Found problems: 85335
Find all positive integers $a,b,c$ and prime $p$ satisfying that
\[ 2^a p^b=(p+2)^c+1.\]
Let $x, y, z \ge 0$ be real numbers. Prove that: \[\frac{x^{3}+y^{3}+z^{3}}{3}\ge xyz+\frac{3}{4}|(x-y)(y-z)(z-x)| .\]
[hide="Additional task"]Find the maximal real constant $\alpha$ that can replace $\frac{3}{4}$ such that the inequality is still true for any non-negative $x,y,z$.[/hide]
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
Let $(x_n)_{n\geq 1}$ be the sequence defined by $x_1\in (0,1)$ and $x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$ for all $n\geq 1$. Find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.
Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers $k$ such that every term of the sequence is a perfect square.
let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $
let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $
suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $
now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $
Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$
Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$-digit integer $ELMO?$
Given a right-angled triangle $ABC$ and two perpendicular lines $x$ and $y$ passing through the vertex $A$ of its right angle. For an arbitrary point $X$ on $x$ define $y_B$ and $y_C$ as the reflections of $y$ about $XB$ and $ XC $ respectively. Let $Y$ be the common point of $y_b$ and $y_c$. Find the locus of $Y$ (when $y_b$ and $y_c$ do not coincide).
Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that
$(1)$ $f(n) \leq n^8$ for any $n \geq 2$,
$(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that
$$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$
Let $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) be real numbers such that \[ a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}. \] Prove that $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$.
Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property.
$$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$
[b]a)[/b] Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $
[b]b)[/b] Prove that $ f $ is constant on the interval $ (0,\infty ) . $
[b]c)[/b] Give an example of a non-monotone such function.
a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$
b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$
Given triangle $ABC$. Suppose $P$ and $P_1$ are points on $BC, Q$ lies on $CA, R$ lies on $AB$, such that
$$\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B}$$
Let $G$ be the centroid of triangle $ABC$ and $K = AP_1 \cap RQ$. Prove that points $P,G$, and $K$ are collinear.
Triangle ABC has incircle w centered as S that touches the sides BC,CA and AB at P,Q and R respectively. AB isn't equal AC, the lines QR and BC intersects at point M, the circle that passes through points B and C touches the circle w at point N, circumcircle of triangle MNP intersects with line AP at L (L isn't equal to P). Then prove that S,L and M lie on the same line
Let $A$ be a $2\,x\,2$ matrix with real numbers. Prove that if $A^3=\mathbb{O}$ then $A^2=\mathbb{O}$.
Let $c_1<c_2<c_3$ be the three smallest positive integer values of $c$ such that the distance between the parabola $y=x^2+2020$ and the line $y=cx$ is a rational multiple of $\sqrt{2}$. Compute $c_1+c_2+c_3$.
A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$.
[i]Proposed by K. Kokhas[/i]
In the figure, the rectangle is formed by $4$ smaller equal rectangles.
If we count the total number of rectangles in the figure we find $10$.
How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?
Octagon $ABCDEFGH$ is equiangular. Given that $AB=1$, $BC=2$, $CD=3$, $DE=4$, and $EF=FG=2$, compute the perimeter of the octagon.
A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]
The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.
Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.
Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*.
Clarification:
* For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$
Prove that for any sequence $a_1, a_2, \ldots , a_{2002}$ of non-negative integers written in the usual decimal notation with $a_1 > 0$ there exists an integer $n$ such that $n^2$ starts with digits $a_1, a_2, \ldots , a_{2002}$ (in this order).
A line parallel to the side $AC$ of triangle $ABC$ cuts the side $AB$ at $K$ and the side $BC$ at $M$. $O$ is the intersection point of $AM$ and $CK$. If $AK = AO$ and $KM = MC$, prove that $AM = KB$.