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At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many players are taking chemistry?
|
16
|
Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.
|
390
|
In a quadrilateral $ABCD$ lying in the plane, $AB=\sqrt{3}$, $AD=DC=CB=1$. The areas of triangles $ABD$ and $BCD$ are $S$ and $T$ respectively. What is the maximum value of $S^{2} + T^{2}$?
|
\frac{7}{8}
|
What is the sum of the greatest common divisor of $45$ and $4410$ and the least common multiple of $45$ and $4410$?
|
4455
|
Find the smallest possible value of $x+y$ where $x, y \geq 1$ and $x$ and $y$ are integers that satisfy $x^{2}-29y^{2}=1$
|
11621
|
What is the median of the following list of $4100$ numbers?
\[1, 2, 3, \ldots, 2050, 1^2, 2^2, 3^2, \ldots, 2050^2\]
A) $1977.5$
B) $2004.5$
C) $2005.5$
D) $2006.5$
E) $2025.5$
|
2005.5
|
John sublets his apartment to 3 people who each pay $400 per month. He rents the apartment for $900 a month. How much profit does he make in a year?
|
He gets 3*$400=$<<3*400=1200>>1200 per month
So he gets 1200*12=$<<1200*12=14400>>14,400 per year
So he pays 12*900=$<<12*900=10800>>10800 a year
That means his profit is 14400-10800=$<<14400-10800=3600>>3600 per year
#### 3600
|
Given that $2+\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 10 = 0\]and that $a$ and $b$ are rational numbers, compute $b.$
|
-39
|
There are 2 teachers, 3 male students, and 4 female students taking a photo together. How many different standing arrangements are there under the following conditions? (Show the process, and represent the final result with numbers)
(1) The male students must stand together;
(2) The female students cannot stand next to each other;
(3) If the 4 female students have different heights, they must stand from left to right in order from tallest to shortest;
(4) The teachers cannot stand at the ends, and the male students must stand in the middle.
|
1728
|
Consider a sequence of positive real numbers where \( a_1, a_2, \dots \) satisfy
\[ a_n = 9a_{n-1} - n \]
for all \( n > 1 \). Find the smallest possible value of \( a_1 \).
|
\frac{17}{64}
|
Cindy leaves school at the same time every day. If she cycles at \(20 \ \text{km/h}\), she arrives home at 4:30 in the afternoon. If she cycles at \(10 \ \text{km/h}\), she arrives home at 5:15 in the afternoon. Determine the speed, in \(\text{km/h}\), at which she must cycle to arrive home at 5:00 in the afternoon.
|
12
|
Mr. Fortchaud turns on his heater on the 1st of November, 2005. The fuel tank was then full and contained 3,000 L. On January 1, 2006, the tank counter indicated that 180 L remained. Mr. Fortchaud again filled his tank completely. On 1 May 2006, Mr. Fortchaud decided to stop heating and he read 1,238 L on the meter. What the volume of fuel oil that was used between 1 November 2005 and 1 May 2006?
|
I calculate consumption between November 1, 2005 and January 1, 2006: 3,000 – 180 = <<3000-180=2820>>2820 L
I calculate consumption between January 1, 2006 and May 1, 2006: 3,000 – 1238 = <<3000-1238=1762>>1762 L
I calculate the total consumption between November 1st and May 1st: 2,820 + 1,762 = <<2820+1762=4582>>4582 L
#### 4,582
|
In the Cartesian coordinate plane, there are four fixed points \(A(-3,0), B(1,-1), C(0,3), D(-1,3)\) and a moving point \(P\). What is the minimum value of \(|PA| + |PB| + |PC| + |PD|\)?
|
3\sqrt{2} + 2\sqrt{5}
|
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents?
|
1009
|
Define a function $A(m, n)$ by \[ A(m,n) = \left\{ \begin{aligned} &n+1& \text{ if } m = 0 \\ &A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\ &A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]Compute $A(2, 1).$
|
5
|
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
[asy]
unitsize(3mm); defaultpen(linewidth(0.8pt));
path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);
path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2,2)--(2,3);
path p5=(1,3)--(1,2)--(0,2);
path p6=(1,1)--(2,2);
path p7=(2,1)--(1,2);
path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7;
for(int i=0; i<3; ++i) {
for(int j=0; j<3; ++j) {
draw(shift(3*i,3*j)*p);
}
}
[/asy]
|
56
|
A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of groups that she needs?
|
3
|
Kenny wants to make sure he does more jumping jacks every week than he did the week before. He recorded that last week he did 324 total jumping jacks. On Saturday of this week, he looks at his records and sees that on Sunday he did 34. On Monday he did 20. On Tuesday he skipped a day. On Wednesday he did 123. On Thursday he did 64. On Friday he did 23. How many does he have to do on Saturday to make sure he beats last week's number?
|
He has already done 264 this week because 34 + 20 + 123 + 64 + 23 = <<34+20+123+64+23=264>>264.
If he does 60 he will have done as many as last week because 324 - 264 = <<60=60>>60
He has to do 61 on Saturday because 60 + 1 = <<60+1=61>>61
#### 61
|
If $\|\mathbf{v}\| = 4,$ then find $\mathbf{v} \cdot \mathbf{v}.$
|
16
|
I have 10 distinguishable socks in my drawer: 4 white, 4 brown, and 2 blue. In how many ways can I choose a pair of socks, provided that I get two socks of the same color?
|
13
|
Kevin Kangaroo begins hopping on a number line at 0. He wants to get to 1, but he can hop only $\frac{1}{3}$ of the distance. Each hop tires him out so that he continues to hop $\frac{1}{3}$ of the remaining distance. How far has he hopped after five hops? Express your answer as a common fraction.
|
\frac{211}{243}
|
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
|
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$.
|
1+3\sqrt{2}
|
Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from more than half of the other 7 balls.
|
\frac{35}{128}
|
The formula for the total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh,$ where $r$ is the radius and $h$ is the height. A particular solid right cylinder of radius 2 feet has a total surface area of $12\pi$ square feet. What is the height of this cylinder?
|
1
|
The sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.
$(1) q > 1$ $(2) T_{2013} > 1$ $(3) S_{2012}a_{2013} < S_{2013}a_{2012}$ $(4)$ The smallest natural number $n$ for which $T_n > 1$ is $4025$ $(5) \min \left( T_n \right) = T_{2012}$
|
(1)(3)(4)(5)
|
How many sequences of 6 digits $x_1, x_2, \ldots, x_6$ can we form, given the condition that no two adjacent $x_i$ have the same parity? Leading zeroes are allowed. (Parity means 'odd' or 'even'; so, for example, $x_2$ and $x_3$ cannot both be odd or both be even.)
|
31,250
|
How many numbers are in the following list: $$-4, -1, 2, 5,\ldots, 32$$
|
13
|
A line that passes through the two foci of the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola.
|
\frac{\sqrt{5} + 1}{2}
|
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$
|
-2006
|
Given that the function $f(x)$ satisfies $f(x+y)=f(x)+f(y)$ for any $x, y \in \mathbb{R}$, and $f(x) < 0$ when $x > 0$, with $f(1)=-2$.
1. Determine the parity (odd or even) of the function $f(x)$.
2. When $x \in [-3, 3]$, does the function $f(x)$ have an extreme value (maximum or minimum)? If so, find the extreme value; if not, explain why.
|
-6
|
Ruth goes to school 8 hours a day and 5 days a week. She is in math class 25% of this time. How many hours per week does she spend in math class?
|
She is in school for 40 hours because 5 x 8 = <<5*8=40>>40
She is in math class for 10 hours a week because 40 x .25 = <<40*.25=10>>10
#### 10
|
Find the coefficient of $x$ when $3(x - 4) + 4(7 - 2x^2 + 5x) - 8(2x - 1)$ is simplified.
|
7
|
Given a sample of size 66 with a frequency distribution as follows: $(11.5, 15.5]$: $2$, $(15.5, 19.5]$: $4$, $(19.5, 23.5]$: $9$, $(23.5, 27.5]$: $18$, $(27.5, 31.5]$: $11$, $(31.5, 35.5]$: $12$, $[35.5, 39.5)$: $7$, $[39.5, 43.5)$: $3$, estimate the probability that the data falls in [31.5, 43.5).
|
\frac{1}{3}
|
An ellipse with equation
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 +y^2 = 1.$ Then the smallest possible area of the ellipse can be expressed in the form $k \pi.$ Find $k.$
|
\frac{3 \sqrt{3}}{2}
|
Tim drops a ball off the roof of a 96-foot tall building. The ball bounces to half the height from which it fell on each bounce. How high will it bounce on the fifth bounce?
|
On the first bounce, it will reach a height of 96 / 2 = <<96/2=48>>48 feet.
On the second bounce, it will reach a height of 48 / 2 = <<48/2=24>>24 feet.
On the third bounce, it will reach a height of 24 / 2 = <<24/2=12>>12 feet.
On the fourth bounce, it will reach a height of 12 / 2 = <<12/2=6>>6 feet.
On the fifth bounce, it will reach a height of 6 / 2 = <<6/2=3>>3 feet.
#### 3
|
Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$
|
28
|
If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____.
|
-1
|
Given that the sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ is $S\_n$, with $a\_2 = 4$ and $S\_{10} = 110$, find the minimum value of $\frac{S\_n + 64}{a\_n}$.
|
\frac{17}{2}
|
If $3+a=4-b$ and $4+b=7+a$, what is $3-a$?
|
4
|
Josh built his little brother a rectangular sandbox. The perimeter of the sandbox is 30 feet and the length is twice the width. What is the width of the sandbox?
|
There are four sides to a rectangle, the width is W and so the length is 2W, and we can put that in this equation: W + W + 2W + 2W = 30 feet.
If we add the sides together, we get 6W = 30 feet.
To find W, we divide both sides by 6, like this: 6W / 6 = 30 feet / 6, so W = 5 feet.
#### 5
|
Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$
|
22
|
Find the phase shift of the graph of $y = \sin (3x - \pi).$
|
-\frac{\pi}{3}
|
For how many integers $n$ between 1 and 150 is the greatest common divisor of 21 and $n$ equal to 3?
|
43
|
If the inequality \( ab + b^2 + c^2 \geq \lambda(a + b)c \) holds for all positive real numbers \( a, b, c \) that satisfy \( b + c \geq a \), then the maximum value of the real number \( \lambda \) is \(\quad\) .
|
\sqrt{2} - \frac{1}{2}
|
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ are perpendicular. If $AD= 15$ and $BE = 20$, then what is the area of $\triangle ABC$?
|
200
|
Cupcakes are sold in packages of 10 and 15. Jean bought 4 packs of 15 cupcakes. If she will give one cupcake each to 100 children in the orphanage, how many packs of 10 cupcakes should she need to buy?
|
Jean already bought 15 cupcakes x 4 packs = <<15*4=60>>60 cupcakes.
She needs 100 - 60 = <<100-60=40>>40 more cupcakes.
Thus, she needs to buy 40/10 = <<40/10=4>>4 packs of 10 cupcakes.
#### 4
|
Let \(ABCDEF\) be a regular hexagon and let point \(O\) be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn't exist any equilateral triangle with vertices of all the same color?
|
6
|
In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively. Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-sqrt(2),2+sqrt(2)); G=(-sqrt(2),sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--E);
pair M=(B+C)/2; pair N=(F+G)/2;
draw(M--N);
label("$A$",A,N); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,E);
label("$E$",E,S); label("$F$",F,S); label("$G$",G,W); label("$H$",H,W);
label("$M$",M,NE); label("$N$",N,SW);
label("$O$",(1,2.4),E);
[/asy]
|
\frac{3}{5}
|
All vertices of a regular tetrahedron \( A B C D \) are located on one side of the plane \( \alpha \). It turns out that the projections of the vertices of the tetrahedron onto the plane \( \alpha \) are the vertices of a certain square. Find the value of \(A B^{2}\), given that the distances from points \( A \) and \( B \) to the plane \( \alpha \) are 17 and 21, respectively.
|
32
|
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\).
|
42
|
Marina solved the quadratic equation $9x^2-18x-720=0$ by completing the square. In the process, she came up with the equivalent equation $$(x+r)^2 = s,$$where $r$ and $s$ are constants.
What is $s$?
|
81
|
Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$ . Adam says, " $n$ leaves a remainder of $2$ when divided by $3$ ." Bendeguz says, "For some $k$ , $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$ . Then $2n - s = 20$ ." Dennis says, "For some $m$ , if I have $m$ marbles, there are $n$ ways to choose two of them." If exactly one of them is lying, what is $n$ ?
|
210
|
Find all integers $n$, not necessarily positive, for which there exist positive integers $a, b, c$ satisfying $a^{n}+b^{n}=c^{n}$.
|
\pm 1, \pm 2
|
In the diagram, $ABCD$ is a parallelogram with an area of 27. $CD$ is thrice the length of $AB$. What is the area of $\triangle ABC$?
[asy]
draw((0,0)--(2,3)--(10,3)--(8,0)--cycle);
draw((2,3)--(0,0));
label("$A$",(0,0),W);
label("$B$",(2,3),NW);
label("$C$",(10,3),NE);
label("$D$",(8,0),E);
[/asy]
|
13.5
|
Let $H$ be the orthocenter of triangle $ABC.$ For all points $P$ on the circumcircle of triangle $ABC,$
\[PA^2 + PB^2 + PC^2 - PH^2\]is a constant. Express this constant in terms of the side lengths $a,$ $b,$ $c$ and circumradius $R$ of triangle $ABC.$
|
a^2 + b^2 + c^2 - 4R^2
|
A rectangle has dimensions $8 \times 12$, and a circle centered at one of its corners has a radius of 10. Calculate the area of the union of the regions enclosed by the rectangle and the circle.
|
96 + 75\pi
|
Marcus had 18 pebbles. He skipped half of them across the lake, but Freddy gave him another 30 pebbles. How many pebbles does Marcus have now?
|
Marcus had 18/2 = <<18/2=9>>9 pebbles left.
After Freddy gave him more, he has 9 + 30 = <<9+30=39>>39 pebbles.
#### 39
|
If $\log_2(\log_2(\log_2(x)))=2$, then how many digits are in the base-ten representation for x?
|
5
|
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $c\sin A= \sqrt {3}a\cos C$.
(I) Find $C$;
(II) If $c= \sqrt {7}$ and $\sin C+\sin (B-A)=3\sin 2A$, find the area of $\triangle ABC$.
|
\frac {3 \sqrt {3}}{4}
|
In the equation $w^3+x^3+y^3=z^3$, $w^3$, $x^3$, $y^3$, and $z^3$ are distinct, consecutive positive perfect cubes listed in ascending order. What is the smallest possible value of $z$?
|
6
|
Giselle is in charge of the relay run on track and field day. Last year, the race was 300 meters. This year, it will be 4 times as long. Giselle needs to set up 6 tables for the relay run. The distance between the tables must be the same and the last table will be at the finish line. What is the distance between table 1 and table 3 in meters?
|
The relay run is 4 times the length of last year’s race, so 300 meters x 4 = <<300*4=1200>>1200 meters.
With 6 tables along the route, the distance between tables is 1200 meters / 6 tables = <<1200/6=200>>200 meters.
The distance between tables 1 and 3 is 200 meters x 2 = <<200*2=400>>400 meters.
#### 400
|
If $2^n$ divides $5^{256} - 1$ , what is the largest possible value of $n$ ? $
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 11
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ \text{None of above}
$
|
10
|
From the set $\{ -3, 0, 0, 4, 7, 8\}$, find the probability that the product of two randomly selected numbers is $0$.
|
\frac{3}{5}
|
50 people, consisting of 30 people who all know each other, and 20 people who know no one, are present at a conference. Determine the number of handshakes that occur among the individuals who don't know each other.
|
1170
|
The average of the numbers $1, 2, 3,\dots, 148, 149,$ and $x$ is $50x$. What is $x$?
|
\frac{11175}{7499}
|
Find $x$ if
\[1 + 5x + 9x^2 + 13x^3 + \dotsb = 85.\]
|
\frac{4}{5}
|
Find the maximum value of $\cos x + 2 \sin x,$ over all angles $x.$
|
\sqrt{5}
|
In the expansion of $(1+x)^3+(1+x)^4+\ldots+(1+x)^{19}$, the coefficient of the $x^2$ term is \_\_\_\_\_\_.
|
1139
|
Let $x$ and $y$ be real numbers such that
\[4x^2 + 8xy + 5y^2 = 1.\]Let $m$ and $M$ be the minimum and maximum values of $2x^2 + 3xy + 2y^2,$ respectively. Find the product $mM.$
|
\frac{7}{16}
|
Given a sequence $\{a_n\}$ where all terms are positive integers, let $S_n$ denote the sum of the first $n$ terms. If $a_{n+1}=\begin{cases} \frac{a_n}{2},a_n \text{ is even} \\\\ 3a_n+1,a_n \text{ is odd} \end{cases}$ and $a_1=5$, calculate $S_{2015}$.
|
4725
|
If $8 \cdot 2^x = 5^{y + 8}$, then when $y = -8$, $x =$
|
-3
|
In $\triangle ABC$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$.
(1) Find the measure of angle $A$;
(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, if $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of $\triangle ABC$.
|
\frac{\sqrt{3}}{6}
|
A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));
fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));
fill((7,1)--(8,1)--(8,2)--(7,2)--cycle,gray(.8));
fill((1,4)--(2,4)--(2,5)--(1,5)--cycle,gray(.8));
fill((3,3)--(6,3)--(6,6)--(3,6)--cycle,gray(.8));
fill((7,4)--(8,4)--(8,5)--(7,5)--cycle,gray(.8));
fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,gray(.8));
fill((4,7)--(5,7)--(5,8)--(4,8)--cycle,gray(.8));
fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,gray(.8));
label("T",(1.5,7),S);
label("S",(6,4.5),W);
[/asy] If the ratios $9:\text{S}$ and $\text{S}:\text{T}$ are both equal to 3 and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?
|
17
|
The user has three computer disks from companies $\mathrm{K}$, $\mathrm{L}$, and $\mathrm{M}$, one disk from each of these companies, but the company stamps on the disks are absent. Two out of the three disks are defective. What is the probability that the defective disks are from companies $\mathrm{L}$ and $\mathrm{M}$, given that the defect rates for companies $\mathrm{K}$, $\mathrm{L}$, and $\mathrm{M}$ are $10\%$, $20\%$, and $15\%$, respectively?
|
0.4821
|
A supplier is packing cartons of canned juice. Each carton has 20 cans of juice. Out of the 50 cartons that have been packed, only 40 cartons have been loaded on a truck. How many cans of juice are left to be loaded on the truck?
|
There are 50 - 40 = <<50-40=10>>10 cartons of canned juice that are left to be loaded on the truck.
This is equal to 10 x 20 = <<10*20=200>>200 cans of juice.
#### 200
|
Inside an angle of $60^{\circ}$, there is a point located at distances $\sqrt{7}$ and $2 \sqrt{7}$ from the sides of the angle. Find the distance of this point from the vertex of the angle.
|
\frac{14 \sqrt{3}}{3}
|
At the beginning of an academic year, there were 15 boys in a class and the number of girls was 20% greater. Later in the year, transfer students were admitted such that the number of girls doubled but the number of boys remained the same. How many students are in the class now?
|
The original number of girls was 20% greater than 15 (the number of boys) giving 15+(20/100)*15 = <<15+(20/100)*15=18>>18
After the admission of transfer students, the number of girls doubled to become 18*2 = <<18*2=36>>36
In addition to the 15 boys, there are now 15+36 = <<15+36=51>>51 students in the class
#### 51
|
Suppose that $n, n+1, n+2, n+3, n+4$ are five consecutive integers.
Determine a simplified expression for the sum of these five consecutive integers.
|
5n+10
|
If eight movie tickets cost 2 times as much as one football game ticket, and each movie ticket is sold at $30, calculate the total amount of money Chandler will pay if he buys eight movie tickets and five football game tickets.
|
If each movie ticket is sold at $30, the cost of buying eight movie tickets is 8*$30 = $<<8*30=240>>240
Since eight movie tickets cost 2 times as much as one football game ticket, the cost of buying one football game ticket is $240/2 = $<<240/2=120>>120
Buying five football game tickets will cost 5*$120 = $<<5*120=600>>600
The total cost of buying eight movie tickets and five football game tickets is $600+$240 = $840
#### 840
|
Each of $a, b$ and $c$ is equal to a number from the list $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}$. There are $N$ triples $(a, b, c)$ with $a \leq b \leq c$ for which each of $\frac{ab}{c}, \frac{ac}{b}$ and $\frac{bc}{a}$ is equal to an integer. What is the value of $N$?
|
86
|
If the wages of 15 workers for 6 days is $9450. What would be the wages for 19 workers for 5 days?
|
This is equivalent to one worker working for 15 * 6 = <<15*6=90>>90 days.
The pay per worker per day would be $9450 / 90 = $<<9450/90=105>>105.
19 workers working for 5 days is equivalent to one worker working for 19 * 5 = <<19*5=95>>95 days.
It would cost $105 * 95 = $<<105*95=9975>>9975 to pay the workers for their time.
#### 9975
|
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $j+k$?
|
59
|
What is the smallest positive integer with exactly 20 positive divisors?
|
240
|
Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.
|
3 + 2\sqrt{2}
|
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$.
|
f(x) = x
|
A random point \(N\) on a line has coordinates \((t, -2-t)\), where \(t \in \mathbb{R}\). A random point \(M\) on a parabola has coordinates \(\left( x, x^2 - 4x + 5 \right)\), where \(x \in \mathbb{R}\). The square of the distance between points \(M\) and \(N\) is given by \(\rho^2(x, t) = (x - t)^2 + \left( x^2 - 4x + 7 + t \right)^2\). Find the coordinates of points \(M\) and \(N\) that minimize \(\rho^2\).
When the point \(M\) is fixed, \(\rho^2(t)\) depends on \(t\), and at the point of minimum, its derivative with respect to \(t\) is zero: \[-2(x - t) + 2\left( x^2 - 4x + 7 + t \right) = 0.\]
When point \(N\) is fixed, the function \(\rho^2(x)\) depends on \(x\), and at the point of minimum, its derivative with respect to \(x\) is zero:
\[2(x - t) + 2\left( x^2 - 4x + 7 + t \right)(2x - 4) = 0.\]
We solve the system:
\[
\begin{cases}
2 \left( x^2 - 4x + 7 + t \right) (2x - 3) = 0 \\
4(x - t) + 2 \left( x^2 - 4x + 7 + t \right) (2x - 5) = 0
\end{cases}
\]
**Case 1: \(x = \frac{3}{2}\)**
Substituting \(x = \frac{3}{2}\) into the second equation, we get \(t = -\frac{7}{8}\). Critical points are \(N^* \left( -\frac{7}{8}, -\frac{9}{8} \right)\) and \(M^* \left( \frac{3}{2}, \frac{5}{4} \right)\). The distance between points \(M^*\) and \(N^*\) is \(\frac{19\sqrt{2}}{8}\). If \(2r \leq \frac{19\sqrt{2}}{8}\), then the circle does not intersect the parabola (it touches at points \(M^*\) and \(N^*\)).
**Case 2: \(x \neq \frac{3}{2}\)**
Then \(\left( x^2 - 4x + 7 + t \right) = 0\) and from the second equation \(x = t\). Substituting \(x = t\) into the equation \(\left( x^2 - 4x + 7 + t \right) = 0\), we get the condition for \(t\), i.e. \(t^2 - 3t + 7 = 0\). Since this equation has no solutions, case 2 is not realizable. The maximum radius \(r_{\max} = \frac{19\sqrt{2}}{16}\).
|
\frac{19\sqrt{2}}{8}
|
How many three-digit whole numbers contain at least one 6 or at least one 8?
|
452
|
If $x = 2$ and $y = 1,$ what is the value of $2\times x - 3 \times y?$
|
1
|
Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$.
|
-\frac{3}{5}
|
Put ping pong balls in 10 boxes. The number of balls in each box must not be less than 11, must not be 17, must not be a multiple of 6, and must be different from each other. What is the minimum number of ping pong balls needed?
|
174
|
Find the value of \(\cos ^{5} \frac{\pi}{9}+\cos ^{5} \frac{5 \pi}{9}+\cos ^{5} \frac{7 \pi}{9}\).
|
\frac{15}{32}
|
Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your answer as a decimal to the nearest tenth. [asy]
size(80);
import graph;
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle, linetype("2 4"));
draw(Arc((0,0),1,0,90),linewidth(.8));
draw(Arc((0,3),1,0,-90),linewidth(.8));
draw(Arc((3,0),1,90,180),linewidth(.8));
draw(Arc((3,3),1,180,270),linewidth(.8));
draw((1,0)--(2,0),linewidth(.8));draw((3,1)--(3,2),linewidth(.8));
draw((1,3)--(2,3),linewidth(.8));draw((0,1)--(0,2),linewidth(.8));
[/asy]
|
30.8
|
In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
|
\frac{10}{3}
|
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$ . Compute the eighth perfect square in the sequence.
*Proposed by Lewis Chen*
|
64
|
If $x$ and $y$ are integers with $2x^{2}+8y=26$, what is a possible value of $x-y$?
|
26
|
Let $r$, $s$, and $t$ be the three roots of the equation \[8x^3 + 1001x + 2008 = 0.\] Find $(r + s)^3 + (s + t)^3 + (t + r)^3$.
|
753
|
A right triangular prism $ABC-A_{1}B_{1}C_{1}$ has all its vertices on the surface of a sphere. Given that $AB=3$, $AC=5$, $BC=7$, and $AA_{1}=2$, find the surface area of the sphere.
|
\frac{208\pi}{3}
|
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2002}+1\right)^{10/7}$?
|
428
|
A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow jelly bean from this jar?
|
0.3
|
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