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Gryphon Gladiatorial Mathematics practice problems Rules 1. Teams of 4 students will compete against one another to answer mathematical problems and puzzles rapidly and accurately. 2. One player from each team will work each problem on a white board or chalk board at the same time with points for the problem accruing to the team that produces a correct answer first. 3. Answers must be written on the board and circled before the referee can judge them. 4. Team members may help verbally as long as the next step or exact answer is not given. Violation of this rule is a foul that awards the points for the problem to the other team. 5. Each team member must come to the board once before any team member returns to the board. 6. Ranking of teams is by total points earned. 7. Problems come in categories with stated point values and the teams chose categories in alternation with a coin flip determining first choice. 8. The referee may always require a contestant to explain their reasoning or show their work. A correct answer only counts if it is well supported in the opinion of the referee. Guessing answers may be cool but it is not mathematics. For the teacher. The following sheets contain two problems per sheet. They need to be cut, possibly ten sheets at a time with a good paper cutter. The questions are arranged on a table not easily visible from the perspective of the students. You do not need to use all the categories in a given practice session - choose several categories appropriate to your students and arrange them by topic and point value. The actual questions at the Olympics will be different and some of the categories will be different. These categories are intended for practice for both junior and senior students. √ Calculators are not used: an answer like 8 is left in that form and most of the answers will work out to whole numbers or other values that can be computed without calculators. Many of the questions require that students recognize patterns rather than compute numerical values. Trigonometry 1 Find the tangent of θ if: 6 ! 11 6 Answer: 11 Trigonometry 2 Find H if: H Answer: ! 5 12 52 + 122 = √ 169 = 23 Trigonometry 3 Suppose that sin(θ) = 0.4, what is cos(θ)? " ! cos2(θ) Answer: = 1− √ √ 1 − 0.16 = 0.84 . sin2(θ) " so 1 − 0.42 = Trigonometry 4 Suppose that sin(θ) = 0.337, what is cos(φ)? " ! Answer: cos(φ) = sin(90 − φ) = sin(θ) = 0.337 Trigonometry 1 What are the angles of an equilateral triangle? ! ! ! π, π Answer: 60,60,60 (degrees) or π 3 3 3 (radians). Trigonometry 2 suppose that X = 8 what are Y and Z? Z Y 45 o X √ Answer Y = 8, Z = 8 2 . Trigonometry 3 Give an example of three numbers that could be the side lengths of a triangle that is neither and equilateral or a right triangle. Answer: any three positive numbers a ≤ b < c, not all equal, with a2 + b2 $= c and a + b ≥ c. Trigonometry 4 √ suppose that X = 5 3 what are Y and Z? Z Y 30 o X Answer Y = 5, Z = 10 . Trigonometry 1 If A = 67◦ and C = 22◦ then what is B? A B C Answer: 180 − 67 − 22 = 91◦ Trigonometry 2 If A > B > C then which of a, b, or c has the smallest cosine? a C B b A Answer: a . c Trigonometry 3 Demonstrate the following is or is not a right triangle: (4,4) (2,2) (7,2) Answer: It is not . The slopes of the sides are 0, and so no slope is a negative reciprocal 1, and −2 5 of the other. This means none of the three angles are right angles. Trigonometry 4 Can you have a right equilateral triangle in the plane? If not, say why, if yes, say where. Answer: No 3 × 90 = 270 > 180 and every triangle in the plane has angles summing to 180 degrees. Patterns 1 What comes next? 1, 8, 15, 22, 29, 36, ? Answer: 36+7= 43 . Patterns 2 What comes next? 1, 3, 7, 15, 31, ? Answer: (double and add one) 63 . Patterns 3 What comes next? 2, 4, 6, 10, 16, 26, ? Answer: 16+26= 42 . Patterns 4 What comes next? 2, 3, 5, 9, 17, 33, Answer:(double and subtract one) 65 . Patterns 1 What comes next? 1, 2, 3, 4, 5, 6, 7, ? Answer: 8 . Patterns 2 What comes next? 1, 4, 9, 16, 25, 36, ? Answer: (sequence of squares) 49 . Patterns 3 What comes next? 1, 2, 4, 5, 7, 8, 10, 11, ? Answer: add 1, add 2, repeat, 11+2= 13 . Patterns 4 What comes next? 1, 3, 6, 18, 36, 108, 216, ? Answer: 648 . (triple then double, repeat) Patterns 1 What comes next? 1, 3, 2, 4, 5, 7, 6, 8, ? Answer: two odds, two evens and repeat, always increasing = 9 . Patterns 2 What comes next? 2, 6, 18, 54, 162,? Answer: triple the previous number 3*162= 486 . Patterns 3 2, 3, 4, 8, 9, 16, 27, ? Answer: powers of 2 and 3 32 . Patterns 4 Where does 8 go? 1 → 4, 2 → 1, 3 → 10, 4 → 2, 5 → 15, 6 → 3, 7 → 22, 8 →?, Answer: The hailstone sequence. 8 → 4 Even numbers are divided by two, odd numbers you triple and add one. Patterns 1 Fill in the square so that there is an A, B, C , or D in each square but never the same thing twice in the same row or column. A B D C B C Answer, e.g. A D D A C B C D B A Patterns 2 Where does the missing star go? * . . . . . . . . . . * . . . . * . . . . . * . . . . . . . . . . . . . . . . . . * . . . . . * . Answer: Row 5, Column 4 Patterns 3 Each symbol may appear at most once in a row or column. Fill in the blanks! A B C B D A C D D B D C A B C D B D A C C A D B D C B A A Answer: Patterns 4 Which does not belong? Up, Down, Left and Right. 1. 2. 3. 4. 5. URDRUULL RULURRDD UUURDDLU UULLDDLL UURDDRUU Answer: Number 4 , it does not cover a 3x3 grid. Arithmetic 1 Compute: 8-7+6-5+4-3+2-1=? Answer: 4 . Arithmetic 2 26 × 35 =? 3 2 4 ×9 Answer: Cancel powers of 2 and 3 to get 3. Arithmetic 3 Simplify the fraction 6×7 . 2 1 2 +2 +1 Answer: 6 1 or 6 . Arithmetic 4 Find a positive number so that the reciprocal of the number plus two is equal to the number. √ ∼ Answer: 1 + 2 = 2.4142136 If 2 + 1 x = x then x2 − 2x − 1 = 0 and use the quadratic formula. Arithmetic 1 Compute: 1+2+4+8+16=? Answer: 31 . Arithmetic 2 45 × 37 =? 10 2 2 ×9 Answer: Cancel to 33 = 27 . Arithmetic 3 Simplify the fraction 1+2+3+4+5+6+7+8 . 4×5×6 3 . Answer: 10 Arithmetic 4 ((1 + 2) × 3)4 =? Answer: 6561 Arithmetic 1 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 4 + 4 =? Answer: 1+4+9+16= 30 . Arithmetic 2 Compute: If we multiply 2 by itself four times and subtract one what is the largest prime divisor of the resulting number? Answer: 24 − 1 = 15 = 3 × 5 so 5 . Arithmetic 3 Simplify 24 − 22 =? 25 − 23 Answer: cancel) 1/2 . (factor out a 2 and Arithmetic 4 Find the largest prime divisor of 63 − 1 Answer: 63 − 1 = 215 so 5 . Geometry 1 Suppose that we draw four line segments, what is the largest possible number of intersections? Answer:6 See above diagram. Geometry 2 What is the smallest value that the largest angle in a quadrilateral can take on? Answer: 90◦ or π 2 radians, as in a square or rectangle. Geometry 3 If one angle of a isosceles triangle is 30◦ when what other values can its angles take on? Answer: 120◦ or 75◦ depending if there are one or two 30◦ angles. Both answers are required. Geometry 4 The points (1,2), (3,0), (5,2), and (3,4) are: a) b) c) d) e) A triangle. A non-square rectangle. A non-rectangular quadrilateral. A square. None of the above Answer: d, a square. Geometry 1 If the grid is made of 1 × 1 cm squares, find the area of the shape: 8 6 4 2 0 0 2 4 6 8 Answer:Base 6 (vertically), height 4, 1 6 2 × 4= 12cm2 Geometry 2 If the grid is made of 1 × 1 cm squares, find the area of the shape: 8 6 4 2 0 0 2 4 6 8 Answer: break into 7 2 × 2 rectangle for a total of 28cm2 . Geometry 3 If the grid is made of 1 × 1 cm squares, find the perimeter of the shape: 8 6 4 2 0 0 2 4 6 8 Answer: Add it up, 32cm2 . Geometry 4 If the grid is made of 1 × 1 cm squares, find the area of the shape: 8 6 4 2 0 0 2 4 6 8 Put the diamond in a 4 × 6 rectangle and subtract the right triangle corners. Area is 12cm2 . Geometry 1 If angle a is 44◦ then what is angle b? b a Answer: 180 − 44 = 136◦. Geometry 2 Give points with wholenumber coordinates (a, b), (c, d), (e, f ) so that the points are the vertices of a triangle with area 8. Answer: E.g.: (0,0) (0,4) (4,0), check the answer directly. Geometry 3 Suppose L is a line through (1,3) and (3,9) while M is a line through (1,5) and (2,6). Demonstrate that L and M are or are not parallel. Answer: 9−3 = 3 while 6−5 = 1 so the lines have 3−1 2−1 different slopes and are not parallel. Geometry 4 Suppose a rectangle can be cut in half to yield two rectangles exactly the same shape but with half the area: What is the ratio of the longer side to the shorter side? Answer √ 2. Algebra 1 Solve for x 3x + 1 = x + 7 Answer: x = 3 Algebra 2 Solve BC + CD − AB = 0 for C. AB Answer: C = B+D Algebra 2 How many real solutions does x2 − 3x + 5 = 0 have? You may only give an answer once. Answer: Compute the discriminant (−3)2 − 4 × 1 × 5 = −11, so it has no real solutions. Algebra 4 Find a, b, c all not zero so that a2 + bx + c = 0 has no solutions. Answer: Any values so that b2 < 4ac. Algebra 1 Solve for x: 3x + 5 = 8 Answer: x = 1 . Algebra 2 Compute and simplify: (2 − 2x)(4 − x) Answer: 2x2 − 10x + 8 Algebra 3 Solve for y: x= y+1 y−1 Answer: y = x+1 x−1 . Algebra 4 Compute and simplify: (x2 + x + 2)2 Answer: x4 + 2x3 + 5x2 + 4x + 4 . Algebra 1 If ab + ac − bc = 3 solve for a. Answer: a = 3+bc b+c Algebra 2 What is the value of (x − 1)(x + 1)(x + 2) if x = 3? Answer: 2 × 4 × 5 = 40 Algebra 3 Suppose that y = xn + axn−1 + · · ·+ bx + c. If the graph of this function is: Y X then what is the smallest value n could have? Answer: 4 , this is the shape of a quartic (4thpower) curve. Algebra 4 In (x2 − x + 3)2 what is the coefficient of x2? Answer: 7 .