Year 1: Fall. HSA Topics - 2 Year Cycle

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1 Year 1: Fall Primes and divisibility Derive divisibility rules for 2,3,5,6,9,10,11 Optional:Derive divisibility rules for larger primes such as 7, 13, 17, etc Show Euclid s proof of the infinitude of primes Prove the existence of an arbitrarily long prime gap using factorials Solve divisibility problems Factorization of Integers Count the number of factors of a positive integer Solve problems on counting the number of factors of a particular type (e.g. Find the number of factors that are squares, odd etc) Demonstrate Fermat s and Euler s factorization methods The Euclidean Algorithm and Linear Diophantine Equations Use the Euclidean algorithm to find GCD of two numbers Use the Euclidean algortihm to derive Bezout s identity (GCD of a and b can be written as their linear combination) Solve Problems on linear Diophantine equations GCD/LCM Problems Solve GCD/LCM problems for two and more numbers Use Venn diagrams to prove that ab = gcd(ab) lcm(a, b) Derive general solutions to a linear Diophantine equation Optional: Frobenius number for two integers Quadratics Factor quadratic expressions over integers Solve quadratic equations by factoring Zero-product property and Factor Theorem Solve quadratic equations by completing the square Use completing the square to derive the quadratic formula Solve problems that lead to quadratic equations

2 Graphs of Quadratic Functions Graph quadratic functions Solve optimizaton problems using quadratics Find the domain and range of a quadratic function Use the discriminant to determine the nature of the roots Find tangents to parabolas using the discriminant Triangles Solve problems with triangles Derive Heron s formula and use it to find the area of a given triangle Year 1: Spring Segments in a Triangle Define a cevian and identify special cevians such as angle bisectors and medians Derive Stewart s Theorem Find the lengths of an angle bisector and a median in terms of the sides of a triangle. Solve triangle problems using certain cevians. Greatest Integer Function Define and graph the greatest integer function Solve equations containing greatest integer function Derive Legendre s Theorem Solve problems on the greatest integer function Modular Arithmetic Prove basic properties State/demonstrate Fermat s Little Theorem Solve divisibility and remainder problems using modular arithmetic Advanced Modular Arithmetic Review of the previous topic Wilson s and Euler s Theorems Coordinate Geometry Area of a polygon as the sum of determinants

3 Surveyor s formula (Gauss s formula/shoelace formula) Find the coordinates of a point that partitions a segment in a given ratio Combinatorics Multiplicative Counting Principle Permutations and Combinations Permutations with Indistinguishable Items (Multinomial Coefficients) Binomial Theorem Practice expansion of binomials Find the coefficient of a given term Pascal s Triangle and is patterns

4 Year 2: Fall Factorization of Polynomials Basic factoring techniques Use Sophie Germain Identity to factor expressions of the type A 4 + 4B 4 Problems that require factoring of polynomials Factor Theorem Factorization of A n ± B n Rational Root Theorem (RRT) Viette s Formulas Sequences and Series Arithmetic and Geometric sequences and series Topics in Geometry Triangle Trigonometry Year 2: Spring Multinomial Theorem Practice expansion of multinomials Count terms in multinomial expansion Find coefficient of given term Remainder Theorem More Counting Problems Recursive Functions Absolute Value Equations Topics in Geometry Triangle Trigonometry

5 Sample Problems 1. Derive a divisibility property of Find five consecutive composite numbers. Are there six consecutive composite numbers? 3. Find all ordered pairs (A, B) such athat A23774B is divisionbe by Find the number of positive divisors of Given a multiplication table, how many times does the number 288 appear? 6. How many divisors of 2016 are perfect squares? 7. Find the prime factorization of Find the prime factorization of if you are given that = Find the greatest common divisor of 3234 and 5264 using the Euclidean algorithm. 10. Given two unmarked containers with volumes 32 and 52 gallons, how can 8 gallons of water be obtained? 11. Solve 5x + 13y = 19 over integers. 12. Find all pairs of numbers a and b if lcm(a, b) = 1989 and gcf(a, b) = Factor over integers 6x 2 37xy + 35y Solve by factoring 456x = 366x 15. Solve by completing the square x 2 5x = The product of two numbers is -336 and the sum is 10. Find the two numbers. 17. Graph y = x 2 5x The difference between two numbers is 9. Find the minimum and the maximum value of the product of these numbers. 19. A farmer wants to build a rectanglular pen partitioned into two equal sections (pictured below) with 400 meters of fencing. What is the maximum area of the pen? 20. Denest Find the equation of a tangent line at x = 8 to a parabola given by the equation y = x 2 3x + 11.

6 22. In trapezoid ABCD, the bases BC and AD have lengths 4 and 9 inches respectively. The difference of the lengths of the sides of the trapezoid is 2.5 inches and one of its diagonals is 6 inches. Find the sides of the trapezoid. 23. Given a triangle ABC with AB = 7, AC = 8. Find the area of the triangle if the median to side BC is In triangle ABC, AB = 32, BC = 10, andac = 26. If CH and CM are the altitude and median to AB, compute the length of HM. (NYSML 1988). 25. How many ending zeros does the number 2017! have? 26. Solve for x: 8x 1 2 = What is the highest power of 12 that divides 2017!? 28. Evaluate Prove that is divisble by Find the remainder when is divided by Prove that the equation x 2 + y 2 = 59 4n+59 has no solution in integers. 32. How many positive numbers are less than and relatively prime to it? 33. How many positive numbers n are less than and gcd(n, 12600) = 72? 34. Determine if the number is prime. (Ustinov) 35. The vertices of a pentagon ABCDE are given by the points A(1, 7), B(4, 12), E(7, 2), D(8, 11), andc(6, 10). Find the area of the pentagon. ( ) ( ) ( ) If + = and x > 996, compute x (NYSML 1991) x 37. Two boys and four girls need to be picked from a group of six boys and seven girls to participate in a school play. How many different selections can be made? ( 38. Find a constant coefficient in the expansion of x ) 20 x 39. Given a three-digit number 4A1 raised to the power of 1A4, for which the tens digit is 2, compute all possible values for the digit A (ARML) 40. Compute Factor over the integers x y Factor over the integers x 6 9x 4 x x Factor over the integers x y Solve x 3 79x = Given a cubic equation 4x 3 + 3x 2 + 2x + 1 = 0 with roots r 1, r 2, r 3, find the value of r 2 1r 2 + r 1 r r 2 2r 3 + r 2 r r 2 1r 3 + r 1 r 2 3

7 46. In an arithmetic progression, the ratio of the sum of the first r terms to the sum of the first s terms is equal to the ratio of r 2 to s 2 (r s). Compute the ratio of the 8 th term to the 23 rd term.(arml) 47. Prove that the sum of the squares of the distances from the vertex of the right angle, in a right triangle, to the trisection points along the hypotenuse is equal to 5 the square of the 9 measure of the hypotenuse. (Challenging Problems in Geometry 2) 48. In quadrilateral ABCD, no pair of opposite sides is parallel. The acute angle between the diagonals of the quadrilateral is 75. Find the exact area of ABCD if the lengths of the diagonals are 8 cm and 13 cm. 49. How many terms does a completely simplified expansion of (a + b + c) 11 contain? 50. What is the coefficient of the term containing ab 2 c 3 in the expansion of (a + 2b 3c) 6? 51. Apolynomial P (x) leaves the remainder of -11 when divided by x 4 and the remainder of 16 when divided by x + 5. Find the remainder left after dividing P (x) by (x 4)(x + 5) 52. A grid, 14 squares wide and 20 squares long, is drawn in a plane. How many rectangles whose sides are the edges of the grid can be formed on this grid? 53. How many numbers can be made using only digits 1,2, or 3 such that the sum of the digits is 10? 54. Evaluate x = Find the number of ways to tile a 10x1 strip using only 1x1 squares or 2x1 dominoes. 56. Solve 2x + 3 3x 5 = A circle has two parallel chords of length x that are x units apart. If the part of the circle included between the chords has area 2 + π, find x. (HMMT) 58. In parallelogram ABCD, angle A is acute and AB = 5. Point E is on AD with AE = 4 and BE = 3. A line through B, perpendicular to CD, intersects CD at F. If BF = 5, find EF