Found problems: 1340
Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by
$$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.
Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:
a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$.
b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.
[b]p1.[/b] You are on a flat planet. There are $100$ cities at points $x = 1, ..., 100$ along the line $y = -1$, and another $100$ cities at points $x = 1, ... , 100$ along the line $y = 1$. The planet’s terrain is scalding hot, and you cannot walk over it directly. Instead, you must cross archways from city to city. There are archways between all pairs of cities with different $y$ coordinates, but no other pairs: for instance, there is an archway from $(1, -1)$ to $(50, 1)$, but not from $(1, -1)$ to $(50, -1)$. The amount of “effort” necessary to cross an archway equals the square of the distance between the cities it connects. You are at $(1, -1)$, and you want to get to $(100, -1)$. What is the least amount of effort this journey can take?
[b]p2.[/b] Let $f(x) = x^4 + ax^3 + bx^2 + cx + 25$. Suppose $a, b, c$ are integers and $f(x)$ has $4$ distinct integer roots. Find $f(3)$.
[b]p3.[/b] Frankenstein starts at the point $(0, 0, 0)$ and walks to the point $(3, 3, 3)$. At each step he walks either one unit in the positive $x$-direction, one unit in the positive $y$-direction, or one unit in the positive $z$-direction. How many distinct paths can Frankenstein take to reach his destination?
[b]p4.[/b] Let $ABCD$ be a rectangle with $AB = 20$, $BC = 15$. Let $X$ and $Y$ be on the diagonal $\overline{BD}$ of $ABCD$ such that $BX > BY$ . Suppose $A$ and $X$ are two vertices of a square which has two sides on lines $\overline{AB}$ and $\overline{AD}$, and suppose that $C$ and $Y$ are vertices of a square which has sides on $\overline{CB}$ and $\overline{CD}$. Find the length $XY$ .
[img]https://cdn.artofproblemsolving.com/attachments/2/8/a3f7706171ff3c93389ff80a45886e306476d1.png[/img]
[b]p5.[/b] $n \ge 2$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they realize that each kid (except the first to leave) is friends with at least one kid who left before him. In how many possible orders could they have left the haunted house?
[b]p6.[/b] Call a set $S$ sparse if every pair of distinct elements of S differ by more than $1$. Find the number of sparse subsets (possibly empty) of $\{1, 2,... , 10\}$.
[b]p7.[/b] How many ordered triples of integers $(a, b, c)$ are there such that $1 \le a, b, c \le 70$ and $a^2 + b^2 + c^2$ is divisible by $28$?
[b]p8.[/b] Let $C_1$, $C_2$ be circles with centers $O_1$, $O_2$, respectively. Line $\ell$ is an external tangent to $C_1$ and $C_2$, it touches $C_1$ at $A$ and $C_2$ at $B$. Line segment $\overline{O_1O_2}$ meets $C_1$ at $X$. Let $C$ be the circle through $A, X, B$ with center $O$. Let $\overline{OO_1}$ and $\overline{OO_2}$ intersect circle $C$ at $D$ and $E$, respectively. Suppose the radii of $C_1$ and $C_2$ are $16$ and $9$, respectively, and suppose the area of the quadrilateral $O_1O_2BA$ is $300$. Find the length of segment $DE$.
[b]p9.[/b] What is the remainder when $5^{5^{5^5}}$ is divided by $13$?
[b]p10.[/b] Let $\alpha$ and $\beta$ be the smallest and largest real numbers satisfying
$$x^2 = 13 + \lfloor x \rfloor + \left\lfloor \frac{x}{2} \right\rfloor +\left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{4} \right\rfloor .$$ Find $\beta - \alpha$ .
($\lfloor a \rfloor$ is defined as the largest integer that is not larger than $a$.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find all real numbers $ x$ such that:
$ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$
(Here $ [x]$ denotes the largest integer not exceeding $ x$.)
Prove that: $ \sum_{i\equal{}1}^{n^2} \lfloor \frac{i}{3} \rfloor\equal{} \frac{n^2(n^2\minus{}1)}{6}$
For all $ n \in N$.
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?
Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel
has $k$ sheets of paper lying next to each other on a table, where $k$ is a
positive integer. On each of the sheets, he writes some of the numbers
from $1$ up to $n$ (he is allowed to write no number at all, or all numbers).
On the back of each of the sheets, he writes down the remaining numbers.
Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is
allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making
all of the numbers from $1$ up to n visible at least once, then he wins.
Determine the smallest $k$ for which Merlijn can always win, regardless of
Daniel’s actions.
For a natural number $ n\ge 2, $ calculate the integer part of $ \sqrt[n]{1+n}-\sqrt {2/n} . $
[i]Dan Nedeianu[/i]
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ .
(a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ .
(b) Can such an $A$ be a rational number?
(I. Varge, Romania)
While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$.
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.
Prove that $\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right]$ is a positive integer.
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
\[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\]
where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$
Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition
\[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\]
Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
Let $ f(x) \equal{} \log_{10} (\sin (\pi x)\cdot\sin (2\pi x)\cdot\sin (3\pi x) \cdots \sin (8\pi x))$. The intersection of the domain of $ f(x)$ with the interval $ [0,1]$ is a union of $ n$ disjoint open intervals. What is $ n$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36$
There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]
Find all reals $x$ satisfying $0 \le x \le 5$ and
$\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.
Let $x > 1$ be a non-integer number. Prove that
\[\biggl( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \biggr) + \biggl( \frac{x+[x]}{ \{x \} } - \frac{ \{ x \}}{x+[x]} \biggr) > \frac 92 \]
Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation
$$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$
for any real numbers $ x. $
[i]Marius Perianu[/i]
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 $S_i(m)=S_i(n)$.
Suppose that the set $M=\{1,2,\cdots,n\}$ is split into $t$ disjoint subsets $M_{1}$, $\cdots$, $M_{t}$ where the cardinality of $M_i$ is $m_{i}$, and $m_{i} \ge m_{i+1}$, for $i=1,\cdots,t-1$. Show that if $n>t!\cdot e$ then at least one class $M_z$ contains three elements $x_{i}$, $x_{j}$, $x_{k}$ with the property that $x_{i}-x_{j}=x_{k}$.