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1,800 | The reciprocal of $\left( \frac{1}{2}+\frac{1}{3}\right)$ is | \frac{6}{5} | 82.8125 |
1,801 | Rectangles $R_1$ and $R_2,$ and squares $S_1, S_2,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units? | 651 | 14.84375 |
1,802 | John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by $50\%$. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was: | 1:4 | 0 |
1,803 | Ryan got $80\%$ of the problems correct on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? | 84 | 85.9375 |
1,804 | What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$? | 1010 | 38.28125 |
1,805 | The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97,$ then the shorter base is: | 91 | 85.15625 |
1,806 | What value of $x$ satisfies
\[x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?\] | \frac{5}{6} | 83.59375 |
1,807 | If the sum of the first $10$ terms and the sum of the first $100$ terms of a given arithmetic progression are $100$ and $10$, respectively, then the sum of first $110$ terms is: | -110 | 97.65625 |
1,808 | Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$? | 7 | 50 |
1,809 | Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor$? | \frac{1}{3} | 29.6875 |
1,810 | The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle? | 150 | 48.4375 |
1,811 | 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 |
1,812 | Of the $500$ balls in a large bag, $80\%$ are red and the rest are blue. How many of the red balls must be removed so that $75\%$ of the remaining balls are red? | 100 | 99.21875 |
1,813 | The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^{2}$, $8\text{ in}^{2}$, and $6\text{ in}^{2}$ respectively is: | $24\text{ in}^{3}$ | 0 |
1,814 | What is the sum of the digits of the square of $111111111$? | 81 | 76.5625 |
1,815 | A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black? | \frac{49}{512} | 46.875 |
1,816 | The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? | 4 | 4.6875 |
1,817 | A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? | 4 | 84.375 |
1,818 | The base of isosceles $\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides? | 13 | 94.53125 |
1,819 | An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint? | 33 | 9.375 |
1,820 | What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal? | 26 | 65.625 |
1,821 | The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total? | $0 | 0 |
1,822 | Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race? | P and S | 0 |
1,823 | A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is | 3 | 67.96875 |
1,824 | A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its own division? | 48 | 85.15625 |
1,825 | If $\theta$ is an acute angle, and $\sin 2\theta=a$, then $\sin\theta+\cos\theta$ equals | $\sqrt{a+1}$ | 0 |
1,826 | 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? | 12.5 | 96.09375 |
1,827 | If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are $\frac{a + b + c}{d^2}$, $\frac{a + b + d}{c^2}$, $\frac{a + c + d}{b^2}$, $\frac{b + c + d}{a^2}$ is | 3x^4 - bx^3 - 1 = 0 | 60.9375 |
1,828 | What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? | 48 | 23.4375 |
1,829 | A two-digit positive integer is said to be $cuddly$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? | 1 | 78.125 |
1,830 | The difference of the roots of $x^2-7x-9=0$ is: | \sqrt{85} | 93.75 |
1,831 | A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$. What is $\log_{a}{b}$? | \pi | 31.25 |
1,832 | While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? | 8 | 87.5 |
1,833 | Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride? | 8 | 78.90625 |
1,834 | A barn with a roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is: | 490 | 2.34375 |
1,835 | If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $a/b$, to the nearest integer, is: | 14 | 87.5 |
1,836 | A six digit number (base 10) is squarish if it satisfies the following conditions:
(i) none of its digits are zero;
(ii) it is a perfect square; and
(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish numbers are there? | 2 | 100 |
1,837 | All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have? | 4 | 67.1875 |
1,838 | Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?\] | \frac{1}{4} | 25 |
1,839 | Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is | 171 | 88.28125 |
1,840 | All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? | -88 | 17.96875 |
1,841 | A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac{1}{3}$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac{1}{4}$. How many cards were in the deck originally? | 12 | 76.5625 |
1,842 | A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet? | 4000 | 44.53125 |
1,843 | How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\] | 6 | 84.375 |
1,844 | A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$? | 8078 | 79.6875 |
1,845 | Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches.
Then the area of circle $II$, in square inches, is: | 16 | 95.3125 |
1,846 | If a whole number $n$ is not prime, then the whole number $n-2$ is not prime. A value of $n$ which shows this statement to be false is | 9 | 67.1875 |
1,847 | In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? | 65\pi | 0 |
1,848 | Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is: | $32\sqrt{3}+21\frac{1}{3}\pi$ | 0 |
1,849 | Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? | 22.5 | 10.15625 |
1,850 | In $\triangle ABC$, $AB=5$, $BC=7$, $AC=9$, and $D$ is on $\overline{AC}$ with $BD=5$. Find the ratio of $AD:DC$. | 19/8 | 7.8125 |
1,851 | Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$? | 9000 | 3.90625 |
1,852 | A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ | 20 | 52.34375 |
1,853 | In the country of East Westmore, statisticians estimate there is a baby born every $8$ hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year? | 700 | 80.46875 |
1,854 | An integer $N$ is selected at random in the range $1 \leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$? | \frac{4}{5} | 75 |
1,855 | The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: | 108 | 81.25 |
1,856 | Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. | \pi | 20.3125 |
1,857 | Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? | 40 | 72.65625 |
1,858 | Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal? | 21 | 99.21875 |
1,859 | Equilateral $\triangle ABC$ has side length $2$, $M$ is the midpoint of $\overline{AC}$, and $C$ is the midpoint of $\overline{BD}$. What is the area of $\triangle CDM$? | \frac{\sqrt{3}}{2} | 85.9375 |
1,860 | Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $\frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: | \frac{3}{4} | 96.875 |
1,861 | The logarithm of $27\sqrt[4]{9}\sqrt[3]{9}$ to the base $3$ is: | $4\frac{1}{6}$ | 0 |
1,862 | The graphs of $y=\log_3 x$, $y=\log_x 3$, $y=\log_{\frac{1}{3}} x$, and $y=\log_x \dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs? | 3 | 20.3125 |
1,863 | When you simplify $\left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$, the result is: | a^4 | 83.59375 |
1,864 | Twenty percent less than 60 is one-third more than what number? | 36 | 99.21875 |
1,865 | The sum of two prime numbers is $85$. What is the product of these two prime numbers? | 166 | 82.8125 |
1,866 | Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$? | 1/5 | 46.875 |
1,867 | A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards? | 2 | 77.34375 |
1,868 | In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$? | 2\sqrt{31} | 95.3125 |
1,869 | Consider the statements:
(1) p and q are both true
(2) p is true and q is false
(3) p is false and q is true
(4) p is false and q is false.
How many of these imply the negative of the statement "p and q are both true?" | 3 | 95.3125 |
1,870 | Find the area of the shaded region. | 6\dfrac{1}{2} | 0 |
1,871 | $P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals: | $3\sqrt{2}$ | 0 |
1,872 | In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle? | 306 | 96.09375 |
1,873 | Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? | 25 | 63.28125 |
1,874 | There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is | 6 | 92.1875 |
1,875 | Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year? | 200 | 60.9375 |
1,876 | The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more? | 30 \% | 0 |
1,877 | What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers? | -5 | 20.3125 |
1,878 | Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$, what is the probability that the ball numbered $2$ was drawn all three times? | \frac{1}{7} | 86.71875 |
1,879 | In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$? | 2 | 75.78125 |
1,880 | If $x,y>0$, $\log_y(x)+\log_x(y)=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$ | 13\sqrt{3} | 71.09375 |
1,881 | For what value of $x$ does $10^{x} \cdot 100^{2x}=1000^{5}$? | 3 | 82.03125 |
1,882 | Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk? | 2.8 | 89.0625 |
1,883 | Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time? | 14 | 24.21875 |
1,884 | The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
$\begin{array}{c@{}c@{}c@{}c@{}c}
& & 6 & 4 & 1 \\
& & 8 & 5 & 2 \\
& + & 9 & 7 & 3 \\
\hline
& 2 & 4 & 5 & 6
\end{array}$ | 7 | 20.3125 |
1,885 | A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is | .08 | 0 |
1,886 | The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$? | 37 | 53.125 |
1,887 | In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? | 32 | 91.40625 |
1,888 | In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$? | \frac{8}{15}\sqrt{5} | 0 |
1,889 | There is a positive integer $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$? | 10 | 89.0625 |
1,890 | The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations:
\[x=x^2+y^2\] \[y=2xy\]
is | 4 | 92.1875 |
1,891 | Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:
i. The actual attendance in Atlanta is within $10 \%$ of Anita's estimate.
ii. Bob's estimate is within $10 \%$ of the actual attendance in Boston.
To the nearest 1,000, the largest possible difference between the numbers attending the two games is | 22000 | 39.0625 |
1,892 | A set of $25$ square blocks is arranged into a $5 \times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column? | 600 | 92.1875 |
1,893 | Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\$1A2. What is the missing digit $A$ of this $3$-digit number? | 3 | 85.9375 |
1,894 | Given the series $2+1+\frac {1}{2}+\frac {1}{4}+\cdots$ and the following five statements:
(1) the sum increases without limit
(2) the sum decreases without limit
(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small
(4) the difference between the sum and 4 can be made less than any positive quantity no matter how small
(5) the sum approaches a limit
Of these statments, the correct ones are: | 4 and 5 | 0 |
1,895 | What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$? | 64 | 92.96875 |
1,896 | Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana? | \frac{5}{3} | 75.78125 |
1,897 | All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? | -88 | 14.84375 |
1,898 | Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$? | 19 | 59.375 |
1,899 | Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC, \triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? | 12+10\sqrt{3} | 0 |
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