## Proving exactly one town which cannot be swept away by other one

Solution IMO 2015 (combinatoric shortlist problem) Problem 1: In Line land there are towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left)...

## Proving the line is tangent to the circle (solution APMO 2016)

Proving the line is tangent to the circle (solution APMO 2016) Problem 3. Let AB and AC be two distinct rays not lying on the same line, and let be a circle with Center O that is tangent to...

## Finding the smallest positive integer from the fancy number

Finding the smallest positive integer from the fancy number Problem 2. A positive integer is called fancy if it can be expressed in the form where a1,a2, … , a100 are non-negative integers that are not necessarily distinct. Find...

## Solve equations with exponents using the odd root property and the even root property

Another type of equation we can solve is one with exponents. As you might expect we can clear exponents by using roots. This is done with very few unexpected results when the exponent is odd. We solve these problems...

## Solve equations with radicals and check for extraneous solution

Here we look at equations that have roots in the problem. As you might expect, to clear a root we can raise both sides to an exponent. So to clear a square root we can rise both sides to...

## Add, subtract, multiply, rationalize, and simplify expressions using complex numbers

World View Note: When mathematics was first used, the primary purpose was for counting. Thus they did not originally use negatives, zero, fractions or irrational numbers. However, the ancient Egyptians quickly developed the need for “a part” and so...

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