Found problems: 85335
Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that
$ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
Giiven $\Delta ABC$, $\angle CAB=75^{\circ}$ and $\angle ACB=45^{\circ}$. $BC$ is extended to $T$ so that $BC=CT$. Let $M$ be the midpoint of the segment $AT$. Find $\angle BMC$.
Let $A_1,\ldots,A_n$ be different collinear points. Every point is dyed by one of four colors and every of these colors is used at least once. Show that there is a line segment where two colors are used exactly once and the other two are used at least once.
The vault of a bank has $N$ locks. To open it, they must be operated simultaneously. Five executives have some of the keys, so any trio can open the vault, but no pair can do it. Determine $N$.
Given is triangle $ABC$ with $AB > AC$. Circles $O_B$, $O_C$ are inscribed in angle $BAC$ with $O_B$ tangent to $AB$ at $B$ and $O_C$ tangent to $AC$ at $C$. Tangent to $O_B$ from $C$ different than $AC$ intersects $AB$ at $K$ and tangent to $O_C$ from $B$ different than $AB$ intersects $AC$ at $L$. Line $KL$ and the angle bisector of $BAC$ intersect $BC$ at points $P$ and $M$, respectively. Prove that $BP = CM$.
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that
\[f(f(x) - y) = f(xy) + f(x)f(-y)\]
for any two real numbers $x, y$.
[i]Proposed by Pablo Valeriano[/i]
Parallelograms $ABGF$, $CDGB$ and $EFGD$ are drawn so that $ABCDEF$ is a convex hexagon, as shown. If $\angle ABG = 53^o$ and $\angle CDG = 56^o$, what is the measure of $\angle EFG$, in degrees?
[img]https://cdn.artofproblemsolving.com/attachments/9/f/79d163662e02bc40d2636a76b73f632e59d584.png[/img]
Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by [i]Morteza Saghafian[/i]
Let $a$, $b$, and $c$ be nonzero real numbers. Prove that $a+b+c$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ cannot both be $0$.
On the side $BC$ of the triangle $\Delta ABC$ is choosen point $K$,such that $2\angle BAK=3\angle KAC$.Prove that $AB^2AC^3>AK^5$
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.
A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$.
(Karl Czakler)
Which of the following can be used to distinguish a solid ball from a hollow sphere of the same radius and mass?
$\textbf{(A)}$ Measurements of the orbit of a test mass around the object.
$\textbf{(B)}$ Measurements of the time it takes the object to roll down an inclined plane.
$\textbf{(C)}$ Measurements of the tidal forces applied by the object to a liquid body.
$\textbf{(D)}$ Measurements of the behavior of the object as it oats in water.
$\textbf{(E)}$ Measurements of the force applied to the object by a uniform gravitational field.
Find all values of the parameter $a$ for which the inequality
\[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\]
has a unique solution.
A square array of dots with $7$ rows and $7$ columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of difference colors, they are then joined by a purple line segment. There are $20$ red line segments and $19$ blue line segments. Find the positive difference between the maximum and minimum number of red dots.
[asy]
size(4cm);
for (int i = 0; i <= 7; ++i) {
for (int j = 0; j <= 7; ++j) {
dot((i,j));
}
}
[/asy]
$ \mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 8$
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
Find all positive integers $A,B$ satisfying the following properties:
(i) $A$ and $B$ has same digit in decimal.
(ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$)
The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that
$f(x+1)\leq f(2x+1)$ and $f(3x+1)\geq f(6x+1)$ for $\forall$ $x\in \mathbb{R}$.
If $f(3)=2$, prove that there exist at least 2013 distinct values of $x$, for which $f(x)=2$.
A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$
\[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\]
Proposed by Navid Safaei, Iran
Given an $8\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?
[i]Proposed by James Lin.[/i]
Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$.
Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$
A part of a forest park which is located between two roads has caught fire. The fire is spreading at a speed of $10$ kilometers per hour. If the distance between the starting point of the fire and both roads is $10$ kilometers, what is the area of the burned region after two hours in kilometers squared? (Assume that the roads are long, straight parallel lines and the fire does not enter the roads)
$\textbf{(A)}\ 200\sqrt 3\qquad\textbf{(B)}\ 100 \sqrt 3\qquad\textbf{(C)}\ 400\sqrt 3 + 400 \frac{\pi}{3} \qquad\textbf{(D)}\ 200\sqrt 3 + 400 \frac{\pi}{3} \qquad\textbf{(E)}\ 400\sqrt 3 $