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Resources tagged with epsilons similar to Speedy Summations:

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Broad Topics > Collections > epsilons

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Reciprocal Triangles

Age 16 to 18 Challenge Level:

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

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Summats Clear

Age 16 to 18 Challenge Level:

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

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Binary Squares

Age 16 to 18 Challenge Level:

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

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OK! Now Prove It

Age 16 to 18 Challenge Level:

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

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Seriesly

Age 16 to 18 Challenge Level:

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

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Be Reasonable

Age 16 to 18 Challenge Level:

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

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Fixing It

Age 16 to 18 Challenge Level:

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

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Summit

Age 16 to 18 Challenge Level:

Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.

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Growing

Age 16 to 18 Challenge Level:

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

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Shape and Territory

Age 16 to 18 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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Em'power'ed

Age 16 to 18 Challenge Level:

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

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Big, Bigger, Biggest

Age 16 to 18 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

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Remainder Hunt

Age 16 to 18 Challenge Level:

What are the possible remainders when the 100-th power of an integer is divided by 125?

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Ab Surd Ity

Age 16 to 18 Challenge Level:

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

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BT.. Eat Your Heart Out

Age 16 to 18 Challenge Level:

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?

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Exhaustion

Age 16 to 18 Challenge Level:

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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Absurdity Again

Age 16 to 18 Challenge Level:

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

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Three Ways

Age 16 to 18 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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Tetra Inequalities

Age 16 to 18 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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Golden Powers

Age 16 to 18 Challenge Level:

You add 1 to the golden ratio to get its square. How do you find higher powers?

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Biggest Bendy

Age 16 to 18 Challenge Level:

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?

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Reciprocals

Age 16 to 18 Challenge Level:

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

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Wrapping Gifts

Age 16 to 18 Challenge Level:

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

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Mod 7

Age 16 to 18 Challenge Level:

Find the remainder when 3^{2001} is divided by 7.

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Reach for Polydron

Age 16 to 18 Challenge Level:

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

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Upsetting Pitagoras

Age 14 to 18 Challenge Level:

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

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Set Square

Age 16 to 18 Challenge Level:

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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Just Touching

Age 16 to 18 Challenge Level:

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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Kissing

Age 16 to 18 Challenge Level:

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

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Shades of Fermat's Last Theorem

Age 16 to 18 Challenge Level:

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

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Purr-fection

Age 16 to 18 Challenge Level:

What is the smallest perfect square that ends with the four digits 9009?

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Real(ly) Numbers

Age 16 to 18 Challenge Level:

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?

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Cut Cube

Age 16 to 18 Challenge Level:

Find the shape and symmetries of the two pieces of this cut cube.

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Knock-out

Age 16 to 18 Challenge Level:

Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

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Plane to See

Age 16 to 18 Challenge Level:

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

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Mean Geometrically

Age 16 to 18 Challenge Level:

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

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Baby Circle

Age 16 to 18 Challenge Level:

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

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Some Cubes

Age 16 to 18 Challenge Level:

The sum of the cubes of two numbers is 7163. What are these numbers?