Daily Practice

Solution

It is well known that $text\{Distance\}=text\{Speed\}\; cdot\; text\{Time\}$. In the question, we want distance. From the question, we have that the time is $60$ minutes or $1$ hour. By the equation derived from $text\{Distance\}=text\{Speed\}\; cdot\; text\{Time\}$, we have $text\{Speed\}=frac\{text\{Distance\}\}\{text\{Time\}\}$, so the speed is $1$ mile per $x$ minutes. Because we want the distance, we multiply the time and speed together yielding $60text\{\; mins\}cdot\; frac\{1text\{\; mile\}\}\{xtext\{\; mins\}\}$. The minutes cancel out, so now we have $dfrac\{60\}\{x\}$ as our distance for the first day. The distance for the following days are: dfrac{60}{x},dfrac{60}{x+5},dfrac{60}{x+10},dfrac{60}{x+15}. We know that $x,x+5,x+10,x+15$ are all factors of $60$, therefore, $x=5$ because the factors have to be in an arithmetic sequence with the common difference being $5$ and $x=5$ is the only solution. dfrac{60}{5}+dfrac{60}{10}+dfrac{60}{15}+dfrac{60}{20}=12+6+4+3=boxed{textbf{(C)} 25}. == Video Solution by OmegaLearn == https://youtu.be/XixU0JZ5FLk?t=394 ~ pi_is_3.14 ~Hipparachus

Solution

We group the addends inside the parentheses two at a time: begin{align*} -1 + 2 - 3 + 4 - 5 + 6 - 7 + ldots + 1000 &= (-1 + 2) + (-3 + 4) + (-5 + 6) + ldots + (-999 + 1000) \ &= underbrace{1+1+1+ldots + 1}_{text{500 1's}} \ &= 500. end{align*} Then the desired answer is $4\; times\; 500\; =\; boxed\{textbf\{(E)\}\; 2000\}$. ==Video Solution== https://youtu.be/4Gy1CrYTDHA ~savannahsolver ==Video Solution by OmegaLearn== https://youtu.be/TkZvMa30Juo?t=3074 ~ pi_is_3.14

Solution

There are 61 vertical columns with a length of 32 toothpicks, and there are 33 horizontal rows with a length of 60 toothpicks, because 32 and 60 are the number of intervals. You can verify this by trying a smaller case, i.e. a 3 times 4 grid of toothpicks, with 3 times 3 and 2 times 4. Thus, our answer is 61cdot 32 + 33 cdot 60 = boxed{textbf{(E)} 3932}. ~Note by Theraccoon: The person who posted this answer did not include their name. Minor edit by ~NXC

Solution

Given the information above, we start with the equation $frac\{t\}\{4\}+frac\{2t\}\{7\}\; +\; 15\; +\; x\; =\; t$, where $t$ is the total number of points scored and $xle\; 14$ is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation $x+15\; =\; frac\{13\}\{28\}t$, or $28x+28cdot\; 15=13t$. Since $t$ is necessarily divisible by 28, let $t=28u$ where $u\; ge\; 0$ and divide by 28 to obtain $x\; +\; 15\; =\; 13u$. Then, it is easy to see $u=2$ ($t=56$) is the only candidate remaining, giving $x=boxed\{textbf\{(B)\}\; 11\}$. -scrabbler94 ~Minor Edits by Wrenmath

Solution

$B$ is on the top, and $R$ is on the side, and $G$ is on the right side. That means that (image $2$) $W$ is on the left side. From the third image, you know that $P$ must be on the bottom since $G$ is sideways. That leaves us with the back, so the back must be $A$. The front is opposite of the back, so the answer is $boxed\{textbf\{(A)\}\; R\}$.