Are you over 18 and want to see adult content?
More Annotations
A complete backup of michaelkors.com.se
Are you over 18 and want to see adult content?
A complete backup of mattermatters.com
Are you over 18 and want to see adult content?
A complete backup of erfolg-im-beruf.de
Are you over 18 and want to see adult content?
A complete backup of onekindesign.com
Are you over 18 and want to see adult content?
A complete backup of christywhitman.com
Are you over 18 and want to see adult content?
A complete backup of optometrists.asn.au
Are you over 18 and want to see adult content?
Favourite Annotations
A complete backup of thecarspacer.com
Are you over 18 and want to see adult content?
A complete backup of lizziebennet.com
Are you over 18 and want to see adult content?
A complete backup of rockclimbing.com
Are you over 18 and want to see adult content?
A complete backup of gaziantepakinsoft.com
Are you over 18 and want to see adult content?
A complete backup of aquasunozone.com
Are you over 18 and want to see adult content?
A complete backup of kitexchange.com.au
Are you over 18 and want to see adult content?
A complete backup of warisabitchshort.com
Are you over 18 and want to see adult content?
Text
MATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
As the maximum sum of the last two digits is 18, the greatest sum will be 20. Therefore we can obtain digital sums of 4, 9, or 16, with the last two digits adding to 2, 7, or 14 respectively. 4: 2002, 2011, 2020. 9: 2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070. 16: 2059, 2068, 2077, 2086, 2095. That is, there are sixteen years during theMATHSCHALLENGE.NET
Solution. If the average score after four tests is 85, the total of all four test scores must be 85 4 = 340. As the maximum score in any one test would be 100, we shall assume that they scored 100 in three tests, making a total of 300. Hence the minimum score in any one test could be 340 300 = 40%. What would the student need to score on theMATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Traditional Roman numerals are made up of the following denominations: I = 1. V = 5. X = 10. L = 50. C = 100. D = 500. M = 1000. You will read about many different rules concerning Roman numerals, but the truth is that the Romans only had one simple rule:MATHSCHALLENGE.NET
The proper divisors of a positive integer, n, are all the divisors excluding n itself. For example, the proper divisors of 6 are 1, 2, and 3. A number, n, is said to be multiplicatively perfect if the product of its proper divisors equals n. The smallest such example issix: 6 = 1 × 2
MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
As the maximum sum of the last two digits is 18, the greatest sum will be 20. Therefore we can obtain digital sums of 4, 9, or 16, with the last two digits adding to 2, 7, or 14 respectively. 4: 2002, 2011, 2020. 9: 2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070. 16: 2059, 2068, 2077, 2086, 2095. That is, there are sixteen years during theMATHSCHALLENGE.NET
Solution. If the average score after four tests is 85, the total of all four test scores must be 85 4 = 340. As the maximum score in any one test would be 100, we shall assume that they scored 100 in three tests, making a total of 300. Hence the minimum score in any one test could be 340 300 = 40%. What would the student need to score on theMATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Traditional Roman numerals are made up of the following denominations: I = 1. V = 5. X = 10. L = 50. C = 100. D = 500. M = 1000. You will read about many different rules concerning Roman numerals, but the truth is that the Romans only had one simple rule:MATHSCHALLENGE.NET
The proper divisors of a positive integer, n, are all the divisors excluding n itself. For example, the proper divisors of 6 are 1, 2, and 3. A number, n, is said to be multiplicatively perfect if the product of its proper divisors equals n. The smallest such example issix: 6 = 1 × 2
MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Mathematical. Nick's Puzzles range over geometry, probability, number theory, algebra, calculus, and logic. Problems and solutions are presented clearly and provide an excellent challenge. Purdue University Problem of the Week. Canadian Mathematics Competition. TheMATHSCHALLENGE.NET
The intersection of the circle with the perpendicular bisector produces points C and D and it should be possible to see that quadrilateral ACBD is square.. The construction of a pentagon is a little more difficult. Begin, as before, bisecting segment AB to locate point O and drawing circle OA, producing points C and D.. Then bisect segment OB to locate point E.MATHSCHALLENGE.NET
Numerals must be arranged in descending order of size. For example, three ways that sixteen could be written are XVI, XIIIIII, VVVI; the first being the preferred form as it uses the least number of numerals. The "descending size" rule was introduced to allow the use of subtractive combinations. For example, four can be written IVbecause it is
MATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. By ensuring that the row and column in the grid have the same total, how many different totals can this be done with?MATHSCHALLENGE.NET
In a semi-circle the triangle will be right-angled, so using the Pythagorean Theorem, a 2 + b 2 = c 2. Multiplying through by π /8 we get, ½ π (a/2) 2 + ½ π (b/2) 2 = ½ π (c/2) 2.In other words, if semi-circles are drawn on the sides of a right-angle triangle, the area of the semi-circles on the shorter sides will be equal to area of the semi-circle on the hypotenuse.MATHSCHALLENGE.NET
The proper divisors of a positive integer, n, are all the divisors excluding n itself. For example, the proper divisors of 6 are 1, 2, and 3. A number, n, is said to be multiplicatively perfect if the product of its proper divisors equals n. The smallest such example issix: 6 = 1 × 2
MATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Prove that the volume of an open-top box is maximised iff the area of the base is equal to the area of the four sides.MATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
Determine the nature of all multiplicatively perfect numbers. Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value. Prove that 6 n + 8 n is divisible by 7 iff n isodd.
MATHSCHALLENGE.NET
The engine will scan through the problem description, details, solution, and a set of topic keywords for every problem. You can further refine your search by requiring an exact match of every word in your list (AND) or any of your keywords (OR). You can also select the difficulty level of the problem, for which guidance is givenbelow.
MATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Solution. Consider the following diagrams. So there would be 1 + 4 + 9 + 4 + 2 = 20 squares that can be drawn on a 4 4 grid.MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
Determine the nature of all multiplicatively perfect numbers. Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value. Prove that 6 n + 8 n is divisible by 7 iff n isodd.
MATHSCHALLENGE.NET
The engine will scan through the problem description, details, solution, and a set of topic keywords for every problem. You can further refine your search by requiring an exact match of every word in your list (AND) or any of your keywords (OR). You can also select the difficulty level of the problem, for which guidance is givenbelow.
MATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Solution. Consider the following diagrams. So there would be 1 + 4 + 9 + 4 + 2 = 20 squares that can be drawn on a 4 4 grid.MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Determine the nature of all multiplicatively perfect numbers. Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value. Prove that 6 n + 8 n is divisible by 7 iff n isodd.
MATHSCHALLENGE.NET
As the maximum sum of the last two digits is 18, the greatest sum will be 20. Therefore we can obtain digital sums of 4, 9, or 16, with the last two digits adding to 2, 7, or 14 respectively. 4: 2002, 2011, 2020. 9: 2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070. 16: 2059, 2068, 2077, 2086, 2095. That is, there are sixteen years during theMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Problem. A square is split into four smallers squares and exactly two of these smaller squares are shaded. For example, the top left and bottom right squares could be shaded.MATHSCHALLENGE.NET
That is, p = 6k±1, where k is a natural number. However, it is very important to appreciate that although this formula generates every prime, p > 3, not every number it generates is prime; for example, for k = 4, 6 × 4 + 1 = 25, which is clearly not prime.MATHSCHALLENGE.NET
Solution. Consider the diagram, with regions identified by the numbers 1 to 6. In a semi-circle the triangle will be right-angled, so using the Pythagorean Theorem, a2 + b2 = c2. Multiplying through by π /8 we get, ½ π ( a /2) 2 + ½ π ( b /2) 2 = ½ π ( c /2) 2. In other words, if semi-circles are drawn on the sides of aMATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Prove that the volume of an open-top box is maximised iff the area of the base is equal to the area of the four sides.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Frequently Asked Questions How do you work out the sum of divisors? Imagine you wish to work out the sum of divisors of the number 72.MATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
Determine the nature of all multiplicatively perfect numbers. Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value. Prove that 6 n + 8 n is divisible by 7 iff n isodd.
MATHSCHALLENGE.NET
The engine will scan through the problem description, details, solution, and a set of topic keywords for every problem. You can further refine your search by requiring an exact match of every word in your list (AND) or any of your keywords (OR). You can also select the difficulty level of the problem, for which guidance is givenbelow.
MATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Solution. Consider the following diagrams. So there would be 1 + 4 + 9 + 4 + 2 = 20 squares that can be drawn on a 4 4 grid.MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
Determine the nature of all multiplicatively perfect numbers. Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value. Prove that 6 n + 8 n is divisible by 7 iff n isodd.
MATHSCHALLENGE.NET
The engine will scan through the problem description, details, solution, and a set of topic keywords for every problem. You can further refine your search by requiring an exact match of every word in your list (AND) or any of your keywords (OR). You can also select the difficulty level of the problem, for which guidance is givenbelow.
MATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Solution. Consider the following diagrams. So there would be 1 + 4 + 9 + 4 + 2 = 20 squares that can be drawn on a 4 4 grid.MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Determine the nature of all multiplicatively perfect numbers. Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value. Prove that 6 n + 8 n is divisible by 7 iff n isodd.
MATHSCHALLENGE.NET
As the maximum sum of the last two digits is 18, the greatest sum will be 20. Therefore we can obtain digital sums of 4, 9, or 16, with the last two digits adding to 2, 7, or 14 respectively. 4: 2002, 2011, 2020. 9: 2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070. 16: 2059, 2068, 2077, 2086, 2095. That is, there are sixteen years during theMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Problem. A square is split into four smallers squares and exactly two of these smaller squares are shaded. For example, the top left and bottom right squares could be shaded.MATHSCHALLENGE.NET
That is, p = 6k±1, where k is a natural number. However, it is very important to appreciate that although this formula generates every prime, p > 3, not every number it generates is prime; for example, for k = 4, 6 × 4 + 1 = 25, which is clearly not prime.MATHSCHALLENGE.NET
Solution. Consider the diagram, with regions identified by the numbers 1 to 6. In a semi-circle the triangle will be right-angled, so using the Pythagorean Theorem, a2 + b2 = c2. Multiplying through by π /8 we get, ½ π ( a /2) 2 + ½ π ( b /2) 2 = ½ π ( c /2) 2. In other words, if semi-circles are drawn on the sides of aMATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Prove that the volume of an open-top box is maximised iff the area of the base is equal to the area of the four sides.MATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Frequently Asked Questions How do you work out the sum of divisors? Imagine you wish to work out the sum of divisors of the number 72.MATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
mathschallenge.net is run by one person (Colin Hughes) in his spare time, so please be patient when waiting for replies. I remain strongly committed to providing mathematical problems for educational and recreational purposes in an entirely non-commercial setting. Consequently I will not accept business proposals or requests to feature advertising.MATHSCHALLENGE.NET
Solution. Consider the following diagrams. So there would be 1 + 4 + 9 + 4 + 2 = 20 squares that can be drawn on a 4 4 grid.MATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
The proper divisors of a positive integer, n, are all the divisors excluding n itself. For example, the proper divisors of 6 are 1, 2, and 3. A number, n, is said to be multiplicatively perfect if the product of its proper divisors equals n. The smallest such example issix: 6 = 1 × 2
MATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
Prove that the roots of the polynomial, x n + c n-1 x n-1 + + c 2 x 2 + c 1 x + c 0 = 0, are irrational or integer. Hops And Slides But Never Square Problem ID: 372 (07 Aug 2010) Prove that the minimum number of moves to completely reverse the positions of theMATHSCHALLENGE.NET
Solution. Consider the two diagrams below. The area of the square is 4, so the area of the large triangle is 2 (half of the square) and the area of the small triangle is 1 (quarter of the square). Hence the area of the arrow is 2 1 = 1 square unit. What would be the area of a similar arrow, drawn in a 10 10 square? Can you generalise for an n nMATHSCHALLENGE.NET
mathschallenge.net is run by one person (Colin Hughes) in his spare time, so please be patient when waiting for replies. I remain strongly committed to providing mathematical problems for educational and recreational purposes in an entirely non-commercial setting. Consequently I will not accept business proposals or requests to feature advertising.MATHSCHALLENGE.NET
Solution. Consider the following diagrams. So there would be 1 + 4 + 9 + 4 + 2 = 20 squares that can be drawn on a 4 4 grid.MATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
Suppose you wish to find the number of divisors of 48. Starting with 1 we can work through the set of natural numbers and test divisibility in each case, noting that divisors can be listed in factor pairs. 48 = 1×48 = 2×24 = 3×16 = 4×12 = 6×8. Hence we can see that 48 has exactly ten divisors. It should also be clear that, using thisMATHSCHALLENGE.NET
The proper divisors of a positive integer, n, are all the divisors excluding n itself. For example, the proper divisors of 6 are 1, 2, and 3. A number, n, is said to be multiplicatively perfect if the product of its proper divisors equals n. The smallest such example issix: 6 = 1 × 2
MATHSCHALLENGE.NET
Problem. It can be seen that 22 − 1 = 3 is prime. Find the next example of a prime which is one less than a perfect square. Problem ID: 281 (15 Jul 2006) Difficulty: 2 Star.MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Solution. If we let the number of birds be represented by b and the number of rabbits be represented by r then we get the following two equations: b + r = 16 ( 1) 2 b + 4 r = 38 ( 2) Dividing the second equation by two gives: b + 2 r = 19 ( 3) If we now subtract equation ( 1) from equation ( 3) we get r = 3, and as b + r = 16 it follows thatMATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Contact mathschallenge.net. mathschallenge.net is run by one person (Colin Hughes) in his spare time, so please be patient when waiting for replies. I remain strongly committed to providing mathematical problems for educational and recreational purposes in an entirely non-commercial setting.MATHSCHALLENGE.NET
History of Mathematics. Tools and Resources. Programming Problems. Revision. The links are checked regularly, but if you discover a broken link or if you would like to contribute a link that you have found particularly useful, please email me at the addred below. Click icon to contact mathschallenge.net.MATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Problem. Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, , ÷, and parentheses), express all of the integers from 1 to 25. For example, 1 = 2 3 – (1 + 4)MATHSCHALLENGE.NET
A website dedicated to the puzzling world of mathematics. Frequently Asked Questions. This section has been produced in response to the continued FAQ received, often relating to one of the published solutions or just general curiosity.MATHSCHALLENGE.NET
If we let the radius of the circle be r then the area of the circle is given by π r 2 and the area of the outside square will be ( 2 r) 2 = 4 r 2. It should also be clear that the red square is exactly half the area of the outside square, so its area will be 2 r 2. ∴ (Area ofCircle) / (Area of
MATHSCHALLENGE.NET
Solution. If the average score after four tests is 85, the total of all four test scores must be 85 4 = 340. As the maximum score in any one test would be 100, we shall assume that they scored 100 in three tests, making a total of 300. Hence the minimum score in any one test could be 340 300 = 40%. What would the student need to score on theMATHSCHALLENGE.NET
Problem. A square is split into four smallers squares and exactly two of these smaller squares are shaded. For example, the top left and bottom right squares could be shaded.MATHSCHALLENGE.NET
Traditional Roman numerals are made up of the following denominations: I = 1. V = 5. X = 10. L = 50. C = 100. D = 500. M = 1000. You will read about many different rules concerning Roman numerals, but the truth is that the Romans only had one simple rule:MATHSCHALLENGE.NET
Problem. Sarah started school at the age of five. She spent one quarter of her life being educated, and went straight into work. After working for one half of herMATHSCHALLENGE.NET
Solution. Consider the diagram, with regions identified by the numbers 1 to 6. In a semi-circle the triangle will be right-angled, so using the Pythagorean Theorem, a2 + b2 = c2. Multiplying through by π /8 we get, ½ π ( a /2) 2 + ½ π ( b /2) 2 = ½ π ( c /2) 2. In other words, if semi-circles are drawn on the sides of aMATHSCHALLENGE.NET
* Home
* Latest
* Problems
* FAQ
* Links
RSS
WELCOME TO MATHSCHALLENGE.NET A WEBSITE DEDICATED TO THE PUZZLING WORLD OF MATHEMATICS Click icon to contactmathschallenge.net
© mathschallenge.netDetails
Copyright © 2024 ArchiveBay.com. All rights reserved. Terms of Use | Privacy Policy | DMCA | 2021 | Feedback | Advertising | RSS 2.0