Found problems: 5923
Given any integer $ n \geq 2,$ assume that the integers $ a_1, a_2, \ldots, a_n$ are not divisible by $ n$ and, moreover, that $ n$ does not divide $ \sum^n_{i\equal{}1} a_i.$ Prove that there exist at least $ n$ different sequences $ (e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $ \sum^n_{i\equal{}1} e_i \cdot a_i$ is divisible by $ n.$
The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.
[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$?
[b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors?
[b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$?
[b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain?
[b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$)
[b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$.
[b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides?
[b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent?
[b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$.
[b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$?
[b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$?
[b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon?
[b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important?
[b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$.
[b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/
[b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles?
[b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ .
[b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer?
[b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals?
[b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$.
\\
\\
Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.
Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc.
(a) Prove the relation
\[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \]
where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that
\[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\]
(b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.
Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?
From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.
Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.
The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$.
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?
Show that the sequence
$$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$
converges and evaluate the limit.
For any pair $(x,y)$ of real numbers, a sequence $(a_{n}(x,y))$ is defined as follows:
$$a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0$$
Find the area of the region $\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}$.
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define
\[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \}
\]
and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
\[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)
\]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
[i]Author: Michael Albert, New Zealand[/i]
We have $p$ players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers $s_1 \le s_2 \le s_3 \le\cdots\le s_p$ is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if
\[(i)\displaystyle\sum_{i=1}^{p} s_i =\frac{1}{2}p(p-1)\]
\[\text{and}\]
\[(ii)\text{ for all }k < p,\displaystyle\sum_{i=1}^{k} s_i \ge \frac{1}{2} k(k - 1).\]
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the $1000^{th}$ number in $S$. Find the remainder when $N$ is divided by $1000$.
Let $\{x_n\}_{n\geq 0}$ be a sequence of real numbers which satisfy \[ (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0, \quad n\geq 0. \] a) Prove that the sequence is bounded;
b) Is it possible that the sequence is not convergent?
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 $x$. )
Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$.
a. Find the order of $1996$ in the sequence.
b. Find the $1996$th term in the sequence.
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.
Find all real constants c for which there exist strictly increasing sequence $a$ of positive integers such that $(a_{2n-1}+a_{2n})/{a_n}=c$ for all positive intеgers n.
Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences
\begin{eqnarray*}
P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\
Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014)
\end{eqnarray*}
are permutations of each other.
(a) Prove that there exist distinct block-similar polynomials of degree $n + 1$.
(b) Prove that there do not exist distinct block-similar polynomials of degree $n$.
[i]Proposed by David Arthur, Canada[/i]
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 $x$. )
Since guts has 36 questions, they will be combined into posts.
1.[b][5][/b] Farmer Yang has a $2015$ × $2015$ square grid of corn plants. One day, the plant in the very center
of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased.
After how many days will all of Yang's corn plants be diseased?
2. [b][5][/b] The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of
the hypotenuse to the length of the shorter leg.
3. [b][5][/b] A parallelogram has $2$ sides of length $20$ and $15$. Given that its area is a positive integer, find the
minimum possible area of the parallelogram.
4. [b][6][/b] Eric is taking a biology class. His problem sets are worth $100$ points in total, his three midterms are
worth $100$ points each, and his final is worth $300$ points. If he gets a perfect score on his problem sets
and scores $60\%$,$70\%$, and $80\%$ on his midterms respectively, what is the minimum possible percentage
he can get on his final to ensure a passing grade? (Eric passes if and only if his overall percentage is
at least $70\%$).
5. [b][6][/b] James writes down three integers. Alex picks some two of those integers, takes the average of them,
and adds the result to the third integer. If the possible final results Alex could get are $42$, $13$, and $37$,
what are the three integers James originally chose?
6. [b][6][/b] Let $AB$ be a segment of length $2$ with midpoint $M$. Consider the circle with center $O$ and radius
$r$ that is externally tangent to the circles with diameters $AM$ and $BM$ and internally tangent to the
circle with diameter $AB$. Determine the value of $r$.
7. [b][7][/b] Let n be the smallest positive integer with exactly $2015$ positive factors. What is the sum of
the (not necessarily distinct) prime factors of n? For example, the sum of the prime factors of $72$ is
$2 + 2 + 2 + 3 + 3 = 14$.
8. [b][7][/b] For how many pairs of nonzero integers $(c, d)$ with $-2015 \le c,d \le 2015$ do the equations $cx = d$
and $dx = c$ both have an integer solution?
9. [b][7][/b] Find the smallest positive integer n such that there exists a complex number z, with positive real
and imaginary part, satisfying $z^n = (\overline{z})^n$.