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2,800 | A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? | 15 | 92.1875 |
2,801 | Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. What is the probability that $BD > 5\sqrt2$? | \frac{3-\sqrt3}{3} | 3.125 |
2,802 | The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined? | 2010 | 37.5 |
2,803 | Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes $5$ hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)? | 10 | 63.28125 |
2,804 | A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is: | 5 | 81.25 |
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 |
2,806 | Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$? | 15 | 14.84375 |
2,807 | Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$ | 208 | 13.28125 |
2,808 | What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?
[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy] | 6 | 14.0625 |
2,809 | What is the sum of the distinct prime integer divisors of $2016$? | 12 | 68.75 |
2,810 | Let $F=\log\dfrac{1+x}{1-x}$. Find a new function $G$ by replacing each $x$ in $F$ by $\dfrac{3x+x^3}{1+3x^2}$, and simplify.
The simplified expression $G$ is equal to: | 3F | 60.15625 |
2,811 | Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number? | \frac{37}{56} | 15.625 |
2,812 | The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning at 2 is: | 27\frac{1}{2} | 0 |
2,813 | The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals | 6 | 69.53125 |
2,814 | Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color? | \frac{253}{256} | 2.34375 |
2,815 | Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. The area of the square $ABCD$ is | 64 | 17.1875 |
2,816 | Two candles of the same height are lighted at the same time. The first is consumed in $4$ hours and the second in $3$ hours.
Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second? | 2\frac{2}{5} | 0 |
2,817 | A $25$ foot ladder is placed against a vertical wall of a building. The foot of the ladder is $7$ feet from the base of the building. If the top of the ladder slips $4$ feet, then the foot of the ladder will slide: | 8 | 89.0625 |
2,818 | A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is | 32 | 41.40625 |
2,819 | The lines $x=\frac{1}{4}y+a$ and $y=\frac{1}{4}x+b$ intersect at the point $(1,2)$. What is $a+b$? | \frac{9}{4} | 85.9375 |
2,820 | In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$? | 130 | 0.78125 |
2,821 | Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$? | 165 | 32.03125 |
2,822 | In $\triangle ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\triangle ABC$? | 420 | 80.46875 |
2,823 | The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and $x$ are all equal. What is the value of $x$? | 11 | 62.5 |
2,824 | Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? | 132 | 64.0625 |
2,825 | If a number eight times as large as $x$ is increased by two, then one fourth of the result equals | 2x + \frac{1}{2} | 18.75 |
2,826 | Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: | 79 | 0 |
2,827 | Find the smallest whole number that is larger than the sum
\[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\] | 16 | 96.09375 |
2,828 | For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is: | -20.5 | 0 |
2,829 | The value of $x + x(x^x)$ when $x = 2$ is: | 10 | 99.21875 |
2,830 | A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | $3 \pi \sqrt{7}$ | 0 |
2,831 | A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have? | 9 | 57.8125 |
2,832 | Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat? | 27 | 35.9375 |
2,833 | If $P(x)$ denotes a polynomial of degree $n$ such that $P(k)=\frac{k}{k+1}$ for $k=0,1,2,\ldots,n$, determine $P(n+1)$. | $\frac{n+1}{n+2}$ | 0 |
2,834 | In a narrow alley of width $w$ a ladder of length $a$ is placed with its foot at point $P$ between the walls.
Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground.
Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ is equal to | $h$ | 0 |
2,835 | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted? | \frac{145}{147} | 3.125 |
2,836 | Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\$90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price of the computer, in dollars? | 750 | 50 |
2,837 | How many of the following are equal to $x^x+x^x$ for all $x>0$?
$\textbf{I:}$ $2x^x$ $\qquad\textbf{II:}$ $x^{2x}$ $\qquad\textbf{III:}$ $(2x)^x$ $\qquad\textbf{IV:}$ $(2x)^{2x}$ | 1 | 97.65625 |
2,838 | The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$? | 54 | 2.34375 |
2,839 | 100 \times 19.98 \times 1.998 \times 1000= | (1998)^2 | 0 |
2,840 | A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is: | 3r^2 | 73.4375 |
2,841 | Let $S$ be the set of the $2005$ smallest positive multiples of $4$, and let $T$ be the set of the $2005$ smallest positive multiples of $6$. How many elements are common to $S$ and $T$? | 668 | 93.75 |
2,842 | The product, $\log_a b \cdot \log_b a$ is equal to: | 1 | 97.65625 |
2,843 | In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. What is the percent growth of the town's population during this twenty-year period? | 62 | 61.71875 |
2,844 | The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? | 10 | 71.875 |
2,845 | A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square? | \frac{5}{8} | 32.8125 |
2,846 | A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? | 45 | 2.34375 |
2,847 | At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$? | 2 | 88.28125 |
2,848 | A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth? | 0.4 | 48.4375 |
2,849 | Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had correct answers to $90\%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers? | 93 | 80.46875 |
2,850 | Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? | 36 | 0 |
2,851 | Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say? | 365 | 16.40625 |
2,852 | $X, Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets
$X, Y, Z, X \cup Y, X \cup Z$ and $Y \cup Z$ are $37, 23, 41, 29, 39.5$ and $33$ respectively.
Find the average age of the people in set $X \cup Y \cup Z$. | 34 | 42.96875 |
2,853 | If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\overarc {AC}$ and $\overarc{BC}$, and to $\overline{AB}$. If the length of $\overarc{BC}$ is $12$, then the circumference of the circle is | 27 | 0 |
2,854 | What is the maximum number of possible points of intersection of a circle and a triangle? | 6 | 99.21875 |
2,855 | For how many integers $x$ does a triangle with side lengths $10, 24$ and $x$ have all its angles acute? | 4 | 54.6875 |
2,856 | In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers will expect in the fall of 1999 is | 2700 | 86.71875 |
2,857 | If $8^x = 32$, then $x$ equals: | \frac{5}{3} | 100 |
2,858 | The expression
\[\frac{P+Q}{P-Q}-\frac{P-Q}{P+Q}\]
where $P=x+y$ and $Q=x-y$, is equivalent to: | \frac{x^2-y^2}{xy} | 69.53125 |
2,859 | A contractor estimated that one of his two bricklayers would take $9$ hours to build a certain wall and the other $10$ hours.
However, he knew from experience that when they worked together, their combined output fell by $10$ bricks per hour.
Being in a hurry, he put both men on the job and found that it took exactly 5 hours to build the wall. The number of bricks in the wall was | 900 | 96.875 |
2,860 | A fair coin is tossed 3 times. What is the probability of at least two consecutive heads? | \frac{1}{2} | 1.5625 |
2,861 | For $x \ge 0$ the smallest value of $\frac {4x^2 + 8x + 13}{6(1 + x)}$ is: | 2 | 82.03125 |
2,862 | If $A$ and $B$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is
$\begin{array}{cccc} 9 & 8 & 7 & 6 \\ & A & 3 & 2 \\ & B & 1 \\ \hline \end{array}$ | 5 | 97.65625 |
2,863 | How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four? | 76 | 73.4375 |
2,864 | What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?$ | 167 | 92.1875 |
2,865 | A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16.$ The set of all possible values of $r$ is an interval $[a,b).$ What is $a+b?$ | \sqrt{15}+8 | 0.78125 |
2,866 | In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$? | 7 | 6.25 |
2,867 | For real numbers $x$, let
\[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\]
where $i = \sqrt{-1}$. For how many values of $x$ with $0\leq x<2\pi$ does
\[P(x)=0?\] | 0 | 8.59375 |
2,868 | If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio of $y:x$ is: | 23:17 | 0 |
2,869 | Two boys $A$ and $B$ start at the same time to ride from Port Jervis to Poughkeepsie, $60$ miles away. $A$ travels $4$ miles an hour slower than $B$. $B$ reaches Poughkeepsie and at once turns back meeting $A$ $12$ miles from Poughkeepsie. The rate of $A$ was: | 8 | 82.8125 |
2,870 | Suppose that $\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\frac{1}{2}$ of $5$ bananas? | 3 | 72.65625 |
2,871 | A value of $x$ satisfying the equation $x^2 + b^2 = (a - x)^2$ is: | \frac{a^2 - b^2}{2a} | 95.3125 |
2,872 | All the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a 3x3 array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18. What is the number in the center? | 7 | 56.25 |
2,873 | In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$? | 13 | 50.78125 |
2,874 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? | 500 | 12.5 |
2,875 | Chloe chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0, 4034]$. What is the probability that Laurent's number is greater than Chloe's number? | \frac{3}{4} | 38.28125 |
2,876 | Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
[asy]
draw(circle((0,0),3));
draw(circle((0,0),1));
draw(circle((1,sqrt(3)),1));
draw(circle((-1,sqrt(3)),1));
draw(circle((-1,-sqrt(3)),1));
draw(circle((1,-sqrt(3)),1));
draw(circle((2,0),1));
draw(circle((-2,0),1));
[/asy] | \sqrt{2} | 78.125 |
2,877 | What is the least possible value of
\[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number? | 2018 | 96.875 |
2,878 | If the sum of two numbers is $1$ and their product is $1$, then the sum of their cubes is: | -2 | 96.09375 |
2,879 | A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test? | 93 | 100 |
2,880 | If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is: | \frac{3}{5} | 45.3125 |
2,881 | In triangle $ABC$ we have $AB = 25$, $BC = 39$, and $AC=42$. Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$. What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$? | \frac{19}{56} | 62.5 |
2,882 | Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible? | 300 | 52.34375 |
2,883 | The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers? | 4 | 92.96875 |
2,884 | On hypotenuse $AB$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $BC=1$, $AC=b$, and $AD=2$, then $BD$ equals: | \sqrt{b^2-3} | 93.75 |
2,885 | Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey? | 11 | 74.21875 |
2,886 | Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students? | 76 | 79.6875 |
2,887 | Let \(z=\frac{1+i}{\sqrt{2}}.\)What is \(\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right)?\) | 36 | 44.53125 |
2,888 | A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$? | \frac{37}{35} | 12.5 |
2,889 | Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$? | 1 \frac{1}{5} | 0 |
2,890 | In $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $2$, $3$, and $6$, respectively. The length of the shortest side of $\triangle ABC$ is | 2\sqrt{10} | 0 |
2,891 | Two parabolas have equations $y= x^2 + ax +b$ and $y= x^2 + cx +d$, where $a, b, c,$ and $d$ are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common? | \frac{31}{36} | 86.71875 |
2,892 | What is the greatest number of consecutive integers whose sum is $45?$ | 90 | 53.125 |
2,893 | Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$ | 1 + \left\lceil \frac{n}{2} \right\rceil | 0 |
2,894 | In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$. | 1 | 71.875 |
2,895 | Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there? | 2(p!)^2 | 0 |
2,896 | There are $2022$ equally spaced points on a circular track $\gamma$ of circumference $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.
Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labellings of the $2022$ points.
[i]Kevin Cong[/i] | 2042222 | 0 |
2,897 | For a pair $ A \equal{} (x_1, y_1)$ and $ B \equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \equal{} |x_1 \minus{} x_2| \plus{} |y_1 \minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \leq 2$. Determine the maximum number of harmonic pairs among 100 points in the plane. | 3750 | 0 |
2,898 | Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array? | 561 | 0 |
2,899 | Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends. | 6 | 70.3125 |
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