Found problems: 85335
Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer.
Prove that the equation cannot have five integer solutions of the form
$ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$.
Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]
Let $G$ be a simple graph on $n$ vertices with at least one edge, and let us consider those $S:V(G)\to\mathbb R^{\ge 0}$ weighings of the vertices of the graph for which $\sum_{v\in V(G)} S(v)=1$. Furthermore define
\[f(G)=\max_S\min_{(v,w)\in E(G)}S(v)S(w),\]
where $S$ runs through all possible weighings.
Prove that $f(G)=\frac1{n^2}$ if and only if the vertices of $G$ can be covered with a disjoint union of edges and odd cycles.
($V(G)$ denotes the vertices of graph $G$, $E(G)$ denotes the edges of graph $G$.)
In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$.
(1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$.
Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal.
(2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.
For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.
Let $a,$ $b,$ $c,$ $d$ be four positive reals such that $d=a+b+c+2\sqrt{ab+bc+ca}.$
Prove that $a=b+c+d-2\sqrt{bc+cd+db}.$
Darij Grinberg
A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.
Answer either (i) or (ii):
(i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line.
(II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?
The real numbers $a, b, c, d$ satisfy $a > b > c > d$ and
$$a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004.$$
Answer, with proof, to the following questions:
a) What is the smallest possible value of $a$?
b) What is the number of possible values of $a$?
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.
Let $f:\mathbb{R}\backslash0 \rightarrow \mathbb{R}\backslash0$ be a non-constant, continuous function defined such that $f(3^x2^y)=\frac{y}{x}f(3^y)$ for any $x,y \neq 0.$ Compute $\frac{f(1296)}{f(6)}.$
[i]Proposed by Richard Chen and Zachary Perry[/i]
Compute \[ \left\lfloor \dfrac {2005^3}{2003 \cdot 2004} - \dfrac {2003^3}{2004 \cdot 2005} \right\rfloor \]
A function $f$ defined on integers such that
$f (n) =n + 3$ if $n$ is odd
$f (n) = \frac{n}{2}$ if $n$ is even
If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$.
Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?
A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.
A fair die is rolled six times. The probability of rolling at least a five at least five times is
$ \textbf{(A)}\ \frac{13}{729} \qquad\textbf{(B)}\ \frac{12}{729} \qquad\textbf{(C)}\ \frac{2}{729} \qquad\textbf{(D)}\ \frac{3}{729} \qquad\textbf{(E)}\ \text{none of these} $
The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$
.
Let $n$ be a positive integer. Consider $n^2+1$ (closed, i.e. including endpoints) segments on a single line. Show that at least one of the following statements holds:
a) there are $n+1$ segments with non-empty intersection,
b) there are $n+1$ segments among which two of them are disjoint.
Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition:
The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$. Determine the least value of $f(K)$.
$\forall n\in \mathbb{Z_{+}}-\{1,2\}$ find the maximal length of a sequence with elements from a set $\{1,2,\ldots,n\}$, such that any two consecutive elements of this sequence are different and after removing all elements except for the four we do not receive a sequence in form $x,y,x,y$ ($x\neq y$).
Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$
[i]Radu Bumbăcea & Călin Popescu[/i]
Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$
Straight lines are drawn containing the sides of an unequal triangle $ABC$, its incircle $I$ circle and a its circumcircle, the center of which is not marked. Using only a ruler (without divisions), construct the symedian of the triangle (a straight line symmetrical to the median relative to the corresponding bisector), drawing no more than six lines.