Found problems: 85335
In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously:
[list]
– for any three people, at least two do not know each other;
– for any four people there are at least two who know each other.
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Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$?
An infinite triangular lattice is given, such that the distance between any two adjacent points is always equal to $1$.
Points $A$, $B$, and $C$ are chosen on the lattice such that they are the vertices of an equilateral triangle of side length $L$, and the sides of $ABC$ contain no points from the lattice. Prove that, inside triangle $ABC$, there are exactly $\frac{L^2-1}{2}$ points from the lattice.
A set of three nonnegative integers $\{x, y, z \}$ with $x<y<z$ is called historic if $\{z-y, y-x\}=\{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.
Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to $1$. A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of $good$ parabolas?
Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.
We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black.
Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black?
[It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]
Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
(A. Zaslavsky)
Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$
Let $ABCD$ be a tetragonal.
[b](a)[/b] If the plane $(P)$ cuts $ABCD,$ find the necessary and sufficient condition such that the area formed from the intersection of the plane $(P)$ and the tetragonal be a parallelogram. Prove that the problem has three solutions in this case.
[b](b)[/b] Consider one of the solutions of [b](a)[/b]. Find the situation of the plane $(P)$ for which the parallelogram has maximum area.
[b](c)[/b] Find a plane $(P)$ for which the parallelogram be a lozenge and then find the length side of his lozenge in terms of the length of the edges of $ABCD.$
Determine all pairs $(a,b)$ of positive integers satisfying
\[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]
Compute the largest prime factor of $357!+358!+359!+360!$.
[i]Proposed by Luke Robitaille
Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set
\[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}.
\]What is the minimum possible value of
\[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2
\]$ \textbf{(A)}\ 30\qquad
\textbf{(B)}\ 32\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 40\qquad
\textbf{(E)}\ 50$
Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular. For $R$ the maximum possible radius of this sphere, find $R^2$.
Let $ f(n,k)$ be the number of ways of distributing $ k$ candies to $ n$ children so that each child receives at most $ 2$ candies. For example $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$. Determine the value of $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$.
Consider triangle $ABC$ on the coordinate plane with $A = (2, 3)$ and $C =\left( \frac{96}{13} , \frac{207}{13} \right)$. Let $B$ be the point with the smallest possible $y$-coordinate such that $AB = 13$ and $BC = 15$. Compute the coordinates of the incenter of triangle $ABC$.
In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?
Raul's class has $15$ students, all with different heights. The Mathematics teacher wants to place them in a queue so that, at the beginning of the queue, they are ordered in ascending order of heights, from then on, they are ordered in descending order and Raul, who He is the tallest in the class, he cannot be at the extremes. In how many different ways is it possible to form this queue?
Find all subsets of $\mathbb N$ like $S$ such that
\[\forall m,n \in S \implies \dfrac{m+n}{\gcd(m,n)} \in S \]
Let $A_0,A_1, \ldots , A_n$ be points in a plane such that
(i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and
(ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$
Prove that the four segments connecting the vertices of the tetrahedron with the centers of gravity of the opposite faces have a common point.
In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color.
[b]IMO Shortlist 2007 Problem C5 as it appears in the official booklet:[/b]
In the Cartesian coordinate plane define the strips $ S_n \equal{} \{(x,y)|n\le x < n \plus{} 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color.
([i]Edited by Orlando Döhring[/i])
[i]Author: Radu Gologan and Dan Schwarz, Romania[/i]
Let A$BC$ be a triangle with $AB = 5, AC = 4, BC = 6$. The internal angle bisector of $C$ intersects the side $AB$ at $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM\parallel AC$ and $DN \parallel BC$. If $(MN)^2 =\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers then what is the sum of the digits of $|p - q|$?
A splitting of a planar polygon is a fi nite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.