problem
stringlengths 10
2.37k
| original_solution
stringclasses 890
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stringlengths 0
253
| source
stringclasses 1
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int64 6
40.3k
| domain
stringclasses 1
value |
|---|---|---|---|---|---|
Given $\sin \alpha - \cos \alpha = \frac{\sqrt{10}}{5}$, $\alpha \in (\pi, 2\pi)$,
$(1)$ Find the value of $\sin \alpha + \cos \alpha$;
$(2)$ Find the value of $\tan \alpha - \frac{1}{\tan \alpha}$.
|
-\frac{8}{3}
|
deepscale
| 17,766
| ||
If \( 20 \times 21 \times 22 \times \ldots \times 2020 = 26^{k} \times m \), where \( m \) is an integer, what is the maximum value of \( k \)?
|
165
|
deepscale
| 7,460
| ||
An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$?
|
1. **Setup the coordinate system and define points**:
- Let $B = (0, 0)$, $A = (0, -8)$.
- The possible points of $C$ create a semi-circle of radius $5$ centered at $B$.
2. **Define the circles**:
- The circle centered at $B$ with radius $5$ is described by the equation $x^2 + y^2 = 25$.
- The circle centered at $A$ with radius $7$ is described by the equation $x^2 + (y+8)^2 = 49$.
3. **Find the intersection point $O$**:
- Solve the system of equations:
\[
\begin{cases}
x^2 + y^2 = 25 \\
x^2 + (y+8)^2 = 49
\end{cases}
\]
- Simplify the second equation:
\[
x^2 + y^2 + 16y + 64 = 49 \implies 16y + 64 = 24 \implies y = -\frac{5}{2}
\]
- Substitute $y = -\frac{5}{2}$ into $x^2 + y^2 = 25$:
\[
x^2 + \left(-\frac{5}{2}\right)^2 = 25 \implies x^2 + \frac{25}{4} = 25 \implies x^2 = \frac{75}{4} \implies x = \pm \frac{5\sqrt{3}}{2}
\]
- Choose $x = \frac{5\sqrt{3}}{2}$ (since we are considering the clockwise direction), so $O = \left(\frac{5\sqrt{3}}{2}, -\frac{5}{2}\right)$.
4. **Analyze the triangle $BDO$**:
- Recognize that $\triangle BDO$ is a $30-60-90$ triangle:
- $BO = 5$ (radius of the semi-circle),
- $BD = \frac{5\sqrt{3}}{2}$ (horizontal component),
- $DO = \frac{5}{2}$ (vertical component).
- Therefore, $\angle CBO = 30^\circ$ and $\angle ABO = 60^\circ$.
5. **Calculate the probability**:
- The probability that $AC < 7$ is the ratio of the angle $\angle ABO$ to $180^\circ$:
\[
\frac{\angle ABO}{180^\circ} = \frac{60^\circ}{180^\circ} = \frac{1}{3}
\]
Thus, the probability that $AC < 7$ is $\boxed{\textbf{(D) } \frac{1}{3}}$. $\blacksquare$
|
\frac{1}{3}
|
deepscale
| 1,739
| |
Given a cubic function $f(x)=ax^{3}+bx^{2}+cx+d(a\neq 0)$, define the "nice point" of the function as the point $(x_{0},f(x_{0}))$ where $x_{0}$ is a real root of the equation $f''(x)=0$. It has been observed that every cubic function has a "nice point," a symmetry center, and that the "nice point" is the symmetry center. Based on this observation, find the value of $g(\frac{1}{2011})+g(\frac{2}{2011})+g(\frac{3}{2011})+g(\frac{4}{2011})+\cdots+g(\frac{2010}{2011})$ for the function $g(x)=\frac{1}{3}x^{3}-\frac{1}{2}x^{2}+3x-\frac{5}{12}$.
|
2010
|
deepscale
| 8,893
| ||
In an election, there are two candidates, A and B, who each have 5 supporters. Each supporter, independent of other supporters, has a \(\frac{1}{2}\) probability of voting for his or her candidate and a \(\frac{1}{2}\) probability of being lazy and not voting. What is the probability of a tie (which includes the case in which no one votes)?
|
63/256
|
deepscale
| 11,996
| ||
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is
|
1. **Set up the equation based on the problem statement:**
The sum of the first $3n$ positive integers is given by the formula $\frac{3n(3n+1)}{2}$, and the sum of the first $n$ positive integers is given by $\frac{n(n+1)}{2}$. According to the problem, the sum of the first $3n$ integers is $150$ more than the sum of the first $n$ integers. Therefore, we can write the equation:
\[
\frac{3n(3n+1)}{2} = \frac{n(n+1)}{2} + 150
\]
2. **Simplify the equation:**
Multiply through by $2$ to clear the fraction:
\[
3n(3n+1) = n(n+1) + 300
\]
Expanding both sides:
\[
9n^2 + 3n = n^2 + n + 300
\]
Rearrange to form a quadratic equation:
\[
9n^2 + 3n - n^2 - n - 300 = 0 \Rightarrow 8n^2 + 2n - 300 = 0
\]
Simplify the quadratic equation:
\[
4n^2 + n - 150 = 0
\]
3. **Solve the quadratic equation:**
Use the quadratic formula where $a = 4$, $b = 1$, and $c = -150$:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1 + 2400}}{8} = \frac{-1 \pm \sqrt{2401}}{8}
\]
Simplifying further:
\[
n = \frac{-1 \pm 49}{8} \Rightarrow n = \frac{48}{8} = 6 \quad \text{or} \quad n = \frac{-50}{8} = -\frac{25}{4}
\]
Since $n$ must be a positive integer, we choose $n = 6$.
4. **Calculate the sum of the first $4n$ positive integers:**
Now, $4n = 4 \times 6 = 24$. The sum of the first $24$ positive integers is:
\[
\frac{24 \times 25}{2} = 300
\]
5. **Conclude with the final answer:**
The sum of the first $4n$ positive integers is $\boxed{300}$. This corresponds to choice $\text{(A)}$.
|
300
|
deepscale
| 503
| |
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots?
|
The probability is \(1/99\). In fact, we show by induction on \(n\) that after \(n\) shots, the probability of having made any number of shots from \(1\) to \(n-1\) is equal to \(1/(n-1)\). This is evident for \(n=2\). Given the result for \(n\), we see that the probability of making \(i\) shots after \(n+1\) attempts is \[\frac{i-1}{n} \frac{1}{n-1} + \left( 1 - \frac{i}{n} \right) \frac{1}{n-1} = \frac{(i-1) + (n-i)}{n(n-1)} = \frac{1}{n},\] as claimed.
|
\(\frac{1}{99}\)
|
deepscale
| 5,742
| |
In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is:
|
Let's denote the first term of the arithmetic sequence as $a = 2$, the common difference as $d$, and the number of terms as $n$. The last term, which is also the $n$-th term, is given as $29$. The sum of all terms is $155$.
1. **Expression for the $n$-th term:**
The $n$-th term of an arithmetic sequence can be expressed as:
\[
a_n = a + (n-1)d
\]
Plugging in the values for $a_n = 29$, $a = 2$, we get:
\[
29 = 2 + (n-1)d
\]
Simplifying, we find:
\[
27 = (n-1)d \quad \text{(1)}
\]
2. **Sum of the arithmetic sequence:**
The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} (a + a_n)
\]
Substituting $S_n = 155$, $a = 2$, and $a_n = 29$, we have:
\[
155 = \frac{n}{2} (2 + 29)
\]
Simplifying, we find:
\[
155 = \frac{n}{2} \cdot 31
\]
\[
310 = 31n
\]
\[
n = 10 \quad \text{(2)}
\]
3. **Finding the common difference $d$:**
Substituting $n = 10$ into equation (1):
\[
27 = (10-1)d
\]
\[
27 = 9d
\]
\[
d = \frac{27}{9} = 3
\]
Thus, the common difference $d$ is $3$.
### Conclusion:
The common difference of the arithmetic sequence is $\boxed{3}$, which corresponds to choice $\text{(A) } 3$.
|
3
|
deepscale
| 1,279
| |
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
*Author: Ray Li*
|
256
|
deepscale
| 18,672
| ||
A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\pi$).
|
144\pi
|
deepscale
| 27,252
| ||
If there are exactly $3$ integer solutions for the inequality system about $x$: $\left\{\begin{array}{c}6x-5≥m\\ \frac{x}{2}-\frac{x-1}{3}＜1\end{array}\right.$, and the solution to the equation about $y$: $\frac{y-2}{3}=\frac{m-2}{3}+1$ is a non-negative number, find the sum of all integers $m$ that satisfy the conditions.
|
-5
|
deepscale
| 31,390
| ||
In triangle $XYZ,$ points $G,$ $H,$ and $I$ are on sides $\overline{YZ},$ $\overline{XZ},$ and $\overline{XY},$ respectively, such that $YG:GZ = XH:HZ = XI:IY = 2:3.$ Line segments $\overline{XG},$ $\overline{YH},$ and $\overline{ZI}$ intersect at points $S,$ $T,$ and $U,$ respectively. Compute $\frac{[STU]}{[XYZ]}.$
|
\frac{9}{55}
|
deepscale
| 24,720
| ||
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$.
|
172822
|
deepscale
| 31,035
| ||
Let \( M \) be the centroid of \( \triangle ABC \), and \( AM = 3 \), \( BM = 4 \), \( CM = 5 \). Find the area of \( \triangle ABC \).
|
18
|
deepscale
| 15,336
| ||
Consider a large square where each side is divided into four equal parts. At each division, a point is placed. An inscribed square is constructed such that its corners are at these division points nearest to the center of each side of the large square. Calculate the ratio of the area of the inscribed square to the area of the large square.
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\sqrt{2}$
D) $\frac{3}{4}$
E) $1$
|
\frac{1}{2}
|
deepscale
| 21,413
| ||
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 180 = 0$ has integral solutions?
|
90
|
deepscale
| 10,055
| ||
Find the constant term in the expansion of \\((x+ \frac {2}{x}+1)^{6}\\) (Answer with a numerical value)
|
581
|
deepscale
| 22,412
| ||
Consider a memorable $9$-digit telephone number defined as $d_1d_2d_3d_4-d_5d_6d_7d_8d_9$. A number is memorable if the prefix sequence $d_1d_2d_3d_4$ is exactly the same as either of the sequences $d_5d_6d_7d_8$ or $d_6d_7d_8d_9$. Each digit $d_i$ can be any of the ten decimal digits $0$ through $9$. Find the number of different memorable telephone numbers.
A) 199980
B) 199990
C) 200000
D) 200010
E) 200020
|
199990
|
deepscale
| 12,707
| ||
The segments connecting the feet of the altitudes of an acute-angled triangle form a right triangle with a hypotenuse of 10. Find the radius of the circumcircle of the original triangle.
|
10
|
deepscale
| 15,799
| ||
A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.
Compute $G(G(G(G(G(1)))))$.
[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(-5,7,-4,10);
real f(real x) {return ((x-1)*(x-1)/2 - 3);}
draw(graph(f,1-sqrt(2*13),1+sqrt(2*13),operator ..), red);
[/asy]
|
5
|
deepscale
| 34,477
| ||
Two dice are differently designed. The first die has faces numbered $1$, $1$, $2$, $2$, $3$, and $5$. The second die has faces numbered $2$, $3$, $4$, $5$, $6$, and $7$. What is the probability that the sum of the numbers on the top faces of the two dice is $3$, $7$, or $8$?
A) $\frac{11}{36}$
B) $\frac{13}{36}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
|
\frac{13}{36}
|
deepscale
| 10,614
| ||
One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.
|
Given the problem, we have to find the number in position 210 under the constraints provided. We have 268 numbers written in a circle, denoted as \( a_1, a_2, \ldots, a_{268} \), and we know that the sum of any 20 consecutive numbers is 75.
This implies:
\[
a_i + a_{i+1} + \cdots + a_{i+19} = 75
\]
for all \( i \). Given the circular nature of the arrangement, indices wrap around. For example, \( a_{269} = a_1 \).
Given:
- \( a_{17} = 3 \)
- \( a_{83} = 4 \)
- \( a_{144} = 9 \)
We need to find \( a_{210} \).
Firstly, consider the sum relation:
\[
a_{k} + a_{k+1} + \cdots + a_{k+19} = 75
\]
Since every group of 20 numbers sums to 75, moving one position forward effectively means:
\[
a_{k+1} + a_{k+2} + \cdots + a_{k+20} = 75
\]
Considering overlapping sections and the constant sum, observe:
\[
a_{k+20} = a_k \text{ since the numbers repeat cyclically under constant sum conditions}
\]
Thus, all sets of 20 consecutive numbers sum to 75 implies the structure or behavior of \( a_i \) repeats after every 20 positions based on given information.
Now compute necessary differences:
The positions 17, 83, and 144 give specific values. Translating position numbers to mod 20 to exploit regular intervals within circle constraints:
- Position 210 \( \equiv 10 \pmod{20} \)
- Position 17 \( \equiv 17 \pmod{20} \)
- Position 83 \( \equiv 3 \pmod{20} \)
- Position 144 \( \equiv 4 \pmod{20} \)
Given that information is not directly useful in finding a pattern due to unknown explicit values.
However, via the given problem's specific placements and queries, solve by adding a small trial:
Set cyclic differences based on revealed positioning up-to identical modular intervals.
Thus, translating to closely examine \( a_{210} = a_{10} \):
Re-calculate visibly recurring calculations attributable through vicious iterations & breaks on initial constants reduction resulting in:
\[
a_{210} = \boxed{-1}
\]
This should be the sought number due to integer frameworks from assumed uniform distribution adjustments. Adjust results into continuity expectation via rational number simplification.
|
-1
|
deepscale
| 6,101
| |
Let $z$ be a complex number. In the complex plane, the distance from $z$ to 1 is 2 , and the distance from $z^{2}$ to 1 is 6 . What is the real part of $z$ ?
|
Note that we must have $|z-1|=2$ and \left|z^{2}-1\right|=6$, so $|z+1|=\frac{\left|z^{2}-1\right|}{|z-1|}=3$. Thus, the distance from $z$ to 1 in the complex plane is 2 and the distance from $z$ to -1 in the complex plane is 3 . Thus, $z, 1,-1$ form a triangle with side lengths $2,3,3$. The area of a triangle with sides $2,2,3$ can be computed to be \frac{3 \sqrt{7}}{4}$ by standard techniques, so the length of the altitude from $z$ to the real axis is \frac{3 \sqrt{7}}{4} \cdot \frac{2}{2}=\frac{3 \sqrt{7}}{4}$. The distance between 1 and the foot from $z$ to the real axis is \sqrt{2^{2}-\left(\frac{3 \sqrt{7}}{4}\right)^{2}}=\frac{1}{4}$ by the Pythagorean Theorem. It is clear that $z$ has positive imaginary part as the distance from $z$ to -1 is greater than the distance from $z$ to 1 , so the distance from 0 to the foot from $z$ to the real axis is $1+\frac{1}{4}=\frac{5}{4}$. This is exactly the real part of $z$ that we are trying to compute.
|
\frac{5}{4}
|
deepscale
| 4,483
| |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
draw((-5,i)--(5,i),gray(0.7));
}
draw((-5,0)--(5,0),Arrows(6));
draw((0,-5)--(0,5),Arrows(6));
label("$x$", (5,0), E);
label("$y$", (0,5), N);
draw(graph(func,-3,3),red);
label("$y = f(x)$", (3,-2), UnFill);
[/asy]
Which is the graph of $y = f(|x|)$?
[asy]
unitsize(0.5 cm);
picture[] graf;
int i, n;
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
real funca(real x) {
return(func(abs(x)));
}
real funcb(real x) {
real y = max(0,func(x));
return(y);
}
real funcd(real x) {
return(abs(func(x)));
}
real funce(real x) {
return(func(-abs(x)));
}
for (n = 1; n <= 5; ++n) {
graf[n] = new picture;
for (i = -5; i <= 5; ++i) {
draw(graf[n],(i,-5)--(i,5),gray(0.7));
draw(graf[n],(-5,i)--(5,i),gray(0.7));
}
draw(graf[n],(-5,0)--(5,0),Arrows(6));
draw(graf[n],(0,-5)--(0,5),Arrows(6));
label(graf[n],"$x$", (5,0), E);
label(graf[n],"$y$", (0,5), N);
}
draw(graf[1],graph(funca,-3,3),red);
draw(graf[2],graph(funcb,-3,3),red);
draw(graf[3],reflect((0,0),(0,1))*graph(func,-3,3),red);
draw(graf[4],graph(funcd,-3,3),red);
draw(graf[5],graph(funce,-3,3),red);
label(graf[1], "A", (0,-6));
label(graf[2], "B", (0,-6));
label(graf[3], "C", (0,-6));
label(graf[4], "D", (0,-6));
label(graf[5], "E", (0,-6));
add(graf[1]);
add(shift((12,0))*(graf[2]));
add(shift((24,0))*(graf[3]));
add(shift((6,-12))*(graf[4]));
add(shift((18,-12))*(graf[5]));
[/asy]
Enter the letter of the graph of $y = f(|x|).$
|
\text{A}
|
deepscale
| 37,435
| ||
Given the function $f(x)=x^{2}-3x$. If for any $x_{1}$, $x_{2}$ in the interval $[-3,2]$, we have $|f(x_{1})-f(x_{2})| \leqslant m$, then the minimum value of the real number $m$ is _______.
|
\frac{81}{4}
|
deepscale
| 10,616
| ||
In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\angle LMN$ is $60^{\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12.
|
\frac{4 \sqrt{3}}{3}
|
deepscale
| 28,423
| ||
Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\]
|
\frac{1}{3}
|
deepscale
| 22,212
| ||
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$
|
-80
|
deepscale
| 39,998
| ||
Find the total number of sets of positive integers \((x, y, z)\), where \(x, y\) and \(z\) are positive integers, with \(x < y < z\) such that
$$
x + y + z = 203.
$$
|
3333
|
deepscale
| 13,284
| ||
When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\frac{n}{6^{7}}$, where $n$ is a positive integer. What is $n$?
|
To find the number of ways to get a sum of $10$ when rolling $7$ fair $6$-sided dice, we can use the stars and bars method, considering the constraints that each die must show at least $1$ and at most $6$.
1. **Initial Setup**: Each die must show at least $1$. Therefore, we start by assigning $1$ to each die, reducing the problem to distributing $10 - 7 = 3$ additional points among the $7$ dice.
2. **Constraints**: Each die can show a maximum of $6$. Since we have already assigned $1$ to each die, the maximum additional points a die can receive is $5$. However, since we only have $3$ points to distribute, this constraint does not affect our calculation.
3. **Using Stars and Bars**: We need to distribute $3$ points among $7$ dice. Using the stars and bars method, we consider the $3$ points as stars and the separations between different dice as bars. There are $6$ bars needed to separate $7$ dice.
4. **Calculation**:
- The number of ways to arrange $3$ stars and $6$ bars is given by the combination formula $\binom{n+k-1}{k-1}$, where $n$ is the number of stars and $k$ is the number of bars (or groups).
- Here, $n = 3$ and $k = 7$, so we need to calculate $\binom{3+7-1}{7-1} = \binom{9}{6}$.
5. **Evaluating the Combination**:
\[
\binom{9}{6} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
\]
Thus, the number of ways to achieve a sum of $10$ when rolling $7$ dice, each showing at least $1$, is $\boxed{\textbf{(E) } 84}$.
|
84
|
deepscale
| 2,448
| |
Determine the binomial coefficient and the coefficient of the 4th term in the expansion of $\left( \left. x^{2}- \frac{1}{2x} \right. \right)^{9}$.
|
- \frac{21}{2}
|
deepscale
| 21,453
| ||
There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, and the final number displayed on the computer was 12. What is the smallest possible initial number on the computer?
|
27
|
deepscale
| 15,765
| ||
Given the quadratic equation $ax^{2}+bx+c=0$ with $a > 0$ and $b, c \in \mathbb{R}$, and the roots of the equation lying in the interval $(0, 2)$, determine the minimum value of the real number $a$ given that $25a+10b+4c \geqslant 4$ for $c \geqslant 1$.
|
\frac{16}{25}
|
deepscale
| 12,316
| ||
A large chest contains 10 smaller chests. In each of the smaller chests, either 10 even smaller chests are placed or nothing is placed. In each of those smaller chests, either 10 smaller chests are placed or none, and so on. After this, there are exactly 2006 chests with contents. How many are empty?
|
18054
|
deepscale
| 29,017
| ||
Given the function $f(x) = -x^2 + ax + 3$.
1. When $a=2$, find the interval over which $f(x)$ is monotonically increasing.
2. If $f(x)$ is an even function, find the maximum and minimum values of $f(x)$ on the interval $[-1,3]$.
|
-6
|
deepscale
| 18,403
| ||
The altitude of an equilateral triangle is $\sqrt6$ units. What is the area of the triangle, in square units? Express your answer in simplest radical form.
|
2\sqrt{3}
|
deepscale
| 38,698
| ||
A certain circle's area is $x$ square units, and its circumference is $y$ units. The value of $x + y$ is $80\pi$. What is the radius of the circle, in units?
|
8
|
deepscale
| 36,321
| ||
Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\sum_{k=1}^{2023} f(k) =$ ____.
|
1012
|
deepscale
| 30,962
| ||
Find $ 8^8 \cdot 4^4 \div 2^{28}$.
|
16
|
deepscale
| 38,923
| ||
There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers?
|
10890
|
deepscale
| 13,050
| ||
An equivalent of the expression
$\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right)+\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right)$, $xy \not= 0$,
is:
|
1. **Start by rewriting the expression**:
\[
\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right)+\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right)
\]
2. **Simplify each term separately**:
- For the first term:
\[
\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right) = \frac{(x^2+1)(y^2+1)}{xy}
\]
- For the second term:
\[
\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right) = \frac{(x^2-1)(y^2-1)}{xy}
\]
3. **Combine the terms over a common denominator**:
\[
\frac{(x^2+1)(y^2+1) + (x^2-1)(y^2-1)}{xy}
\]
4. **Expand both numerators**:
- Expanding $(x^2+1)(y^2+1)$:
\[
x^2y^2 + x^2 + y^2 + 1
\]
- Expanding $(x^2-1)(y^2-1)$:
\[
x^2y^2 - x^2 - y^2 + 1
\]
5. **Add the expanded expressions**:
\[
(x^2y^2 + x^2 + y^2 + 1) + (x^2y^2 - x^2 - y^2 + 1) = 2x^2y^2 + 2
\]
6. **Place the result over the common denominator**:
\[
\frac{2x^2y^2 + 2}{xy}
\]
7. **Simplify the expression**:
- Split the fraction:
\[
\frac{2x^2y^2}{xy} + \frac{2}{xy}
\]
- Simplify each term:
\[
2xy + \frac{2}{xy}
\]
8. **Conclude with the final answer**:
\[
\boxed{D}
\]
|
2xy+\frac{2}{xy}
|
deepscale
| 2,473
| |
With four standard six-sided dice in play, Vivian rolls all four and can choose to reroll any subset of them. To win, Vivian needs the sum of the four dice after possibly rerolling some of them to be exactly 12. Vivian plays optimally to maximize her chances of winning. What is the probability that she chooses to reroll exactly three of the dice?
**A)** $\frac{1}{72}$
**B)** $\frac{1}{12}$
**C)** $\frac{1}{10}$
**D)** $\frac{1}{8}$
**E)** $\frac{1}{6}$
|
\frac{1}{8}
|
deepscale
| 13,151
| ||
In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), two identical pieces are placed, with at most one piece per cell. There are ___ distinct ways to arrange the pieces. (If two arrangements can be made to coincide by rotation, they are considered the same arrangement).
|
10
|
deepscale
| 7,434
| ||
The real numbers \(x_{1}, x_{2}, \cdots, x_{2001}\) satisfy \(\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001\). Let \(y_{k}=\frac{1}{k}\left(x_{1}+ x_{2} + \cdots + x_{k}\right)\) for \(k=1, 2, \cdots, 2001\). Find the maximum possible value of \(\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|\).
|
2000
|
deepscale
| 27,582
| ||
In $\triangle ABC$, $\angle ABC = \angle ACB = 40^\circ$, and $P$ is a point inside the triangle such that $\angle PAC = 20^\circ$ and $\angle PCB = 30^\circ$. Find the measure of $\angle PBC$.
|
20
|
deepscale
| 15,682
| ||
A club has 99 members. Find the smallest positive integer $n$ such that if the number of acquaintances of each person is greater than $n$, there must exist 4 people who all know each other mutually (here it is assumed that if $A$ knows $B$, then $B$ also knows $A$).
|
66
|
deepscale
| 11,263
| ||
What is the value of $(2^0 - 1 + 5^2 - 0)^{-1} \times 5?$
|
1. **Evaluate the expression inside the parentheses**:
\[
2^0 - 1 + 5^2 - 0
\]
- \(2^0 = 1\) because any non-zero number raised to the power of 0 is 1.
- \(5^2 = 25\) because squaring 5 gives 25.
- Therefore, the expression simplifies to:
\[
1 - 1 + 25 - 0 = 25
\]
2. **Apply the exponent of \(-1\) to the result**:
\[
25^{-1}
\]
- \(25^{-1}\) is the reciprocal of 25, which is \(\frac{1}{25}\).
3. **Multiply by 5**:
\[
\frac{1}{25} \times 5
\]
- Multiplying \(\frac{1}{25}\) by 5 can be done by:
\[
\frac{1}{25} \times 5 = \frac{5}{25} = \frac{1}{5}
\]
4. **Conclude with the final answer**:
\[
\boxed{\textbf{(C)} \, \frac{1}{5}}
\]
|
\frac{1}{5}
|
deepscale
| 1,920
| |
The medians \( A M \) and \( B E \) of triangle \( A B C \) intersect at point \( O \). Points \( O, M, E, C \) lie on the same circle. Find \( A B \) if \( B E = A M = 3 \).
|
2\sqrt{3}
|
deepscale
| 15,245
| ||
Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles right triangle where one side of the square lies on the hypotenuse of the triangle. What is $\dfrac{x}{y}$?
A) $\frac{1}{\sqrt{2}}$
B) $1$
C) $\sqrt{2}$
D) $\frac{\sqrt{2}}{2}$
|
\sqrt{2}
|
deepscale
| 21,118
| ||
If 600 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?
|
22
|
deepscale
| 16,896
| ||
Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=2$, $a_{n+2}=(1+\cos ^{2} \frac {nπ}{2})a_{n}+\sin ^{2} \frac {nπ}{2}$, find the sum of the first 12 terms of the sequence.
|
147
|
deepscale
| 22,654
| ||
Sandy's daughter has a playhouse in the back yard. She plans to cover the one shaded exterior wall and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 8-foot by 12-foot sections that cost $\$27.30$ each. If Sandy can cut the siding when she gets home, how many dollars will be the cost of the siding Sandy must purchase?
[asy]
import three;
size(101);
currentprojection=orthographic(1/3,-1,1/2);
real w = 1.5;
real theta = pi/4;
string dottedline = "2 4";
draw(surface((0,0,0)--(8,0,0)--(8,0,6)--(0,0,6)--cycle),gray(.7)+opacity(.5));
draw(surface((0,0,6)--(0,5cos(theta),6+5sin(theta))--(8,5cos(theta),6+5sin(theta))--(8,0,6)--cycle),gray(.7)+opacity(.5));
draw(surface((0,5cos(theta),6+5sin(theta))--(8,5cos(theta),6+5sin(theta))--(8,10cos(theta),6)--(0,10cos(theta),6)--cycle),gray
(.7)+opacity(.5));
draw((0,0,0)--(8,0,0)--(8,0,6)--(0,0,6)--cycle,black+linewidth(w));
draw((0,0,6)--(0,5cos(theta),6+5sin(theta))--(8,5cos(theta),6+5sin(theta))--(8,0,6)--cycle,black+linewidth(w));
draw((8,0,0)--(8,10cos(theta),0)--(8,10cos(theta),6)--(8,5cos(theta),6+5sin(theta)),linewidth(w));
draw((0,0,0)--(0,10cos(theta),0)--(0,10cos(theta),6)--(0,0,6),linetype(dottedline));
draw((0,5cos(theta),6+5sin(theta))--(0,10cos(theta),6)--(8,10cos(theta),6)--(8,0,6),linetype(dottedline));
draw((0,10cos(theta),0)--(8,10cos(theta),0),linetype(dottedline));
label("8' ",(4,5cos(theta),6+5sin(theta)),N);
label("5' ",(0,5cos(theta)/2,6+5sin(theta)/2),NW);
label("6' ",(0,0,3),W);
label("8' ",(4,0,0),S);
[/asy]
|
\$ 54.60
|
deepscale
| 39,549
| ||
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.
|
\frac{3}{4}
|
deepscale
| 31,122
| ||
A dart board is shaped like a regular dodecagon divided into regions, with a square at its center. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
**A)** $\frac{2+\sqrt{3}}{6}$
**B)** $\frac{\sqrt{3}}{6}$
**C)** $\frac{2 - \sqrt{3}}{6}$
**D)** $\frac{3 - 2\sqrt{3}}{6}$
**E)** $\frac{2\sqrt{3} - 3}{6}$
|
\frac{2 - \sqrt{3}}{6}
|
deepscale
| 19,178
| ||
Simplify: $$\sqrt[3]{2744000}$$
|
140
|
deepscale
| 33,340
| ||
Let $a_1,$ $a_2,$ $a_3$ be the first three terms of a geometric sequence. If $a_1 = 1,$ find the smallest possible value of $4a_2 + 5a_3.$
|
-\frac{4}{5}
|
deepscale
| 36,911
| ||
Four vertices of a cube are given as \(A=(1, 2, 3)\), \(B=(1, 8, 3)\), \(C=(5, 2, 3)\), and \(D=(5, 8, 3)\). Calculate the surface area of the cube.
|
96
|
deepscale
| 26,621
| ||
An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?
|
We have an \(8 \times 8\) array filled with the numbers from 1 to 64, where consecutive numbers are adjacent either along a row or along a column. Our task is to find the minimum possible value of the sum of the numbers along a diagonal of this array.
### Analysis
Let's denote the elements of the array by \( a_{ij} \), where \(1 \leq i, j \leq 8\). The diagonal we are interested in is given by the elements \( a_{11}, a_{22}, a_{33}, \ldots, a_{88} \).
### Constraint
The constraint given is that consecutive numbers must be adjacent along a row or a column. Therefore, this array can be seen as some sort of path (like a Hamiltonian path) through the array starting from 1 and ending with 64, with each step moving to an adjacent cell either horizontally or vertically.
### Construction
To minimize the diagonal sum, we should try to place the smallest possible numbers on the diagonal. A reasonable strategy is to start the path at \(1\) and wrap around the rectangle in a spiral-like or zigzag manner to attempt to keep smaller numbers along the diagonal.
### Example Arrangement
Consider this specific arrangement to understand the spiral pattern:
\[
\begin{array}{cccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
16 & 17 & 18 & 19 & 20 & 21 & 22 & 9 \\
15 & 24 & 25 & 26 & 27 & 28 & 23 & 10 \\
14 & 32 & 33 & 34 & 35 & 29 & 30 & 11 \\
13 & 31 & 40 & 41 & 36 & 37 & 38 & 12 \\
44 & 43 & 42 & 39 & 46 & 47 & 48 & 20 \\
45 & 58 & 57 & 56 & 55 & 54 & 53 & 19 \\
64 & 63 & 62 & 61 & 60 & 59 & 52 & 21 \\
\end{array}
\]
### Calculating the Diagonal Sum
In this arrangement, the diagonal consists of the numbers:
- \(1, 17, 25, 34, 36, 47, 53, 64\).
Calculate the sum of these numbers:
\[
1 + 17 + 25 + 34 + 36 + 47 + 53 + 64 = 288.
\]
This setup is not optimal, but by continuing this logic and adjustments to reduce crossings over higher number positions, careful adjustments can lead to optimizing to the minimal sum.
### Proven Minimum
Through systematic construction and testing swaps along the array path to maintain consecutive adjacency, the minimum value that can be achieved for the diagonal sum without violating row or column adjacency turns out to be:
\[
\boxed{88}.
\]
This minimum exploits optimal intermediate number placement and diagonal construction alignment.
|
88
|
deepscale
| 6,181
| |
In $\triangle ABC$, $\angle A = 60^\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
|
\sqrt{3}
|
deepscale
| 25,858
| ||
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse defined by \( x^2 + 4y^2 = 5 \). The centers of the circles are located along the x-axis. Find the value of \( r \).
|
\frac{\sqrt{15}}{4}
|
deepscale
| 29,415
| ||
What is the smallest positive integer with exactly 12 positive integer divisors?
|
288
|
deepscale
| 27,284
| ||
The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?
A) 41
B) 49
C) 51
D) 59
E) 61
|
59
|
deepscale
| 24,283
| ||
A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has one vertex tangent to the edge of the circle at point $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.
|
9(\sqrt{3} - \pi)
|
deepscale
| 29,477
| ||
Given a moving circle $E$ that passes through the point $F(1,0)$, and is tangent to the line $l: x=-1$.
(Ⅰ) Find the equation of the trajectory $G$ for the center $E$ of the moving circle.
(Ⅱ) Given the point $A(3,0)$, if a line with a slope of $1$ intersects with the line segment $OA$ (not passing through the origin $O$ and point $A$) and intersects the curve $G$ at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$.
|
\frac{32 \sqrt{3}}{9}
|
deepscale
| 17,124
| ||
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
|
We need to count the number of distinct patterns formed by shading exactly two of the nine squares on a 3x3 grid, considering that patterns which can be matched by flips and/or turns are not considered different.
#### Case 1: At least one square is a vertex
Without loss of generality (WLOG), assume one square is in the upper-left corner. We analyze the possible positions for the second shaded square:
- **Diagonal positions**: There are two other vertices on the diagonal from the upper-left corner (bottom-right and center). Shading any of these creates a symmetric pattern.
- **Non-diagonal positions**: We only consider squares on one side of the diagonal due to symmetry. These squares are the upper-center, center-left, and center. Each of these creates a unique pattern when combined with the upper-left corner.
Thus, there are $2$ (diagonal) + $3$ (non-diagonal) = $5$ distinct patterns when one square is a vertex.
#### Case 2: At least one square is on an edge, but no square is on a vertex
Consider the middle square on any edge. We analyze the possible positions for the second shaded square:
- **Edge-Edge combinations**: If we choose another edge square (not adjacent to the first or on the same edge), there are two distinct patterns (one with squares on adjacent edges and one with squares on opposite edges).
- **Edge-Center combination**: Shading the center square and an edge square forms another distinct pattern.
Thus, there are $2$ (edge-edge) + $1$ (edge-center) = $3$ distinct patterns when no square is a vertex but at least one is on an edge.
#### Total distinct patterns
Adding the distinct patterns from both cases, we have $5$ (from Case 1) + $3$ (from Case 2) = $8$ distinct patterns.
Therefore, the total number of different patterns is $\boxed{8}$.
|
8
|
deepscale
| 2,409
| |
In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?
|
50
|
deepscale
| 17,556
| ||
Given positive integers \(a, b,\) and \(c\) that satisfy \(2017 \geq 10a \geq 100b \geq 1000c\), determine how many such ordered triples \((a, b, c)\) exist.
|
574
|
deepscale
| 11,018
| ||
For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides
$\prod_{n=2}^{5300}\text{pow}(n)$?
|
To solve this problem, we need to determine how many times the prime factor $67$ appears in the product $\prod_{n=2}^{5300}\text{pow}(n)$, where $\text{pow}(n)$ is defined as the largest power of the largest prime that divides $n$.
1. **Identify the largest prime factor of 2010**: The prime factorization of $2010$ is $2 \cdot 3 \cdot 5 \cdot 67$. The largest prime factor is $67$.
2. **Determine the range of $n$ where $67$ is the largest prime factor**: We need to find all integers $n$ such that $67$ is the largest prime factor and $n \leq 5300$. This includes numbers of the form $67^k \cdot m$ where $m$ is a product of primes less than $67$ and $k \geq 1$.
3. **Count the contributions of $67$ to the product**:
- For $n = 67$, $\text{pow}(67) = 67^1$.
- For $n = 67^2$, $\text{pow}(67^2) = 67^2$.
- For $n = 67^3, 67^4, \ldots$ up to the largest power of $67$ that is $\leq 5300$, $\text{pow}(67^k) = 67^k$.
- For each $n = 67^k \cdot m$ where $m$ is a product of primes less than $67$, $\text{pow}(n) = 67^k$.
4. **Calculate the maximum power of $67$ in the range**:
- The largest $k$ such that $67^k \leq 5300$ is found by solving $67^k \leq 5300$. We find that $k = 2$ since $67^3 = 300763$ exceeds $5300$.
- For $k = 1$, $n = 67 \cdot m$ where $m$ ranges over products of primes less than $67$. The number of such $n$ is the count of integers from $1$ to $\lfloor \frac{5300}{67} \rfloor = 79$ excluding multiples of $67$ greater than $1$.
- For $k = 2$, $n = 67^2 \cdot m$ where $m$ ranges over products of primes less than $67$. The number of such $n$ is the count of integers from $1$ to $\lfloor \frac{5300}{67^2} \rfloor = 1$.
5. **Summing up the contributions**:
- For $k = 1$, we have contributions from $n = 67, 67 \cdot 2, \ldots, 67 \cdot 79$ except for $67^2$.
- For $k = 2$, we have a contribution from $n = 67^2$ which counts twice.
Thus, the total count is $79 - 1 + 2 = 80$.
6. **Conclusion**: The largest integer $m$ such that $2010^m$ divides $\prod_{n=2}^{5300}\text{pow}(n)$ is determined by the power of $67$ in the product, which is $80$. However, the options provided in the problem statement do not include $80$. Rechecking the calculation, we realize that the correct count should be $79 - 1 + 1 = 79$ (since $67^2$ is counted once as part of the sequence and once more for its square). Therefore, the correct answer is $\boxed{77} \Rightarrow \boxed{D}$.
|
77
|
deepscale
| 1,984
| |
Circles of radius 3 and 4 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$.
|
24\pi
|
deepscale
| 17,251
| ||
The area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\frac{4}{3}$.
|
\frac{8}{3}
|
deepscale
| 30,749
| ||
Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers.
|
13p-30
|
deepscale
| 38,521
| ||
The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) can be simplified by converting each fraction to a decimal, and then calculating the sum.
|
0.333
|
deepscale
| 17,393
| ||
Given a $9 \times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how many unit squares will contain an even number of marks?
|
56. Consider the rectangles which contain the square in the $i$th row and $j$th column. There are $i$ possible positions for the upper edge of such a rectangle, $10-i$ for the lower edge, $j$ for the left edge, and $10-j$ for the right edge; thus we have $i(10-i) j(10-j)$ rectangles altogether, which is odd iff $i, j$ are both odd, i.e. iff $i, j \in\{1,3,5,7,9\}$. There are thus 25 unit squares which lie in an odd number of rectangles, so the answer is $81-25=56$.
|
56
|
deepscale
| 3,776
| |
Given the circle \(\Gamma: x^{2} + y^{2} = 1\) with two intersection points with the \(x\)-axis as \(A\) and \(B\) (from left to right), \(P\) is a moving point on circle \(\Gamma\). Line \(l\) passes through point \(P\) and is tangent to circle \(\Gamma\). A line perpendicular to \(l\) passes through point \(A\) and intersects line \(BP\) at point \(M\). Find the maximum distance from point \(M\) to the line \(x + 2y - 9 = 0\).
|
2\sqrt{5} + 2
|
deepscale
| 9,988
| ||
Formulas for shortened multiplication (other).
Common fractions
|
198719871987
|
deepscale
| 25,524
| ||
It costs 2.5 cents to copy a page. How many pages can you copy for $\$20$?
|
800
|
deepscale
| 38,648
| ||
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$
|
1
|
deepscale
| 36,384
| ||
One endpoint of a line segment is $(4,3)$ and its midpoint is $(2,9)$. What is the sum of the coordinates of the other endpoint?
|
15
|
deepscale
| 34,063
| ||
Let $A$ be a point on the circle $x^2 + y^2 + 4x - 4y + 4 = 0$, and let $B$ be a point on the parabola $y^2 = 8x$. Find the smallest possible distance $AB$.
|
\frac{1}{2}
|
deepscale
| 13,163
| ||
Compute \( 105 \times 95 \).
|
9975
|
deepscale
| 16,931
| ||
Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89.
|
1383
|
deepscale
| 13,767
| ||
When four lines intersect pairwise, and no three of them intersect at the same point, find the total number of corresponding angles formed by these lines.
|
48
|
deepscale
| 7,747
| ||
Given a geometric sequence {a_n} satisfies a_1 = 3, and a_1 + a_3 + a_5 = 21, find the value of a_3 + a_5 + a_7.
|
42
|
deepscale
| 20,125
| ||
Let the set $I = \{1,2,3,4,5\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$?
|
49
|
deepscale
| 32,461
| ||
Given the set $A=\{x|1 < x < k\}$ and the set $B=\{y|y=2x-5, x \in A\}$, if $A \cap B = \{x|1 < x < 2\}$, find the value of the real number $k$.
|
3.5
|
deepscale
| 17,848
| ||
Triangle $PAB$ is formed by three tangents to circle $O$ and $\angle APB = 40^\circ$. Find $\angle AOB$.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, O, P, R, S, T;
R = dir(115);
S = dir(230);
T = dir(270);
P = extension(R, R + rotate(90)*(R), T, T + rotate(90)*(T));
A = extension(S, S + rotate(90)*(S), T, T + rotate(90)*(T));
B = extension(R, R + rotate(90)*(R), S, S + rotate(90)*(S));
draw(Circle((0,0),1));
draw((R + 0.1*(R - P))--P--(T + 0.1*(T - P)));
draw(A--B--O--cycle);
label("$A$", A, dir(270));
label("$B$", B, NW);
label("$O$", O, NE);
label("$P$", P, SW);
label("$R$", R, NW);
//label("$S$", S, NE);
label("$T$", T, dir(270));
[/asy]
|
70^\circ
|
deepscale
| 35,754
| ||
What is the largest three-digit multiple of 9 whose digits' sum is 18?
|
990
|
deepscale
| 38,245
| ||
The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30^{\circ}$ larger than the other. What is the degree measure of the largest angle in the triangle?
|
1. **Identify the given information and the problem statement**: We are given that the sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these angles is $30^\circ$ larger than the other. We need to find the degree measure of the largest angle in the triangle.
2. **Convert the fraction of the right angle to degrees**: A right angle is $90^\circ$. Therefore, $\frac{6}{5}$ of a right angle is:
\[
\frac{6}{5} \times 90^\circ = 108^\circ
\]
This means the sum of the two angles is $108^\circ$.
3. **Set up an equation for the two angles**: Let $x$ be the measure of the smaller angle. Then, the measure of the larger angle, which is $30^\circ$ more than the smaller angle, is $x + 30^\circ$. The equation representing their sum is:
\[
x + (x + 30^\circ) = 108^\circ
\]
Simplifying this equation:
\[
2x + 30^\circ = 108^\circ \implies 2x = 108^\circ - 30^\circ = 78^\circ \implies x = \frac{78^\circ}{2} = 39^\circ
\]
Therefore, the smaller angle is $39^\circ$.
4. **Calculate the measure of the second angle**: Since the second angle is $30^\circ$ larger than the first:
\[
x + 30^\circ = 39^\circ + 30^\circ = 69^\circ
\]
So, the second angle is $69^\circ$.
5. **Use the Triangle Sum Theorem to find the third angle**: The Triangle Sum Theorem states that the sum of the angles in a triangle is $180^\circ$. Thus, the third angle is:
\[
180^\circ - (39^\circ + 69^\circ) = 180^\circ - 108^\circ = 72^\circ
\]
Therefore, the third angle is $72^\circ$.
6. **Determine the largest angle**: Comparing the angles $39^\circ$, $69^\circ$, and $72^\circ$, the largest angle is $72^\circ$.
7. **Conclusion**: The degree measure of the largest angle in the triangle is $\boxed{\mathrm{(B) \ } 72}$.
|
72
|
deepscale
| 2,788
| |
Wang Lei and her older sister walk from home to the gym to play badminton. It is known that the older sister walks 20 meters more per minute than Wang Lei. After 25 minutes, the older sister reaches the gym, and then realizes she forgot the racket. She immediately returns along the same route to get the racket and meets Wang Lei at a point 300 meters away from the gym. Determine the distance between Wang Lei's home and the gym in meters.
|
1500
|
deepscale
| 18,085
| ||
Let $p,$ $q,$ $r,$ $s,$ $t,$ $u$ be positive real numbers such that $p + q + r + s + t + u = 8.$ Find the minimum value of
\[\frac{1}{p} + \frac{9}{q} + \frac{16}{r} + \frac{25}{s} + \frac{36}{t} + \frac{49}{u}.\]
|
84.5
|
deepscale
| 27,162
| ||
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$ ?
|
16
|
deepscale
| 19,584
| ||
Find the sum of the first seven prime numbers that have a units digit of 7.
|
379
|
deepscale
| 38,432
| ||
The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\circ}$ to the base, is $9 \mathrm{~cm}^{3}$. Find the total surface area of the pyramid.
|
9 \sqrt{3} (1 + \sqrt{2})
|
deepscale
| 14,485
| ||
There are 2016 points arranged on a circle. We are allowed to jump 2 or 3 points clockwise at will.
How many jumps must we make at least to reach all the points and return to the starting point again?
|
2017
|
deepscale
| 12,453
| ||
How many positive integers $k$ are there such that $$\frac{k}{2013}(a+b)=\operatorname{lcm}(a, b)$$ has a solution in positive integers $(a, b)$?
|
First, we can let $h=\operatorname{gcd}(a, b)$ so that $(a, b)=(h A, h B)$ where $\operatorname{gcd}(A, B)=1$. Making these substitutions yields $\frac{k}{2013}(h A+h B)=h A B$, so $k=\frac{2013 A B}{A+B}$. Because $A$ and $B$ are relatively prime, $A+B$ shares no common factors with neither $A$ nor $B$, so in order to have $k$ be an integer, $A+B$ must divide 2013, and since $A$ and $B$ are positive, $A+B>1$. We first show that for different possible values of $A+B$, the values of $k$ generated are distinct. In particular, we need to show that $\frac{2013 A B}{A+B} \neq \frac{2013 A^{\prime} B^{\prime}}{A^{\prime}+B^{\prime}}$ whenever $A+B \neq A^{\prime}+B^{\prime}$. Assume that such an equality exists, and cross-multiplying yields $A B\left(A^{\prime}+B^{\prime}\right)=A^{\prime} B^{\prime}(A+B)$. Since $A B$ is relatively prime to $A+B$, we must have $A+B$ divide $A^{\prime}+B^{\prime}$. With a similar argument, we can show that $A^{\prime}+B^{\prime}$ must divide $A+B$, so $A+B=A^{\prime}+B^{\prime}$. Now, we need to show that for the same denominator $A+B$, the values of $k$ generated are also distinct for some relatively prime non-ordered pair $(A, B)$. Let $n=A+B=C+D$. Assume that $\frac{2013 A B}{n}=\frac{2013 C D}{n}$, or equivalently, $A(n-A)=C(n-C)$. After some rearrangement, we have $(C+A)(C-A)=n(C-A)$ This implies that either $C=A$ or $C=n-A=B$. But in either case, $(C, D)$ is some permutation of $(A, B)$. Our answer can therefore be obtained by summing up the totients of the factors of 2013 (excluding 1) and dividing by 2 since $(A, B)$ and $(B, A)$ correspond to the same $k$ value, so our answer is $\frac{2013-1}{2}=$ 1006.
|
1006
|
deepscale
| 3,313
| |
Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - 9 a b}{\lambda^3} \).
|
\frac{3\sqrt{3}}{2}
|
deepscale
| 27,795
| ||
Suppose that $y = \frac34x$ and $x^y = y^x$. The quantity $x + y$ can be expressed as a rational number $\frac {r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r + s$.
|
Substitute $y = \frac34x$ into $x^y = y^x$ and solve. \[x^{\frac34x} = \left(\frac34x\right)^x\] \[x^{\frac34x} = \left(\frac34\right)^x \cdot x^x\] \[x^{-\frac14x} = \left(\frac34\right)^x\] \[x^{-\frac14} = \frac34\] \[x = \frac{256}{81}\] \[y = \frac34x = \frac{192}{81}\] \[x + y = \frac{448}{81}\] \[448 + 81 = \boxed{529}\]
|
529
|
deepscale
| 6,973
| |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $b=\sqrt{2}a$. Find the maximum value of the area of $\triangle ABC$.
|
2\sqrt{2}
|
deepscale
| 16,471
| ||
A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?
A) 6
B) 10
C) 12
D) 18
E) 24
|
10
|
deepscale
| 31,102
| ||
Given the table showing the weekly reading times of $30$ students, with $7$ students reading $6$ hours, $8$ students reading $7$ hours, $5$ students reading $8$ hours, and $10$ students reading $9$ hours, find the median of the weekly reading times for these $30$ students.
|
7.5
|
deepscale
| 12,912
|
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