All the points in the plane are coloured using 3 colours. Prove that there exists a triangle with vertices having the same color such that either it is isosceles or its angles are in geometric progression
123406
Prove:  If we color the plain with three different colours, then there will always be an equilateral triangle which has three vertices of the same colour.  I have proved it for two colours but I just can't solve this one.
22092
Q) Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let $F_{1}$ be the set of all pairs of circles $(S_{1}$, $S_{2}$) such that T is tangent to $S_{1}$ at P and tangent to $S_{2}$ at Q, and also such that $S_{1}$ and $S_{2}$ touch each other at a unique point, say, M. Let $E_{1}$ be the set representing the locus of M as the pair ($S_{1}$, $S_{2}$) varies in $F_{1}$. Let the set of all straight line segments joining a pair of distinct points of $E_{1}$ and passing through the point R(1, 1) be $F_{2}$. Let $E_{2}$ be the set of the mid-points of the line segments in the set $F_{2}$. Let $C$ be the circle $x^2+y^2+6(2y+7x)=53$. The number of times $C$ intersects $E_{1}$ and $E_{2}$ is (are):
54992
I'm able to prove that, for a triangle $PQR$,  $$ PQ+ QR+ RP = 0. $$ I am not sure how to show that for  $A \perp PQ$, $B\perp QR$, and $C \perp RP$, where $A$ has the same length as $PQ$, $B$ has the same length as $QR$, and $C$ has the same length as $RP$, we have  $$A+B+C=0.$$  I've made a sketch of the triangle and perpendicular vectors, but I'm not sure how to proceed.
119929
Let $a,b,c\in\Bbb R^+$. For any given $n\in\Bbb N$ there exists a triangle with sides $a^n,b^n,c^n$. Show that at least two of $a,b,c$ are equal.   I know that you can create inequalities for $a^n,b^n,c^n$, but what's next?
46290
Prove that a triangle can be inscribed in the hyperbola $xy=c^2$ whose sides touch the parabola $y^2=4ax$. I have no idea how to start this problem. In my first attempt I assumed any three points on the given rectangular hyperbola and wrote down the equations of each of the three chords. Then I tried proving that each of the three sides are tangents to the given parabola...then I put the value of $x$  from the equation of the chord in the equation of the parabola and tried to prove that the discriminant of the quadratic is equal to zero...but I am unable to proceed after that. Need help...thanks!!
73885
Can anyone give me a vectorial solution to the following problem:     Prove that if each pair of opposite edges of the tetrahedron $ABCD$ is perpendicular (that is, $AB \perp CD$ and $AC \perp BD$ and $AD \perp BC$), then the heights of the tetrahedron are concurrent.   Here, the heights (also known as the altitudes) of tetrahedron $ABCD$ are the perpendicular from $A$ to the plane $BCD$, and three other similarly defined perpendiculars.  Tetrahedra satisfying the condition of this problem are called , and this appears to be a known result.
67768
Given a $10×10$ grid with $9$ red blocks and $91$ white blocks, in each step we color one red block black and after that one white block red until there are $91$ black blocks and $9$ red blocks. Prove that there exits a step in which a black block shares an edge with a white block. I ran a few trials and couldn't get past the fifth step without adjacency. But I don't know how to build a proof.
20421
How many distinct ways are there to $2$-color the $8$ vertices of a cube, with colorings only considered distinct up to rotation?
34743
If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.! then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the answer..!! But i want a complete proof!
56620
If the edges of the complete graph $K_{17}$ (on 17 vertices with no three collinear) are each colored one of three colours can it be proven to have two or more monochromatic triangles?
7307
I was reading up on a proof of Ramsey's Theorem and I can't understand this part of the proof:      Pick a vertex $v$ from the graph, and partition the remaining vertices into two sets $M$ and $N$, such that for every vertex $w$, $w$ is in $M$ if $(v, w)$ is blue, and $w$ is in $N$ if $(v, w)$ is red. Because the graph has $R(r - 1, s) + R(r, s - 1) = |M| + |N| + 1$ vertices, it follows that either $|M| \geq R(r − 1, s)$ or $|N| \geq R(r, s − 1)$.   Why is that $|M| \geq R(r − 1, s)$ or $|N| \geq R(r, s − 1)$? I know from the proof of $R(3,3) = 6$ that given a vertex, at least $3$ are guaranteed to be either red or blue, so what is the pigeonhole argument for the general case?
4903
That's my problem: Can every triangle be divided into five isosceles triangles?  I've got to give evidence why this is true or not true...  (sorry for possible language mistakes - I'm from Germany)  Thanks in advance  Markus
85540
There are $n$ points on the plane. Any $3$ of them are inside of a unit circle. How to show that all points are inside of unit circle?  It is needed to prove that if there is a unit circle for each $3$  points which contains them, there is also a unit circle which contains all $n$ points.
35509
This is again some homework translated (hopefully not too badly) from my book  The graph $K_{n}$ is colored using $n$ different colors, in a way that each color is used at least once. Prove that there exists a triangle with its edges colored in different colors.
61841
I need a synthetic proof on this problem without the use of trigonometry.     Question: Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ the foot of perpendicular drawn from $D$ to $AC$ and $F$ the midpoint of $DE$, prove that $AF\perp BE$.    The proof is very easy using coordinate geometry.    I'm stuck at: Since point $F$ is midpoint of $DE$, it is fixed and is not a trivial information. But I can't see a way to use this information.
27759
Show that, in any finite graph with at least two vertices, there must be two vertices with the same degree.  HINTS ONLY!
24652
Let $H$ be the orthocentre of an acute triangle $ABC$. Consider the circumcenters of triangle $ABH,BCH,CAH$ .Prove that they are the vertices of a triangle that is congruent to $ABC$.   My Work:   Observe that $AH$ is perpendicular to $BC$ &amp; $O_BO_C$. (Let $O_B$ be the circumcentre of $AHC$ and the others are defined same).  Clearly,$AH\perp BC$ &amp; $AH\perp O_BO_C$. Therefore $BC||O_BO_C$  Similarly $AC||O_AO_C$ &amp; $AB||O_AO_B$  But, I failed to show $O_BO_C=BC$  You may also show that  $AO_C=AO_B$