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2,100 | In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is: | \frac{\sqrt{5}-1}{2} | 3.90625 |
2,101 | Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$ | 108 | 58.59375 |
2,102 | The product of all real roots of the equation $x^{\log_{10}{x}}=10$ is | 1 | 99.21875 |
2,103 | Two integers have a sum of $26$. When two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers? | 1 | 78.125 |
2,104 | Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\frac{2}{3}$ of its games and team $B$ has won $\frac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has team $A$ played? | 42 | 85.9375 |
2,105 | Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain? | 24 | 59.375 |
2,106 | In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$? | 14 | 25.78125 |
2,107 | The edges of a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$. | \frac{\sqrt{2}}{2} | 31.25 |
2,108 | For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$? | 1 | 90.625 |
2,109 | The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins? | 22 | 97.65625 |
2,110 | The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: | -12 | 50 |
2,111 | A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is | \frac{2}{x+1} | 37.5 |
2,112 | Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$? | 36 | 96.09375 |
2,113 | The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? | 991 | 54.6875 |
2,114 | The lengths of the sides of a triangle are integers, and its area is also an integer.
One side is $21$ and the perimeter is $48$. The shortest side is: | 10 | 80.46875 |
2,115 | Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less? | 4 | 67.1875 |
2,116 | What is the volume of a cube whose surface area is twice that of a cube with volume 1? | 2\sqrt{2} | 94.53125 |
2,117 | If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals | 60^{\circ} | 85.9375 |
2,118 | If $x = (\log_82)^{(\log_28)}$, then $\log_3x$ equals: | -3 | 99.21875 |
2,119 | Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is: | \frac{xy}{x-1} | 82.8125 |
2,120 | A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? | \frac{4}{9} | 4.6875 |
2,121 | In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\angle GDA$ is | 120^{\circ} | 35.15625 |
2,122 | Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? | \frac{2\sqrt{3}}{3} | 0 |
2,123 | Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which
\begin{align*} \label{eq:1} \log_{10}(x+y) = z \\
\log_{10}(x^{2}+y^{2}) = z+1 \end{align*} There are real numbers $a$ and $b$ such that for all ordered triples $(x,y.z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a+b?$ | \frac{29}{2} | 92.96875 |
2,124 | Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks? | 4 | 77.34375 |
2,125 | Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? | 190 | 78.125 |
2,126 | Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded? | \frac{7}{16} | 0 |
2,127 | The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$? | \frac{8k-168}{k-12} | 30.46875 |
2,128 | How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$. | 4 | 0 |
2,129 | There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square? | \frac{2}{5} | 28.90625 |
2,130 | Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be | 11 | 0 |
2,131 | While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\%$ of his bill and Joe tipped $20\%$ of his bill. What was the difference, in dollars between their bills? | 10 | 92.1875 |
2,132 | A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three digit number having an equal chance of being chosen. The probability that $\log_2 N$ is an integer is | \frac{1}{300} | 38.28125 |
2,133 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? | 6 | 47.65625 |
2,134 | Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? | 480 | 0 |
2,135 | Consider this histogram of the scores for $81$ students taking a test:
The median is in the interval labeled | 70 | 0.78125 |
2,136 | Assume every 7-digit whole number is a possible telephone number except those that begin with $0$ or $1$. What fraction of telephone numbers begin with $9$ and end with $0$? | \frac{1}{80} | 95.3125 |
2,137 | If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that: | $|r_1+r_2|>4\sqrt{2}$ | 0 |
2,138 | What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? | 10 | 98.4375 |
2,139 | What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$? | \frac{2}{7} | 71.875 |
2,140 | The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is: | \frac{10}{81} | 88.28125 |
2,141 | The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $1996$ was | 84 | 80.46875 |
2,142 | Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square? | 29\sqrt{3} | 0 |
2,143 | Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.) | 12 | 94.53125 |
2,144 | \frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10} + \frac{5}{10} + \frac{6}{10} + \frac{7}{10} + \frac{8}{10} + \frac{9}{10} + \frac{55}{10}= | 11 | 0.78125 |
2,145 | In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to | 2x+1 | 0 |
2,146 | Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost? | $1.11 | 0 |
2,147 | What is the area of the region defined by the inequality $|3x-18|+|2y+7| \le 3$? | 3 | 45.3125 |
2,148 | In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$,
$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. Find the area of $\triangle BEC$. | 4.5 | 28.125 |
2,149 | Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? | \frac{4}{7} | 92.1875 |
2,150 | Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits? | 119 | 52.34375 |
2,151 | A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is | 19 | 31.25 |
2,152 | In the set of equations $z^x = y^{2x}$, $2^z = 2 \cdot 4^x$, $x + y + z = 16$, the integral roots in the order $x,y,z$ are: | 4,3,9 | 66.40625 |
2,153 | Label one disk "$1$", two disks "$2$", three disks "$3$", ..., fifty disks "$50$". Put these $1+2+3+ \cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is | 415 | 0.78125 |
2,154 | How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have? | 18 | 93.75 |
2,155 | The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? | 8 | 79.6875 |
2,156 | A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars? | 100 | 91.40625 |
2,157 | A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possible salary, in dollars, for a single player? | 400000 | 97.65625 |
2,158 | 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 | 56.25 |
2,159 | The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ | 9 | 76.5625 |
2,160 | Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$. | 9 | 39.84375 |
2,161 | If the margin made on an article costing $C$ dollars and selling for $S$ dollars is $M=\frac{1}{n}C$, then the margin is given by: | \frac{1}{n+1}S | 0 |
2,162 | A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is | 17 | 8.59375 |
2,163 | Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end? | 103 | 98.4375 |
2,164 | What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$? | 1120 | 95.3125 |
2,165 | In the obtuse triangle $ABC$ with $\angle C>90^\circ$, $AM=MB$, $MD\perp BC$, and $EC\perp BC$ ($D$ is on $BC$, $E$ is on $AB$, and $M$ is on $EB$). If the area of $\triangle ABC$ is $24$, then the area of $\triangle BED$ is | 12 | 6.25 |
2,166 | Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$? | 11 | 39.84375 |
2,167 | How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) | 2 | 13.28125 |
2,168 | In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was $161$. What is the correct value of the product of $a$ and $b$? | 224 | 66.40625 |
2,169 | A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall? | 353 | 4.6875 |
2,170 | Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is: | \frac{135}{2} | 3.125 |
2,171 | There are unique integers $a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$ such that
\[\frac {5}{7} = \frac {a_{2}}{2!} + \frac {a_{3}}{3!} + \frac {a_{4}}{4!} + \frac {a_{5}}{5!} + \frac {a_{6}}{6!} + \frac {a_{7}}{7!}\]where $0\leq a_{i} < i$ for $i = 2,3,\ldots,7$. Find $a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}$. | 9 | 68.75 |
2,172 | One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon ($128$ ounces) of soda? | 11 | 97.65625 |
2,173 | A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter? | 2.5 | 3.125 |
2,174 | How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? | 65 | 93.75 |
2,175 | If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals | 1 | 82.03125 |
2,176 | Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? | 12 | 25.78125 |
2,177 | 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} | 4.6875 |
2,178 | In $\triangle ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$? | 140 | 54.6875 |
2,179 | The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded, the arithmetic mean of the remaining set of numbers is: | 36.5 | 0 |
2,180 | $\left(\frac{(x+1)^{2}(x^{2}-x+1)^{2}}{(x^{3}+1)^{2}}\right)^{2}\cdot\left(\frac{(x-1)^{2}(x^{2}+x+1)^{2}}{(x^{3}-1)^{2}}\right)^{2}$ equals: | 1 | 92.1875 |
2,181 | The number of ordered pairs of integers $(m,n)$ for which $mn \ge 0$ and
$m^3 + n^3 + 99mn = 33^3$
is equal to | 35 | 59.375 |
2,182 | Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is: | 22 | 97.65625 |
2,183 | An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
(a) the selection of four red marbles;
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color.
What is the smallest number of marbles satisfying the given condition? | 21 | 44.53125 |
2,184 | If $n$ is a real number, then the simultaneous system
$nx+y = 1$
$ny+z = 1$
$x+nz = 1$
has no solution if and only if $n$ is equal to | -1 | 84.375 |
2,185 | As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles? | 6\sqrt{3} - 3\pi | 97.65625 |
2,186 | A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $\frac{a}{b}$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals: | $\sqrt{a}:\sqrt{a-b}$ | 0 |
2,187 | For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? | 6 | 94.53125 |
2,188 | Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \cdot CD = AB \cdot DA$. What is the largest possible value of $BD$? | \sqrt{\dfrac{425}{2}} | 0 |
2,189 | The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10$. What is $b - a$? | 20 | 94.53125 |
2,190 | The coefficient of $x^7$ in the expansion of $\left(\frac{x^2}{2}-\frac{2}{x}\right)^8$ is: | -14 | 78.90625 |
2,191 | 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 = | 945 | 81.25 |
2,192 | $(1+x^2)(1-x^3)$ equals | 1+x^2-x^3-x^5 | 19.53125 |
2,193 | Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: | \frac{40}{3} | 7.03125 |
2,194 | How many pairs of positive integers $(a,b)$ with $a+b\le 100$ satisfy the equation
\[\frac{a+b^{-1}}{a^{-1}+b}=13?\] | 7 | 78.90625 |
2,195 | The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$? | \frac{8a^2}{a^2+1} | 0.78125 |
2,196 | In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\%$ of the juniors and $10\%$ of the seniors are on the debate team. How many juniors are in the program? | 8 | 90.625 |
2,197 | A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth? | 13 | 69.53125 |
2,198 | A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 < x < 2$, $-2 < y < 2$ at each step? | 1698 | 14.84375 |
2,199 | Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}.$ For how many such quadruples is it true that $a \cdot d-b \cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0 \cdot 1-3 \cdot 1 = -3$ is odd.) | 96 | 92.1875 |
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