# TA Advanced Problem Solving Skills

High schools can opt for advance problem solving exercise. Even suitable for middle schoolers who have been practicing problem solving. Students with mathematical knowledge with the Advanced level of mathematical concepts select this course.It consists of around 700 Problems. Counting Techniques, Number Theory, Number Sense, Prime Factor and more concepts are covered.

Farmer Bob has 96 square inches of wrapping paper. Find the volume of the largest rectangular box he can wrap with the paper. Answer :64 Solution : Let the box have dimensions x, y, and z. The amount of wrapping paper should equal the surface area of the box, or 2(xy+yz+zx) = 96 => (xy+yz+zx) = 48 From the AM-GM inequality, we have So the maximum value of xyz, the volume of the box, is 64. This value can be achieved by letting x = y = z = 4. There is a box of books. Alex takes one more than half the number of the books, Bob takes 2 more than half of the remaining books in the box; Carrie takes 3 more than half of the remaining books in the box. There are still 3 books left in the box. How many books were there in the beginning? Answer :50 Solution : Going backwards, we have 3+2 = 5 → 5×2=10 → 10+2=12 → 12×2=24 → 24+1=25 → 25×2= 50. There were 50 books in the beginning. A thief stole 5/7 of Brendon’s money and spent 5/7 of the amount stolen. The thief was then caught, and the remaining money was returned to Brendon. The remaining amount was \\$40\$40 less than the amount Brendon had after being robbed. How many dollars did Brendon have before the theft? Answer :490 Solution : Let x be the amount of money that brendoa had originally any y be the amount of that was returned to brendon. Based on our back calculation figure below, we can write the equation: we also know that Solve for x using (1) and (2), we get: x = 490. There are three storage rooms named A, B, and C each with the same capacity used for storing wheat. Assume that at the beginning each storage room is fully filled with wheat. One moving machine and 12 workers can move all the wheat in storage room A in 5 hours. One moving machine and 28 workers can move all the wheat in storage room B in 3 hours. How many workers are needed in order to move all the wheat in room C with two moving machines in 2 hours? Answer :36

Workers are paving a road from Greenville to Kinston. On the first day, they paved 2 miles more than half of the total length of the road. On the second day they paved 1 mile less than half of the remaining length of the road. On the third day they finished the job by paving 20 miles. How long is the road? Answer :80 Bob harvested his crops. On the first day he harvested 5 acres less than half of his crops. On the second day he harvested 2 acres more than half of the remaining crops. On the third day he harvested 20 acres of the land. He still had 5 acres left to be harvested. How many acres does he have on his land? Answer :98Bottle A contains more Diet Coke than Bottle B. Now do the following: a. Pour from Bottle A into B as much Diet Coke as B already contains. b. Pour from B into A as much Diet Coke as A now contains. c. Pour from A into B as much Diet Coke as B now contains. Both bottles now have 64 ounces. How many more ounces were in A than in B at the beginning? Answer :48 A thief stole 5/7 of Brendon’s money and spent 5/7 of the amount stolen. The thief was then caught, and the remaining money was returned to Brendon. The remaining amount was \\$40\$40 less than the amount Brendon had after being robbed. How many dollars did Brendon have before the theft? Answer :490 Solution : Let x be the amount of money that brendoa had originally any y be the amount of that was returned to brendon. Based on our back calculation figure below, we can write the equation: we also know that Solve for x using (1) and (2), we get: x = 490. Alfreda hiked half her planned trip the first day. She only covered one-third the remaining distance the second day; rain slowed her to one-fourth the remaining distance the third day, and the fourth day she hiked three miles, which was one-fifth of what she had left. How many miles did she hike during the four days? (1993 Mathcounts State Target). Answer :48

Students require Open minded thought process, right attitude, analytical skills and questioning capabilities to be proficient in problem solving. All the skills can be developed with practicing and being creative with thoughts and ideas. But students need to note that contents are available in abundance, students need to utilize all the resources to the max. Learning is fun when used the correct techniques, and Talent academy makes sure to use the correct techniques to teach all concepts to the students.

Prime Factorization

• Greatest Common Factor and Least Common Multiple

Visuals

• Back Calculation
• Shortest and Longest Path
• Math Grids
• Net

Logic and Reasoning

• Magic Square
• Tiling Problems
• Folding Paper
• Balls and Boxes
• Probability Problems
• Possible ways to walk
• Problems on Clocks
• Harder Clock Problems with Proportion
• Permutations of Digits and Sum of N-Digit Numbers

Number Sense

• Special numbers
• Advance Number Sense

Number Theory

• INTEGER SOLUTIONS (PART 1)
• INTEGER SOLUTIONS (PART 2)
• Divisibilty and Divisors
• Cyclicity of Remainders
• Diophantine Equations

Counting Techniques

• PIE (PRINCIPLE OF INCLUSION AND EXCLUSION)
• How Many Faces Are Painted
• How Many Triangles With Integer Lengths
• How Many Ways To Walk

Inequality

• Maximal Inequality

Arithmetic Sequence

• Newton’s Little Formula
• Page Numbers
• Arithmetic Sequence and Common Terms
• Modular Arithmetic

Sequences

• Solving Recurrence Relations

Algebra

• Pythagorean Triples
• Absolute-Value Inequalities
• Consecutive Integers
• Work Word Problems

Geometry

• Heronian Triangle
• Dividing regions
• Counting Rectangles
• Geometry Potpourri
• Analytic Geometry