MathProblemSolver101

$3(-4) + 2(7) - 5(-1) + 8(3) - 4(-5) = -12 + 14 + 5 + 24 + 20 = 51$

First thing that comes to mind is a marathon, but marathons are held inside central america. Iirc, the answer is decalthon.

$-9x - 27y = 36$

$3x - 7y = 36$

$3x - 7y = -9x - 27y$

$12x = -20y$

$3x = -5y$

$x = - \frac{5}{3} y$

$3x + 9y = -12$

$16y = -48$

$y = -3$

$x = 5$

$(x,y) = (5, -3)$

I also have had this problem for alcumus- it was a very challenging one ! I eventually solved it, so I can give you a hint:

Consider the possible ways to make a square. Find a pattern in each of every one of them. Add them up to form an equation, and the answer is your answer.

I will post a solution later

$2^{10} \cdot 4^{20} \cdot 8^{40} = 2^n$

$2^{10} \cdot \left(2^2 \right)^{20} \cdot \left(2^3 \right)^{40} = 2^n$

$2^{10} \cdot 2^{40} \cdot 2^{120} = 2^n$

$2^{10 + 40 + 120} = 2^n$

$n = 10 + 40 + 120$

$n = \boxed{170}$

Monday = 1 (mod 7)

21 (mod 7) = 0 mod 7 is obviously Sunday given that information.

The simplest polynomial is $ax^2 + bx + c$

Let $x = k \cdot c$ for $k \in \mathbb{Z}$

We have \(\begin{align*} ax^2 + bx + c &= a(kc)^2 + b(kc) + c\\ &= ak^2c^2 + bkc + c\\ &= c(ak^2c + bk + 1) \end{align*}\)

Because there is are at least two factors for $ax^2 + bx + c,$ $P(x)$ is composite.

Let $\angle{A} = x.$

Let the change in angles be $n.$

We have: $x+(x+n)+(x+2n)+(x+3n) = 360$

$x+n = 72$

We want to find $x+3n.$

Rewrite first equation: $3x+5n + 72 = 360$

$3x + 5n = 288$

Rewrite third equation: $3x + 3n = 216$

$2n = 72$

$3n = \frac{3}{2} \cdot 72 = 108$

$n = 36$

$x + 36 = 72$

$x = 36$

$x + 3n = 36 + 108 = \boxed{144}$