problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that $|x|=3$, $y^{2}=4$, and $x < y$, find the value of $x+y$. | -1 |
The point $P$ $(4,5)$ is reflected over the $y$-axis to $Q$. Then $Q$ is reflected over the line $y=-x$ to $R$. What is the area of triangle $PQR$? | 36 |
Let $x,$ $y,$ and $z$ be angles such that
\begin{align*}
\cos x &= \tan y, \\
\cos y &= \tan z, \\
\cos z &= \tan x.
\end{align*}Find the largest possible value of $\sin x.$ | \frac{\sqrt{5} - 1}{2} |
In the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system using the same unit of length. The parametric equations of line $l$ are given by $\begin{cases}x=2+t \\ y=1+t\end{cases} (t \text{ is a parameter})$, and the p... | \sqrt{2} |
To pass the time while she is waiting somewhere Carla likes to count things around her. While she is waiting for school to start on Monday she counts the tiles on the ceiling--38. While she is waiting for everyone to finish their tests after she has handed in hers, she counts the books in the room--75. On Tuesday Carla... | When Carla was waiting on Monday she counted 38 ceiling tiles and on Tuesday she counted them twice, 38 ceiling tiles x 2 = <<38*2=76>>76 times she counted a ceiling tile.
She also counted 75 books in the room on Monday and counted them three times in a row on Tuesday, 75 books x 3 = <<75*3=225>>225 times she counted a... |
Given the parabola $y^{2}=4x$, a line with a slope of $\frac{\pi}{4}$ intersects the parabola at points $P$ and $Q$, and $O$ is the origin of the coordinate system. Find the area of triangle $POQ$. | 2\sqrt{2} |
Express the quotient $1121_5 \div 12_5$ in base $5$. | 43_5. |
Calculate the value of $1.000 + 0.101 + 0.011 + 0.001$. | 1.113 |
$\triangle KWU$ is an equilateral triangle with side length $12$ . Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$ . If $\overline{KP} = 13$ , find the length of the altitude from $P$ onto $\overline{WU}$ .
*Proposed by Bradley Guo* | \frac{25\sqrt{3}}{24} |
Two teams of 20 people each participated in a relay race from Moscow to Petushki. Each team divided the distance into 20 segments, not necessarily of equal length, and assigned them to participants such that each member runs exactly one segment (each participant maintains a constant speed, but the speeds of different p... | 38 |
Person A and Person B start walking towards each other from points $A$ and $B$ respectively, which are 10 kilometers apart. If they start at the same time, they will meet at a point 1 kilometer away from the midpoint of $A$ and $B$. If Person A starts 5 minutes later than Person B, they will meet exactly at the midpoin... | 10 |
An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ | 3 |
There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done? | 144 |
Let $l,$ $m,$ and $n$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(l,0,0),$ the midpoint of $\overline{AC}$ is $(0,m,0),$ and the midpoint of $\overline{AB}$ is $(0,0,n).$ Find
\[\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2}.\] | 8 |
The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? | 12 |
A bookseller sells 15 books in January, 16 in February and a certain number of books in March. If the average number of books he sold per month across all three months is 16, how many books did he sell in March? | He sold an average of 16 books per month across all 3 months so he must have sold a total of 3*16 = <<16*3=48>>48 books in those 3 months
He sold 15+16 = <<15+16=31>>31 in January and February
He must have sold 48-31 = <<48-31=17>>17 books in March
#### 17 |
The matrix
\[\begin{pmatrix} a & \frac{15}{34} \\ c & \frac{25}{34} \end{pmatrix}\]corresponds to a projection. Enter the ordered pair $(a,c).$ | \left( \frac{9}{34}, \frac{15}{34} \right) |
If $f(x)=4^x$ then $f(x+1)-f(x)$ equals: | 3f(x) |
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter o... | \frac{38}{7} |
Let $\alpha$ and $\beta$ be reals. Find the least possible value of $(2 \cos \alpha+5 \sin \beta-8)^{2}+(2 \sin \alpha+5 \cos \beta-15)^{2}$. | 100 |
There are 12 students in a classroom; 6 of them are Democrats and 6 of them are Republicans. Every hour the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change his/her political alignment to that of the ma... | \frac{341}{55} |
Carefully observe the arrangement pattern of the following hollow circles ($β$) and solid circles ($β$): $ββββββββββββββββββββββββββββ¦$. If this pattern continues, a series of $β$ and $β$ will be obtained. The number of $β$ in the first $100$ of $β$ and $β$ is $\_\_\_\_\_\_\_$. | 12 |
John climbs 3 staircases. The first staircase has 20 steps. The next has twice as many steps as the first. The final staircase has 10 fewer steps than the second one. Each step is 0.5 feet. How many feet did he climb? | The second flight had 20*2=<<20*2=40>>40 steps
The third had 40-10=<<40-10=30>>30 steps
So in total, he climbed 20+40+30=<<20+40+30=90>>90 steps
So he climbed 90*.5=<<90*.5=45>>45 feet
#### 45 |
Find the number of natural numbers \( k \), not exceeding 333300, such that \( k^{2} - 2k \) is exactly divisible by 303. | 4400 |
Given the function $y=\sin(\omega x)$ ($\omega>0$) is increasing in the interval $[0, \frac{\pi}{3}]$ and its graph is symmetric about the point $(3\pi, 0)$, the maximum value of $\omega$ is \_\_\_\_\_\_. | \frac{4}{3} |
The number of female students in the school hall is 4 times as many as the number of male students. If there are 29 male students, and there are 29 benches in the hall, at least how many students can sit on each bench for them to all fit in the hall? | The number of female students is 4 times 29: 4*29 = <<4*29=116>>116
There are a total of 29+116 = <<29+116=145>>145 students in the hall
There are 29 benches so each bench should take 145/29 = 5 students
#### 5 |
We write the following equation: \((x-1) \ldots (x-2020) = (x-1) \ldots (x-2020)\). What is the minimal number of factors that need to be erased so that there are no real solutions? | 1010 |
Leticia, Scarlett, and Percy decide to eat at a Greek restaurant for lunch. The prices for their dishes cost $10, $13, and $17, respectively. If the trio gives the waiter a 10% tip, how much gratuity should the waiter receive in dollars? | The total bill comes to $10 + $13 + $17 = $<<10+13+17=40>>40.
The total gratuity earned comes to $40 * 0.1 = $<<40*0.1=4>>4
#### 4 |
How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles? | 5 |
In how many ways can you fill a $3 \times 3$ table with the numbers 1 through 9 (each used once) such that all pairs of adjacent numbers (sharing one side) are relatively prime? | 2016 |
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \] | f(x) = -1 \text{ and } f(x) = x - 1. |
The weights of Christine's two cats are 7 and 10 pounds. What is her dog's weight if its weight is twice the sum of her two cats' weights? | The sum of the weights of Christine's two cats is 7 + 10 = <<7+10=17>>17 pounds.
Christine's dog weighs 17 x 2 = <<17*2=34>>34 pounds.
#### 34 |
What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 253 + 259$ is divided by $6$? | 2 |
100 participants came to the international StarCraft championship. The game is played in a knockout format, meaning two players participate in each match, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who won exactly two matches. | 49 |
Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are 92 parrots total on Bird Island, how many of those parrots are blue? | 23 |
Rounded to 3 decimal places, what is $\frac{8}{11}$? | 0.727 |
Compute $(\cos 185^\circ + i \sin 185^\circ)^{54}.$ | -i |
Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region. | 18 |
In an archery competition held in a certain city, after ranking the scores, the average score of the top seven participants is 3 points lower than the average score of the top four participants. The average score of the top ten participants is 4 points lower than the average score of the top seven participants. How man... | 28 |
Given the three digits 2, 4 and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? | 6 |
The area of trapezoid $ABCD$ is $164 \text{cm}^2$. The altitude is $8 \text{cm}$, $AB$ is $10 \text{cm}$, and $CD$ is $17 \text{cm}$. What is $BC$, in centimeters? [asy]
/* AMC8 2003 #21 Problem */
size(2inch,1inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)... | 10\text{ cm} |
Sasha and Masha each picked a natural number and communicated them to Vasya. Vasya wrote the sum of these numbers on one piece of paper and their product on another piece, then hid one of the pieces and showed the other (on which the number 2002 was written) to Sasha and Masha. Seeing this number, Sasha said he did not... | 1001 |
What is the largest three-digit integer whose digits are distinct and form a geometric sequence? | 964 |
Peter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in "random" mode, he can listen to the 11 tracks in a ... | 7/11 |
In an isosceles trapezoid, the lengths of the bases are 9 cm and 21 cm, and the height is 8 cm. Find the radius of the circumscribed circle around the trapezoid. | 10.625 |
All people named Barry are nice, while only half of the people named Kevin are nice. Three-fourths of people named Julie are nice, while 10% of people named Joe are nice. If a crowd contains 24 people named Barry, 20 people named Kevin, 80 people named Julie, and 50 people named Joe, how many nice people are in the c... | If all people named Barry are nice, and the crowd contains 24 people named Barry, then 1*24=<<24*1=24>>24 of these people are nice.
If only half of people named Kevin are nice, and the crowd contains 20 people named Kevin, then 0.5*20=<<0.5*20=10>>10 of these people are nice.
If three-fourths of people named Julie are ... |
Define $a$ $\$$ $b$ to be $a(b + 1) + ab$. What is the value of $(-2)$ $\$$ $3$? | -14 |
A convex polyhedron \( P \) has 2021 edges. By cutting off a pyramid at each vertex, which uses one edge of \( P \) as a base edge, a new convex polyhedron \( Q \) is obtained. The planes of the bases of the pyramids do not intersect each other on or inside \( P \). How many edges does the convex polyhedron \( Q \) hav... | 6063 |
Phil has 7 green marbles and 3 purple marbles in a bag. He removes a marble at random, records the color, puts it back, and then repeats this process until he has withdrawn 6 marbles. What is the probability that exactly three of the marbles that he removes are green? Express your answer as a decimal rounded to the nea... | .185 |
How many positive three-digit integers less than 500 have at least two digits that are the same? | 112 |
How many of the first 1000 positive integers can be expressed in the form
$\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$,
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$? | 600 |
Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. | 1440 |
At what value of $a$ do the graphs of $y=ax^2+3x+1$ and $y=-x-1$ intersect at exactly one point? | 2 |
The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.
[asy]
draw((-2,0)--(7,0),linewidth(1),Arrows);
draw((0,-1)--(0,4),linewidth(1),Arrows);
draw((1,.25)--(1,-.25),linewidth(1));
draw((2,.25)... | \frac{2}{3} |
Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\dots$, $C_6$. All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Find the product of the lengths of these twelve chords. | 28672 |
For every $m$ and $k$ integers with $k$ odd, denote by $\left[ \frac{m}{k} \right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that
\[\left[ \frac{n}{k} \right] + \left[ \frac{100 - n}{k} \right] = \left[ \frac{100}{k} \right]\]for an integer $n$ randomly chosen from ... | \frac{34}{67} |
What is the largest integer \( k \) such that \( k+1 \) divides
\[ k^{2020} + 2k^{2019} + 3k^{2018} + \cdots + 2020k + 2021? \ | 1010 |
Ivan and Peter are running in opposite directions on circular tracks with a common center, initially positioned at the minimal distance from each other. Ivan completes one full lap every 20 seconds, while Peter completes one full lap every 28 seconds. In the shortest possible time, when will they be at the maximum dist... | 35/6 |
Find $(21 \div (6 + 1 - 4)) \cdot 5.$ | 35 |
Given a "double-single" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000. | 81 |
Misha is the 50th best as well as the 50th worst student in her grade. How many students are in Misha's grade? | 99 |
Doris earns $20 per hour by babysitting. She needs to earn at least $1200 for her monthly expenses. She can babysit for 3 hours every weekday and 5 hours on a Saturday. How many weeks does it take for Doris to earn enough to cover her monthly expenses? | Doris babysits for a total of 5 weekdays x 3 hours/weekday = <<5*3=15>>15 hours
She babysits for a total of 15 + 5 = <<15+5=20>>20 hours in a week.
She earns 24 x $20 = $<<24*20=480>>480 in a week.
So, it will take $1440/$480 = <<1440/480=3>>3 weeks to earn Doris needed amount.
#### 3 |
Given real numbers $x$ and $y$ satisfy that three of the four numbers $x+y$, $x-y$, $\frac{x}{y}$, and $xy$ are equal, determine the value of $|y|-|x|$. | \frac{1}{2} |
For $\mathbf{v} = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 9 \\ 3 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} -6 \\ -2 \end{pmatrix}.\]Find $y$. | -23 |
An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$? | 9/35 |
One of the asymptotes of a hyperbola has equation $y=3x.$ The foci of the hyperbola have the same $x-$coordinate, which is $5.$ Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$". | y = -3x + 30 |
2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k} is equal to | -2^{-(2k+1)} |
Tim's cat bit him. He decided to get himself and the cat checked out. His doctor's visits $300 and insurance covered 75%. His cat's visit cost $120 and his pet insurance covered $60. How much did he pay? | The insurance covers 300*.75=$<<300*.75=225>>225
So he had to pay 300-225=$<<300-225=75>>75
The cats visit cost 120-60=$<<120-60=60>>60
So in total he paid 75+60=$<<75+60=135>>135
#### 135 |
Triangle $ABC$ is equilateral with side length $12$ . Point $D$ is the midpoint of side $\overline{BC}$ . Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$ . Point $E$ lies on segment $\overline{AD}$ and circle $E$ is tangent to circles $A$ and $D$ . Compute the radius of circle... | 3\sqrt{3} - 6 |
Calculate the radius of the circle where all the complex roots of the equation $(z - 1)^6 = 64z^6$ lie when plotted in the complex plane. | \frac{2}{3} |
What is the least possible value of \((x+1)(x+2)(x+3)(x+4)+2023\) where \(x\) is a real number? | 2022 |
A rectangle has side lengths $6$ and $8$ . There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$ . | 11 |
Find the smallest positive integer \( n \) such that for any given \( n \) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \( R_{1}, R_{2}, R_{3} \) such that \( R_{1} \) can be nested inside \( R_{2} \) and \( R_{2} \) can be nested inside \( R_{3} \). | 101 |
How many of the integers $1,2, \ldots, 2004$ can be represented as $(m n+1) /(m+n)$ for positive integers $m$ and $n$ ? | 2004 |
Given that $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\frac{{y^2}}{4}=1$, and they satisfy $\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respectively. Find the minimum value of ${m^2}+\frac{... | \frac{8}{3} |
Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m_{}$ and $n_{}$ a... | 489 |
Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______. | 5 - 2\sqrt{3} |
A particle moves in the Cartesian plane according to the following rules:
From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$
There are no right angle turns in the particle's path.
How many different paths can the particle take from $(0,0)$ to $(5,5)$? | 83 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\_1|=3|PF\_2|$ and $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}=a^{2}$, calculate the eccentricity of th... | \sqrt{2} |
The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum when written as an irreducible fraction?
Note: We say that the fraction \( p / q \) is irreducible if the integers \( p \) and \( q \) do not have common prime factors in their factorizati... | 168 |
Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$? | 16 |
Mancino is tending 3 gardens that each measure 16 feet by 5 feet. His sister, Marquita, is tilling the soil for two gardens that each measure 8 feet by 4 feet. How many square feet combined are in all their gardens? | 3 * (16 * 5) = <<3*(16*5)=240>>240 square feet
2 * (8 * 4) = <<2*(8*4)=64>>64 square feet
240 + 64 = <<240+64=304>>304 square feet
The gardens have a total of 304 square feet.
#### 304 |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{20} |
Given the vectors $\overrightarrow{a} = (1, 1)$, $\overrightarrow{b} = (2, 0)$, the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{4} |
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ | \pi + 2 |
Compute the integrals:
1) \(\int_{0}^{5} \frac{x \, dx}{\sqrt{1+3x}}\)
2) \(\int_{\ln 2}^{\ln 9} \frac{dx}{e^{x}-e^{-x}}\)
3) \(\int_{1}^{\sqrt{3}} \frac{(x^{3}+1) \, dx}{x^{2} \sqrt{4-x^{2}}}\)
4) \(\int_{0}^{\frac{\pi}{2}} \frac{dx}{2+\cos x}\) | \frac{\pi}{3 \sqrt{3}} |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate t... | 1925 |
ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between | 7 and 8 |
What is the least integer value of $x$ such that $\lvert2x+ 7\rvert\le 16$? | -11 |
For all triples \((x, y, z)\) that satisfy the system
$$
\left\{\begin{array}{l}
2 \cos x = \operatorname{ctg} y \\
2 \sin y = \operatorname{tg} z \\
\cos z = \operatorname{ctg} x
\end{array}\right.
$$
find the minimum value of the expression \(\sin x + \cos z\). | -\frac{5 \sqrt{3}}{6} |
Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8. | 26 |
In right triangle $ABC$, where $\angle A = \angle B$ and $AB = 10$. What is the area of $\triangle ABC$? | 25 |
Let $f(x) = ax+b$, where $a$ and $b$ are real constants, and $g(x) = 2x - 5$. Suppose that for all $x$, it is true that $g(f(x)) = 3x + 4$. What is $a+b$? | 6 |
Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then
\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\]
For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\le N\le 40... | 18 |
Among the three-digit numbers formed by the digits 1, 2, 3, 4, 5, 6 with repetition allowed, how many three-digit numbers have exactly two different even digits (for example: 124, 224, 464, β¦)? (Answer with a number). | 72 |
Given $\sin\theta= \frac {m-3}{m+5}$ and $\cos\theta= \frac {4-2m}{m+5}$ ($\frac {\pi}{2} < \theta < \pi$), calculate $\tan\theta$. | - \frac {5}{12} |
Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$. | -4 |
Find the $r$ that satisfies $\log_{16} (r+16) = \frac{5}{4}$. | 16 |
Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is | B |
What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{55}$ and $a^{-1}$ is also undefined $\pmod{66}$? | 10 |
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