MathDB

1999 Gauss

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Subcontests

(16)
25
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1999 Gauss (7) #25

In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie and no points for a loss, how many games ended in a tie?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 7<spanclass=′latex−bold′>(E)</span> 10<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 10
22
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1999 Gauss (7) #22

Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 73<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 4<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ \dfrac{7}{3} \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4
21
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1999 Gauss (7) #21

A game is played on the board shown. In this game, a player can move three places in any direction (up, down, right or left) and then can move two places in a direction perpendicular to the first move. If a player starts at SS, which position on the board (P,Q,R,TP, Q, R, T, or WW) cannot be reached through any sequence of moves?
\begin{tabular}{|c|c|c|c|c|}\hline & & P & & \\ \hline & Q & & R &\\ \hline & & T & & \\ \hline S & & & & W\\ \hline\end{tabular}
<spanclass=′latex−bold′>(A)</span> P<spanclass=′latex−bold′>(B)</span> Q<spanclass=′latex−bold′>(C)</span> R<spanclass=′latex−bold′>(D)</span> T<spanclass=′latex−bold′>(E)</span> W<span class='latex-bold'>(A)</span>\ P \qquad <span class='latex-bold'>(B)</span>\ Q \qquad <span class='latex-bold'>(C)</span>\ R \qquad <span class='latex-bold'>(D)</span>\ T \qquad <span class='latex-bold'>(E)</span>\ W
20
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1999 Gauss (7) #20

The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of A+EA + E.
\begin{tabular}{|c|c|c|}\hline A & 1 & B \\ \hline 5 & C & 13\\ \hline D & E & 3 \\ \hline\end{tabular}
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 28<spanclass=′latex−bold′>(C)</span> 26<spanclass=′latex−bold′>(D)</span> 24<spanclass=′latex−bold′>(E)</span> 16<span class='latex-bold'>(A)</span>\ 32 \qquad <span class='latex-bold'>(B)</span>\ 28 \qquad <span class='latex-bold'>(C)</span>\ 26 \qquad <span class='latex-bold'>(D)</span>\ 24 \qquad <span class='latex-bold'>(E)</span>\ 16
17
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1999 Gauss (7) #17

In a “Fibonacci” sequence of numbers, each term beginning with the third, is the sum of the previous two terms. The first number in such a sequence is 2 and the third is 9. What is the eighth term in the sequence?
<spanclass=′latex−bold′>(A)</span> 34<spanclass=′latex−bold′>(B)</span> 36<spanclass=′latex−bold′>(C)</span> 107<spanclass=′latex−bold′>(D)</span> 152<spanclass=′latex−bold′>(E)</span> 245<span class='latex-bold'>(A)</span>\ 34 \qquad <span class='latex-bold'>(B)</span>\ 36 \qquad <span class='latex-bold'>(C)</span>\ 107 \qquad <span class='latex-bold'>(D)</span>\ 152 \qquad <span class='latex-bold'>(E)</span>\ 245
15
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1999 Gauss (7) #15

A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green?
<spanclass=′latex−bold′>(A)</span> 19<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 14<spanclass=′latex−bold′>(E)</span> 970<span class='latex-bold'>(A)</span>\ \dfrac{1}{9} \qquad <span class='latex-bold'>(B)</span>\ \dfrac{1}{8} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{1}{5} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{1}{4} \qquad <span class='latex-bold'>(E)</span>\ \dfrac{9}{70}
14
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1999 Gauss (7) #14

Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between 12212^2 and 13213^2?
<spanclass=′latex−bold′>(A)</span> 105<spanclass=′latex−bold′>(B)</span> 147<spanclass=′latex−bold′>(C)</span> 156<spanclass=′latex−bold′>(D)</span> 165<spanclass=′latex−bold′>(E)</span> 175<span class='latex-bold'>(A)</span>\ 105 \qquad <span class='latex-bold'>(B)</span>\ 147 \qquad <span class='latex-bold'>(C)</span>\ 156 \qquad <span class='latex-bold'>(D)</span>\ 165 \qquad <span class='latex-bold'>(E)</span>\ 175
12
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1999 Gauss (7) #12

Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. To decide who goes first in a game, they play “countdown”. Henry starts by saying ‘34’, with Iggy saying ‘33’. If they continue to count down in their circular order, who will eventually say ‘1’?
<spanclass=′latex−bold′>(A)</span> Fred<spanclass=′latex−bold′>(B)</span> Gail<spanclass=′latex−bold′>(C)</span> Henry<spanclass=′latex−bold′>(D)</span> Iggy<spanclass=′latex−bold′>(E)</span> Joan<span class='latex-bold'>(A)</span>\ \text{Fred} \qquad <span class='latex-bold'>(B)</span>\ \text{Gail} \qquad <span class='latex-bold'>(C)</span>\ \text{Henry} \qquad <span class='latex-bold'>(D)</span>\ \text{Iggy} \qquad <span class='latex-bold'>(E)</span>\ \text{Joan}
11
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1999 Gauss (7) #11

The floor of a rectangular room is covered with square tiles. The room is 10 tiles long and 5 tiles wide. The number of tiles that touch the walls of the room is
<spanclass=′latex−bold′>(A)</span> 26<spanclass=′latex−bold′>(B)</span> 30<spanclass=′latex−bold′>(C)</span> 34<spanclass=′latex−bold′>(D)</span> 46<spanclass=′latex−bold′>(E)</span> 50<span class='latex-bold'>(A)</span>\ 26 \qquad <span class='latex-bold'>(B)</span>\ 30 \qquad <span class='latex-bold'>(C)</span>\ 34 \qquad <span class='latex-bold'>(D)</span>\ 46 \qquad <span class='latex-bold'>(E)</span>\ 50
8
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7
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1999 Gauss (7) #7

If the numbers 45,81%\dfrac{4}{5},81\% and 0.8010.801 are arranged from smallest to largest, the correct order is
<spanclass=′latex−bold′>(A)</span> 45,81%,0.801<spanclass=′latex−bold′>(B)</span> 81%,0.801,45<spanclass=′latex−bold′>(C)</span> 0.801,45,81%<spanclass=′latex−bold′>(D)</span> 81%,45,0.801<spanclass=′latex−bold′>(E)</span> 45,0.801,81%<span class='latex-bold'>(A)</span>\ \dfrac{4}{5},81\%,0.801 \qquad <span class='latex-bold'>(B)</span>\ 81\%,0.801,\dfrac{4}{5} \qquad <span class='latex-bold'>(C)</span>\ 0.801,\dfrac{4}{5},81\% \qquad <span class='latex-bold'>(D)</span>\ 81\%,\dfrac{4}{5},0.801 \qquad <span class='latex-bold'>(E)</span>\ \dfrac{4}{5},0.801,81\%
5
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4
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1999 Gauss (7) #4

1+12+14+181+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8} is equal to
<spanclass=′latex−bold′>(A)</span> 158<spanclass=′latex−bold′>(B)</span> 1314<spanclass=′latex−bold′>(C)</span> 118<spanclass=′latex−bold′>(D)</span> 134<spanclass=′latex−bold′>(E)</span> 78<span class='latex-bold'>(A)</span>\ \dfrac{15}{8} \qquad <span class='latex-bold'>(B)</span>\ 1\dfrac{3}{14} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{11}{8} \qquad <span class='latex-bold'>(D)</span>\ 1\dfrac{3}{4} \qquad <span class='latex-bold'>(E)</span>\ \dfrac{7}{8}
3
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1999 Gauss (7) #3

Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed?
<spanclass=′latex−bold′>(A)</span> 36<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 53<spanclass=′latex−bold′>(D)</span> 70<spanclass=′latex−bold′>(E)</span> 71<span class='latex-bold'>(A)</span>\ 36 \qquad <span class='latex-bold'>(B)</span>\ 18 \qquad <span class='latex-bold'>(C)</span>\ 53 \qquad <span class='latex-bold'>(D)</span>\ 70 \qquad <span class='latex-bold'>(E)</span>\ 71
2
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1
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