Paul has 6 boxes, each of which contains 12 trays. Paul also has 4 extra trays. If each tray can hold 8 apples, what is the largest possible number of apples that can be held by the 6 boxes and 4 extra trays?

A rabbit, a skunk and a turtle are running a race. The skunk finishes the race in 6 minutes. The rabbit runs 3 times as quickly as the skunk. The rabbit runs 5 times as quickly as the turtle. How long does the turtle take to finish the race?

A jar contains 6 crayons, of which 3 are red, 2 are blue, and 1 is green. Jakob reaches into the jar and randomly removes 2 of the crayons. What is the probability that both of these crayons are red?

Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$ If the sum of the digits of $a$ is 567, what is the value of $n$?

In the diagram, $ABCDEF$ is a regular hexagon with side length 2. Points $E$ and $F$ are on the $x$ axis and points $A$, $B$, $C$, and $D$ lie on a parabola. What is the distance between the two $x$ intercepts of the parabola? [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.3215445204635294, xmax = 7.383669550094284, ymin = -4.983460515387094, ymax = 6.688676116382409; pen zzttqq = rgb(0.6,0.2,0); pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((2,0)--(4,0)--(5,1.7320508075688774)--(4,3.4641016151377553)--(2,3.4641016151377557)--(1,1.732050807568879)--cycle, linewidth(1)); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true); draw((2,0)--(4,0), linewidth(1)); draw((4,0)--(5,1.7320508075688774), linewidth(1)); draw((5,1.7320508075688774)--(4,3.4641016151377553), linewidth(1)); draw((4,3.4641016151377553)--(2,3.4641016151377557), linewidth(1)); draw((2,3.4641016151377557)--(1,1.732050807568879), linewidth(1)); draw((1,1.732050807568879)--(2,0), linewidth(1)); real f1 (real x) {return -0.58*x^(2)+3.46*x-1.15;} draw(graph(f1,-3.3115445204635297,7.373669550094284), linewidth(1)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /*yes i used geogebra fight me*/ [/asy][/asy]

Suppose that $0^\circ < A < 90^\circ$ and $0^\circ < B < 90^\circ$ and \[\left(4+\tan^2 A\right)\left(5+\tan^2 B\right) = \sqrt{320}\tan A\tan B\]Determine all possible values of $\cos A\sin B$.

Alexandra draws a letter A which stands on the $x$-axis. The left side of the letter A lies along the line with equation $y=3x+6$. What is the $x$-intercept of the line with equation $y=3x+6$? The right side of the letter A lies along the line $L_2$ and the leter is symmetric about the $y$-axis. What is the equation of line $L_2$? Determine the are of the triangle formed by the $x$ axis and the left and right sides of the letter A. Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line $y=c$, as in Figure 2. The area of the shaded region inside the letter A and above the line with equation $y=c$ is $\frac49$ of the total area of the region above the $x$ axis and between the left and right sides. Determine the value of $c$. Figure 1 [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8408113739622465, xmax = 5.491811096383217, ymin = -3.0244242161812847, ymax = 8.241467380517944; /* image dimensions */ pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(8.25),above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((0,6)--(-2,0), linewidth(2)); draw((-2,0)--(2,0), linewidth(2)); draw((2,0)--(0,6), linewidth(2)); label("$y=3x+6$",(-2.874280000573916,3.508459668295191),SE*labelscalefactor); label("$L_2$",(1.3754276283584919,3.5917872688624928),SE*labelscalefactor); label("$O$",(0,0),SW*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Figure 2 [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.707487213054563, xmax = 5.6251352572909, ymin = -3.4577277391312538, ymax = 7.808163857567977; /* image dimensions */ pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--(0,6)--cycle, linewidth(2)); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((0,6)--(-2,0), linewidth(2)); draw((-2,0)--(2,0), linewidth(2)); draw((2,0)--(0,6), linewidth(2)); label("$O$",(0,0),SW*labelscalefactor); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2)); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2)); draw((1.1148845961134441,2.6553462116596678)--(0,6), linewidth(2)); draw((0,6)--(-1.114884596113444,2.6553462116596678), linewidth(2)); fill((0,6)--(-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--cycle,black); label("$y=c$",(1.4920862691527148,3.1251527056856054),SE*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* yes i used geogebra fight me*/ [/asy][/asy]

Determine the positive integer $x$ for which $\dfrac14-\dfrac{1}{x}=\dfrac16.$ Determine all pairs of positive integers $(a,b)$ for which $ab-b+a-1=4.$ Determine the number of pairs of positive integers $(y,z)$ for which $\dfrac{1}{y}-\dfrac{1}{z}=\dfrac{1}{12}.$ Prove that, for every prime number $p$, there are at least two pairs $(r,s)$ of positive integers for which $\dfrac{1}{r}-\dfrac{1}{s}=\dfrac{1}{p^2}.$

A string of length $n$ is a sequence of $n$ characters from a specified set. For example, $BCAAB$ is a string of length 5 with characters from the set $\{A,B,C\}$. A substring of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string $CA$ is a substring of $BCAAB$ but $BA$ is not a substring of $BCAAB$. List all strings of length 4 with characters from the set $\{A,B,C\}$ in which both the strings $AB$ and $BA$ occur as substrings. (For example, the string $ABAC$ should appear in your list.) Determine the number of strings of length 7 with characters from the set $\{A,B,C\}$ in which $CC$ occures as a substring. Let $f(n)$ be the number of strings of length $n$ with characters from the set $\{A,B,C\}$ such that $CC$ occurs as a substring, and if either $AB$ or $BA$ occurs as a substring then there is an occurrence of the substring $CC$ to its left. (for example, when $n\;=\;6$, the strings $CCAABC$ and $ACCBBB$ and $CCABCC$ satisfy the requirements, but the strings $BACCAB$ and $ACBBAB$ and $ACBCAC$ do not). Prove that $f(2097)$ is a multiple of $97$.