Found problems: 85335
There are $n$ distinct points in the plane, no three of which are collinear. Suppose that $A$ and $B$ are two of these points. We say that segment $AB$ is independent if there is a straight line such that points $A$ and $B$ are on one side of the line, and the other $n-2$ points are on the other side. What is the maximum possible number of independent segments?
The NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each worth a set number of points. However, the NEMO is weighted using Fibonacci numbers; that is, the $n^{\text{th}}$ problem is worth $F_n$ points, where $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$. The two problem writers are fair people, so they make sure that each of them is responsible for problems worth an equal number of total points. Compute the number of ways problem writing assignments can be distributed between the two writers.
[i]Proposed by Lewis Chen[/i]
We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum.
(The total distance of the point P from the sides of an equilateral triangle is fixed )
[i]Proposed by Erfan Salavati[/i]
Let $ABC$ be a triangle with no right angle, $E$ on the line $BC$ such that $\angle AEB = \angle BAC$ and $\Delta_A$ the perpendicular to $BC$ at $E$. Let the circle $\gamma$ with diameter $BC$ intersect $BA$ again at $D$. For each point $M$ on $\gamma$ ($M$ is distinct from $B$), the line $BM$ meets $\Delta_A$ at $M'$ and the line $AM$ meets $\gamma$ again at $M''$.
(a) Show that $p(A) = AM' \times DM''$ is independent of the chosen $M$.
(b) Keeping $B,C$ fixed, and let $A$ vary. Show that $\frac{p(A)}{d(A,\Delta_A)}$ is independent of $A$.
Michel Bataille
A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by [i]Petar Filipovski, Macedonia[/i]
Let $A_1,A_2,\ldots,A_m$ be finite sets of size $2012$ and let $B_1,B_2,\ldots,B_m$ be finite sets of size $2013$ such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Find the maximum value of $m$.
Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$
Irish Mathematical Olympiad
The altitude of a quadrangular pyramid $SABCD$ passes through the intersection point of the diagonals of its base $ABCD$. From the tops of the base perpendiculars $AA_1$, $BB_1$, $CC_1$, $DD_1$ are dropped onto lines $SC$, $SD,$ $SA$ and $SB$ respectively. It turned out that the points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are different and lie on the same sphere. Prove that lines $AA_1$, $ BB_1$, $CC_1$, $DD_1$ pass through one point.
$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$.
Prove that
$$ \sum (\frac{x^2-1}{x})^2 \geq 4$$
[i]Proposed by Amin Fathpour[/i]
Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.
Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.
On a blackboard there are $ n \geq 2, n \in \mathbb{Z}^{\plus{}}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $ n$ for which it is possible to yield $ n$ identical number after a finite number of steps.
A four-digit number $n=\overline{a b c d}$, where $a, b, c$ and $d$ are digits, with $a \neq 0$, is said to be [i]guanaco[/i] if the product $\overline{a b} \times \overline{c d}$ is a positive divisor of $n$. Find all guanaco numbers.
Find all non-zero real numbers $x$ such that
\[\min \left\{ 4, x+ \frac 4x \right\} \geq 8 \min \left\{ x,\frac 1x\right\} .\]
A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if
$\bullet$ $p$ is an odd prime number
$\bullet$ $a, b$ and $c$ are distinct, and
$\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$.
(a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$.
(b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$
For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that
$$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$
[i]Mircea Fianu[/i]
We color some points in the plane with red, in such way that if $P,Q$ are red and $X$ is a point such that triangle $\triangle PQX$ has angles $30º, 60º, 90º$ in some order, then $X$ is also red. If we have vertices $A, B, C$ all red, prove that the barycenter of triangle $\triangle ABC$ is also red.
Let $a,b$ be positive real numbers such that $b^3+b\le a-a^3$. Prove that:
i) $b<a<1$
ii) $a^2+b^2<1$
A paper rectangle $ABCD$ has $AB = 8$ and $BC = 6$. After corner $B$ is folded over diagonal $AC$, what is $BD$?
Let $n>d>0$ integers. Batman, Joker, Clark play the following game in an infinite checkered board. Initially, Batman and Joker are in cells with distance $n$ and a candy is in a cell with distance $d$ to Batman. Batman is blindfold, and can only see his cell. Clark and Joker can see the whole board. The following two moves go alternately.
1 - Batman goes to an adjacent cell. If he touches Joker, Batman loses. If he touches the candy, Batman wins. If the cell is empty, Clark chooses to say loudly one of the following two words [b]hot[/b] or [b]cold[/b].
2 - Joker goes to an adjacent cell. If he touches Batman or candy, Joker wins. Otherwise, the game continues.
Determine for each $d$, the least $n$, such that Batman, and Clark can plan an strategy to ensure the Batman's win, regardless of initial positions of the Joker and of the candy.
Note: Two cells are adjacent if its have a common side. The distance between two cells $X$ and $Y$ is the least $p$ such that there exist cells $X=X_0,X_1,X_2,\dots, X_p=Y$ with $X_i$ adjacent to $X_{i-1}$ for all $i=1,2,\dots,p$.
Let $ABCD$ be a isosceles trapezoid with $AB \| CD $ , $AD=BC$, $ AC \cap BD = $ { $O$ }. $ M $ is the midpoint of the side $AD$ . The circumcircle of triangle $ BCM $ intersects again the side $AD$ in $K$. Prove that $OK \| AB $ .
Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.
In the trapezoid $ABCD$ with bases $AB$ and $CD$, let $M$ be the midpoint of side $DA$. If $BC=a$, $MC=b$ and $\angle MCB=150^\circ$, what is the area of trapezoid $ABCD$ as a function of $a$ and $b$?
Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$.
For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$.
[i]Linus Hamilton.[/i]