200708 handbook problems

MATHCOUNTS 2007-2008 19 Warm-Up 1 1._ _________ _What_is_the_value_of_1_+_2_+_3_+_…_+_7_+_8_+_9_+_8_+_7_+_…_+_3_+_2_+_1,_where_all_of_the_ integers_from_1_through_9_and_then_back_down_to_1_are_added_together?__ _ _ 2.__________ _What_is_the_value_of_(2x_+_5)2_when_x_=_3?_ ___ 3.__________ _ A_toy_car_travels_one_foot_in_10_seconds.__At_this_rate,_ how_far_will_the_car_travel_in_two_minutes?_ 4.__________ _ Place_one_member_of_the_set_{1,_2,_3,_4}_into_each_blank_in_the_expression________________ ___(____+____)_+____,_such_that_each_member_of_the_set_is_used_exactly_once_and_ the_value_of_the_expression_is_as_small_as_possible.__What_is_the_value_of_the_ expression?__ _ _ _ _ _ _ _ _ _ _ 5.__________ The floor plan for a single-story house is shown. Each pair of_consecutive_sides_is_perpendicular.__What_is_the_area_of_ the floor for the house whose dimensions are given? 6.__________ _ How_many_degrees_does_the_minute_hand_of_ a clock move during a 10-minute period? _ 7.__________ A recipe uses two eggs, three cups of flour and some other ingredients. The recipe makes enough batter for six servings. How many cups of flour will be needed to make_enough_batter_for_10_servings? 8.__________ Aaron worked 10 hours per week for six weeks at his part-time office job, where he earned $8 per hour. He also worked full-time at his regular job, earning $400 per week. What were Aaron’s total earnings for the six-week period? _ 9.__________ _ A_science_workbook_is_one_centimeter_thick_and_weighs_75_grams.__How_much_will_ a one-meter-high stack of these workbooks weigh, in kilograms? Express your answer_as_a_decimal_to_the_nearest_tenth._ _ _ _ _ _ _ 10._ ________ _ Three_years_ago_Mary_was_three_times_as_old_as_her_sister.__Now_Mary_is_twice_as_ old_as_her_sister.__How_old_is_Mary_now? _ degrees sq_ft kg $ cups years feet 20’ 30’ 40’ 60’ 20’ 10’ 1 foot 10 seco nds MATHCOUNTS 2007-200820 Warm-Up 2 1._ _________ _ Grady_distributed_x__pieces_of_candy_evenly_among_nine_Halloween_bags_such_that_ every_bag_received_the_greatest_possible_number_of_whole_pieces_of_candy,_but_ some_candy_was_left_over.__What_is_the_greatest_possible_number_of_pieces_that_ could_have_been_left_over?__ _ _ _ _ _ _ _ _ _ 2.__________ Noah will mount a 5-inch by 5-inch photograph on an 8-inch by 10-inch mat board. How many square inches of mat board will be visible?_ 3.__________ _When_a_piece_of_paper_is_folded_in_half,_there_are_two_layers_of_paper.__How_many_ layers_would_there_be_if_a_piece_of_paper_is_folded_in_half_a_total_of_10_times_ without_ever_unfolding_it?__ _ _ _ _ _ _ _ _ 4.__________ _ Pi_plates_cost_$24_each.__Shipping_costs_$10_for_orders_under_$100_and_$15_for_ orders_of_$100_or_more.__How_much_more_does_it_cost_to_order_and_have_delivered_ five Pi plates instead of four Pi plates? 5.__________ _ Charlene_considers_herself_to_be_a_great_bargain_shopper_because_she_found_a_prom_ dress_that_cost_her_only_$22_before_tax.__The_dress_was_on_a_rack_labeled_“50%_ off_lowest_marked_price,”_and_the_lowest_marked_price_was_already_a_75%_reduction_ from_the_original_price.__What_was_the_original_price_of_the_dress?_ _ _ _ 6.__________ _ Fisher_wrote_a_book_about_bugs.__He_created_the_front_ cover,_spine_and_back_cover_from_one_continuous_piece_of__ 1 4 -inch thick cardboard. The cardboard fits exactly over the pages, and the final dimensions of the book are 6 inches by_8_inches_by_2_inches,_as_shown.__If_the_entire_cardboard_ piece_(front,_spine_and_back)_were_removed,_what_would_be_ the_sum_of_the_length,_width_and_thickness_of_the_remainder_of_the_book_(the_stack_ of paper pages)? Express your answer as a mixed number. 7.__________ _ Triangle_ABC_has_a_perimeter_of_2007_units.__The_sides_have_lengths_that_are_all_ integer values with AB < BC ≤ AC. What is the smallest possible value of BC – AB? 8.__________ _ For_all_positive_integers_n,_the_expression_n_! denotes the product of the first n_positive_integers.__When_5!_is_expressed_as_an_integer,_what_is_the_ones_digit?_ _ 9.__________ _ A_month_ago_the_ratio_of_nurses_to_doctors_on_a_hospital_staff_was_3:5.__Since_ that time two additional nurses joined the staff, no nurses left and the number of doctors_remained_the_same.__The_ratio_of_nurses_to_doctors_on_the_hospital_staff_is_ now_4:5.__How_many_nurses_are_now_on_the_staff?_ 10._ ________ _ Cooling_a_room_requires_27_BTUs_per_square_foot_per_hour.__Jose_buys_an_air_ conditioner_with_a_power_of_10,800_BTUs_per_hour.__What_is_the_width_of_the_ largest_square_room_that_the_air_conditioner_can_cool?_ feet sq_in layers $ $ inches units nurses pieces BUGS BU G S 6” 8” 2” MATHCOUNTS 2007-2008 21 Workout 1 1._ _________ The stem-and-leaf plot shows the scores on Mrs. Norris’ last quiz_for_her_third_period_class.__(6|3_represents_63_points.)__ What was the mean score on this quiz? Express your answer to_the_nearest_whole_number.__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2.__________ _ Tansy_wanted_to_leave_the_waiter_an_18%_tip,_and_her_bill_was_$21.50.__How_much_ will_the_total_cost_of_her_dinner_and_tip_be_if_she_rounds_the_tip_itself_to_the_next_ higher_dollar?_ 3.__________ _What_is_the_largest_possible_value_of_x__given_that_x_3_=_5x_? Express your answer in simplest_radical_form._ 4.__________ _ The_exterior_dimensions_of_an_empty_cooler_in_the_shape_of_a_rectangular_prism_are_ 18_inches_by_18_inches_by_24_inches.__For_insulation,_the_top,_the_bottom_and_each_ of_the_four_walls_are_two_inches_thick.__What_is_the_volume_of_the_empty_space_ inside_the_cooler?_ 5.__________ This_square_has_an_area_of_49_sq_cm.__What_is_the_area_of_the_inscribed_ circle? Express your answer as a decimal to the nearest tenth. _ _ 6.__________ Joe’s Store guarantees that it will refund what you paid for an item and give you an additional 10% if you find the item for less at a different store. Li paid $137.80 for a DVD player at Joe’s Store. Li then found an ad for the same DVD player at Mike’s Store for less. How much money should Joe’s Store give to Li? 7.__________ _ In_the_circle_with_center_O,_the_shaded_region_is_20%_of_the_area_ of_the_entire_circle.__What_is_the_measure_of_angle_AOB?_ _ _ _ _ _ _ _ _ _ _ 8.__________ _ In_a_triangle_with_three_distinct_angle_measures,_the_smallest_angle_measures_30°.__ The_measures_of_the_other_two_interior_angles_are_each_a_whole_number_of_degrees.__ What_is_the_measure_of_the_largest_possible_angle_in_the_triangle?_ _ _ _ 9.__________ _ A_roll_of_carpet_is_15_feet_wide.__The_carpet_costs_$20_per_ square_yard.__A_customer_must_purchase_the_carpet_in_a_piece_ that_is_15_feet_in_width,_but_the_carpet_can_be_cut_to_any_length.__ Beth wants to use exactly one piece of carpet to cover a floor that_is_trapezoidal_in_shape,_as_shown,_rather_than_putting_ multiple_pieces_together.__However,_she_must_also_pay_for_any_ wasted_carpet.__What_is_the_cost_of_the_shortest_length_of_ carpet_that_she_can_use?___ _ 10._ ________ _What_is_the_value_of_x__such_that_(x,_0)_is_a_solution_of_the_equation_y_=_3x_ – 4? Express your answer as a common fraction. points $ cu_in sq_cm $ degrees degrees $ 6_ 3_5_5 7_ 1_ 7_9_9_9_9 8_ 1_ 3_4_4_5_9_9 9_ 0_1_ 2_2_3_5_8_8 Quiz Scores B A O 15’ 15’ 21’ MATHCOUNTS 2007-200822 Warm-Up 3 1._ _________ _What_is_the_area_of_the_circle_that_is_centered_at_the_origin_ and_is_tangent_to_the_line_y = 7? Express your answer in terms of π. 2.__________ _When_the_expression_3444_+_4333_is_written_as_an_integer,_what_is_the_units_digit?_ 3.__________ The sum of three numbers is 98. The ratio of the first to the second is 2:3, and_the_ratio_of_the_second_to_the_third_is_5:8.__What_is_the_value_of_the_second_ number?_ _ _ 4.__________ What is the value of -(12008) + (-1)2007?_ 5.__________ _ A_cider_recipe_created_by_the_Xpress_Co._combines_nine_cups_of_applesauce_with_ one_cup_of_water.__The_mixture_is_brought_to_a_boil_and_then_simmered_until_the_ volume_is_reduced_by_15%.__If_a_cider_batch_begins_with_36_cups_of_applesauce,_how_ much_cider_will_there_be_after_the_appropriate_amount_of_water_is_added_and_the_ simmering_process_is_complete?_ _ _ _ _ _ _ _ 6.__________ A total of 180 marbles (gray, white and black) are placed in a line. The first five are gray, followed by four white, followed by three black, followed by five gray, followed_by_four_white,_followed_by_three_black,_…_.__If_this_pattern_continues,_what_ is_the_color_of_the_158th_marble_in_this_line?_ _ _ _ _ _ _ 7.__________ _What_is_the_value_of_n_for_which_(3!)(5!)(7!)_=_n_!_?_ 8.__________ _ A_regular_hexagon_is_inscribed_in_a_circle_of_radius_four_inches.__What_is_the_ perimeter_of_the_hexagon?_ 9.__________ If a standard six-sided die is rolled twice, what is the probability that the result of the second roll is not less than the result of the first roll? Express your answer as_a_common_fraction._ 10._ ________ _ If_only_squares_may_be_used,_how_many_squares_must_be_placed_ on_the_right_side_of_the_third_scale_so_that_all_three_scales_are_ balanced? (The distance of the objects from the centers of these_scales_is_not_relevant.)_ _ inches sq_units cups squares ? ... MATHCOUNTS 2007-2008 23 Warm-Up 4 1._ _________ _ The_shipping_cost_of_a_dozen_lemons_is_directly_proportional_ to_how_many_miles_they_are_shipped.__A_dozen_lemons_shipped_ the_200_miles_to_Taraville_cost_$5_to_ship.__How_much_will_it_ cost_to_ship_a_dozen_lemons_1000_miles?_ 2.__________ Four congruent quarter-circles are drawn inside a square of side length_4_centimeters,_as_shown.__What_is_the_area_of_the_shaded_ portion of the square region? Express your answer in terms of π. _ _ _ _ _ _ _ _ _ _ 3.__________ If all of the letters of the word BEEP are used, in how many different ways can the four letters be arranged in a four-letter sequence? The two Es are indistinguishable._ _ _ _ _ 4.__________ _What_is_the_greatest_possible_value_of_a_in_the_system_of_equations_5a_+_2b_=_0_and_ ab = -10? 5.__________ Office Surplus promises delivery of orders within three working hours of receipt of any order. Working hours are 8 a.m. to 5 p.m., Monday through Friday. Clyde’s order_is_received_at_3_p.m._on_a_Friday_and_takes_the_maximum_promised_number_of_ working_hours_to_be_delivered.__How_many_actual_hours_elapse_from_the_time_the_ order_is_placed_until_it_is_delivered?_ _ _ _ _ _ _ _ _ _ 6.__________ _ How_many_positive_integers_less_than_1000_can_be_written_using_only_the_digits_0,_1_ and_2?___Two_such_integers_to_include_are_101_and_12.___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 7.__________ The first day of the year 2006 was a Sunday. How many Friday the 13ths_were_ there_in_2006?_ _ _ _ _ _ _ _ _ _ 8.__________ _What_is_the_value_of_(43)_÷_(22)? Express your answer in the form a_b_where_a_and_b_ are_positive_integers_and_a_has_the_least_possible_value._ 9.__________ _ Several_points_are_plotted_on_a_graph.__For_each_point,_the_x_-coordinate is the length_of_a_side_of_a_square_while_the_y_-coordinate is the perimeter of that same_square.__One_such_point_is_(2,_8)_since_a_square_with_side_length_2_units_ has_a_perimeter_of_8_units.__What_is_the_slope_of_the_line_connecting_the_points?___ Express your answer in simplest form. 10._ ________ _ Triangle_ABC_has_side_lengths_5,_5_and_8_units.___ Triangle_XYZ_has_side_lengths_5,_5_and_6_units.__ What__is_the_difference_of_the_areas_of_these_ two_triangles?_ ways sq_cm hours integers Friday_______ the_13ths $ sq_units LEMO NS 5 5 5 5 8 6CA X Z YB MATHCOUNTS 2007-200824 Workout 2 1._ _________ _ On_a_certain_day,_10_U.S._dollars_were_worth_7.60_euros,_and_1_euro_was_worth__ 155_Japanese_yen.__How_many_yen_were_the_10_U.S._dollars_worth?_ 2.__________ _ If_ + − − =225 64 147 25n ,_what_is_the_value_of_n_?_ _ _ _ _ 3.__________ Lighting experts recommend 150 to 200 watts of illumination for every 50 square feet of floor space. What is the minimum number of watts recommended for a room with a rectangular floor measuring 30 feet by 40 feet? 4.__________ _ By_what_percent_is_the_commercial_red_meat_ production_for_Iowa_(IA)_greater_than_that_for__ Texas (TX), according to the data shown? Express your_answer_to_the_nearest_whole_number._ _ _ _ _ _ _ _ _ _ _ 5.__________ _ Anna_bought_12_pieces_of_gum_consisting_of_only_red_gumballs_and_white_gumballs.__ The_total_cost_was_$1.29.__The_red_gumballs_each_cost_three_cents_more_than_each_ white_gumball,_and_she_bought_fewer_red_gumballs_than_white_gumballs.__How_many_ white_gumballs_did_she_buy?_ _ _ _ _ _ _ _ _ 6.__________ _ If_k_is_an_integer_and_k_>_100,_what_is_the_smallest_possible_integer_value_of_ 23 k ?_ _ 7.__________ Jared wants to estimate the number of deer in a 600-acre state park. When driving along the one-mile road that is one side_of_the_park,_he_can_see_an_average_distance_of_50_yards_ into_the_park_and_counts_7_deer.__An_acre_is_4840_square_ yards,_and_a_mile_is_1760_yards.__If_the_deer_are_evenly_ distributed_throughout_the_park,_what_is_the_best_estimate_ of_the_number_of_deer_in_the_park?_ _ _ _ _ 8.__________ In the figure, BA = AD = DC and point D is on segment BC.__The_measure_of_angle_ADC_is_135_degrees.__What_is_ the_measure_of_angle_ABC?_ _ _ _ _ 9.__________ _ In_forming_his_budget_for_this_year,_Jamal_decided_to_use_his_average_cost_for_ utilities last year to project his expenses for the future. Last year he spent an average_of_$216_per_month_on_utilities,_but_he_anticipates_a_5%_increase_in_the_ annual_cost_of_utilities.__Based_on_this_information,_how_much_should_he_expect_to_ pay_for_utilities_each_month_this_year?_ _ _ _ _ _ _ _ _ _ 10._ ________ Twenty-one congruent circular discs are stacked in a_triangular_arrangement,_as_shown.__Connecting_the_ centers_of_the_three_vertex_discs_forms_an_equilateral_ triangle.__The_circumference_of_each_disc_is_18_cm.__ What_is_the_outside_perimeter_of_the_arrangement?_ _ yen watts % white_ gumballs degrees $ cm deer A B D C 600-acre park 50 yards 1- m ile ro ad 585 583 466 389 238 NEIA KS TX IL Commercial_Red_Meat_Production (millions_of_pounds) Information_for_problem_#4_is_from_USA_TODAY_Snapshots®,_February_8,_2006.__Source:_USDA. MATHCOUNTS 2007-2008 25 Warm-Up 5 1._ _________ Eighty books will be divided among six people. No two people will receive the same number_of_books.__The_person_who_receives_the_most_books_will_receive_20;_the_ person_who_receives_the_fewest_books_will_receive_seven.__What_is_the_maximum_ number of books the person with the third-largest number of books could receive? _ _ 2.__________ _ In_the_circle_with_center_X,_the_measure_of_angle_AXT_is_60°,_and_ the_measure_of_angle_ABC_is_90°.__The_length_of_segment_AT_is__ 5_units,_and_the_length_of_segment_BC_is_6_units.__What_is_the_length_ of_segment_AB?_ 3.__________ A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden_larger,_while_using_the_same_amount_of_fencing,_its_shape_is_changed_to_a_ square.__How_many_square_feet_larger_than_the_old_garden_is_the_new_garden?_ _ 4.__________ _ Rita_is_selecting_a_sandwich_at_the_deli.__The_deli_has_four_types_of_meat,_three_ types_of_cheese_and_two_types_of_bread.__A_deluxe_sandwich_consists_of_exactly_one_ meat_type,_two_different_types_of_cheese_and_one_bread_type.__How_many_different_ deluxe_sandwich_combinations_are_possible?_ _ 5.__________ _ One_zip,_two_zaps_and_three_zups_cost_$1.50.__Two_zips,_three_zaps_and_one_zup_ cost_$1.00.__Three_zips,_one_zap_and_two_zups_cost_$1.40.__What_is_the_total_cost,_in_ cents,_of_one_zip,_one_zap_and_one_zup?_ 6.__________ A rectangular candy wrapper is made from a one-inch by three-inch piece of paper. What is the greatest number of wrappers_that_can_be_cut_from_a_rectangular_piece_of_paper_ measuring_one_foot_by_two_feet?_ _ _ _ _ _ _ _ _ _ 7.__________ The stocks of five companies had the following percent changes in 2006: +9.9%, +5.7%, -7.2%, +1.8% and -0.9%. What is the average percent change in 2006 for these five companies? Express your answer to the nearest hundredth. 8.__________ Audrey’s 300-mile trip took six hours to complete. She spent 23 _of_the_total_time_ of her trip going the first 13 _of_the_distance.__What_is_the_ratio_of_her_average_ speed during that first 23 _of_the_time_to_her_average_speed_during_the_remaining_ 13 _ of the time? Express your answer as a common fraction. 9.__________ A large game field has an area of 400 square feet. A small practice field is made with every linear dimension half as large as the game field’s dimensions. What is the area of the small practice field? 10._ ________ _ A_triangle_has_sides_of_length_5_and_6_units.__The_length_of_the_third_side_is_x_units,_ where_x_is_an_integer.__What_is_the_largest_possible_perimeter_of_the_triangle?_ books units sq_ft combos cents wrappers % sq_ft units A T B C X MATHCOUNTS Munchie Bars MATHCOUNTS 2007-200826 Warm-Up 6 1._ _________ What is the simplified value of (2 + 4 + 6 + … + 48 + 50) – (1 + 3 + 5 + … + 47 + 49)? _ 2.__________ _ For_prime_numbers_p__and_q,_p__+_q_=_102_and_p__>_q.__What_is_the_least_possible_value_ of_p_ – q_?__ 3.__________ Sally is taking a test with 10 questions. Each answer is either True or False. She knows the correct answer for five of the questions and randomly guesses on the other five questions. What is the probability_that_she_will_have_at_least_eight_correct_answers_on_the_ test? Express your answer as a common fraction. 4.__________ _ The_sum_of_two_numbers_is_32,_and_the_product_of_these_two_numbers_is_48.__What_ is the sum of the reciprocals of the two numbers? Express your answer as a common_fraction._ 5.__________ _ Fabric_is_run_through_a_machine_that_fastens_sequins_in_rows_ and_columns_such_that_each_sequin_is_four_inches_vertically_and_ horizontally_from_the_next_closest_sequins,_forming_a_grid_as_shown.__ There is always a four-inch space with no sequins at the beginning, end_and_side_margins_of_the_fabric.__If_a_piece_of_fabric_60_inches_ by_72_inches_is_fed_through_the_machine,_how_many_sequins_will_be_attached?_ _ 6.__________ _What_is_the_value_of_(2x__3)_÷_(2x_)3_when_x = 2007? Express your answer as a common_fraction._ _ _ _ _ _ _ _ _ _ _ _ 7.__________ _What_is_the_area_of_the_quadrilateral_with_vertices_at_(1,_1),_(5,_2),_(4,_4)_and__ (2,_3)?__ 8.__________ A six-pointed star is formed by overlapping two congruent, equilateral_triangles,_placed_such_that_each_side_of_the_original_ triangles is trisected. What fraction of the area of the six-pointed star is the shaded triangle? Express your answer as a common fraction._ _ _ _ _ _ _ _ _ 9.__________ _ Alexander_used_exactly_20_meters_of_fencing_around_three_sides_of_a_rectangular_ flower bed beside his house. He did not fence the fourth side, which is an eight- meter section along the side of the house. What is the area of this flower bed? 10._ ________ _ Circles_with_centers_at_(2,_2)_and_(17,_10)_are_both_tangent_to_the_x_-axis. What is the_distance_between_the_closest_points_of_the_two_circles?_ sequins sq_units sq_meters units 1. F 6. F 2. T 7. F 3. F 8. T 4. F 9. F 5. T 10. T Sally MATHCOUNTS 2007-2008 27 Workout 3 1._ _________ The formula for the total surface area of a cylinder is SA = 2πr_2 + 2πrh,_where_r_is_ the_radius_and_h_is_the_height.__A_particular_solid_right_cylinder_of_radius_2_feet_has_ a total surface area of 12π square feet. What is the height of this cylinder? 2.__________ _ A_cube_varies_in_size_over_time,_with_its_largest_volume_being_twice_its_smallest_ volume.__What_is_its_largest_edge_length_divided_by_its_shortest_edge_length?__ Express your answer as a decimal to the nearest