Problems

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is \(18\), find the area of the small blue triangle.

In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.

Which triangle has the largest area? The dots form a regular grid.

What is the ratio between the red and blue area? All shapes are semicircles and the red ones have equal radii.

In a parallelogram \(ABCD\), point \(E\) belongs to the side \(CD\) and point \(F\) belongs to the side \(BC\). Show that the total red area is the same as the total blue area:

The figure below is a regular pentagram. What is larger, the black area or the blue area?

Eleven sages were blindfolded and on everyone’s head a cap of one of \(1000\) colours was put. After that their eyes were untied and everyone could see all the caps except for their own. Then at the same time everyone shows the others one of the two cards – white or black. After that, everyone must simultaneously name the colour of their caps. Will they succeed?

The recertification of the Council of Sages takes place as follows: the king arranges them in a column one by one and puts on a cap of white or black colours for each. All the sages see the colours of all the caps of the sages standing in front, but they do not see the colour of their own and all those standing behind. Once a minute one of the wise men must shout one of the two colours. (each sage shouts out a colour once). After the end of this process the king executes every sage who shouts a colour different from the colour of his cap. On the eve of the recertification all one hundred members of the Council of Sages agreed and figured out how to minimize the number of those executed. How many of them are guaranteed to avoid execution?

At the elections to the High Government every voter who comes, votes for himself (if he is a candidate) and for those candidates who are his friends. The forecast of the media service of the mayor’s office is considered good if it correctly predicts the number of votes at least for one of the candidates, and not considered good otherwise. Prove that for any forecast voters can show up at the elections in such a way that this forecast will not be considered good.

In this example we will discuss division with remainder. For polynomials \(f(x)\) and \(g(x)\) with \(\deg(f)\geq \deg(g)\) there always exists polynomials \(q(x)\) and \(r(x)\) such that \[f(x)=q(x)g(x)+r(x)\] and \(\deg(r)<\deg(g)\) or \(r(x)=0\). This should look very much like usual division of numbers, and just like in that case, we call \(f(x)\) the dividend, \(g(x)\) the divisor, \(q(x)\) the quotient, and \(r(x)\) the remainder. If \(r(x)=0\), we say that \(g(x)\) divides \(f(x)\), and we may write \(g(x)\mid f(x)\). Let \(f(x)=x^7-1\) and \(g(x)=x^3+x+1\). Is \(f(x)\) divisible by \(g(x)\)?