Clever tricks

Red Question (Brilliant Online): Prove that there is a point such that the line

hits regardless of the value of .

Pink Question: (CEMC vol.8) The equation represents a family of parabolas. There is one point through which all of these parabolas pass. Find the coordinates of this point.

Aqua Question (Brilliant Online modified): The lines

and intersect in Quadrant 1. What is the smallest possible integer value of ?

Some Useful Formulas

Here are some useful formulas:

Paul Zeitz uses an interesting notation for this. He let's

stand for the set of all possible ways of combining symbols. Then writes

Can you figure out what he means by this?

Green Question: (P. Zeitz) Without using any calculus, find the minimum value of

where , , and are real numbers and .

STEP 2004, Math I, #1: (i) Express

in the form where and are integers.

(ii) Find the positive integers

and such that .

(iii) Find the two real solutions of

STEP 2001, Math I, #2: Solve the inequalities

(i)

(ii)

Purple Question (APCM): (i) Prove that if

then

(ii) Prove that if

then

Substitution

STEP 2004, Math II, #1: Find all real values of

that satisfy:

(i) ;

(ii)

;

(iii)

Factoring

Violet Question: (P. Zeitz) Solve

where .

Yellow Question: (P. Zeitz) Find all (x,y) such that

and where .

Brown Question: (P. Zeitz) Find all positive integer solutions of .

Cyan Question: (P. Zeitz) Find all real values of

that satisfy .

Magenta Question: (P. Zeitz) Find all integer solutions of

STEP 2000, Math I, #6: (modified) Find constants

and such that

Use this to indicate by means of a sketch the region in the xy-plane for which

Sketch also the region of the xy-plane for which

A solution to this problem is posted below.

STEP 2003, Math I, #2: The first question on an examination paper is:

Solve for

in the equation .

where (in the question)

and are given non-zero real numbers. One candidate writes as the solution. Show that there are no values of and for which this will give the correct answer.

The next question on the examination paper is:

Solve for

in the equation .

where (in the question)

, , and are given non-zero real numbers. The candidate uses the same technique, giving the answer as . Show that the candidate's answer will be correct if and only if