DeL-TaiseiOzaki/Tengentoppa-DeepScaleR_Difficulty

Unnamed: 0 int64 12 40.3k	problem stringlengths 19 5.15k	ground_truth stringlengths 1 1.22k
2,246	If $\frac{x}{y}=\frac{3}{4}$, then the incorrect expression in the following is:	$\frac{1}{4}$
2,261	The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\frac{1}{2}$. The first term of the progression is:	9 or 3
2,266	In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?	6
2,285	If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then:	$0<x<1$
2,292	Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the fixed monthly fee?	6.24
2,299	Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point at which Andy makes the $2020$th left turn?	$(-1030, -990)$
2,304	The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:	$x=\frac{-p}{2}$
2,307	The school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost?	$5.84
2,309	A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?	8 \frac{3}{4}
2,311	If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S, S'$, and $n$ is	$(S/S')^{\frac{1}{2}n}$
2,313	A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?	$4\sqrt{2}+4\sqrt{3}$
2,315	If the $whatsis$ is $so$ when the $whosis$ is $is$ and the $so$ and $so$ is $is \cdot so$, what is the $whosis \cdot whatsis$ when the $whosis$ is $so$, the $so$ and $so$ is $so \cdot so$ and the $is$ is two ($whatsis, whosis, is$ and $so$ are variables taking positive values)?	$so \text{ and } so$
2,318	Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily	$-a$
2,322	The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is	$90^\circ$
2,326	In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?	\frac{10}{3}
2,333	The set of $x$-values satisfying the inequality $2 \leq \|x-1\| \leq 5$ is:	-4\leq x\leq-1\text{ or }3\leq x\leq 6
2,334	A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was	63 or 12
2,337	The roots of the equation $2\sqrt{x} + 2x^{-\frac{1}{2}} = 5$ can be found by solving:	4x^2-17x+4 = 0
2,352	A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\] [asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (27370,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4r1rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]	\frac{143}{14}
2,364	There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is	11\frac{1}{4}
2,386	If $a$ and $b$ are two unequal positive numbers, then:	\frac {a + b}{2} > \sqrt {ab} > \frac {2ab}{a + b}
2,389	A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle $\theta$ is	$3\sec ^2\theta\csc\theta$
2,396	The bar graph shows the results of a survey on color preferences. What percent preferred blue?	24\%
2,398	Sale prices at the Ajax Outlet Store are $50\%$ below original prices. On Saturdays an additional discount of $20\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\$ 180$?	$72
2,407	The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.) How many different routes can Paula take?	4
2,416	Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?	$3.50
2,419	Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?	3+\sqrt3
2,423	A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if	\frac{1}{4}
2,442	Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost?	$20
2,450	In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$ The ratio $\frac{AD}{AB}$ is	\frac{1}{\sqrt[4]{12}}
2,452	Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum?	164
2,453	If $x-y>x$ and $x+y<y$, then	$x<0,y<0$
2,455	\(\triangle ABC\) is isosceles with base \(AC\). Points \(P\) and \(Q\) are respectively in \(CB\) and \(AB\) and such that \(AC=AP=PQ=QB\). The number of degrees in \(\angle B\) is:	25\frac{5}{7}
2,458	If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6},$ the middle part is:	17\frac{1}{3}
2,469	If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a circular patio whose radius is $12$ feet?	48
2,471	Jack had a bag of $128$ apples. He sold $25\%$ of them to Jill. Next he sold $25\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?	65
2,474	The product $(1.8)(40.3 + .07)$ is closest to	74
2,476	From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are	$3^{1/3},\ 2^{1/2}$
2,479	Circles with radii $1$, $2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?	\frac{6}{5}
2,482	Points $A(11, 9)$ and $B(2, -3)$ are vertices of $ riangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?	$\left( -4, 9 \right)$
2,490	The symbol $\|a\|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to?	\|t\|\sqrt{1+t^2}
2,494	If $t = \frac{1}{1 - \sqrt[4]{2}}$, then $t$ equals	$-(1+\sqrt[4]{2})(1+\sqrt{2})$
2,508	The percent that $M$ is greater than $N$ is:	\frac{100(M-N)}{N}
2,510	If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \cdot d - b \cdot c$, what is the value of $\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$?	$2$
2,512	Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students. What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?	4.36
2,515	If the product $\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \ldots \cdot \frac{a}{b} = 9$, what is the sum of $a$ and $b$?	37
2,526	A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?	200
2,542	If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$, $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$	$f(-x)=f(x)$
2,546	The coordinates of $A, B$ and $C$ are $(5,5), (2,1)$ and $(0,k)$ respectively. The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible is:	2\frac{1}{7}
2,549	$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s speed to $B$'s speed is:	5/6
2,559	If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is:	$y = \frac{1}{3} x + 8$
2,571	In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long. Of the three equations \begin{equation} \label{eq:1} d-s=1, \qquad ds=1, \qquad d^2-s^2=\sqrt{5} \end{equation}those which are necessarily true are	I, II and III
2,576	Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA=1$. The area of the "bat wings" (shaded area) is	3 \frac{1}{2}
2,577	A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:	2/1
2,582	Distinct planes $p_1, p_2, \dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?	20
2,585	Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$	12\sqrt2
2,598	Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?	\frac{85\pi}{8}
2,608	Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?	0
2,612	In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?	\frac{366}{31 \times 24}
2,616	Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$. The area of the shaded "smile" $AEFBDA$, is	$2\pi-\pi \sqrt{2}-1$
2,618	Students from three middle schools worked on a summer project. Seven students from Allen school worked for 3 days. Four students from Balboa school worked for 5 days. Five students from Carver school worked for 9 days. The total amount paid for the students' work was 744. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?	180.00
2,624	A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?	4:40 a.m.
2,625	The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is	20
2,634	Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?	\frac{147}{1024}
2,639	Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is:	$x >\frac{10}{3}$
2,640	An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?	134
2,659	Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?	(13,-22)
2,666	ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between	7 and 8
2,668	When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:	1: 2
2,673	A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?	$\frac{15}{8}$
2,674	The formula expressing the relationship between $x$ and $y$ in the table is: \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline x & 2 & 3 & 4 & 5 & 6\ \hline y & 0 & 2 & 6 & 12 & 20\ \hline \end{tabular}	$y = x^{2}-3x+2$
2,676	When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school?	18 minutes
2,681	Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?	17
2,684	Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$. If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:	$a>9, b>6$
2,689	If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is	$5$
2,691	The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are:	\frac{3}{5} < x \le 2
2,694	Which one of the following is not equivalent to $0.000000375$?	$\frac{3}{8} \times 10^{-7}$
2,698	Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?	\frac{85\pi}{8}
2,702	The symbol $\|a\|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $\|3-x\|<4$ consists of all $x$ such that:	$-1<x<7$
2,705	In $\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio	1/3
2,715	Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$	270
2,725	The area of polygon $ABCDEF$, in square units, is	46
2,730	Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?	16:1
2,732	A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?	2400
2,734	Sides $AB$, $BC$, $CD$ and $DA$ of convex quadrilateral $ABCD$ are extended past $B$, $C$, $D$ and $A$ to points $B'$, $C'$, $D'$ and $A'$, respectively. Also, $AB = BB' = 6$, $BC = CC' = 7$, $CD = DD' = 8$ and $DA = AA' = 9$. The area of $ABCD$ is $10$. The area of $A'B'C'D'$ is	114
2,737	The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$	23
2,746	Which triplet of numbers has a sum NOT equal to 1?	1.1 + (-2.1) + 1.0
2,749	A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?	54\sqrt{3}-18\pi
2,750	An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,...,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. How many cute $6$-digit integers are there?	4
2,756	What expression is never a prime number when $p$ is a prime number?	$p^2+26$
2,762	A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?	$3\sqrt{2}+\sqrt{6}$
2,781	The difference between a $6.5\%$ sales tax and a $6\%$ sales tax on an item priced at $\$20$ before tax is	$0.10
2,783	In the complex plane, let $A$ be the set of solutions to $z^{3}-8=0$ and let $B$ be the set of solutions to $z^{3}-8z^{2}-8z+64=0.$ What is the greatest distance between a point of $A$ and a point of $B?$	$2\sqrt{21}$
2,787	Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB=11$, $BC=5$, $CD=19$, and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$?	$30\sqrt{3}$
2,791	If $y=x+\frac{1}{x}$, then $x^4+x^3-4x^2+x+1=0$ becomes:	$x^2(y^2+y-6)=0$
2,792	Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?	4:1
2,793	What is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$?	$2\sqrt{6}$
2,796	Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$?	\frac{3-\sqrt{3}}{9}
2,805	A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?	18
2,812	The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning at 2 is:	27\frac{1}{2}

2,246

If $\frac{x}{y}=\frac{3}{4}$, then the incorrect expression in the following is:

$\frac{1}{4}$

0

2,261

The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\frac{1}{2}$. The first term of the progression is:

9 or 3

0

2,266

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

6

0

2,285

If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then:

$0<x<1$

0

2,292

Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the fixed monthly fee?

6.24

0

2,299

Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point at which Andy makes the $2020$th left turn?

$(-1030, -990)$

0

2,304

The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:

$x=\frac{-p}{2}$

0

2,307

The school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost?

$5.84

0

2,309

A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?

8 \frac{3}{4}

0

2,311

If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S, S'$, and $n$ is

$(S/S')^{\frac{1}{2}n}$

0

2,313

A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?

$4\sqrt{2}+4\sqrt{3}$

0

2,315

If the $whatsis$ is $so$ when the $whosis$ is $is$ and the $so$ and $so$ is $is \cdot so$, what is the $whosis \cdot whatsis$ when the $whosis$ is $so$, the $so$ and $so$ is $so \cdot so$ and the $is$ is two ($whatsis, whosis, is$ and $so$ are variables taking positive values)?

$so \text{ and } so$

0

2,318

Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily

$-a$

0

2,322

The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is

$90^\circ$

0

2,326

In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

\frac{10}{3}

0

2,333

The set of $x$-values satisfying the inequality $2 \leq |x-1| \leq 5$ is:

-4\leq x\leq-1\text{ or }3\leq x\leq 6

0

2,334

A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was

63 or 12

0

2,337

The roots of the equation $2\sqrt{x} + 2x^{-\frac{1}{2}} = 5$ can be found by solving:

4x^2-17x+4 = 0

0

2,352

A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\] [asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]

\frac{143}{14}

0

2,364

There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is

11\frac{1}{4}

0

2,386

If $a$ and $b$ are two unequal positive numbers, then:

\frac {a + b}{2} > \sqrt {ab} > \frac {2ab}{a + b}

0

2,389

A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle $\theta$ is

$3\sec ^2\theta\csc\theta$

0

2,396

The bar graph shows the results of a survey on color preferences. What percent preferred blue?

24\%

0

2,398

Sale prices at the Ajax Outlet Store are $50\%$ below original prices. On Saturdays an additional discount of $20\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\$ 180$?

$72

0

2,407

The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.) How many different routes can Paula take?

4

0

2,416

Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?

$3.50

0

2,419

Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?

3+\sqrt3

0

2,423

A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if

\frac{1}{4}

0

2,442

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost?

$20

0

2,450

In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$ The ratio $\frac{AD}{AB}$ is

\frac{1}{\sqrt[4]{12}}

0

2,452

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum?

164

0

2,453

If $x-y>x$ and $x+y<y$, then

$x<0,y<0$

0

2,455

$\triangle ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in $\angle B$ is:

25\frac{5}{7}

0

2,458

If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6},$ the middle part is:

17\frac{1}{3}

0

2,469

If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a circular patio whose radius is $12$ feet?

48

0

2,471

Jack had a bag of $128$ apples. He sold $25\%$ of them to Jill. Next he sold $25\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?

65

0

2,474

The product $(1.8)(40.3 + .07)$ is closest to

74

0

2,476

From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are

$3^{1/3},\ 2^{1/2}$

0

2,479

Circles with radii $1$, $2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?

\frac{6}{5}

0

2,482

Points $A(11, 9)$ and $B(2, -3)$ are vertices of $ riangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?

$\left( -4, 9 \right)$

0

2,490

The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to?

|t|\sqrt{1+t^2}

0

2,494

If $t = \frac{1}{1 - \sqrt[4]{2}}$, then $t$ equals

$-(1+\sqrt[4]{2})(1+\sqrt{2})$

0

2,508

The percent that $M$ is greater than $N$ is:

\frac{100(M-N)}{N}

0

2,510

If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \cdot d - b \cdot c$, what is the value of $\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$?

$2$

0

2,512

Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students. What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?

4.36

0

2,515

If the product $\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \ldots \cdot \frac{a}{b} = 9$, what is the sum of $a$ and $b$?

37

0

2,526

A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?

200

0

2,542

If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$, $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$

$f(-x)=f(x)$

0

2,546

The coordinates of $A, B$ and $C$ are $(5,5), (2,1)$ and $(0,k)$ respectively. The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible is:

2\frac{1}{7}

0

2,549

$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s speed to $B$'s speed is:

5/6

0

2,559

If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is:

$y = \frac{1}{3} x + 8$

0

2,571

In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long. Of the three equations \begin{equation*} \label{eq:1} d-s=1, \qquad ds=1, \qquad d^2-s^2=\sqrt{5} \end{equation*}those which are necessarily true are

I, II and III

0

2,576

Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA=1$. The area of the "bat wings" (shaded area) is

3 \frac{1}{2}

0

2,577

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:

2/1

0

2,582

Distinct planes $p_1, p_2, \dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?

20

0

2,585

Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$

12\sqrt2

0

2,598

Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?

\frac{85\pi}{8}

0

2,608

Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?

0

2,612

In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?

\frac{366}{31 \times 24}

0

2,616

Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$. The area of the shaded "smile" $AEFBDA$, is

$2\pi-\pi \sqrt{2}-1$

0

2,618

Students from three middle schools worked on a summer project. Seven students from Allen school worked for 3 days. Four students from Balboa school worked for 5 days. Five students from Carver school worked for 9 days. The total amount paid for the students' work was 744. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?

180.00

0

2,624

A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?

4:40 a.m.

0

2,625

The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is

20

0

2,634

Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?

\frac{147}{1024}

0

2,639

Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is:

$x >\frac{10}{3}$

0

2,640

An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?

134

0

2,659

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?

(13,-22)

0

2,666

ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between

7 and 8

0

2,668

When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:

1: 2

0

2,673

A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?

$\frac{15}{8}$

0

2,674

The formula expressing the relationship between $x$ and $y$ in the table is: \begin{tabular}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6\ \hline y & 0 & 2 & 6 & 12 & 20\ \hline \end{tabular}

$y = x^{2}-3x+2$

0

2,676

When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school?

18 minutes

0

2,681

Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?

17

0

2,684

Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$. If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:

$a>9, b>6$

0

2,689

If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is

$5$

0

2,691

The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are:

\frac{3}{5} < x \le 2

0

2,694

Which one of the following is not equivalent to $0.000000375$?

$\frac{3}{8} \times 10^{-7}$

0

2,698

Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$?

\frac{85\pi}{8}

0

2,702

The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that:

$-1<x<7$

0

2,705

In $\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio

1/3

0

2,715

Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$

270

0

2,725

The area of polygon $ABCDEF$, in square units, is

46

0

2,730

Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?

16:1

0

2,732

A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

2400

0

2,734

Sides $AB$, $BC$, $CD$ and $DA$ of convex quadrilateral $ABCD$ are extended past $B$, $C$, $D$ and $A$ to points $B'$, $C'$, $D'$ and $A'$, respectively. Also, $AB = BB' = 6$, $BC = CC' = 7$, $CD = DD' = 8$ and $DA = AA' = 9$. The area of $ABCD$ is $10$. The area of $A'B'C'D'$ is

114

0

2,737

The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$

23

0

2,746

Which triplet of numbers has a sum NOT equal to 1?

1.1 + (-2.1) + 1.0

0

2,749

A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

54\sqrt{3}-18\pi

0

2,750

An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,...,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. How many cute $6$-digit integers are there?

4

0

2,756

What expression is never a prime number when $p$ is a prime number?

$p^2+26$

0

2,762

A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?

$3\sqrt{2}+\sqrt{6}$

0

2,781

The difference between a $6.5\%$ sales tax and a $6\%$ sales tax on an item priced at $\$20$ before tax is

$0.10

0

2,783

In the complex plane, let $A$ be the set of solutions to $z^{3}-8=0$ and let $B$ be the set of solutions to $z^{3}-8z^{2}-8z+64=0.$ What is the greatest distance between a point of $A$ and a point of $B?$

$2\sqrt{21}$

0

2,787

Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB=11$, $BC=5$, $CD=19$, and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$?

$30\sqrt{3}$

0

2,791

If $y=x+\frac{1}{x}$, then $x^4+x^3-4x^2+x+1=0$ becomes:

$x^2(y^2+y-6)=0$

0

2,792

Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?

4:1

0

2,793

What is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$?

$2\sqrt{6}$

0

2,796

Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$?

\frac{3-\sqrt{3}}{9}

0

2,805

A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?

18

0

2,812

The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning at 2 is:

27\frac{1}{2}

0