Tag: G-8

Categories
Geometry /Shapes

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.

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).

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. Do you know how to do it?

Categories
Numbers

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. This is described as follows.

Function π(x) — for integer x, represents the number of primes less than or equal (i.e. not exceeding) x.

 

For example,

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.]

Categories
Math Contests - Problems and Discussion Math Junior (G9 and under)

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 colors: 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

Categories
Geometry /Shapes Math BASICS Numbers

Protected: The (3-4-5) Pattern for Pythagorean Triplet [Thinking-of sides of a right triangle]

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Categories
Algebra Numbers

Complete Numbers in Fraction Equations

The formula on our face page of “amazing numbers” is rather interesting:
1 – (1 ⁄ 28) = (1 ⁄ 2) + (1 ⁄ 4) + (1 ⁄ 7) + (1 ⁄ 14)

The point of interest is that: if you look at all divisors of 28: they are 1,2,4,7,14,28; with the exception of 28 which is itself, all divisors have appeared in this formula, and they appear in the form of so-called “unit fraction”, where numerator is 1. So (1 ⁄ 2), (1 ⁄ 4), etc. are all unit fractions.

Indeed, we present a fraction equation to make it a bit unusual, but there is a low-pitch but straightforward ways to present number 28. We have that:
28 = 1 + 2 + 4 + 7 + 14
To get to the earlier fraction form, just divide every term by the number 28.

The smallest complete number is 6 (=1+2+3), 28 is the 2nd complete number, and after that, you will not see a complete number until 496. So complete numbers are rare among all positive whole numbers.

Complete numbers 6 also has a nice fraction form, as:
1 – (1⁄6) = (1⁄2) + (1⁄3)

Categories
Numbers

What’s so special about number 8208?

The seemingly trivial, innocent number 8208 actually has something special. It is a member of a niche class of numbers called narcissistic numbers.

What’s so special?

First note it has 4 digits. If we raise every digit 8, 2, 0, 8 to the 4th power, and then add them together:

84 + 24 + 04 +84 = 8208

(To find 84, just think 84 = 212 = 4 x 210 = 4096)

Now read the four digits of the total. It’s 8, 2, 0, 8 again!

See the point? All the digits of the number are applied to generate that same number. Repeat the same procedure once more, you still get the same number. Seems this number loves itself so its ego is repeated. — That speaks for its name: narcissistic.

 

The 3-digit number 153 has similar property.

Since it has 3 digits, we raise every digit 1, 5, 3 to the 3rd power, and add them together!

13 + 53 + 33 = 153

Another time to witness the magic!

** **

Any one-digit number belongs to the class of narcissistic numbers. Reason like this: (if you raise everything to exponent one, it is just itself); this is plain yet may be too trivial to be interesting.

And tell you another secret. There is NOT ANY 2-digit number that is narcissistic (we assume that tens-digit cannot be zero: it has to be between 10 and 99). Some may be close (almost narcissistic!): for example

32 + 52 = 34;  72 + 52 = 74  (note 34 is just one off from 35)

That’s it. Happy reading!

 

Categories
Algebra

Power Mad

Continue reading “Power Mad”