Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $ 2: 1$. The ratio of the rectangle's length to its width is $ 2: 1$. What percent of the rectangle's area is inside the square? $ \textbf{(A)}\ 12.5 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 87.5$

Twenty percent less than $ 60$ is one-third more than what number? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 30\qquad \textbf{(C)}\ 32\qquad \textbf{(D)}\ 36\qquad \textbf{(E)}\ 48$

A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up $36\%$ of the area of the flag, what percent of the area of the flag is blue? [asy] draw((0,0)--(5,0)--(5,5)--(0,5)--cycle); draw((1,0)--(5,4)); draw((0,1)--(4,5)); draw((0,4)--(4,0)); draw((1,5)--(5,1)); label("RED", (1.2,3.7)); label("RED", (3.8,3.7)); label("RED", (1.2,1.3)); label("RED", (3.8,1.3)); label("BLUE", (2.5,2.5)); [/asy] $ \textbf{(A)}\ 0.5 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6 $

The average cost of a long-distance call in the USA in $1985$ was $41$ cents per minute, and the average cost of a long-distance call in the USA in $2005$ was $7$ cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call. $\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 34 \qquad \textbf{(D)}\ 41 \qquad \textbf{(E)}\ 80$

For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$ and $C$ are sets for which \[n(A) + n(B) + n(C) = n(A \cup B \cup C)\quad\text{and}\quad |A| = |B| = 100,\] then what is the minimum possible value of $|A \cap B \cap C|$? $ \textbf{(A)}\ 96\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 98\qquad\textbf{(D)}\ 99\qquad\textbf{(E)}\ 100 $

[asy]draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); draw((0,5)--(-2,7)); label("F",(0,0.5),W); label("D",(0,1.5),W); label("C",(0,2.5),W); label("B",(0,3.5),W); label("A",(0,4.5),W); label("A",(0.5,5),N); label("B",(1.5,5),N); label("C",(2.5,5),N); label("D",(3.5,5),N); label("F",(4.5,5),N); label("0",(0.5,0),N); label("0",(0.5,1),N); label("1",(0.5,2),N); label("1",(0.5,3),N); label("2",(0.5,4),N); label("0",(1.5,0),N); label("0",(1.5,1),N); label("3",(1.5,2),N); label("4",(1.5,3),N); label("2",(1.5,4),N); label("2",(2.5,0),N); label("1",(2.5,1),N); label("5",(2.5,2),N); label("3",(2.5,3),N); label("1",(2.5,4),N); label("1",(3.5,0),N); label("1",(3.5,1),N); label("2",(3.5,2),N); label("0",(3.5,3),N); label("0",(3.5,4),N); label("0",(4.5,0),N); label("1",(4.5,1),N); label("0",(4.5,2),N); label("0",(4.5,3),N); label("0",(4.5,4),N); label("TEST 2",(1,6),N); label("TEST 1",(-2,5),SW);[/asy] The table displays the grade distribution of the $ 30$ students in a mathematics class on the last two tests. For example, exactly one student received a "D" on Test 1 and a "C" on Test 2. What percent of the students received the same grade on both tests? \[ \textbf{(A)}\ 12 \% \qquad \textbf{(B)}\ 25 \% \qquad \textbf{(C)}\ 33 \frac{1}{3} \% \qquad \textbf{(D)}\ 40 \% \qquad \textbf{(E)}\ 50 \% \qquad \]

A jar was filled with jelly beans so that $54\%$ of the beans were red, $30\%$ of the beans were green, and $16\%$ of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now $20\%$ of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $

Mary is about to pay for five items at the grocery store. The prices of the items are $ \$$7.99, $ \$$ 4.99, $ \$$2.99, $ \$$1.99, and $ \$$0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $ \$$20.00 that she will receive in change? $ \textbf{(A) } 5 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 25$

An article costing $ C$ dollars is sold for $ \$100$ at a lostt of $ x$ percent of the selling price. It is then resold at a profit of $ x$ percent of the new selling price $ S'$. If the difference between $ S'$ and $ C$ is $ 1\frac {1}{9}$ dollars, then $ x$ is: $ \textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ \frac {80}{9} \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ \frac {95}{9} \qquad \textbf{(E)}\ \frac {100}{9}$

The table below gives the percent of students in each grade at Annville and Cleona elementary schools: \[\begin{tabular}{rccccccc} & \textbf{\underline{K}} & \textbf{\underline{1}} & \textbf{\underline{2}} & \textbf{\underline{3}} & \textbf{\underline{4}} & \textbf{\underline{5}} & \textbf{\underline{6}} \\ \textbf{Annville:} & 16\% & 15\% & 15\% & 14\% & 13\% & 16\% & 11\% \\ \textbf{Cleona:} & 12\% & 15\% & 14\% & 13\% & 15\% & 14\% & 17\% \end{tabular}\] Annville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6? $\text{(A)}\ 12\% \qquad \text{(B)}\ 13\% \qquad \text{(C)}\ 14\% \qquad \text{(D)}\ 15\% \qquad \text{(E)}\ 28\%$

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$ Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$ Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$ For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$? $\textbf{(A) }\$179.95\qquad \textbf{(B) }\$199.95\qquad \textbf{(C) }\$219.95\qquad \textbf{(D) }\$239.95\qquad \textbf{(E) }\$259.95\qquad$

A canister contains two and a half cups of flour. Greg and Sally have a brownie recipe which calls for one and one third cups of flour. Greg and Sally want to make one and a half recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?

Price of some item has decreased by $5\%$. Then price increased by $40\%$ and now it is $1352.06\$$ cheaper than doubled original price. How much did the item originally cost?

A positive number $ x$ has the property that $ x\%$ of $ x$ is $ 4$. What is $ x$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 40$

A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by members of the audience? $\text{(A)} \ 24 \qquad \text{(B)} \ 27 \qquad \text{(C)} \ 30 \qquad \text{(D)} \ 33 \qquad \text{(E)} \ 36$

A jar contains one quarter red jelly beans and three quarters blue jelly beans. If Chris removes three quarters of the red jelly beans and one quarter of the blue jelly beans, what percent of the jelly beans remaining in the jar will be red?

Price of the book increased by $20\%$, and then decreased by $10\%$. How many percents should we decrease current price so we get a price which is $54\%$ percent of an original one?

What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$

A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed? $ \textbf{(A)}\ 4.5 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

A capacitor is made with two square plates, each with side length $L$, of negligible thickness, and capacitance $C$. The two-plate capacitor is put in a microwave which increases the side length of each square plate by $ 1 \% $. By what percent does the voltage between the two plates in the capacitor change? $ \textbf {(A) } \text {decreases by } 2\% \\ \textbf {(B) } \text {decreases by } 1\% \\ \textbf {(C) } \text {it does not change} \\ \textbf {(D) } \text {increases by } 1\% \\ \textbf {(E) } \text {increases by } 2\% $ [i]Problem proposed by Ahaan Rungta[/i]

In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds: \[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\] [i]Calvin Deng.[/i]

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to $ \textbf{(A)} \ 50 \qquad \textbf{(B)} \ 52 \qquad \textbf{(C)} \ 54 \qquad \textbf{(D)} \ 56 \qquad \textbf{(E)} \ 58 \qquad$ [asy]unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } }[/asy]

Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip? $\textbf{(A) } 20\% \qquad \textbf{(B) } 26\tfrac23\% \qquad \textbf{(C) } 27\tfrac79\% \qquad \textbf{(D) } 33\tfrac13\% \qquad \textbf{(E) } 66\tfrac23\%$