Found problems: 85335
Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$
Solve the system of equations:
$ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$
$ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$
Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies
• $f(p) = 1$ for all primes $p$, and
• $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $.
For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?
Let $ABC$ be an acute-angled triangle with $AB < AC$ and $\Gamma$ the circumference that passes through $A,\ B$ and $C$. Let $D$ be the point diametrically opposite $A$ on $\Gamma$ and $\ell$ the tangent through $D$ to $\Gamma$. Let $P, Q$ and $R$ be the intersection points of $B C$ with $\ell$, of $A P$ with $\Gamma$ such that $Q \neq A$ and of $Q D$ with the $A$-altitude of the triangle $ABC$, respectively. Define $S$ to be the intersection of $AB$ with $\ell$ and $T$ to be the intersection of $A C$ with $\ell$. Show that $S$ and $T$ lie on the circumference that passes through $A, Q$ and $R$.
Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$
Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$.
[i]Proposed by Nikola Velov[/i]
Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that
$\bullet$ the sides of $T$ are parallel to the sides of $U$;
$\bullet$ the interior of $T$ contains exactly one point of $C$;
$\bullet$ the area of $T$ is at least $\mu$.
On the table was a pile of $135$ chocolate chips. Phil ate $\tfrac49$ of the chips, Eric ate $\tfrac4{15}$ of the chips, and Beverly ate the rest of the chips. How many chips did Beverly eat?
Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties:
(i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$;
(ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then
$ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$;
(iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$.
A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties:
If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$;
If $ s_i\geq 0$, then $ f(S)\geq 0$;
$ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$;
$ f(ST)\equal{}f(S)f(T)$,
$ f(S')\equal{}f(S)$
Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.
A number is a palindrome if it does not change when the order of its digits is reversed. For example, $121$ and $23,432$ are palindromes. How many $4$-digit numbers are palindromes?
$\text{(A) }9\qquad\text{(B) }10\qquad\text{(C) }45\qquad\text{(D) }90\qquad\text{(E) }100$
The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon.
(a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.)
(b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter.
(c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths.
It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle.
(d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon
can be increased without changing the perimeter.
(e) Prove that the polygon of maximum area must be equiangular, or have all angles equal.
(f ) Prove that the polygon of maximum area is a regular polygon.
PS. You had better use hide for answers.
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $, $ a $, $ b $, $ c $, $ d \neq 3 $ , and $ d \le 50 $. Find the maximum value of the difference between two of these prime numbers.
Triangle $ABC$ has $AC = 5$. $D$ and $E$ are on side $BC$ such that $AD$ and $AE$ trisect $\angle BAC$, with $D$ closer to $B$ and $DE =\frac32$, $EC =\frac52$ . From $B$ and $E$, altitudes $BF$ and $EG$ are drawn onto side $AC$. Compute $\frac{CF}{CG}-\frac{AF}{AG}$ .
Let $\widehat{xOy}$ be an acute angle , $A$ a point on ray $Oy$ and $B$ a point on ray $Ox$ such that $AB \perp OX$ .Prove that there are two points on $Ox$, each of the equidistant from $A$ and $Ox$.
A circle with area $\tfrac{36}{\pi}$ has the same perimeter as a square with what side length?
$\textbf{(A) }\frac{9}{\pi}\qquad\textbf{(B) }3\qquad\textbf{(C) }\pi\qquad\textbf{(D) }6\qquad\textbf{(E) }\pi^2$
Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.
Let $ABCDEFGHIJKL$ be a regular dodecagon. Find $\frac{AB}{AF} + \frac{AF}{AB}$.
Point $P$ is $\sqrt3$ units away from plane $A$. Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$?
For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$.
[b]abc Conjecture[/b]
Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: \[ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} \]
Now prove each of the following statements by using the $abc$ conjecture :
a) Fermat's last theorem for $n>N$ where $N$ is some natural number.
b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$.
c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong.
d) Prove that there are finitely many rational numbers $\frac pq$ such that: \[ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3} \]
Let $a$ and $b$ be two positive integers and $A$ be a subset of $\{1, 2,…, a + b \}$ that has more than $ \frac {a + b} {2}$ elements. Show that there are two numbers in $A$ whose difference is $a$ or $b$.
Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\]
[i]Nanang Susyanto, Jogjakarta[/i]
If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.
A farmer noticed that, during the last year, there were exactly as many calves born as during the two preceding years together. Even better, the number of pigs born during the last year was one larger than the number of pigs born during the two preceding years together. The farmer promised that if such a trend will continue then, after some years, at least twice as many pigs as calves will be born in his cattle, even though this far this target has not yet ever been reached. Will the farmer be able to keep his promise?
Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.