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ACHS Math Team Solutions to Math League Contest 2 of 28 Nov 2006 Peter S. Simon Problem 2-1 What is the only number less than √ 2006 whose square is 2006? Problem 2-1 What is the only number less than √ 2006 whose square is 2006? There is equal to 2006. They √ numbers whose √ are only two √ square √ are 2006 and − 2006. Since − 2006 < 2006, the answer is √ − 2006. Problem 2-2 A rectangle is divided into two congruent trapezoids, as shown. The lengths of the legs of both trapezoids are 4 and 5. The length of the shorter base of both trapezoids is 3. What is the area of the rectangle? 3 5 4 3 4 Problem 2-2 A rectangle is divided into two congruent trapezoids, as shown. The lengths of the legs of both trapezoids are 4 and 5. The length of the shorter base of both trapezoids is 3. What is the area of the rectangle? 3 3 5 4 3 Draw perpendiculars as shown. Each is of length 4. 4 3 Problem 2-2 A rectangle is divided into two congruent trapezoids, as shown. The lengths of the legs of both trapezoids are 4 and 5. The length of the shorter base of both trapezoids is 3. What is the area of the rectangle? 3 3 3 5 4 3 4 3 3 Draw perpendiculars as shown. Each is of length 4. Middle line segments must be of length 3 by Pythagorean theorem. The area of rectangle is then ((3 + 3 + 3)(4) = 9 × 4 = 36 Problem 2-3 One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2). What is the largest number x for which (2, 2, 2, x) satisfies a2 + b2 + c 2 + d 2 = abcd? Problem 2-3 One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2). What is the largest number x for which (2, 2, 2, x) satisfies a2 + b2 + c 2 + d 2 = abcd? 22 + 22 + 22 + x 2 = 8x Problem 2-3 One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2). What is the largest number x for which (2, 2, 2, x) satisfies a2 + b2 + c 2 + d 2 = abcd? 22 + 22 + 22 + x 2 = 8x 12 + x 2 = 8x Problem 2-3 One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2). What is the largest number x for which (2, 2, 2, x) satisfies a2 + b2 + c 2 + d 2 = abcd? 22 + 22 + 22 + x 2 = 8x 12 + x 2 = 8x x 2 − 8x + 12 = 0 Problem 2-3 One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2). What is the largest number x for which (2, 2, 2, x) satisfies a2 + b2 + c 2 + d 2 = abcd? 22 + 22 + 22 + x 2 = 8x 12 + x 2 = 8x x 2 − 8x + 12 = 0 (x − 2)(x − 6) = 0 Problem 2-3 One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2). What is the largest number x for which (2, 2, 2, x) satisfies a2 + b2 + c 2 + d 2 = abcd? 22 + 22 + 22 + x 2 = 8x 12 + x 2 = 8x x 2 − 8x + 12 = 0 (x − 2)(x − 6) = 0 so x = 2 or x = 6 Problem 2-4 I filled 49 packages with big and small pens. No package was left empty, and all the packages were filled differently. For example, a package could have been filled with 3 pens in only 4 ways: 3 big, 2 big and 1 small, 1 big and 2 small, 3 small. What is the least total number of pens I could have put in these packages? 5 Problem 2-4 I filled 49 packages with big and small pens. No package was left empty, and all the packages were filled differently. For example, a package could have been filled with 3 pens in only 4 ways: 3 big, 2 big and 1 small, 1 big and 2 small, 3 small. What is the least total number of pens I could have put in these packages? There are 2 ways to package 1 pen (1 big, 1 small). 5 Problem 2-4 I filled 49 packages with big and small pens. No package was left empty, and all the packages were filled differently. For example, a package could have been filled with 3 pens in only 4 ways: 3 big, 2 big and 1 small, 1 big and 2 small, 3 small. What is the least total number of pens I could have put in these packages? There are 2 ways to package 1 pen (1 big, 1 small). There are 3 ways to package 2 pens (3 big, 2 big and 1 small, 1 big and 2 small, 3 small). 5 Problem 2-4 I filled 49 packages with big and small pens. No package was left empty, and all the packages were filled differently. For example, a package could have been filled with 3 pens in only 4 ways: 3 big, 2 big and 1 small, 1 big and 2 small, 3 small. What is the least total number of pens I could have put in these packages? There are 2 ways to package 1 pen (1 big, 1 small). There are 3 ways to package 2 pens (3 big, 2 big and 1 small, 1 big and 2 small, 3 small). In general, if a package is to contain n pens, we can place 0, 1, 2, . . . , n − 1, or n big pens in it (with the remaining pens small). So in general there are n + 1 ways to fill a bag with n pens. 5 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 # Pkgs Used 2 # Pens Used 2 Total # Pens 2 Total # Pkgs 2 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 # Pkgs Used 2 3 # Pens Used 2 6 Total # Pens 2 8 Total # Pkgs 2 5 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 # Pkgs Used 2 3 4 # Pens Used 2 6 12 Total # Pens 2 8 20 Total # Pkgs 2 5 9 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 # Pkgs Used 2 3 4 5 # Pens Used 2 6 12 20 Total # Pens 2 8 20 40 Total # Pkgs 2 5 9 14 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 5 # Pkgs Used 2 3 4 5 6 # Pens Used 2 6 12 20 30 Total # Pens 2 8 20 40 70 Total # Pkgs 2 5 9 14 20 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 5 6 # Pkgs Used 2 3 4 5 6 7 # Pens Used 2 6 12 20 30 42 Total # Pens 2 8 20 40 70 112 Total # Pkgs 2 5 9 14 20 27 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 5 6 7 # Pkgs Used 2 3 4 5 6 7 8 # Pens Used 2 6 12 20 30 42 56 Total # Pens 2 8 20 40 70 112 168 Total # Pkgs 2 5 9 14 20 27 35 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 5 6 7 8 # Pkgs Used 2 3 4 5 6 7 8 9 # Pens Used 2 6 12 20 30 42 56 72 Total # Pens 2 8 20 40 70 112 168 240 Total # Pkgs 2 5 9 14 20 27 35 44 Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 5 6 7 8 # Pkgs Used 2 3 4 5 6 7 8 9 # Pens Used 2 6 12 20 30 42 56 72 Total # Pens 2 8 20 40 70 112 168 240 Total # Pkgs 2 5 9 14 20 27 35 44 In order to use all 49 packages, we will fill the remaining 5 packages with 9 pens each, using up 45 more pens. The total number of pens uses is therefore at least Problem 2-4 (Continued) Let’s make a table showing the number of pens used to fill packages with increasing amounts of pens: # Pens/Pkg 1 2 3 4 5 6 7 8 # Pkgs Used 2 3 4 5 6 7 8 9 # Pens Used 2 6 12 20 30 42 56 72 Total # Pens 2 8 20 40 70 112 168 240 Total # Pkgs 2 5 9 14 20 27 35 44 In order to use all 49 packages, we will fill the remaining 5 packages with 9 pens each, using up 45 more pens. The total number of pens uses is therefore at least 240 + 45 = 285. Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 and b3 + ba2 = 90? Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, and b3 + ba2 = 90? b(a2 + b2 ) = 90, Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, and b3 + ba2 = 90? b(a2 + b2 ) = 90, =⇒ b/a = 3 Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, a3 + a(3a)2 = 30 and b3 + ba2 = 90? b(a2 + b2 ) = 90, =⇒ b/a = 3 Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, and b3 + ba2 = 90? b(a2 + b2 ) = 90, =⇒ b/a = 3 a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, and b3 + ba2 = 90? b(a2 + b2 ) = 90, =⇒ b/a = 3 a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 ⇐⇒ a3 = 3 Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, and b3 + ba2 = 90? b(a2 + b2 ) = 90, =⇒ b/a = 3 a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 ⇐⇒ a3 = 3 ⇐⇒ a = p 3 3 Problem 2-5 What is the only pair of real numbers (a, b) that satisfies both a3 + ab2 = 30 a(a2 + b2 ) = 30, and b3 + ba2 = 90? b(a2 + b2 ) = 90, =⇒ b/a = 3 a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 ⇐⇒ a3 = 3 ⇐⇒ a = (a, b) = p p 3 3 3, 3 3 p 3 3 Problem 2-6 What is the area of a semicircular region tangent to two sides of a unit square, with endpoints of its diameter on the other two sides? Problem 2-6 What is the area of a semicircular region tangent to two sides of a unit square, with endpoints of its diameter on the other two sides? The hard part is drawing the figure. . . D r O r B b Problem 2-6 What is the area of a semicircular region tangent to two sides of a unit square, with endpoints of its diameter on the other two sides? Draw perpendiculars from O (the center of the circle) to the sides of the square. C D r O F b r B A E Problem 2-6 What is the area of a semicircular region tangent to two sides of a unit square, with endpoints of its diameter on the other two sides? ∠OBA ≡ ∠DOE =⇒ △OAB ∼ △DEO C OB ≡ OD △OAB ≡ △DEO D AB = OE = 1 − r = OA △OAB is right isosceles! p 1 − r = r/ 2 p p 2−r 2=r √ √ p 1− 2 2 r= √ · √ =2− 2 1+ 2 1− 2 p Area = πr 2 /2 = π(3 − 2 2) r O F b r B A E