MathDB

2016 Latvia National Olympiad 3rd Round Grade10Problem5

Source:

7/22/2016

All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:\\ (a) There is no right triangle\\ (b) There is no acute triangle\\ having all vertices in the vertices of the 2016-gon that are still white?

geometrycombinatorics

2016 Latvia National Olympiad 3rd Round Grade11Problem5

Source:

7/22/2016

Prove that every triangle can be cut into three pieces so that every piece has axis of symmetry. Show how to cut it (a) using three line segments, (b) using two line segments.

symmetrygeometry

2016 Latvia National Olympiad 3rd Round Grade9Problem5

Source:

7/22/2016

The integer sequence

(s_i)

"having pattern 2016'" is defined as follows:\\

\circ

The first member

s_1

is 2.\\

\circ

The second member

s_2

is the least positive integer exceeding

s_1

and having digit 0 in its decimal notation.\\

\circ

The third member

s_3

is the least positive integer exceeding

s_2

and having digit 1 in its decimal notation.\\

\circ

The third member

s_3

is the least positive integer exceeding

s_2

and having digit 6 in its decimal notation.\\ The following members are defined in the same way. The required digits change periodically:

2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots

. The first members of this sequence are the following:

2; 10; 11; 16; 20; 30; 31; 36; 42; 50

. What are the 4 numbers that immediately follow

s_k = 2016

in this sequence?

recursionalgebra

2016 Latvia National Olympiad 3rd Round Grade12Problem5

Source:

7/22/2016

Consider the graphs of all the functions

y = x^2 + px + q

having 3 different intersection points with the coordinate axes. For every such graph we pick these 3 intersection points and draw a circumcircle through them. Prove that all these circles have a common point!

functionanalytic geometrygeometry

5

Problems(4)