MATHCOUNTS 2005 @ National Competition @ Team Round Problems 1-10 State Team Members Captain DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This round consists of ten problems which the team has 20 minutes to solve. Team members may work together and talk during this round. This round assumes the use of calculators, and calculations may also be done on scratch paper, but no other aids are allowed. The team captain must record the answers on his/her problem sheet, and all answers must be complete and legible, Only the team captain’s answer sheet will be graded. { Total Correct Scorer’s Initials = Founding Sponsors National Sponsors ENA Foundation ADC Foundation National Society of Professional Engineers fal Motors Foundation Navional Council of Teachers of Mathematics Lockheed Martin National Aeronautics and Space Adninistaion Raytheon Company Shell it Company Texas Instruments Incorporated 5M Foundation Xerox Corporation (©2005 MATHCOUNTS Foundation, 1420 King Street, Alexandtia, VA223141. Place one positive integer in each of the nine © O shapes to the right such that the number at the bottom of each column is the sum of the numbers placed in the shapes of that column, CCongruent shapes contain equal positive O integers. Whatis the value of S, the sum of the three numbers in the second column? 37 s a In Ms, Pham’s third grade class, an election with two candidates 2 was held in which the losing candidate received 31% of the vote, expressed to the nearest whole percent. Knowing that each student cast a vote for one or the other candidate, what is the minimum number of votes that could have been cast in the election? ‘Square ABCD has sides of length | em. c—£ ‘Triangle CFE isan isosceles right triangle tangent to are BD at G. Are BD is a quarter- circle with its center at A. What is the total area of the two shaded regions? Express your | A answer asa decimal to the nearest thousandth, sqem 3 4, Iftwo distinct members of the set {2, 4, 10, 12, 15,20, 50} are 4. randomly selected and multiplied, what isthe probability that the product is a multiple of 100? Express your answer as acommon faction. collection of nickels, dimes and pennies has an average value of 5 ___ dimes. 7¢ per coin. Ifa nickel were replaced by five pennies, the average would drop to 6¢ per coin. What is the number of dimes in the collection? 6. Zanbas created this iterative rule for generating sequences of whole « & numbers numbers. © [fa number is 25 or less, double the number. © [fa number is greater than 25, subtract 12 from it Let F be the first number in a sequence generated by the rule above. F isasweer number if 16 is nota term in the sequence that starts with F, How many of the whole numbers | through 50 are sweet numbers? ©2005 MATHCOUNTS Foundation: 2005 National Team Round9, 10, In the game Tic-Tae-Toe, the first player writes an “X" inone of the nine locations of the grid, and then the second player writes an “O” inone of the remaining eight places. To analyze the game, Melissa decided to study all of the possible “first-two-moves” combinations. ‘She realized that she should consider symmetric patterns like the three to {21 = the rightas the same “first-two- 7 moves” combination, and therefore only count it once. And she decided that she should count patterns like the two to the leftas two different “first- a TIX two-moves” combinations. Using Melissa’s counting technique, how many different “first- ‘two-moves” combinations are there? The equation MATH = COU + NTS can be made true if each of the letters M T,H, C, O, U, Nand S is replaced by a different digit and the equation is seen as the sum of two three-digit integers resulting in a four-digit integer. IfC =2,a solution can be found in which one of the three-digit integers isa multiple of 23. In that case, what four-digit integer does MATH represent? ‘The sequence of integers ineachof [37 the two rows of squares and in 75 each of the two columns of squares form four distinctarithmetic sequences. What is the least positive integer value forN? A vertical polygonal path will be formed by picking one point from each row of the four by four grid of points below (Fig. 1), and then connecting these points sequentially from top to bottom. The area of the grid to the left of the polygonal path will then be shaded. For how many four-point selections will the vertical polygonal path result in exactly half of the grid’s area being shaded? One example is given in Figure 2. combinations selections ©2005 MATHCOUNTS Foundation: 2005 National Team Round