problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
The graph of $x^{4}=x^{2} y^{2}$ is a union of $n$ different lines. What is the value of $n$? | 3 |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(200).$ | 48 |
In the plane rectangular coordinate system $xOy$, the parameter equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar ... | \frac{{5\sqrt{2}}}{{11}} |
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
| 1972 |
Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$ | 100 |
Catherine had an equal number of pencils and pens. If she had 60 pens and gave eight pens and 6 pencils to each of her seven friends and kept the rest for herself, how many pens and pencils did she have left? | Catherine gave eight pens to each of her seven friends, a total of 8*7 = <<8*7=56>>56 pens.
She remained with 60-56 = <<60-56=4>>4 pens.
When she also gave 6 pencils to each of her seven friends, she gave away 6*7 = <<6*7=42>>42 pencils.
Since she had the same number of pencils as pens, she remained with 60-42 = <<60-4... |
Josh has $300 in his wallet, and $2000 invested in a business. If the business's stock price rises 30% and then he sells all of his stocks, how much money will he have in his wallet? | If the business's stock price rises 30%, then his share on the company is worth 30% multiplied by $2000 plus $2000 that he already had. That results in 30% * $2000 + $2000 = $600 + $2000 = $<<30*.01*2000+2000=2600>>2600.
Since he already had $300, with $2600 more he will have $300 + $2600 = $<<300+2600=2900>>2900.
####... |
Rectangles \( A B C D, D E F G, C E I H \) have equal areas and integer sides. Find \( D G \) if \( B C = 19 \). | 380 |
What is the value of $\sqrt{15 - 6\sqrt{6}} + \sqrt{15 + 6\sqrt{6}}$? | 6 |
Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals? | 27.55 |
What is the 10th term of an arithmetic sequence of 20 terms with the first term being 7 and the last term being 67? | \frac{673}{19} |
Calculate $0.25 \cdot 0.08$. | 0.02 |
Inside a square with side length 12, two congruent equilateral triangles are drawn such that each has one vertex touching two adjacent vertices of the square and they share one side. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles? | 12 - 4\sqrt{3} |
If $$1+6+11+16+21+26+\cdots+91+96+101 \equiv n\pmod{15},$$where $0\le n<15$, what is the value of $n$? | 6 |
Given an ellipse $C$ with one focus at $F_{1}(2,0)$ and the corresponding directrix $x=8$, and eccentricity $e=\frac{1}{2}$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Find the length of the chord cut from the ellipse $C$ by a line passing through the other focus and having a slope of $45^{\circ}$. | \frac{48}{7} |
A gym charges its members $18 twice a month. If it has 300 members how much does it make a month? | Each person is charged 18*2=$<<18*2=36>>36 per month
So they get 36*300=$<<36*300=10800>>10,800 per month
#### 10800 |
In a class, there are 30 students: honor students, average students, and poor students. Honor students always answer questions correctly, poor students always answer incorrectly, and average students alternate between correct and incorrect answers in a strict sequence. Each student was asked three questions: "Are you a... | 10 |
Malik rushed for 18 yards in each of 4 games. Josiah ran for 22 yards in each of 4 games. Darnell had an average of 11 yards rushed for each of the 4 games. How many yards did the 3 athletes run in total? | Malik = 18 * 4 = <<18*4=72>>72 yards
Josiah = 22 * 4 = <<22*4=88>>88
Darnell = 11 * 4 = <<11*4=44>>44 yards
Total yards = 72 + 88 + 44 = <<72+88+44=204>>204 yards
The 3 athletes ran 204 yards.
#### 204 |
Let $s(n)$ be the number of 1's in the binary representation of $n$ . Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$ .
*Author:Anderson Wang* | 1458 |
The difference between two perfect squares is 133. What is the smallest possible sum of the two perfect squares? | 205 |
If $m+\frac{1}{m}=8$, then what is the value of $m^{2}+\frac{1}{m^{2}}+4$? | 66 |
What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,6,Ticks(1.0,NoZero));
yaxis(0,6,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(3,3)--cycle);
fill((5,0)--(6,0)--(6,6)--(0,... | 61.11\% |
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AB}$, $\overline{PA}\perp\overline{AD}$, and $PA = 6$, then what is the volume of $PABCD$? | 64 |
Given a geometric progression $\{a_n\}$ with the first term $a_1=2$ and the sum of the first $n$ terms as $S_n$, and the equation $S_5 + 4S_3 = 5S_4$ holds, find the maximum term of the sequence $\left\{ \frac{2\log_{2}a_n + 1}{\log_{2}a_n - 6} \right\}$. | 15 |
Add 78.621 to 34.0568 and round to the nearest thousandth. | 112.678 |
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 shortes... | \frac{145}{147} |
Cheryl placed 300 strawberries into 5 buckets. After she did that she decided to take 20 out of each bucket so they wouldn't get smashed. How many strawberries were left in each bucket? | Every bucket originally had 300/5= <<300/5=60>>60 strawberries
After she removed 20 strawberries there were 60-20=<<60-20=40>>40 strawberries in each bucket.
#### 40 |
What is the largest possible value for $n$ if the average of the two positive integers $m$ and $n$ is 5? | 9 |
Given the number 105,000, express it in scientific notation. | 1.05\times 10^{5} |
The secret object is a rectangle measuring $200 \times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along the outer perimeter (th... | 150 |
Mike wants to be the best goalkeeper on his soccer team. He practices for 3 hours every weekday, on Saturdays he practices for 5 hours, and he takes Sundays off. How many hours will he practice from now until the next game, if his team has a game in 3 weeks? | Mike practices on weekdays 3 hours/day x 5 days = <<3*5=15>>15 hours in total.
Then each week he will practice 15 hours + 5 hours = <<15+5=20>>20 hours in total.
He will practice before the game 20 hours/week x 3 weeks = <<20*3=60>>60 hours in total.
#### 60 |
What is the value of \( \sqrt{16 \times \sqrt{16}} \)? | 2^3 |
Find the maximum value of
\[\frac{x + 2y + 3}{\sqrt{x^2 + y^2 + 1}}\]over all real numbers $x$ and $y.$ | \sqrt{14} |
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1\!:\!00$ PM and finishes the second task at $2\!:\!40$ PM. When does she finish the third task? | 3:30 PM |
The polynomial $2x^3 + bx + 7$ has a factor of the form $x^2 + px + 1.$ Find $b.$ | -\frac{45}{2} |
How many numbers are in the list $$ 1, 4, 7, \ldots, 2005, 2008 ?$$ | 670 |
Select 5 volunteers from 8 candidates, including A and B, to participate in community service activities from Monday to Friday, with one person arranged for each day, and each person participating only once. If at least one of A and B must participate, and when both A and B participate, their service dates cannot be ad... | 5040 |
Valerie’s cookie recipe makes 16 dozen cookies and calls for 4 pounds of butter. She only wants to make 4 dozen cookies for the weekend. How many pounds of butter will she need? | Her original recipe makes 16 dozen and she only needs 4 dozen so she needs to reduce the recipe by 16/4 = <<16/4=4>>4
For 4 dozen cookies, she needs to reduce her recipe by 4 and the original called for 4 pounds of butter so she now needs 4/4 = 1 pound of butter
#### 1 |
Let the function \( f(x) = \left| \log_{2} x \right| \). Real numbers \( a \) and \( b \) (where \( a < b \)) satisfy the following conditions:
\[
\begin{array}{l}
f(a+1) = f(b+2), \\
f(10a + 6b + 22) = 4
\end{array}
\]
Find \( ab \). | 2/15 |
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? | 4 |
Jen and Tyler are gymnasts practicing flips. Jen is practicing the triple-flip while Tyler is practicing the double-flip. Jen did sixteen triple-flips during practice. Tyler flipped in the air half the number of times Jen did. How many double-flips did Tyler do? | Jen did 16 triple-flips, so she did 16 * 3 = <<16*3=48>>48 flips.
Tyler did half the number of flips, so he did 48 / 2 = <<48/2=24>>24 flips.
A double flip has two flips, so Tyler did 24 / 2 = <<24/2=12>>12 double-flips.
#### 12 |
The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term. | 1 |
Hamza has several empty buckets of different sizes, holding either 3, 5, or 6 liters. She fills the 5-liter bucket and pours as much as she can into the 3-liter bucket. Then, she pours the remainder into the 6-liter bucket. How much more water, in liters, can she put into the 6-liter bucket, without overflowing? | Pouring the 5-liter bucket into the 3-liter bucket leaves 5-3=2 liters, which go into the 6-liter bucket.
Then there are 6-2=<<6-2=4>>4 liters that can be added.
#### 4 |
A bin has 8 black balls and 7 white balls. 3 of the balls are drawn at random. What is the probability of drawing 2 of one color and 1 of the other color? | \frac{4}{5} |
Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$ | \frac{1}{9} |
Add $175_{9} + 714_{9} + 61_9$. Express your answer in base $9$. | 1061_{9} |
A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green aft... | 154 |
In each part of this problem, cups are arranged in a circle and numbered \(1, 2, 3, \ldots\). A ball is placed in cup 1. Then, moving clockwise around the circle, a ball is placed in every \(n\)th cup. The process ends when cup 1 contains two balls.
(a) There are 12 cups in the circle and a ball is placed in every 5t... | 360 |
Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | 0 |
Sam memorized six more digits of pi than Carlos memorized. Mina memorized six times as many digits of pi as Carlos memorized. If Mina memorized 24 digits of pi, how many digits did Sam memorize? | Carlos memorized 24/6=<<24/6=4>>4 digits of pi.
Sam memorized 4+6=10 digits of pi.
#### 10 |
If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
| \frac{16}{3} |
Wei decides to modify the design of his logo by using a larger square and three tangent circles instead. Each circle remains tangent to two sides of the square and to one adjacent circle. If each side of the square is now 24 inches, calculate the number of square inches that will be shaded. | 576 - 108\pi |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \] | \frac{3+\sqrt{5}}{2} |
Sarah decided to pull weeds from her garden. On Tuesday she pulled 25 weeds. The next day she pulled three times the number of weeds she did the day before. On Thursday her allergies bothered her and she could only pull up one-fifth of the weeds she pulled on the day before. Finally, on Friday it rained for half the da... | On Wednesday she pulled 25*3=<<25*3=75>>75 weeds
On Thursday she pulled 75/5=<<75/5=15>>15 weeds
On Friday she pulled 15-10=<<15-10=5>>5 weeds
In total she pulled 25+75+15+5=<<25+75+15+5=120>>120 weeds
#### 120 |
Given that $-1 - 4\sqrt{2}$ is a root of the equation \[x^3 + ax^2 + bx + 31 = 0\]and that $a$ and $b$ are rational numbers, compute $a.$ | 1 |
Ryan builds model mustang cars. A full size mustang is 240 inches long. The mid-size model that Ryan creates is 1/10th the size, and the smallest model that Ryan creates is half the size of the mid-size model. How many inches long is the smallest model mustang? | Mid-size:240/10=<<240/10=24>>24 inches
Small Model:24/2=<<24/2=12>>12 inches
#### 12 |
Solve the equations:
(1) $x(x+4)=-5(x+4)$
(2) $(x+2)^2=(2x-1)^2$ | -\frac{1}{3} |
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$, and $c$. If $P(1,2,3)=1$, compute $P(2,4,8)$. | 56 |
Find the coordinates of the point halfway between the points $(3,7)$ and $(5,1)$. | (4,4) |
Perform the calculations.
$(54+38) \times 15$
$1500-32 \times 45$
$157 \times (70 \div 35)$ | 314 |
The first four terms in an arithmetic sequence are $x + y, x - y, xy,$ and $x/y,$ in that order. What is the fifth term? | \frac{123}{40} |
What is $(-1)^1+(-1)^2+\cdots+(-1)^{2006}$ ? | 0 |
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$ | f(x) = 0 \text{ or } f(x) = x |
A local school is holding a food drive. Mark brings in 4 times as many cans as Jaydon. Jaydon brings in 5 more than twice the amount of cans that Rachel brought in. If there are 135 cans total, how many cans did Mark bring in? | Let x represent the number of cans Rachel brought in.
Jaydon:5+2x
Mark:4(5+2x)=20+8x
Total:x+5+2x+20+8x=135
11x+25=135
11x=110
x=<<10=10>>10
Mark: 20+8(10)=100 cans
#### 100 |
Joy has $30$ thin rods, one each of every integer length from $1 \text{ cm}$ through $30 \text{ cm}$. She places the rods with lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How ma... | 17 |
An equilateral triangle shares a common side with a square as shown. What is the number of degrees in $m\angle CDB$? [asy] pair A,E,C,D,B; D = dir(60); C = dir(0); E = (0,-1); B = C+E; draw(B--D--C--B--E--A--C--D--A); label("D",D,N); label("C",C,dir(0)); label("B",B,dir(0));
[/asy] | 15 |
Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______. | 9+4 \sqrt {2} |
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one. | $f(x)=0$ |
Find the minimum value of
$$
\begin{aligned}
A & =\sqrt{\left(1264-z_{1}-\cdots-z_{n}\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\
& \sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\cdots+\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\
& \sqrt{z_{1}^{2}+\left(948-x_{1}-\cdots-x_{n}\right)^{2}+\left(1185-y_{1}-\cdots-y_{n}\right)^{2}}
\end{al... | 1975 |
Let $\mathbf{v}$ and $\mathbf{w}$ be the vectors such that $\mathbf{v} \cdot \mathbf{w} = -3$ and $\|\mathbf{w}\| = 5.$ Find the magnitude of $\operatorname{proj}_{\mathbf{w}} \mathbf{v}.$ | \frac{3}{5} |
Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $32d$ in base $n$ equals $263$, and the value of the numeral $324$ in base $n$ equals the value of the numeral $11d1$ in base six. What is $n + d$? | 11 |
What is the greatest possible value of $x$ for the equation $$\left(\frac{4x-16}{3x-4}\right)^2+\left(\frac{4x-16}{3x-4}\right)=12?$$ | 2 |
Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probability that both groups will stop flipping the... | \frac{1}{702} |
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$.
| 32 |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | 37 |
What is the area, in square units, of a triangle that has sides of $4,3$ and $3$ units? Express your answer in simplest radical form. | 2\sqrt{5} |
There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \leq n < n+m \leq 30$. Find the remainder when $N$ is divided by $1000$. | 440 |
If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? | -2 |
Six small circles, each of radius 4 units, are tangent to a large circle. Each small circle is also tangent to its two neighboring small circles. Additionally, all small circles are tangent to a horizontal line that bisects the large circle. What is the diameter of the large circle in units? | 20 |
What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* | 12 |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + 3y = 2070$. Find the number of such distinct triangles whose area is a positive integer that is also a multiple of three. | 676 |
Doraemon and Nobita are playing the game "rock, paper, scissors." The rules state that the winner of each round receives two dorayakis, while the loser gets none. If there is a tie, each player receives one dorayaki. Nobita knows that Doraemon can only play "rock," but he still wants to share dorayakis with Doraemon. T... | 10 |
Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length? | 618 |
The teacher wrote a positive number $x$ on the board and asked Kolya, Petya, and Vasya to raise this number to the 3rd, 4th, and 12th powers, respectively. It turned out that Kolya's number had at least 9 digits before the decimal point, and Petya's number had no more than 11 digits before the decimal point. How many d... | 33 |
There is an urn with $N$ slips of paper numbered from 1 to $N$. Two slips are drawn at random. What is the expected value of the ratio of the smaller number to the larger number drawn? | 1/2 |
Every second, Andrea writes down a random digit uniformly chosen from the set $\{1,2,3,4\}$. She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1? | 15/44 |
If $\frac{1}{4}$ of all ninth graders are paired with $\frac{1}{3}$ of all sixth graders, what fraction of the total number of sixth and ninth graders are paired? | \frac{2}{7} |
Suppose $p$ and $q$ are inversely proportional. If $p=25$ when $q=6$, find the value of $p$ when $q=15$. | 10 |
How many strictly positive integers less than or equal to 120 are not divisible by 3, 5, or 7? | 54 |
Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$. | 13 |
Given that point $P$ is on curve $C_1: y^2 = 8x$ and point $Q$ is on curve $C_2: (x-2)^2 + y^2 = 1$. If $O$ is the coordinate origin, find the maximum value of $\frac{|OP|}{|PQ|}$. | \frac{4\sqrt{7}}{7} |
The expression $a^3-a^{-3}$ equals: | \left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right) |
Calculate $(-1)^{47} + 2^{(3^3+4^2-6^2)}$. | 127 |
Find the value of the first term in the geometric sequence $a,b,c,32,64$. | 4 |
Find $120_4\times13_4\div2_4$. Express your answer in base 4. | 1110_4 |
How many triangles exist such that the lengths of the sides are integers not greater than 10? | 125 |
Farmer Pythagoras has now expanded his field, which remains a right triangle. The lengths of the legs of this field are $5$ units and $12$ units, respectively. He leaves an unplanted rectangular area $R$ in the corner where the two legs meet at a right angle. This rectangle has dimensions such that its shorter side run... | \frac{151}{200} |
The expression $3y^2-y-24$ can be written as $(3y + a)(y + b),$ where $a$ and $b$ are integers. What is $a - b$? | 11 |
Given the function $f(x)=x^{2}-2ax+{b}^{2}$, where $a$ and $b$ are real numbers.
(1) If $a$ is taken from the set $\{0,1,2,3\}$ and $b$ is taken from the set $\{0,1,2\}$, find the probability that the equation $f(x)=0$ has two distinct real roots.
(2) If $a$ is taken from the interval $[0,2]$ and $b$ is taken from the ... | \frac{2}{3} |
If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is | \frac{b^2}{f} |
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