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National and Regional Contests
Canada Contests
Gauss
1998 Gauss
1998 Gauss
Part of
Gauss
Subcontests
(25)
25
1
Hide problems
1998 Gauss (7) #25
Two natural numbers,
p
p
p
and
q
q
q
, do not end in zero. The product of any pair, p and q, is a power of 10 (that is,
10
,
100
,
1000
,
10000
10, 100, 1000, 10 000
10
,
100
,
1000
,
10000
, ...). If
p
>
q
p >q
p
>
q
, the last digit of
p
–
q
p – q
p
–
q
cannot be
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<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 9
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9
24
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1998 Gauss (7) #24
On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws?
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<span class='latex-bold'>(A)</span>\ 38 \qquad <span class='latex-bold'>(B)</span>\ 39 \qquad <span class='latex-bold'>(C)</span>\ 54 \qquad <span class='latex-bold'>(D)</span>\ 55 \qquad <span class='latex-bold'>(E)</span>\ 30
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23
1
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1998 Gauss (7) #23
A cube measures
10
cm
×
10
cm
×
10
cm
10 \text{cm} \times 10 \text{cm} \times10 \text{cm}
10
cm
×
10
cm
×
10
cm
. Three cuts are made parallel to the faces of the cube as shown creating eight separate solids which are then separated. What is the increase in the total surface area?
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<span class='latex-bold'>(A)</span>\ 300 \text{cm}^2 \qquad <span class='latex-bold'>(B)</span>\ 800 \text{cm}^2 \qquad <span class='latex-bold'>(C)</span>\ 1200 \text{cm}^2 \qquad <span class='latex-bold'>(D)</span>\ 600 \text{cm}^2 \qquad <span class='latex-bold'>(E)</span>\ 0 \text{cm}^2
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22
1
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Gauss (7) #22
Each time a bar of soap is used, its volume decreases by
10
%
10\%
10%
. What is the minimum number of times a new bar would have to be used so that less than one-half its volume remains?
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<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 9
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21
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Gauss (7) #21
Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.)
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<span class='latex-bold'>(A)</span>\ 9 \qquad <span class='latex-bold'>(B)</span>\ 45 \qquad <span class='latex-bold'>(C)</span>\ 17 \qquad <span class='latex-bold'>(D)</span>\ 66 \qquad <span class='latex-bold'>(E)</span>\ 55
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20
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Gauss (7) #20
Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one red edge. What is the smallest number of red edges?
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ 6
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19
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Gauss (7) #19
Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?
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<span class='latex-bold'>(A)</span>\ 7 \qquad <span class='latex-bold'>(B)</span>\ 8 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ 11
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Gauss (7) #18
The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown. 1. AUSSG 9981 2. USSGA 9819 3. SSGAU 8199 etc. If the pattern continues in this way, what number will appear in front of GAUSS 1998?
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<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 9 \qquad <span class='latex-bold'>(D)</span>\ 16 \qquad <span class='latex-bold'>(E)</span>\ 20
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Gauss (7) #17
Claire takes a square piece of paper and folds it in half four times without unfolding, making an isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the paper would look like
16
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Gauss (7) #16
Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two digit number is subtracted from the three digit number, what is the smallest difference?
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<span class='latex-bold'>(A)</span>\ 269 \qquad <span class='latex-bold'>(B)</span>\ 278 \qquad <span class='latex-bold'>(C)</span>\ 484 \qquad <span class='latex-bold'>(D)</span>\ 271 \qquad <span class='latex-bold'>(E)</span>\ 261
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Gauss (7) #15
The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of
n
n
n
?
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<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ 11
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1998 Gauss (7) #14
A cube has a volume of
12
5
3
125 ^3
12
5
3
cm . What is the area of one face of the cube?
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<span class='latex-bold'>(A)</span>\ 20^2 \qquad <span class='latex-bold'>(B)</span>\ 25^2 \qquad <span class='latex-bold'>(C)</span>\ 41\frac{2}{3}^2 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ 75
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1998 Gauss (7) #13
The pattern of figures
△
\triangle
△
∙
\bullet
∙
□
\square
□
▲
\blacktriangle
▲
∘
\circ
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is repeated in the sequence
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,
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,
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\triangle,\bullet, \square, \blacktriangle, \circ, \triangle, \bullet, \square, \blacktriangle, \circ
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,
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The 214th figure in the sequence is (A)
△
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△
(B)
∙
\bullet
∙
(C)
□
\square
□
(D)
▲
\blacktriangle
▲
(E)
∘
\circ
∘
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1998 Gauss (7) #12
Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees?
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<span class='latex-bold'>(A)</span>\ 1~1/4 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ 12~1/2
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1998 Gauss (7) #11
Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts were made parallel to the sides of the original rectangle. In comparing R to S (A) the area and perimeter both decrease (B) the area decreases and the perimeter increases (C) the area and perimeter both increase (D) the area increases and the perimeter decreases (E) the area decreases and the perimeter stays the same
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1998 Gauss (7) #10
At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by
0.25
0.25
0.25
seconds. If Bonnie’s time was exactly
7.80
7.80
7.80
seconds, how long did it take for Wendy to go down the slide?
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<span class='latex-bold'>(A)</span>\ 7.80~ \text{seconds} \qquad <span class='latex-bold'>(B)</span>\ 8.05~ \text{seconds} \qquad <span class='latex-bold'>(C)</span>\ 7.55~ \text{seconds} \qquad <span class='latex-bold'>(D)</span>\ 7.15~ \text{seconds} \qquad
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<span class='latex-bold'>(E)</span>\ 7.50~ \text{seconds}
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1998 Gauss (7) #9
Two numbers have a sum of
32
32
32
. If one of the numbers is
–
36
– 36
–36
, what is the other number?
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<span class='latex-bold'>(A)</span>\ 68 \qquad <span class='latex-bold'>(B)</span>\ -4 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 72 \qquad <span class='latex-bold'>(E)</span>\ -68
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8
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1998 Gauss (7) #8
Tuesday’s high temperature was 4°C warmer than that of Monday’s. Wednesday’s high temperature was 6°C cooler than that of Monday’s. If Tuesday’s high temperature was 22°C, what was Wednesday’s high temperature?
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<span class='latex-bold'>(A)</span>\ 20 \qquad <span class='latex-bold'>(B)</span>\ 24 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 32 \qquad <span class='latex-bold'>(E)</span>\ 16
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7
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Hide problems
1998 Gauss (7) #7
A rectangular field is 80 m long and 60 m wide. If fence posts are placed at the corners and are 10 m apart along the 4 sides of the field, how many posts are needed to completely fence the field?
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<span class='latex-bold'>(A)</span>\ 24 \qquad <span class='latex-bold'>(B)</span>\ 26 \qquad <span class='latex-bold'>(C)</span>\ 28 \qquad <span class='latex-bold'>(D)</span>\ 30 \qquad <span class='latex-bold'>(E)</span>\ 32
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1998 Gauss (7) #6
In the multiplication question, the sum of the digits in the four boxes is http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy83L2NmMTU0MzczY2FhMGZhM2FjMjMwZDcwYzhmN2ViZjdmYjM4M2RmLnBuZw==&rn=U2NyZWVuc2hvdCAyMDE3LTAyLTI1IGF0IDUuMzguMjYgUE0ucG5n
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<span class='latex-bold'>(A)</span>\ 13 \qquad <span class='latex-bold'>(B)</span>\ 12 \qquad <span class='latex-bold'>(C)</span>\ 27 \qquad <span class='latex-bold'>(D)</span>\ 9 \qquad <span class='latex-bold'>(E)</span>\ 22
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1998 Gauss (7) #5
If a machine produces
150
150
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items in one minute, how many would it produce in
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seconds?
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<span class='latex-bold'>(A)</span>\ 10 \qquad <span class='latex-bold'>(B)</span>\ 15 \qquad <span class='latex-bold'>(C)</span>\ 20 \qquad <span class='latex-bold'>(D)</span>\ 25 \qquad <span class='latex-bold'>(E)</span>\ 30
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1998 Gauss (7) #4
Jean writes five tests and achieves the marks shown on the graph. What is her average mark on these five tests? [asy] draw(origin -- (0, 10.1)); for(int i = 0; i < 11; ++i) { draw((0, i) -- (10.5, i)); label(string(10*i), (0, i), W); } filldraw((1, 0) -- (1, 8) -- (2, 8) -- (2, 0) -- cycle, black); filldraw((3, 0) -- (3, 7) -- (4, 7) -- (4, 0) -- cycle, black); filldraw((5, 0) -- (5, 6) -- (6, 6) -- (6, 0) -- cycle, black); filldraw((7, 0) -- (7, 9) -- (8, 9) -- (8, 0) -- cycle, black); filldraw((9, 0) -- (9, 8) -- (10, 8) -- (10, 0) -- cycle, black); label("Test Marks", (5, 0), S); label(rotate(90)*"Marks out of 100", (-2, 5), W); [/asy]
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<span class='latex-bold'>(A)</span>\ 74 \qquad <span class='latex-bold'>(B)</span>\ 76 \qquad <span class='latex-bold'>(C)</span>\ 70 \qquad <span class='latex-bold'>(D)</span>\ 64 \qquad <span class='latex-bold'>(E)</span>\ 79
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1998 Gauss (7) #3
If
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S = 6 \times10 000 +5\times 1000+ 4 \times 10+ 3 \times 1
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<span class='latex-bold'>(A)</span>\ 6543 \qquad <span class='latex-bold'>(B)</span>\ 65043 \qquad <span class='latex-bold'>(C)</span>\ 65431 \qquad <span class='latex-bold'>(D)</span>\ 65403 \qquad <span class='latex-bold'>(E)</span>\ 60541
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60541
2
1
Hide problems
1998 Gauss (7) #2
The number
4567
4567
4567
is tripled. The ones digit (units digit) in the resulting number is
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<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 1
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Hide problems
1998 Gauss (7) #1
The value of
1998
−
998
1000
\frac{1998- 998}{1000}
1000
1998
−
998
is
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<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 1000 \qquad <span class='latex-bold'>(C)</span>\ 0.1 \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ 0.001
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