NL-NWO 2018

We call a positive integer a shuffle number if the following hold: (1) All digits are nonzero. (2) The number is divisible by $11$ . (3) The number is divisible by $12$ . If you put the digits in any other order, you again have a number that is divisible by $12$ . How many $10$ -digit shuffle numbers are there?

The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules: • The number $15$ is red. • If numbers $x$ and $y$ have different colours and $x + y \le 15$ , then $x + y$ is blue. • If numbers $x$ and $y$ have different colours and $x \cdot y \le 15$ , then $x \cdot y$ is red.

Determine all triples $(x, y,z)$ consisting of three distinct real numbers, that satisfy the following system of equations: $\begin {cases}x^2 + y^2 = -x + 3y + z y^2 + z^2 = x + 3y - z x^2 + z^2 = 2x + 2y - z \end {cases}$

In triangle $ABC, \angle A$ is smaller than $\angle C$ . Point $D$ lies on the (extended) line $BC$ (with $B$ between $C$ and $D$ ) such that $|BD| = |AB|$ . Point $E$ lies on the bisector of $\angle ABC$ such that $\angle BAE = \angle ACB$ . Line segment $BE$ intersects line segment $AC$ in point $F$ . Point $G$ lies on line segment $AD$ such that $EG$ and $BC$ are parallel. Prove that $|AG| =|BF|$ .

unitsize (1.5 cm);

real angleindegrees(pair A, pair B, pair C) {
real a, b, c;
a = abs(B - C);
b = abs(C - A);
c = abs(A - B);
return(aCos((a^2 + c^2 - b^2)/(2*a*c)));
};

pair A, B, C, D, E, F, G;

B = (0,0);
A = 2*dir(190);
D = 2*dir(310);
C = 1.5*dir(310 - 180);
E = extension(B, incenter(A,B,C), A, rotate(angleindegrees(A,C,B),A)*(B));
F = extension(B,E,A,C);
G = extension(E, E + D - B, A, D);

filldraw(anglemark(A,C,B,8),gray(0.8));
filldraw(anglemark(B,A,E,8),gray(0.8));
draw(C--A--B);
draw(E--A--D);
draw(interp(C,D,-0.1)--interp(C,D,1.1));
draw(interp(E,B,-0.2)--interp(E,B,1.2));
draw(E--G);

dot(" $A$ ", A, SW);
dot(" $B$ ", B, NE);
dot(" $C$ ", C, NE);
dot(" $D$ ", D, NE);
dot(" $E$ ", E, N);
dot(" $F$ ", F, N);
dot(" $G$ ", G, SW);

At a quiz show there are three doors. Behind exactly one of the doors, a prize is hidden. You may ask the quizmaster whether the prize is behind the left-hand door. You may also ask whether the prize is behind the right-hand door. You may ask each of these two questions multiple times, in any order that you like. Each time, the quizmaster will answer 'yes' or 'no'. The quizmaster is allowed to lie at most $10$ times. You have to announce in advance how many questions you will be asking (but which questions you will ask may depend on the answers of the quizmaster). What is the smallest number you can announce, such that you can still determine with absolute certainty the door behind which the prize is hidden?