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300 | Let $a, a', b,$ and $b'$ be real numbers with $a$ and $a'$ nonzero. The solution to $ax+b=0$ is less than the solution to $a'x+b'=0$ if and only if | $\frac{b'}{a'}<\frac{b}{a}$ | 0 |
301 | Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$ | \frac{1}{2}\csc{\frac{1}{4}} | 0 |
302 | If $\frac{\frac{x}{4}}{2}=\frac{4}{\frac{x}{2}}$, then $x=$ | \pm 8 | 0 |
303 | How many positive even multiples of $3$ less than $2020$ are perfect squares? | 7 | 79.6875 |
304 | Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area $12$ square units? | 8 | 33.59375 |
305 | $\left(\frac{1}{4}\right)^{-\frac{1}{4}}=$ | \sqrt{2} | 90.625 |
306 | Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline? | 16 | 92.1875 |
307 | A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples.
What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? | 6 | 74.21875 |
308 | Let $x$ and $y$ be two-digit positive integers with mean $60$. What is the maximum value of the ratio $\frac{x}{y}$? | \frac{33}{7} | 55.46875 |
309 | How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$? | 1002 | 0 |
310 | Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | 2016 | 0.78125 |
311 | There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? | 78 | 6.25 |
312 | If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is | 18 | 0.78125 |
313 | A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? | 173 | 1.5625 |
314 | Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? | \( \frac{2\sqrt{3}-3}{2} \) | 0 |
315 | Three-digit powers of $2$ and $5$ are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
\[\begin{array}{lcl} \textbf{ACROSS} & & \textbf{DOWN} \\ \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n \end{array}\] | 6 | 7.03125 |
316 | The parabolas $y=ax^2 - 2$ and $y=4 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$? | 1.5 | 0 |
317 | A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? | \frac{7}{27} | 86.71875 |
318 | How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ | 50 | 9.375 |
319 | Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$.
Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF\cdot XG$? | 17 | 0.78125 |
320 | -15 + 9 \times (6 \div 3) = | 3 | 67.1875 |
321 | How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two? | 4 | 88.28125 |
322 | Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$? | 492 | 31.25 |
323 | Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? | 29 | 100 |
324 | If $x=t^{\frac{1}{t-1}}$ and $y=t^{\frac{t}{t-1}},t>0,t \ne 1$, a relation between $x$ and $y$ is: | $y^x=x^y$ | 0 |
325 | The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$ | 2 | 56.25 |
326 | A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? | 26 | 78.125 |
327 | Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\frac{A}{B}$? | \frac{15}{8} | 99.21875 |
328 | Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? | $20\%$ | 0 |
329 | If $\left(a + \frac{1}{a}\right)^2 = 3$, then $a^3 + \frac{1}{a^3}$ equals: | 0 | 96.09375 |
330 | What is the $100\text{th}$ number in the arithmetic sequence: $1,5,9,13,17,21,25,...$? | 397 | 96.09375 |
331 | The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \begin{tabular}{|c|c|c|}\hline 1 & &\\ \hline & 2 & A\\ \hline & & B\\ \hline\end{tabular} | 4 | 61.71875 |
332 | The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? | 279 | 24.21875 |
333 | Evaluate $(x^x)^{(x^x)}$ at $x = 2$. | 256 | 99.21875 |
334 | Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number? | \frac{5}{12} | 73.4375 |
335 | A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad? | 64 | 92.1875 |
336 | What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order? | \frac{5}{19} < \frac{7}{21} < \frac{9}{23} | 21.09375 |
337 | The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is | 81 | 0 |
338 | Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? | 15 | 72.65625 |
339 | At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets? | 100 | 88.28125 |
340 | A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles? | \frac{32}{3} | 0 |
341 | $\frac{1000^2}{252^2-248^2}$ equals | 500 | 90.625 |
342 | Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | 13 | 22.65625 |
343 | How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit? | 60 | 91.40625 |
344 | A fair $6$ sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number? | \frac{7}{12} | 53.90625 |
345 | If $r$ is the remainder when each of the numbers $1059$, $1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals | 15 | 94.53125 |
346 | A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$? | 2.4 | 64.84375 |
347 | Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$ | 19 | 10.9375 |
348 | Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? | 170 | 0 |
349 | Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from the side of $CD$? | \frac{1}{2} s(2-\sqrt{3}) | 0 |
350 | Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt 2$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers? | 31 \text{ and } 32 | 74.21875 |
351 | Let $1$; $4$; $\ldots$ and $9$; $16$; $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$? | 3722 | 56.25 |
352 | Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is | 16 | 22.65625 |
353 | The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term. | 1 | 71.875 |
354 | The ratio of the areas of two concentric circles is $1: 3$. If the radius of the smaller is $r$, then the difference between the radii is best approximated by: | 0.73r | 8.59375 |
355 | The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is: | \frac{(n-2)180}{n} | 0 |
356 | Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents? | 64 | 89.0625 |
357 | Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$% discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $\$6.00$, is paying for everyone. How many dollars must he pay? | 36 | 35.15625 |
358 | How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? | 432 | 66.40625 |
359 | A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | 100 | 38.28125 |
360 | One of the factors of $x^4+4$ is: | $x^2-2x+2$ | 0 |
361 | How many ordered pairs $(m,n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$? | 4 | 84.375 |
362 | Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score? | 40 | 100 |
363 | The number of positive integers less than $1000$ divisible by neither $5$ nor $7$ is: | 686 | 95.3125 |
364 | First $a$ is chosen at random from the set $\{1,2,3,\cdots,99,100\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is | \frac{3}{16} | 43.75 |
365 | The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ (on the number line) is | \frac{4}{15} | 93.75 |
366 | Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: | \frac{4}{3} | 96.875 |
367 | Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ of the remaining pages plus $18$ pages. She then realized that there were only $62$ pages left to read, which she read the next day. How many pages are in this book? | 240 | 77.34375 |
368 | Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? | 112 | 82.8125 |
369 | Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? | 3120 | 68.75 |
370 | John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered? | 9 | 6.25 |
371 | A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? | 8 | 91.40625 |
372 | A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible? | 5 | 80.46875 |
373 | On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. What is the ratio of the area of the smaller sector $COD$ to the area of the circle? | \frac{7}{24} | 50.78125 |
374 | 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
[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] | 56 | 3.125 |
375 | The base three representation of $x$ is $12112211122211112222$. The first digit (on the left) of the base nine representation of $x$ is | 5 | 57.03125 |
376 | 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. | 80 | 0 |
377 | At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place? | 441 | 87.5 |
378 | Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA}, \overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$ | 24\sqrt{5} | 5.46875 |
379 | The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit | 0 | 89.0625 |
380 | In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$,
and sides $BC$ and $ED$. Each side has length $1$. Also, $\angle FAB = \angle BCD = 60^\circ$.
The area of the figure is | \sqrt{3} | 53.125 |
381 | $\diamondsuit$ and $\Delta$ are whole numbers and $\diamondsuit \times \Delta =36$. The largest possible value of $\diamondsuit + \Delta$ is | 37 | 95.3125 |
382 | What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? | 2003 | 98.4375 |
383 | Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die? | \frac{5}{24} | 32.03125 |
384 | In the base ten number system the number $526$ means $5 \times 10^2+2 \times 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. Jones purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is: | 8 | 92.1875 |
385 | A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? | 15 | 0.78125 |
386 | The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams? | 88 | 78.125 |
387 | If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then | x(y-1)=0 | 0 |
388 | If $x_{k+1} = x_k + \frac{1}{2}$ for $k=1, 2, \dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \dots + x_n$. | \frac{n^2+3n}{4} | 42.96875 |
389 | Define $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$? | h | 86.71875 |
390 | The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$?
[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0)); [/asy] | 100 | 1.5625 |
391 | Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? | 4 | 95.3125 |
392 | Indicate in which one of the following equations $y$ is neither directly nor inversely proportional to $x$: | $3x + y = 10$ | 0 |
393 | Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display | -0.25 | 0.78125 |
394 | The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? | \frac{7}{9} | 0 |
395 | $\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$ | .246 | 75.78125 |
396 | Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals | 1+\sqrt{3(r^2-1)} | 0 |
397 | For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? | 16 | 94.53125 |
398 | A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is | 97 | 84.375 |
399 | The domain of the function $f(x)=\log_{\frac{1}{2}}(\log_4(\log_{\frac{1}{4}}(\log_{16}(\log_{\frac{1}{16}}x))))$ is an interval of length $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 271 | 24.21875 |
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