Tag: G-7

Sample Questions for Gauss Contests


Questions chosen from previous Gauss contests
Gauss contests are organized by the Centre of Education for
Math and Computing, University of Waterloo
Problem 1

In the addition shown, P and Q each represent single digits, and the sum is 1PP7. What is P + Q?

(A) 9 (B) 12 (C) 14 (D) 15 (E) 13

 

Problem 2

In the right-angled triangle PQR, we have that PQ = QR. The three segments QS, TU and VW are perpendicular to PR, and the segments ST and UV are perpendicular to QR, as shown. What fraction of triangle PQR is shaded?

(A) 3 ⁄ 16 (B) 3 ⁄ 8 (C) 5 ⁄ 16 (D) 5 ⁄ 32 (E) 7 ⁄ 32

 

Problem 3

A box contains a total of 400 tickets that come in five colours: blue, green, red, yellow, and orange. The ratio of blue to green to red tickets is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of the same colour have been selected?

(A) 50 (B) 246 (C) 148 (D) 196 (E) 115

 

Problem 4

Greg, Charlize, and Azarah run at different but constant speeds. Each pair ran a race on a track that measured 100 m from start to finish. In the first race, when Azarah crossed the finish line, Charlize was 20 m behind. In the second race, when Charlize crossed the finish line, Greg was 10 m behind. In the third race, when Azarah crossed the finish line, how many metres was Greg behind?

(A) 20 (B) 25 (C) 28 (D) 32 (E) 40

 

Problem 5

In right-angled, isosceles triangle FGH, segment FH = √̅8. Arc FH is part of the circumference of a circle with centre G and radius GH. The area of the shaded region is

(A) π – 2; (B) 4 π – 2 (C) 4 π – (1 ⁄ 2) √̅8 ; (D) 4 π – 4 (E) π – √̅8

Number Sense – Activity: Tsunami Numbers in the News

About a decade ago, there was a great Tsunami happening in Asia.
What do you know about the Asian tsunami?

Read through the article first. Use the following numbers to fill in the blanks in the story.Think about which numbers make sense.

500  20  8,000;  2004  110,000  30,000  9.0

A tsunami triggered by a very large earthquake off the coast of the
Indonesian island of Sumatra on December 26, ____, has left
more than 150,000 people dead and millions homeless. Countries hit hardest by the disaster include
Sri Lanka, Indonesia, India, Thailand, and the Maldives. Almost 75% of the deaths occurred in
Indonesia, estimated at ____. Sir Lanka was second highest with about 20% of the estimated deaths, or
______ people lost that day. The rest of the deaths, approximately ____, occurred in the other nine
countries affected by the tsunami.
The ____ foot wall of water, higher than a two-story building,
swallowed entire villages. The tsunami waves were not only very high, they moved at a much faster speed
than normal. These waves were comparable in size to those you see on some of the surfing movies;
but those waves travel at 30 miles an hour, and the tsunami waves
were moving more than fifteen times as fast at ____ miles an hour.
The velocity of the force is what caused the destruction—a massive force that swept away everything in its path.

The earthquake causing this Tsunami was a destructive earthquake measuring ______ on the Richter scale,
the fourth worst earthquake in recorded history. Earthquakes are measured on a Richter scale that has
a range from 0 to 12; a 6.0 on the scale is a pretty bad earthquake.

(Story constructed from January 2005 news reports)

Prime numbers

Prime numbers are those that have 1 (one) and itself as the only two divisors. Examples of primes are 2, 3, 5, 7, 11. None of 4, 6, 9 is a prime since 4 = 2 × 2, 6 = 2 × 3, and 9 = 3 × 3.

If a number greater than one is not a prime, then it is a composite number, and can be factored into the product of primes — called prime factorization. We have given the prime factorization of 4, 6, 9 as above. For a couple of more examples:

12 = 2 × 2 × 3

36 = 2 × 3 × 3 × 3

28 = 2 × 2 × 7

So all natural numbers are divided into three classes: the number 1, the prime numbers, and the composite numbers.

A bonus point: π, besides representing in a circle, the ratio of circumference to diameter, also stands for a special function related to prime numbers. Function π(x) — for every integer x, represent the number of primes less than or equal (i.e. not exceeding) x. For example, we have:

π(2) = 1, π(3) = 2, π(10) = 4, π(20) = 8 etc.
[To find why π(10) = 4, recall the 4 prime numbers not exceeding 10: they are 2,3,5, and 7.]

Find the Center of a Circle — Do you know how to do it?

Using a compass, you can draw a circle at any place, with any radius.

Now let’s reverse the problem. Given a circle, do you know how to find its center? (Of course, once the circle is found, there shall be no problem at all to tell its diameter, or radius.) You only see the circle itself, there is no explicit indication on where the center is.

With two set of restrictions on what kind of tools you can use, there are actually two questions. In general, you can find the center using any convenient method, including copy-and-paste the circle onto a paper, and then fold it. In particular (from classical Euclidean geometry), where it’s required to do so with a ruler (with which you are allowed to draw lines and line segments only) and a compass (with which you are allowed to draw circles only).

See the following article on how to do it in general.

How to find the center of a given circle?

If you attempt to solve this problem with a ruler and a compass, then you are required to know how to make a perpendicular bisector. This will be discussed in another posting.