Found problems: 85335
[b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that:
$$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds.
[i]Proposed by dangerousliri [/i]
Twenty percent less than $ 60$ is one-third more than what number?
$ \textbf{(A)}\ 16\qquad
\textbf{(B)}\ 30\qquad
\textbf{(C)}\ 32\qquad
\textbf{(D)}\ 36\qquad
\textbf{(E)}\ 48$
Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]
The incircle of the triangle ABC touches the sides BC, AC, and AB at points A', B', and C', respectively. It is known that AA'=BB'=CC'. Does the triangle ABC have to be equilateral?
(I am very interested in ingenious solution of this problem, because I found an ugly one using Stewart's theorem and tons of algebra during the contest).
Prove that,if $a,b,c$ are positive real numbers,
\[ \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}\]
Evaluate $\lim_{n\to \infty} cos\frac{x}{2}cos\frac{x}{2^2} cos\frac{x}{2^3}...cos\frac{x}{2^n}$
Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.
Prove that each positive integer is equal to a difference of two positive integers with the same number of the prime divisors.
A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$
$x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$.
Prove that if $a, b$ and $c$ are positive numbers such that
$$a^2 + b^2 - ab = c^2,$$
then $(a - c)(b - c) < 0.$
(A Egorov)
Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$
(1) Express $y$ in terms of $x$.
(2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$.
(3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.
Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that
[list]
[*] [b](a)[/b] the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$;
the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$;
[*] [b](b)[/b] the board is [i]totally non-symmetric[/i]: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.[/list]
Ana baked 15 pasties. She placed them on a round plate in a circular
way: 7 with cabbage, 7 with meat and 1 with cherries in that exact order
and put the plate into a microwave. She doesn’t know how the plate has been rotated
in the microwave. She wants to eat a pasty with cherries. Is it possible for Ana, by trying no more than three pasties, to find exactly where the pasty with cherries is?
Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Consider a sequence of polynomials such that $P_0(x)=2,P_1(x)=x$ and for all $n\ge1$ \[P_{n+1}(x)+P_{n-1}(x)=xP_n(x).\]
a) Determine the polynomial \[Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x)\] for $n=1972.$
b) Express the polynomial \[\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2\] in terms of $P_n(x),Q_n(x).$
A drunken person walks randomly on a tree. Each time, he chooses uniformly at random a neighbouring node and walks there. Show that wherever his starting point and goal are, the expected number of steps the person takes to reach the goal is always an integer.
[i]houkai[/i]
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
The center of circle $\omega_1$ of radius $6$ lies on circle $\omega_2$ of radius $6$. The circles intersect at points $K$ and $W$. Let point $U$ lie on the major arc $\overarc{KW}$ of $\omega_2$, and point $I$ be the center of the largest circle that can be inscribed in $\triangle KWU$. If $KI+WI=11$, find $KI\cdot WI$.
[i]Proposed by Bradley Guo[/i]
Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.
In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\angle{B}-\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\triangle{ABC}$. Find $BC$ and $AC$.
Prove that for any natural numbers $a,b,c$ and $d$ there exist infinetly natural numbers $n$ such that $a^n+b^n+c^n+d^n$ is composite.