problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying
the real and imaginary part of $z$ are both integers;
$|z|=\sqrt{p},$ and
there exists a triangle whose three side lengths are $p,$ the real part of $z^{3},$ and the imaginary part of $z^{3}.$ | 349 |
Sides $\overline{AB}$ and $\overline{EF}$ of regular hexagon $ABCDEF$ are extended to meet at point $P$. What is the degree measure of angle $P$? | 60^\circ |
To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$? | 79 |
An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are positive integers with no common divisor larger than 1. What is the value of \( p+q \)?
(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \( r \), height \( h \), and slant height \( s \) has lateral surface area equal to \( \pi r s \).) | 59 |
In the textbook, students were once asked to explore the coordinates of the midpoint of a line segment: In a plane Cartesian coordinate system, given two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, the midpoint of the line segment $AB$ is $M$, then the coordinates of $M$ are ($\frac{{x}_{1}+{x}_{2}}{2}$, $\frac{{y}_{1}+{y}_{2}}{2}$). For example, if point $A(1,2)$ and point $B(3,6)$, then the coordinates of the midpoint $M$ of line segment $AB$ are ($\frac{1+3}{2}$, $\frac{2+6}{2}$), which is $M(2,4)$. Using the above conclusion to solve the problem: In a plane Cartesian coordinate system, if $E(a-1,a)$, $F(b,a-b)$, the midpoint $G$ of the line segment $EF$ is exactly on the $y$-axis, and the distance to the $x$-axis is $1$, then the value of $4a+b$ is ____. | 4 \text{ or } 0 |
Given circles $P$, $Q$, and $R$ each have radius 2, circle $P$ and $Q$ are tangent to each other, and circle $R$ is tangent to the midpoint of $\overline{PQ}$. Calculate the area inside circle $R$ but outside circle $P$ and circle $Q$. | 2\pi |
What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{129}$? | 7 |
Given the function $f(x) = \cos^4x + 2\sin x\cos x - \sin^4x$
(1) Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
(2) When $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. | -1 |
A club has between 300 and 400 members. The members gather every weekend and are divided into eight distinct groups. If two members are absent, the groups can all have the same number of members. What is the sum of all possible numbers of members in the club? | 4200 |
Cynthia has four times as many water balloons as her husband, Randy. Randy has only half as many water balloons as his daughter, Janice. If Janice throws all 6 of her water balloons at her father, how many water balloons does Cynthia have, which she could also choose to throw at Randy? | Randy has only half as many water balloons as Janice’s 6, for a total of (½)*6=3 water balloons.
Cynthia has 4 times as many water balloons as Randy, for a total of 4*3=<<4*3=12>>12 water balloons
#### 12 |
Given the function \( y = \sqrt{a x^2 + b x + c} \) (where \(a, b, c \in \mathbb{R}\) and \(a < 0\)), the domain is \( D \). If the points \( (s, f(t)) \) (where \( s, t \in D \)) form a square, then the real number \( a \) equals ______. | -4 |
Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$. | \frac{9\sqrt{5}}{10} |
The graph of the equation \[\sqrt{x^2+y^2} + |y-1| = 3\]consists of portions of two different parabolas. Compute the distance between the vertices of the parabolas. | 3 |
Suzanne wants to raise money for charity by running a 5-kilometer race. Her parents have pledged to donate $10 for her first kilometer and double the donation for every successive kilometer. If Suzanne finishes the race, how much money will her parents donate? | For the 2nd kilometer, the donation will be $10 * 2 = $<<10*2=20>>20.
For the 3rd kilometer, the donation will be $20 * 2 = $<<20*2=40>>40.
For the 4th kilometer, the donation will be $40 * 2 = $<<40*2=80>>80.
For the final kilometer, the donation will be $80 * 2 = $<<80*2=160>>160.
For the entire race the donation will be $10 + $20 + $40 + $80 + $160 = $<<10+20+40+80+160=310>>310.
#### 310 |
Football tickets cost $\$13.50$ each. What is the maximum number of tickets Jane can buy with $\$100.00$? | 7 |
The volume of a cylinder is $54\pi$ $\text{cm}^3$. How many cubic centimeters are in the volume of a cone with the same radius and height as the cylinder? Express your answer in terms of $\pi$. [asy]
import solids; currentprojection=orthographic(0,100,25); defaultpen(linewidth(0.8));
revolution cyl = cylinder((5,0,0),1,5,Z);
revolution cone = cone((0,0,0),1,5,Z);
draw(cyl,backpen=dashed);
draw(cone,backpen=dashed);
[/asy] | 18\pi |
Vasya wrote consecutive natural numbers \(N\), \(N+1\), \(N+2\), and \(N+3\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.
The sum of the numbers in the first and second circles equals 200, and the sum of the numbers in the third and fourth circles equals 105. What is the sum of the numbers in the second and third circles? | 103 |
Jeff will pick a card at random from ten cards numbered 1 through 10. The number on this card will indicate his starting point on the number line shown below. He will then spin the fair spinner shown below (which has three congruent sectors) and follow the instruction indicated by his spin. From this new point he will spin the spinner again and follow the resulting instruction. What is the probability that he ends up at a multiple of 3 on the number line? Express your answer as a common fraction. [asy]
import graph;
size(10cm);
defaultpen(linewidth(0.7)+fontsize(8));
xaxis(-2,13,Ticks(OmitFormat(-1),1.0,begin=false,end=false,beginlabel=false,endlabel=false),Arrows(4));
label("-1",(-1,-0.98));
real r=3.5;
pair center=(17,0);
draw(circle(center,r));
int i;
for(i=1;i<=3;++i)
{
draw(center--center+r*dir(120*i-30));
}
label("$\parbox{1cm}{move \\ 1 space \\ left}$",center+r/2*dir(150));
label("$\parbox{1cm}{move \\ 1 space \\ right}$",center+r/2*dir(270));
label("$\parbox{1cm}{move \\ 1 space \\ right}$",center+r/2*dir(30));
draw(center--center+3*r/4*dir(80),EndArrow(4));[/asy] | \frac{31}{90} |
Let \( A \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + x - 110 < 0 \) and \( B \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + 10x - 96 < 0 \). Suppose that the set of integer solutions of the inequality \( x^{2} + ax + b < 0 \) is exactly the set of integers contained in \( A \cap B \). Find the maximum value of \( \lfloor |a - b| \rfloor \). | 71 |
Pascal High School organized three different trips. Fifty percent of the students went on the first trip, $80 \%$ went on the second trip, and $90 \%$ went on the third trip. A total of 160 students went on all three trips, and all of the other students went on exactly two trips. How many students are at Pascal High School? | 800 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\frac{c}{b\sin B}$$. | \frac{2\sqrt{3}}{3} |
Use each of the digits 3, 4, 6, 8 and 9 exactly once to create the greatest possible five-digit multiple of 6. What is that multiple of 6? | 98,634 |
Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that
\[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \]
attains its minimum. | \left( \frac{1}{n+1}, \frac{1}{n+1}, \ldots, \frac{1}{n+1} \right) |
Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=1$, $a_{2}=2$, and $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ for all $n$, and $a_{n+1} a_{n+2} \neq 1$, find the sum $S_{1999}=\sum_{n=1}^{1999} a_{n}$. | 3997 |
What is the digit in the thousandths place of the decimal equivalent of $\frac{3}{16}$? | 7 |
If the two roots of the quadratic $7x^2+3x+k$ are $\frac{-3\pm i\sqrt{299}}{14}$, what is $k$? | 11 |
Given the function $f(x)=a^{x}-(k+1)a^{-x}$ where $a > 0$ and $a\neq 1$, which is an odd function defined on $\mathbb{R}$.
1. Find the value of $k$.
2. If $f(1)= \frac {3}{2}$, and the minimum value of $g(x)=a^{2x}+a^{-2x}-2mf(x)$ on $[0,+\infty)$ is $-6$, find the value of $m$. | 2 \sqrt {2} |
Evelyn’s family watched 10 hours of television last week. The week before, they watched 8 hours of television. If they watch 12 hours of television next week, what is the average number of hours of television that they watch per week? | Over the three weeks, they will have watched a total of 10 + 8 + 12 = <<10+8+12=30>>30 hours of television.
The average number of hours per week is 30 hours / 3 = <<30/3=10>>10 hours of television.
#### 10 |
Anna goes trick-or-treating in a subdivision where she gets 14 pieces of candy per house. Her brother Billy goes trick-or-tricking in a neighboring subdivision where he gets 11 pieces of candy per house. If the first subdivision has 60 houses and the second subdivision has 75 houses, how many more pieces of candy does Anna get? | First find the total number of pieces of candy Anna gets: 14 pieces/house * 60 houses = 840 pieces
Then find the total number of pieces of candy Billy gets: 11 pieces/house * 75 houses = <<11*75=825>>825 pieces
Then subtract the number of pieces Billy gets from the number Anna gets to find the difference: 840 pieces - 825 pieces = <<840-825=15>>15 pieces
#### 15 |
Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with side lengths as labeled. What is the area of the shaded $\text L$-shaped region? [asy]
/* AMC8 2000 #6 Problem */
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
draw((1,5)--(1,1)--(5,1));
draw((0,4)--(4,4)--(4,0));
fill((0,4)--(1,4)--(1,1)--(4,1)--(4,0)--(0,0)--cycle);
label("$A$", (5,5), NE);
label("$B$", (5,0), SE);
label("$C$", (0,0), SW);
label("$D$", (0,5), NW);
label("1",(.5,5), N);
label("1",(1,4.5), E);
label("1",(4.5,1), N);
label("1",(4,.5), E);
label("3",(1,2.5), E);
label("3",(2.5,1), N);
[/asy] | 7 |
Find the coordinates of the foci and the eccentricity of the hyperbola $x^{2}-2y^{2}=2$. | \frac{\sqrt{6}}{2} |
Steve finds 100 gold bars while visiting Oregon. He wants to distribute his gold bars evenly to his 4 friends. If 20 gold bars were lost on the way back to San Diego, how many gold bars will each of his 4 friends get when he returns? | He only has 100 - 20 = <<100-20=80>>80 gold bars after losing 20 of them.
He then gives each of his friends 80 ÷ 4 = <<80/4=20>>20 gold bars.
#### 20 |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-4,3,-2,9);
real f(real x) {return 8-(x+1)^2;}
draw(graph(f,-3.9,2.16,operator ..), red);
[/asy] | 11 |
Let $C$ be the circle with equation $x^2+12y+57=-y^2-10x$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$? | -9 |
Rachel earned $200 babysitting. She spent 1/4 of the money on lunch. She spent 1/2 of the money on a DVD. How much did Rachel have left? | Rachel spent 1/4*200 = $<<200*1/4=50>>50 on lunch.
Rachel spent 1/2*200 = $<<200*1/2=100>>100 on a DVD.
Rachel has 200-50-100 = $<<200-50-100=50>>50 left.
#### 50 |
The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression? | 32 |
Christopher uses 1 packet of a sugar substitute in his coffee. He has 2 coffees a day. The packets come 30 to a box and cost $4.00 a box. How much will it cost him to have enough sugar substitutes to last him 90 days? | He uses 1 packet per cup of coffee and he has 2 cups a day so that's 1*2 = <<1*2=2>>2 packets
He wants enough to last him 90 days and he uses 2 packets a day so that's 90*2 = <<90*2=180>>180 packets
The packets come 30 to a box and he needs 180 packets so that's 180/30 = <<180/30=6>>6 boxes of sugar substitute
Each box costs $4.00 and he needs 6 boxes so that's 4*6 = $<<4*6=24.00>>24.00
#### 24 |
Let $a$ and $b$ be the roots of $x^2 - 4x + 5 = 0.$ Compute
\[a^3 + a^4 b^2 + a^2 b^4 + b^3.\] | 154 |
A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded? | 33.33\% |
The expression $\frac{k^{2}}{1.001^{k}}$ reaches its maximum value with which natural number $k$? | 2001 |
John starts climbing a very steep hill. He travels 1 foot vertically for every two feet horizontally. His elevation increases from 100 feet to 1450 feet. How far does he move horizontally, in feet? | He travels 1450-100=<<1450-100=1350>>1350 feet vertically
So he travels 1350*2=<<1350*2=2700>>2700 feet horizontally
#### 2700 |
Let $x=2001^{1002}-2001^{-1002}$ and $y=2001^{1002}+2001^{-1002}$. Find $x^{2}-y^{2}$. | -4 |
Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits. | 252 |
Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum. | 70000000 |
A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle? | 6\sqrt{2}+6 |
Given two vectors in space, $\overrightarrow{a} = (x - 1, 1, -x)$ and $\overrightarrow{b} = (-x, 3, -1)$. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find the value of $x$. | -1 |
In triangle $ABC$, where $AB = 6$ and $AC = 10$. Let $M$ be a point on $BC$ such that $BM : MC = 2:3$. If $AM = 5$, what is the length of $BC$?
A) $7\sqrt{2.2}$
B) $5\sqrt{6.1}$
C) $10\sqrt{3.05}$
D) $15 - 3\sqrt{6.1}$ | 5\sqrt{6.1} |
For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor of $8!$ and $10!$. | 40320 |
In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\frac{a-\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$. | 56 |
Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, $\overrightarrow{a}\cdot (\overrightarrow{b}-\overrightarrow{a})=0$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} |
If I have a $4\times 4$ chess board, in how many ways can I place four distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 576 |
The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;
for(int i = -4; i <= 5; ++i) {
draw((i,-1)--(i,6), dashed+mediumgrey);
}
for(int i = 1; i <= 6; ++i) {
draw((-4,i)--(5,i), dashed+mediumgrey);
}
Label laxis; laxis.p=fontsize(10);
xaxis("$x$",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis("$y$",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,1)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2)); // draw((min,(-(-9)-(0)*xmin)/3)--(xmax,(-(-9)-(0)*xmax)/3),linetype("6pt 6pt"));
label("$f(x)$",(-3.52,4.6),SE*lsf);
//dot((-1,1),ds); dot((3,5),ds); dot((-3,3),ds); dot((1,3),ds); dot((5,3),ds);
dot((-4.32,4.32),ds); dot((5.56,2.44),ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy] | 2 |
The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left(\frac{x}{4}\right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$
(v) $g\left(\frac{1}{2}\right) = \frac{1}{3}.$
Find $g\left(\frac{3}{16}\right).$ | \frac{2}{9} |
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | 120 |
Given that $\text{1 mile} = \text{8 furlongs}$ and $\text{1 furlong} = \text{40 rods}$, the number of rods in one mile is | 320 |
How many triangles can be formed using the vertices of a regular dodecagon (a 12-sided polygon)? | 220 |
Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$? | 572 |
A regular polygon has interior angles of 162 degrees. How many sides does the polygon have? | 20 |
Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to $90^\circ$ ?
*Proposed by Anton Trygub* | 2,042,220 |
Let $g_0(x) = x + |x - 150| - |x + 150|$, and for $n \geq 1$, let $g_n(x) = |g_{n-1}(x)| - 2$. For how many values of $x$ is $g_{100}(x) = 0$? | 299 |
If
\[\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta} = 1,\]then find the sum of all possible values of
\[\frac{\sin^4 \beta}{\sin^2 \alpha} + \frac{\cos^4 \beta}{\cos^2 \alpha}.\] | 1 |
In the rectangular coordinate system on a plane, establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The parametric equations of the curve $C$ are $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter, } \alpha \in \left[ 0,\pi \right])$, and the polar equation of the line $l$ is $\rho = \frac{4}{\sqrt{2}\sin \left( \theta - \frac{\pi }{4} \right)}$.
(I) Write the Cartesian equation of curve $C$ and the polar equation of line $l$.
(II) Let $P$ be any point on curve $C$ and $Q$ be any point on line $l$. Find the minimum value of $|PQ|$. | \frac{5 \sqrt{2}}{2}-1 |
Lisa is a member of the photography club at school. Every weekend the club will go anywhere to take photos. Leslie took 10 photos of animals to share with the club. She also took 3 times as many photos of flowers as animals and took 10 fewer scenery than the flowers. If Lisa took 15 fewer photos last weekend, How many photos did she take then? | Lisa took 10 x 3 = <<10*3=30>>30 photos of flowers.
She took 30 - 10 = <<30-10=20>>20 photos of the scenery.
So, Lisa took 10 + 30 + 20 = <<10+30+20=60>>60 photos this week.
Therefore, Lisa took 60 - 15 = <<60-15=45>>45 photos last weekend.
#### 45 |
The length of the longer side of rectangle $R$ is $10$ percent more than the length of a side of square $S.$ The length of the shorter side of rectangle $R$ is $10$ percent less than the length of a side of square $S.$ What is the ratio of the area of rectangle $R$ to the area of square $S?$ Express your answer as a common fraction. | \frac{99}{100} |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | -3 |
Given the numbers $1, 2, \cdots, 20$, calculate the probability that three randomly selected numbers form an arithmetic sequence. | \frac{1}{38} |
The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$. | (991,997,1009) |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(3000,0),(3000,2000),$ and $(0,2000)$. What is the probability that $x > 5y$? Express your answer as a common fraction. | \frac{3}{20} |
How many positive divisors do 8400 and 7560 have in common? | 32 |
The complex number $z$ traces a circle centered at the origin with radius 2. Then $z + \frac{1}{z}$ traces a:
(A) circle
(B) parabola
(C) ellipse
(D) hyperbola
Enter the letter of the correct option. | \text{(C)} |
A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.
How many additional rods will the farmer need? | 21 |
Given the expressions $(2401^{\log_7 3456})^{\frac{1}{2}}$, calculate its value. | 3456^2 |
By multiplying a natural number by the number that is one greater than it, the product takes the form $ABCD$, where $A, B, C, D$ are different digits. Starting with the number that is 3 less, the product takes the form $CABD$. Starting with the number that is 30 less, the product takes the form $BCAD$. Determine these numbers. | 8372 |
There are two equilateral triangles with a vertex at $(0, 1)$ , with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$ . Find the area of the larger of the two triangles. | 26\sqrt{3} + 45 |
Voldemort had his dinner and ate a piece of cake that has 110 calories. He also ate 1 pack of chips that contained 310 calories and drank a 500 ml bottle of coke that has 215 calories. His caloric intake for breakfast and lunch is 560 and 780 calories, respectively. If he has a 2500 calorie intake limit per day, how many calories can he still take? | For the night he was able to intake 110 + 310 + 215 = <<110+310+215=635>>635 calories
For breakfast and lunch, he was able to intake 560 + 780 = <<560+780=1340>>1340 calories
So, for the entire day Voldemort was able to intake 1340 + 635 = <<1340+635=1975>>1975 calories
Therefore, he can still intake 2500 - 1975 = <<525=525>>525 calories.
#### 525 |
Someone, when asked for the number of their ticket, replied: "If you add all the six two-digit numbers that can be made from the digits of the ticket number, half of the resulting sum will be exactly my ticket number." Determine the ticket number. | 198 |
Cara is sitting at a round table with her eight friends. How many different pairs of friends could Cara be potentially sitting between? | 28 |
Given that $F$ is the right focus of the hyperbola $C: x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$, when the perimeter of $\triangle APF$ is minimized, the ordinate of point $P$ is ______. | 2 \sqrt {6} |
Betty has 3 red beads for every 2 blue beads that she has. How many blue beads does Betty have if she has 30 red beads? | Betty has 30/3 = <<30/3=10>>10 sets of 3 red beads.
So, she has 10 x 2 = <<10*2=20>>20 blue beads.
#### 20 |
Select 5 people from 4 boys and 5 girls to participate in a math extracurricular group. How many different ways are there to select under the following conditions?
(1) Select 2 boys and 3 girls, and girl A must be selected;
(2) Select at most 4 girls, and boy A and girl B cannot be selected at the same time. | 90 |
Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?
[asy]
pair A,B,C,D,E,F,G,H,J;
A=(20,20(2+sqrt(2)));
B=(20(1+sqrt(2)),20(2+sqrt(2)));
C=(20(2+sqrt(2)),20(1+sqrt(2)));
D=(20(2+sqrt(2)),20);
E=(20(1+sqrt(2)),0);
F=(20,0);
G=(0,20);
H=(0,20(1+sqrt(2)));
J=(10(2+sqrt(2)),10(2+sqrt(2)));
draw(A--B);
draw(B--C);
draw(C--D);
draw(D--E);
draw(E--F);
draw(F--G);
draw(G--H);
draw(H--A);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(J);
label("$A$",A,NNW);
label("$B$",B,NNE);
label("$C$",C,ENE);
label("$D$",D,ESE);
label("$E$",E,SSE);
label("$F$",F,SSW);
label("$G$",G,WSW);
label("$H$",H,WNW);
label("$J$",J,SE);
size(4cm);
[/asy] | 1152 |
Expand the product ${(2x+3)(x+5)}$. | 2x^2 + 13x + 15 |
There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube? | 90 |
In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the number \( M \) have? List all possible answers. | 29 |
If lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+3=0$ are parallel, find the value of $a$. | -1 |
A city does not have electric lighting yet, so candles are used in houses at night. In João's house, one candle is used per night without burning it completely, and with four of these candle stubs, João makes a new candle. How many nights can João light up his house with 43 candles? | 57 |
Find the product of all possible real values for $k$ such that the system of equations $$ x^2+y^2= 80 $$ $$ x^2+y^2= k+2x-8y $$ has exactly one real solution $(x,y)$ .
*Proposed by Nathan Xiong* | 960 |
[asy]
draw((-7,0)--(7,0),black+linewidth(.75));
draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));
draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75));
[/asy]
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each vertex. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is | 12\sqrt{3} |
My co-worker Larry only likes numbers that are divisible by 4, such as 20, or 4,004. How many different ones digits are possible in numbers that Larry likes? | 5 |
Find the curve defined by the equation
\[r = \frac{1}{\sin \theta - \cos \theta}.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | \text{(A)} |
Given \(\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+}\) and \(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=\pi\), find the minimum value of \(\left(2 \sin ^{2} \theta_{1}+\frac{1}{\sin ^{2} \theta_{1}}\right)\left(2 \sin ^{2} \theta_{2}+\frac{1}{\sin ^{2} \theta_{2}}\right)\left(2 \sin ^{2} \theta_{3}+\frac{1}{\sin ^{2} \theta_{3}}\right)\left(2 \sin ^{2} \theta_{4}+\frac{1}{\sin ^{2} \theta_{4}}\right)\). | 81 |
Simplify $2a(2a^2 + a) - a^2$. | 4a^3 + a^2 |
There are four points that are $7$ units from the line $y = 20$ and $10$ units from the point $(10, 20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | 120 |
If a positive integer \( N \) can be expressed as \( \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor \) for some real number \( x \), then we say that \( N \) is "visible"; otherwise, we say that \( N \) is "invisible". For example, 8 is visible since \( 8 = \lfloor 1.5 \rfloor + \lfloor 2(1.5) \rfloor + \lfloor 3(1.5) \rfloor \), whereas 10 is invisible. If we arrange all the "invisible" positive integers in increasing order, find the \( 2011^{\text{th}} \) "invisible" integer. | 6034 |
Colleen is making a batch of 48 cookies. She's going to add 108 chocolate chips and one-third as many M&Ms to the whole batch. What are the average number of chocolate pieces in each cookie? | First find the total number of M&Ms: 108 chocolate chips / 3 chocolate chips / 1 M&M = <<108/3/1=36>>36 M&Ms
Then add that number to the number of chocolate chips to find the total number of pieces of chocolate: 36 M&Ms + 108 chocolate chips = <<36+108=144>>144 chocolates
Then divide the total number of chocolates by the number of cookies to find the number of pieces of chocolate per cookie: 144 chocolates / 48 cookies = <<144/48=3>>3 chocolates/cookie
#### 3 |
Nathan and his two younger twin sisters' ages multiply to 72. Find the sum of their three ages. | 14 |
Jaylen’s dog eats 1 cup of dog food in the morning and 1 cup of dog food in the evening. If she buys a bag that has 32 cups of dog food, how many days can she feed her dog with it? | Every day, Jaylen’s dog eats 1 x 2 = <<1*2=2>>2 cups of dog food.
The bag will feed her dog for 32 / 2 = <<32/2=16>>16 days.
#### 16 |
Janessa has a plan to give her brother Dexter his first collection of baseball cards. She currently has 4 cards in addition to the 13 that her father gave her. She ordered a collection of 36 cards from eBay. After inspecting the cards she found 4 cards are in bad shape and decides to throw them away. Janessa ended up giving Dexter 29 cards. How many cards did Janessa keep for herself? | Janessa starts with 4 cards + 13 cards = <<4+13=17>>17 cards.
She adds 17 cards + 36 cards = <<17+36=53>>53 cards from the eBay purchase.
Janessa decides to throw away 53 cards - 4 cards = <<53-4=49>>49 cards due to damage.
After giving Dexter the present we know that Janessa kept 49 cards - 29 cards = <<49-29=20>>20 cards for herself.
#### 20 |
Tedra is harvesting his tomatoes. He harvests 400 kg on Wednesday, and half as much on Thursday. He harvests a total of 2000 kg on Wednesday, Thursday, and Friday. How many tomatoes of the number he harvested on Friday are remaining if he gives away 700kg of them to his friends? | Tedra harvests 1/2 * 400 kg = <<1/2*400=200>>200 kg of tomatoes on Thursday.
Tedra harvests 2000 kg - 400 kg - 200 kg = <<2000-400-200=1400>>1400 kg of tomatoes on Friday.
After giving away tomatoes to his friends, he is left with 1400 kg - 700 kg = <<1400-700=700>>700 kg of tomatoes.
#### 700 |
To calculate $31^2$, Emily mentally figures the value $30^2$ and adds 61. Emily subtracts a number from $30^2$ to calculate $29^2$. What number does she subtract? | 59 |
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