Show that all integers greater than or equal to \(8\) can be written as a sum of some \(3\)s and \(5\)s. e.g. \(11=3+3+5\). Note that there’s no way to write \(7\) in such a way.
Show that \(R(4,3)\le9\). That is, no matter how you colour the edge of \(K_9\), there must be a red \(K_4\) or a blue \(K_3\).
Show that \(R(4,4)\le18\) - that is, no matter how you colour the edges of \(K_{18}\), there must be a monochromatic \(K_4\).
By considering \(k-1\) copies of \(K_{k-1}\), show that \(R(k,k)\ge(k-1)^2\).
Let \(s>2\) and \(t>2\) be integers. Show that \(R(s,t)\le R(s-1,t)+R(s,t-1)\).
The sum of digits of a positive integer \(n\) is the same as the number of digits of \(n\). What are the possible products of the digits of \(n\)?
Find, with proof, all integer solutions of \(a^3+b^3=9\).
Consider an \(n\)-dimensional simplex \(\mathcal{A} = A_1A_2...A_{n+1}\), namely a body spanned over vertices \((0,0,...,0), (1,0,0,...,0), (0,1,0,0...,0), ... (0,0,...0,1)\). \[\mathcal{A} = \{\sum_{i=0}^{n}a_i(0,0,...,1,...,0), \,\,\, a_i \geq 0, \,\,\,\, \sum_{i=1}^{n+1}a_i = 1\}.\] Where next to \(a_i\) there is a point with coordinate where \(1\) is in \(i\)-th place. The point \((0,0,...,0)\) belongs to the simplex as well.
A simplicial subdivision of an \(n\)-dimensional simplex \(\mathcal{A}\) is a partition of \(\mathcal{A}\) into small simplices (cells)
of the same dimension, such that any two cells are either disjoint, or
they share a full face of a certain dimension.
Define a Sperner’s coloring of a simplicial subdivision as an assignment
of \(n+1\) colors to the vertices of
the subdivision, so that the vertices of \(\mathcal{A}\) receive all different colors,
and points on each face of \(\mathcal{A}\) use only the colors of the
vertices defining the respective face of \(\mathcal{A}\).
Consider a simplicial subdivision given by pairwise connected middles of
all the segments in the original simplex. Assign the numbers \(0,1,2...,n\) to the subdivision vertices in
such a way as to conduct a Sperner’s coloring in such a way that you
will have only one rainbow simplex.
(USO 1974) Let \(a,b,c\) be three distinct integers, and let \(P(x)\) be a polynomial whose coefficients are all integers. Prove that it is not possible that the following three conditions hold at the same time: \(P(a)=b, P(b)=c,\) and \(P(c)=a\).
For a polynomial \(P(x)=ax^2+bx+c\), consider the following two kinds of transformations:
Swap coefficients \(a\) and \(c\). Hence the polynomial \(P(x)\) becomes \(cx^2+bx+a\) after this transformation.
For any number \(t\) of your choice, change the variable \(x\) into \(x+t\). For example, with the choice of \(t=1\), after this transformation, the polynomial \(x^2+x+1\) becomes \((x+1)^2+(x+1)+1=x^2+3x+3\).
Is it possible, using only a sequence of these two transformations, to change the polynomial \(x^2-x-2\) into the polynomial \(x^2-x-1\)?