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Resources tagged with Long problems similar to A Method of Defining Coefficients in the Equations of Chemical Reactions:

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How Many Balls?

Stage: 5 Challenge Level: Challenge Level:1

A bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.

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Absurdity Again

Stage: 5 Challenge Level: Challenge Level:1

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

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Roots and Coefficients

Stage: 5 Challenge Level: Challenge Level:1

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?

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Orbiting Billiard Balls

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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Unusual Long Division - Square Roots Before Calculators

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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Just Touching

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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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Dating Made Easier

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If a sum invested gains 10% each year how long before it has doubled its value?

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Put Out

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

After transferring balls back and forth between two bags the probability of selecting a green ball from bag 2 is 3/5. How many green balls were in bag 2 at the outset?

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Powerful Factors

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

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Old Nuts

Stage: 5 Challenge Level: Challenge Level:1

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?

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Pythagoras for a Tetrahedron

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . .

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Ab Surd Ity

Stage: 5 Challenge Level: Challenge Level:1

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

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Circles Ad Infinitum

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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Xtra

Stage: 4 and 5 Challenge Level: Challenge Level:1

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

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Telescoping Series

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find S_r = 1^r + 2^r + 3^r + ... + n^r where r is any fixed positive integer in terms of S_1, S_2, ... S_{r-1}.

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Equal Temperament

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

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Halving the Triangle

Stage: 5 Challenge Level: Challenge Level:1

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

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Be Reasonable

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Rational Round

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

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Sixty-seven Squared

Stage: 5 Challenge Level: Challenge Level:1

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

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Golden Ratio

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

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Factorial Fun

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

How many divisors does factorial n (n!) have?

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The Dodecahedron

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

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Baby Circle

Stage: 5 Challenge Level: Challenge Level:1

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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Thasan's Flight

Stage: 5 Challenge Level: Challenge Level:1

An aircraft flies on a bearing of 070 degrees at 350 km/hour with wind blowing at 40 km/hour from 340 degrees. Find the actual speed and bearing of the aircraft.

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Real(ly) Numbers

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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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Logosquares

Stage: 5 Challenge Level: Challenge Level:1

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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Shades of Fermat's Last Theorem

Stage: 5 Challenge Level: Challenge Level:1

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

Stage: 5 Challenge Level: Challenge Level:1

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

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Pythagoras Mod 5

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

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Chord

Stage: 5 Challenge Level: Challenge Level:1

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

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Eyes Down

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

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Substitution Cipher and Frequency Analysis

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Investigate some text to find the frequency distribution for ordinary English and use that to help you crack the coded text below.

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Janusz Asked

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Pentakite

Stage: 4 and 5 Challenge Level: Challenge Level:1

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

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Tin Tight

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

What's the most efficient proportion for a 1 litre tin of paint?

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Pythagoras’ Comma

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

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Points in Pairs

Stage: 4 Challenge Level: Challenge Level:1

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

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Swimmers

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Swimmers in opposite directions cross at 20m and at 30m from each end of a swimming pool. How long is the pool ?

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Snow and Cholera

Stage: 4 Challenge Level: Challenge Level:1

What information helped medical pioneers decide on the cause of a disease? Especially in a time before microscopes were as powerful as they are today ?

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A Change in Code

Stage: 4 Challenge Level: Challenge Level:1

There are two sets of numbers. The second is the result of the first after an increase by a constant percentage. How can you find that percentage if one set of numbers is in code?

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Peaches in General

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

It's like 'Peaches Today, Peaches Tomorrow' but interestingly generalized.

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Three by One

Stage: 5 Challenge Level: Challenge Level:1

NRICH has always had good solutions from Madras College in St Andrew's, Scotland but the solutions to this problem were truly exceptional.

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Cola Can

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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Funnel

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

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A Scale for the Solar System

Stage: 4 Challenge Level: Challenge Level:1

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

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Some Cubes

Stage: 5 Challenge Level: Challenge Level:1

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

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Two Shapes & Printer Ink

Stage: 4 Challenge Level: Challenge Level:1

If I print this page which shape will require the more yellow ink?

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BT.. Eat Your Heart Out

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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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Mean Geometrically

Stage: 5 Challenge Level: Challenge Level:1

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?