In y = ax +b when are a, -b/a, b in arithmetic progression. The
polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2
and c be in arithmetic progression?
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational
symmetry. Do graphs of all cubics have rotational symmetry?
The square ABCD is split into three triangles by the lines BP and
CP. Find the radii of the three inscribed circles to these
triangles as P moves on AD.
The sum of any two of the numbers 2, 34 and 47 is a perfect square.
Choose three square numbers and find sets of three integers with
this property. Generalise to four integers.
Find relationships between the polynomials a, b and c which are
polynomials in n giving the sums of the first n natural numbers,
squares and cubes respectively.
Explore a number pattern which has the same symmetries in different bases.
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Show that the arithmetic mean, geometric mean and harmonic mean of
a and b can be the lengths of the sides of a right-angles triangle
if and only if a = bx^3, where x is the Golden Ratio.
If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of
these numbers must be 1. Now for the complexity! When are the other
numbers real and when are they complex?
Given the nets of 4 cubes with the faces coloured in 4 colours,
build a tower so that on each vertical wall no colour is repeated,
that is all 4 colours appear.
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
Label this plum tree graph to make it totally magic!
In turn 4 people throw away three nuts from a pile and hide a
quarter of the remainder finally leaving a multiple of 4 nuts. How
many nuts were at the start?
In this 'mesh' of sine graphs, one of the graphs is the graph of
the sine function. Find the equations of the other graphs to
reproduce the pattern.
Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.
1. LATE GRIN (2 solutions)
Four circles all touch each other and a circumscribing circle. Find
the ratios of the radii and prove that joining 3 centres gives a
Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.
Try out this geometry problem involving trigonometry and number theory
What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?
Sketch the graphs for this implicitly defined family of functions.
Three triangles ABC, CBD and ABD (where D is a point on AC) are all
isosceles. Find all the angles. Prove that the ratio of AB to BC is
equal to the golden ratio.
A quadrilateral changes shape with the edge lengths constant. Show
the scalar product of the diagonals is constant. If the diagonals
are perpendicular in one position are they always perpendicular?
Two brothers belong to a club with 10 members. Four are selected
for a match. Find the probability that both brothers are selected.