2022年大联盟(Math League)国际夏季六年级数学挑战活动二（含答案）.docx

Speed Questions (Grade 6)2022 Math League International Summer Challenge (Unofficial version, for reference only)Note:Listed below are the four types of coins currently being minted in United States.a. Penny: 1¢b. Nickel: 5¢c. Dime: 10¢d. Quarter: 25¢1.(-1)100 =A) -1B) 1C) -100D) 100Answer: B2.(x + 2) + (2x + 4) + (3x + 6) + (4x + 8) =A) x + 20B) 9x + 20C) 10x + 20D) 20x + 20Answer: C3.(y + 2)(y 2) =A) y2 4B) y2 2y + 4C) y2 + 2y + 4D) y2 + 4 Answer: A154.1 + 2 + 3 + . + 100 =2 + 4 + 6 + . + 200A) 1 200B) 1 100C) 1 2D) 100Answer: C5. (x2)(x3)(x4) =A) x5B) x9C) x24D) x234Answer: B6.x2 10x + 24 = A) (x 12)(x + 2)B) (x 6)(x 4)C) (x + 12)(x 2)D) (x + 6)(x + 4)Answer: B7.If n < 0 and p > 0, which inequality is always true?A) n > pB) -n > pC) 1 > 1npD) 1 < 1npAnswer: D8.I am thinking of a number. When I subtract 2 from the number, then take 300% of the result, the number I get is my original number. What is my original number?A) 0B) 2C) 3D) 6Answer: C9.x(x(x(x + 1) + 1) + 1) + 1 =A) 4x + 4B) x4 + 2x3 + 3x2 + 2x + 1C) x4 + 4D) x4 + x3 + x2 + x + 1Answer: D10.How many solutions does the equation |x + 2| = 2x have?A) 0B) 1C) 2D) 3 Answer: B11.How many different numbers satisfy the equation (x 2)2(x 1)3(x + 1)4(x + 2)5 = 0?A) 4B) 5C) 14D) 120Answer: A12.Triangles ABC and DCB are both isosceles right triangles, as shown. If AC = 12, what is the area of triangle BCE?Answer: 1813.In a certain circle, the length of a radius is equal to the reciprocal of the length of a diameter. What is the area of this circle?A) 1p 2B) C) 2D) 2pAnswer: A14.The sum of 20 positive numbers is what percent of the average of these same 20 numbers?Note: This is the way to enter your answer. If, for example, your answer is 5%, please enter 5.Answer: 200015.What are the three ordered pairs of non-negative integers (a, b) for whicha + b= a + b ?Note: This is the way to enter your answer. If your three ordered pairs are (a1, b1), (a2, b2), and (a3, b3), please enter the value of a1 + b1 + a2 + b2 +a3 + b3.Answer: 216.I hiked into the woods at 4 km/h and returned by the same route at 3 km/h. If theround trip took 3 12Answer: 6hours, how many km into the woods did I hike?17.How many two-digit whole numbers contain no 0 and no more than one 1? Answer: 8018.All but two of my math books are titled Algebra, all but two are titled Trigonometry, and all but two are titled Geometry. I have at least one math book titled Algebra. How many math books do I have?Answer: 319.In the entire history of Oz, each president has served exactly one 4-year term. What is the maximum number of presidents that could have held office in Oz during some 10-year period?Answer: 420.What are all values of x for which x and17 are both integers?xNote: This is the way to enter your answer. If your answer contains three values of x: x1, x2, and x3, please enter the value of x1 + x2 + x3, similar for other numbers of values of x.Answer: 021.While reviewing the inventory of his bicycle shop, Pat noticed that, between the new bicycles and the new tricycles, there were a total of 176 wheels and 152 pedals. How many new tricycles were in Pats shop?Answer: 2422.In a right triangle, the lengths of the legs are 333 and 444. What is the length of the hypotenuse?Answer: 55523.Ann and Bob take turns in a game. At a turn, a player takes one or two coins from a cup that at first holds six coins. The player taking the last coin wins. If Ann goes first and both play perfectly, who will win? Note: Write “1” if the answer is Ann. Write “2” if the answer is Bob.Answer: 224.All real values of x which satisfy x + |x| = 0 are in the interval x p. What is the value of p?Answer: 025.Simultaneously, two particles start at point Q, on circle O, and move in different directions around the circle. If the speed of one particle is 4 times that of the other, then the particles will next meet at point P on the circle. If POQ is acute, what is mPOQ, in degrees?Answer: 7226.I pay for a $1 burger with 48 coins and get no change. What is the largest number of nickels I could use?Answer: 1327.What is the only positive value of b for which (x + b)2 = x2 + 2bx + b3 is an identity for all real values of x?Answer: 128.Express, as a fraction in lowest terms, the value of the following product of 99 factors:æ1 - 1 öæ1 - 1 öæ1 - 1 ö ´. ´æ1 -1 ö ´. ´æ1 -1 ö .ç÷ç÷ç÷234èèøèøèøçn + 1÷ç100 ÷øøèAnswer:110029.Figure below, the blue circle is the incircle of triangle ABC. (The incircle is the circle which lies inside a triangle and touches all sides.) O is the center the incircle. D is the center of the circle which touches AC and the extensions of BA and BC. E is the center of the circle which touches AB and the extensions of CA and CB. F is the center of the circle which touches BC and the extensions of AB and AC. Is it true that E is always the orthocenter of triangle ODF for any triangle ABC? (An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. The three altitudes of any triangle intersect in a single point, called the orthocenter of the triangle.)Choices:(a) Yes(b) No Answer: (a)30.When all 8 positive integral factors of 30 are multiplied together, the product is 30k. What is the value of k?Answer: 431.In square ABCD, drawn below, EB = 3 and FC = 4. IfACand EFbisect each other, what is the area of square ABCD?Answer: 4932.Smalltown has only two people who have the same (two-letter) initials. What is the largest population that Smalltown could possibly have?Note:Initials: the first letter of each word in a full name. For example, DW is the initials of David Warren.Answer: 67733.The average of 10 numbers is 63. The average of 6 of these numbers is 57. What is the average of the other 4?Answer: 7234.xçIf æ 2 -èx ö22÷ø= 0 , what is the value of x6?Answer: 6435.Winkin said “Were all liars,” Blinkin replied, “Only you are a liar,” and Finkin said “Youre both liars.” If each of them always tells the truth or always lies, how many of the three are liars?Answer: 236.To express 10 as a sum of different powers of 2, we would write 10 = 23 + 21. The sum of the exponents of these powers is 3 + 1, or 4. If 100 were expressed as a sum of different powers of 2, what would be the sum of the exponents of these powers?Answer: 1337.If, for all values of x, (x a)2 = (x + a)2, what is the value of a? Answer: 038.If x is real, and x64 = 64, what is the value of x32? Answer: 839.The population of the town of Decrease, now 11 800, is decreasing at the rate of 120 people each week. The population of the town of Increase, now 2200, is increasing at the rate of 80 people each week. In how many weeks will these two towns have equal populations?Answer: 4840.What are both values of x that satisfy in ascending order.)Answer: -1, 1æ x +çè1 ö2x÷ø= 4 ? (Note: Please enter your answers41.The number 4 938 271 603 950 617 283 will be a perfect square if one of its digits is increased by 1 (and all the other digits are left unchanged). What is the value of this digit before the increase?Answer: 342.What is the only ordered pair of positive integers (x, y) for which x2 y2 = 7? (Note: Please enter the value of x first.)Answer: (4, 3)43.Two shaded squares are drawn inside a rectangle, as shown. If each square has an area of 64, what is the area of the rectangle?Answer: 25644.Al and Pat each take coins from a piggy bank to buy some gum, but Al is 10¢ short and Pat is 15¢ short of the price of the gum. If they combine their resources, they still cant afford to buy the gum. What is the greatest possible (integral) price of the gum, in cents?Answer: 2445.In an Almost Pythagorean triangle, the square of the longest side is 1 more than the sum of the squares of the other two sides. If all of its sides have integral lengths, what is the least perimeter of such a triangle?Answer: 746.What is the sum of all integers x for which9 equals an integer?xAnswer: 047.Although four students tried to find the sum of the first 21 positive primes, only one got the correct answer. Pat got 709, Lee got 711, Sandy got 712, and Dale got 713. Who got the correct answer? Note: Enter “1” if Pat did; enter “2” if Lee did; enter “3” if Sandy did; enter “4” if Dale did.Answer: 348.For which n > 0 does(32 + 42) + (32 + 42) + (32 + 42) + (32 + 42) = n2?Answer: 1049.What is the largest possible area of a rectangle with integer sides and perimeter 22? Answer: 3050.The sum of the squares of the lengths of the four legs of two right triangles is 100. If the length of the hypotenuse of one of the triangles is 6, how long is the hypotenuse of the other triangle?Answer:851.The values of x which satisfy both |x 8| < 6 and |x 3| > 5 are precisely the values of x that satisfy a < x < b. What is a + b?Answer: 2252.Disregarding order, we can write 4 as a sum of one or more integers, each a power of 2, in only four ways: 4, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. In at most how many different ways can 5 be written as such a sum?Answer: 453.A plane is partitioned into 2 regions by 1 line and into 4 regions by 2 intersecting lines. Into how many disjoint regions do 5 coplanar lines partition the plane, if no 2 of the lines are parallel and no 3 of them are concurrent?Answer: 1654.Two noncongruent circles are externally tangent. Each base of an isosceles trapezoid is a diameter of one of the circles. If the distance between the centers of the circles is8, what is the area of the trapezoid? Answer: 6455.The 4th term of a sequence is 4 and the 6th term is 6. Every term after the 2nd is the sum of the 2 preceding terms. What is the 8th term of this sequence?Answer: 1456.One diagonal of a square serves as the shorter base of a trapezoid, and a line through one of the vertices of the square contains the other base. The legs of the trapezoid are extensions of two sides of the square. If the area of the square is 2800, what is the area of the trapezoid?Answer: 420057.Figure below, the blue circle is the incircle of triangle ABC. (The incircle is the circle which lies inside a triangle and touches all sides.) O is the center the incircle. D is the center of the circle which touches AC and the extensions of BA and BC. Is it true that points B, O, and D are always collinear, i.e., lying on the same straight line, for any triangle ABC?Choices:(a) Yes(b) No Answer: (a)58.For any integer n 1, 1 + 2 + 3 + . . . + (n 1) + n = (n + 2) + . . . + (2n 1) + (2n) =2n2 + nn(n + 1) . Then, n + (n + 1) +2A)23n2 - nB)23n2 + nC)23n2 + 3nD)2Answer: D59.A series of 7 books was published at 9-year intervals. When the 7th book was published, the sum of the publication years was 13 601. In what year was the 4th book published?Answer: 194360.Semicircles drawn on each side of a triangle have areas of 9, 16, and 25. What is the area of the triangle?Answer: 48