Found problems: 85335
For any positive integer $n$, let $ \left< n \right> $ denote the closest integer to $ \sqrt {n} $. Evaluate: \[ \displaystyle\sum_{n=1}^{\infty} \dfrac {2^{\left< n \right>} + 2^{- \left< n \right>}}{2^n} \]
Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.
For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$.
let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $
such that
$gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $
Let $a$ and $b$ be real numbers such that $a + b = 1$. Prove the inequality
$$\sqrt{1+5a^2} + 5\sqrt{2+b^2} \geq 9.$$
[i]Proposed by Baasanjav Battsengel[/i]
$AD$, $BE$, $CF$ are the heights of the acute—angled triangle $ABC$. A perpendicular is drawn to the segment $DE$ at point $E$. It intersects the height of $AD$ at point $G$. The point $J$ is chosen on the segment $BD$ in such a way that $BJ = CD$. The circumscribed circle of a triangle $BD$ intersects the segment $BE$ at point $Q$. Prove that the points $J$, $Q$, and $G$ are collinear.
A red equilateral triangle $T$ with side length $1$ is drawn on a plane. For a positive real $c$, we place three blue equilateral triangle shaped paper with side length $c$ on a plane to cover $T$ completely. Find the minimum value of $c$. As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from $T$. Folding or tearing the paper is not allowed.
Determine the natural numbers that cannot be written as $\lfloor n + \sqrt{n} + \frac{1}{2} \rfloor$ for any $n \in \mathbb{N}$.
A paper rectangle $ABCD $ of area $1$ is folded along a straight line so that $C$ coincides with $A$. Prove that the area of the pentagon thus obtained is less than $3/4$.
Let $ABCD$ be a quadrilateral satisfying $\angle BCD=\angle CDA$. Suppose rays $AD$ and $BC$ meet at $E$, and let $\Gamma$ be the circumcircle of $ABE$. Let $\Gamma_1$ be a circle tangent to ray $CD$ past $D$ at $W$, segment $AD$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_2$ be a circle tangent to ray $DC$ past $C$ at $Y$, segment $BC$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $WX$ and $YZ$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $ABE$, and $AB=544$, $AE=2197$, $BE=2299$, then find $m+n$, where $FP=\tfrac{m}{n}$ with $m,n$ relatively prime positive integers.
[i]Proposed by Michael Kural[/i]
Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers.
(Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)
If $a,b,c>0,a+b+c=1$,then:
$\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$
Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$.
[i]Proposed by Evan Chang (squareman), USA[/i]
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?
[asy]
pair A,B,C;
A=origin; B=(10,0); C=6.5*dir(15);
dot(A); dot(B); dot(C);
draw(B--A--C);
draw(B--C,dashed);
label("$6.5$",3.25*dir(15),NNW);
label("$10$",(5,0),S);
label("$s$",(8,1),NE);
[/asy]
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$
Kelvin the Frog writes 2020 words on a blackboard, with each word chosen uniformly randomly from the set $\{\verb|happy|, \verb|boom|, \verb|swamp|\}$. A multiset of seven words is [i]merry[/i] if its elements can spell $``\verb|happy happy boom boom swamp swamp swamp|."$ For example, the eight words
\[\verb|swamp|, \verb|happy|, \verb|boom|, \verb|swamp|, \verb|swamp|, \verb|boom|, \verb|swamp|, \verb|happy|\]
contain four merry multisets. Determine the expected number of merry multisets contained in the words on the blackboard.
[size=6][url]http://www.hpmor.com/chapter/12[/url][/size]
Determine, with proof, the maximum and minimum among the numbers
$$\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3
\sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor $$
The $53$-digit number
$$37,984,318,966,591,152,105,649,545,470,741,788,308,402,068,827,142,719$$
can be expressed as $n^21$ where $n$ is a positive integer. Find $n$.
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.
Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence?
(Alexandr Shapovalov)
Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define
\[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor.
\]
Prove that for all natural numbers $ p$, $ q$, and $ r$,
\[ (p*q)*r \equal{} p*(q*r).
\]
Suppose that $7$ boys and $13$ girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $GBBGGGBGBGGGBGBGGBGG$ we have $S=12$. The average value of $S$ (if all possible orders of the 20 people are considered) is closest to
$ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13 $
The nodes of a three dimensional unit cube lattice with all three coordinates even are coloured red and blue otherwise. A convex polyhedron with all vertices red is given. Assuming the number of red points on its border is $n$. How many blue vertices can be on its border?
Find the sum of all real numbers $x$ such that $x^2 = 5x + 6\sqrt{x} - 3$.
Triangle $ABC$ is isoceles with $AB = AC$. Point $D$ lies on $AB$ such that the inradius of $ADC$ and the inradius of $BDC$ both equal $\frac{3-\sqrt3}{2}$ . The inradius of $ABC$ equals $1$. What is the length of $BD$?
Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$).