### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### How Many Solutions?

Find all the solutions to the this equation.

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

# Parabolas Again

##### Stage: 4 and 5 Challenge Level:

This problem was solved by Ali Abu Hijleh, age 13, Alex Liao, age 14 and Prateek Mehrotra, age 14 from Riccarton High School, Christchurch, New Zealand and Andrei Lazanu, age: 12 from School No. 205, Bucharest, Romania. Well done all of you. This is Andrei's solution:

I observed that the equations of the parabolas given in the problem are of the form: $y = a(x + b)^2 + c$

where positive corresponds to parabolas with the opening towards the top, and negative to parabolas with the opening towards the bottom; are the coordinates of the peak. I identified the parabolas in your pattern and I verified them with the help of a small program in Matlab:

x=-8:0.1:8

y1=-2*(x+6).*(x+6)-2

y2=-2*(x+4).*(x+4)-1

y3=-2*(x+2).*(x+2);

y4=-2*x.*x+1;

y5=-2*(x-2).*(x-2)+2;

y6=-2*(x-4).*(x-4)+3;

y7=-2*(x-6).*(x-6)+4;

y8=2*(x+6).*(x+6)+2;

y9=2*(x+4).*(x+4)+1;

y10=2*(x+2).*(x+2);

y11=2*x.*x-1;

y12=2*(x-2).*(x-2)-2;

y13=2*(x-4).*(x-4)-3;

y14=2*(x-6).*(x-6)-4;

plot(x,y1,x,y2,x,y3,x,y4,x,y5,x,y6,x,y7,x,y8,x,y9,x,y10,x,y11,x,y12,x,y13,x,y14)

hold

axis([-8 8 -4 4]);

grid

Consequently, the equations of the 14 parabolas in your pattern are:

$y = -2(x + 6)^2 - 2$

$y = -2(x + 4)^2 - 1$

$y = -2(x + 2)^2$

$y = -2x^2 + 1$

$y = -2(x - 2)^2 + 2$

$y = -2(x - 4)^2 + 3$

$y = -2(x - 6)^2 + 4$

$y = 2(x + 6)^2 + 2$

$y = 2(x + 4)^2 + 1$

$y = 2(x + 2)^2$

$y = 2x^2 - 1$

$y = 2(x - 2)^2 - 2$

$y = 2(x - 4)^2 - 3$

$y = 2(x - 6)^2 - 4$

I created my own pattern with a Matlab program and here are the graphs:

Andrei supplied the Matlab program but in order to challenge you to find his equations it is not reproduced here