### Collected Problems 3

I'm collecting various problems especially the short lived ones from twitter here for future use. My taste runs towards geometry as is probably evident. For ease of use I've hidden my notes. Click the button below to expose them.  [Previous Page]

#### Problem 1 [***]

There are unique integers $a_2, a_3, ... a_7$ such that

$$\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!}$$

where $0 <= a_i < i$ for i = 2,3...,7

Find  $a_2, a_3 ... a_7$

[AHSME]

1. Remove the fractions.
2. Look at this as a remainder problem first what's the  remainder when divided by 7 etc.

#### Problem 2 [*]

Prove:  log(1 + 2 + 3) = log (1) + log(2) + log(3)

[@mathematicsprof]

Remember the log rules.

#### Problem 3 [**]

$$(6x + 5)^2(3x +2)(x+1) = 35$$

[@mathematician_3]

1. Substitution is the key  $(6x + 5)^2 = 36x^2 + 60x + 25 = 12(3x^2 + 5x + 2) + 1 = 12(3x + 2)(x + 1) + 1$

#### Problem 4 [*]

The perimeter of the rectangle ABCD is 28 units E is the midpoint of the side AC and the two arc are tangent to each other. What are the dimensions of the rectangle?

[@five_triangles originally]

Find the implicit triangle and then apply the Pythagorean theorem.

#### Problem 5 [**]

This is a pair for comparison:

[@matematik_man]

[@eylem_99]

Both depend on forming 30-60-90 triangles. Look especially at the square root 3 side.

#### Problem 6[**]

[UWSIM program]

1. Take logs to get rid of the exponents  i.e. $\log{x} \cdot \log{3} = \log{y} \cdot \log{4}$
2.  Then you can use substitution for log x or log y.
3. Its hard to see the simplification so it might be useful to substitute things in but essentially this reduces to   $\log{x} \cdot (\frac{\log{4}^2 - \log{3}^2}{\log{4}}) = \log{3}^2 - \log{4}^2$
4. The difference of squares terms cancel out and you're left with $\log{x} = -\log{4}$

#### Problem 7 [**]

A,F,B,E,C on circle O. AC=BD=4, CG=3, AC//FE.
Find the area of AEF

[@five_triangles]

1. Note ACG is a 3-4-5 and BEG is similar to it.
2. Since EF is parallel to AC   triangle DEG is also similar to the 3-4-5
3. Set  x = DG then EG = 5/3X  but x + 4 = 5/3(EG) . Solve and  DG = x = 9/4.
4. Drop and altitude from A to EF. This divides into a 3-4-5 similar triangle where you have one side AE and a triangle similar to ABC where you also know all the sides.
5. This gives the altitude and base for AEF.

#### Problem 8 [***]

Let p be a 50 digit prime number,  when squared its remainder when divided by 120 is not 1.  What is  $p^2\mod 120$ ?

[AOPS]

1. Factor 120
2. Observe the behavior of squares mod each of the factors.
3. If you're prime then you can never be 0 mod any of the factors.
4. Only 2 cases left.

#### Problem 9 [*]

1. Find a polynomial with integer coefficients that has $3 + \sqrt{5}$ as one root.
2. Find a polynomial with integer coefficients that has $\sqrt{5} + \sqrt{7}$ as one root.
3. Find a polynomial with integer coefficients that has $3 + \sqrt{5} + \sqrt{7}$ as one root.

[Mike Lawler]

Use conjugates or square the sides enough to remove the radicals. Also don't forget about shifting a function to the right or left.

#### Problem 10 [*]

Find the smallest positive integer N such that  $\frac{N - 13}{5N + 6}$ is a reducible fraction.
[AHSME]

Try using the Euclidean Algorithm to find the gcd between the numerator and denominator.

#### Problem 11 [*]

Find the value of $\sqrt{100 \cdot 101 \cdot 102 \cdot 103 + 1}$ then find a general formula.
[@matematik_man]

There are a variety of methods that work. Start by looking at much smaller examples i.e. n = 1, 2 etc. Curve fit, use a linear system assume the expression is a square etc.