , n - 1.$
Prove that \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.\]
On the circumference of a circle there are red and blue points. One may add a red point and change the colour of both its neighbours (to the other colour) or remove a red point and change the colour of both its previous neighbours. Initially there are two red points. Prove that there is no sequence of allowed operations which leads to the configuration consisting of two blue points.
(K Kazarnovskiy, Moscow)
Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is [i]nice [/i] if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,
Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).
Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off.
Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on.
Determine $ \frac {N}{M}$.
[i]Author: Bruno Le Floch and Ilia Smilga, France[/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 $x_{i+1} = x_i - b$.
\[\begin{tabular}{ccccccccccccc}
& & & & & & C & & & & & & \\
& & & & & C & O & C & & & & & \\
& & & & C & O & N & O & C & & & & \\
& & & C & O & N & T & N & O & C & & & \\
& & C & O & N & T & E & T & N & O & C & & \\
& C & O & N & T & E & S & E & T & N & O & C & \\
C & O & N & T & E & S & T & S & E & T & N & O & C
\end{tabular}\]
For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram above, is the word CONTEST spelled out as the path is traversed from beginning to end?
$\textbf{(A) }63\qquad\textbf{(B) }128\qquad\textbf{(C) }129\qquad\textbf{(D) }255\qquad \textbf{(E) }\text{none of these}$
p1. Two integers $m$ and $n$ are said to be [i]coprime [/i] if there are integers $a$ and $ b$ such that $am + bn = 1$. Show that for each integer $p$, the pair of numbers formed by $21p + 4$ and $14p + 3$ are always coprime.
p2. Two farmers, Person $A$ and Person $B$ intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of their respective origins.
[img]https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png[/img]
p3. The system of equations of four variables is given: $\left\{\begin{array}{l}
23x + 47y - 3z = 434 \\
47x - 23y - 4w = 183 \\
19z + 17w = 91
\end{array} \right. $
where $x, y, z$, and $w$ are positive integers.
Determine the value of $(13x - 14y)^3 - (15z + 16w)^3$
p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png[/img]
Initially the vehicle contains $ 3$ liters of fuel. After two hours, in the journey of fuel remains $ 1$ liter.
a. If in $ 1$ liter he can cover a distance of $32$ km, what is the distance taken as a whole? Explain why you answered like that?
b. After two hours of travel, is there any acceleration or deceleration? Explain your answer.
c. Determine what the average speed of the vehicle is.
p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below.
[img]https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png[/img]
He made a rule that the bottom four circles would be filled with positive numbers less than $10$ that can be taken from the numbers on the date of his birth, i.e. $26 \,\, - \,\, 12 \,\, - \,\,1961$ without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath.
a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain.
b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as $8$ circles. He no longer cares whether the numbers are repeated or not .
i. In order to get the smallest value in the top circle, how should the numbers be arranged?
ii. How many arrays are worth considering to produce the smallest value?
A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity.
THERE MIGHT BE A TYPO!
Consider the sequence of real numbers $a_n$ satisfying the recurrence
$$a_na_{n+2}-a_{n+1}^2-(n+1)a_na_{n+1}=0.$$
Given that $a_1=1$ and $a_2=2018$, compute
$$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$
Define the sequences $a_{0}, a_{1}, a_{2}, ...$ and $b_{0}, b_{1}, b_{2}, ...$ by $a_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}$. Show that the two sequences converge to the same limit, and find the limit.
A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there?
[i]Proposed by Evan Chen[/i]
Find all positive integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots +
\frac{a_{n-1}}{a_n},
\]
where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.
Let $a_1,a_2,a_3,...$ be a strictly increasing sequence of positive integers. A number $a_n$ in the sequence is said to be [i]lucky [/i] if it is the sum of several (not necessarily distinct) smaller terms of the sequence, and [i]unlucky [/i]otherwise. (For example, in the sequence $4,6,14,15,25,...$ numbers $4,6,15$ are [i]unlucky[/i], while $14 = 4+4+6$ and $25 = 4+6+15$ are [i]lucky[/i].) Prove that there are only finitely many [i]unlucky [/i]numbers in the sequence.
How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression?
$ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 59 \qquad \textbf{(D) } 89 \qquad \textbf{(E) } 178$
In a class there are n students with unequal heights.
$\textbf{(a)}$ Find the number of orderings of the students such that the shortest person
is not at the front and the tallest person is not at the end.
$\textbf{(b)}$ Define the [i]badness[/i] of an ordering as the maximum number $k$ such that there
are $k$ many people with height greater than in front of a person. For example:
the sequence $66, 61, 65, 64, 62, 70$ has [i]badness [/i] $3$ since there are $3$ numbers greater
than $62$ in front of it. Let $f_k(n)$ denote the number of orderings of $n$ with [i]badness[/i] $k$. Find $f_k(n)$.
[hide=hint](Hint: Consider $g_k(n)$ as the number of orderings of n with [i]badness [/i]less than
or equal to $k$)[/hide]
Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that
\begin{align*}
b_k+b_{\ell}=m.
\end{align*}
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties
\[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \]
Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.
How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$?
$
\textbf{(A) }253 \qquad
\textbf{(B) }504 \qquad
\textbf{(C) }505 \qquad
\textbf{(D) }506 \qquad
\textbf{(E) }1009 \qquad
$
An infinite number of lilypads grow in a line, numbered $\dots$, $-2$, $-1$, $0$, $1$, $2$, $\dots$ Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad $0$ or pad $96$. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad $0$ or pad $96$, regardless of her frog's decisions?
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]
Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$.
(Malik Talbi)
Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$. Prove that $ a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$
A positive integer is called [i]downhill[/i] if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?
Define the sequence ($x_n$) as follows: the first term is $1$, the next two are $2,4$, the next three are $5,7,9$, the next four are $10,12,14,16$, and so on. Express $x_n$ as a function of $n$.
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
$\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$
[asy]
size(3.85cm);
label("$X$",(2.5,2.1),N);
for (int i=0; i<=5; ++i)
draw((i,0)--(i,5), linewidth(.5));
for (int j=0; j<=5; ++j)
draw((0,j)--(5,j), linewidth(.5));
void draw_num(pair ll_corner, int num)
{
label(string(num), ll_corner + (0.5, 0.5), p = fontsize(19pt));
}
draw_num((0,0), 17);
draw_num((4, 0), 81);
draw_num((0, 4), 1);
draw_num((4,4), 25);
void foo(int x, int y, string n)
{
label(n, (x+0.5,y+0.5), p = fontsize(19pt));
}
foo(2, 4, " ");
foo(3, 4, " ");
foo(0, 3, " ");
foo(2, 3, " ");
foo(1, 2, " ");
foo(3, 2, " ");
foo(1, 1, " ");
foo(2, 1, " ");
foo(3, 1, " ");
foo(4, 1, " ");
foo(2, 0, " ");
foo(3, 0, " ");
foo(0, 1, " ");
foo(0, 2, " ");
foo(1, 0, " ");
foo(1, 3, " ");
foo(1, 4, " ");
foo(3, 3, " ");
foo(4, 2, " ");
foo(4, 3, " ");
[/asy]