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The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased?
|
1598
|
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| 24,684
| ||
Given the function $f(x)= \begin{cases}x-1,0 < x\leqslant 2 \\ -1,-2\leqslant x\leqslant 0 \end{cases}$, and $g(x)=f(x)+ax$, where $x\in[-2,2]$, if $g(x)$ is an even function, find the value of the real number $a$.
|
-\dfrac{1}{2}
|
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| 18,803
| ||
A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if
|
To solve the inequality $\sqrt{x} < 2x$, we start by squaring both sides, assuming $x \geq 0$ since we are dealing with a square root and $x$ is given as positive:
1. Squaring both sides:
\[
(\sqrt{x})^2 < (2x)^2
\]
\[
x < 4x^2
\]
2. Rearrange the inequality:
\[
0 < 4x^2 - x
\]
\[
0 < x(4x - 1)
\]
3. Analyze the factors:
- The expression $x(4x - 1) = 0$ when $x = 0$ or $4x - 1 = 0$.
- Solving $4x - 1 = 0$ gives $x = \frac{1}{4}$.
4. Determine the sign of $x(4x - 1)$:
- We test intervals determined by the roots $x = 0$ and $x = \frac{1}{4}$.
- For $x > \frac{1}{4}$, both $x > 0$ and $4x - 1 > 0$, hence $x(4x - 1) > 0$.
- For $0 < x < \frac{1}{4}$, $x > 0$ but $4x - 1 < 0$, hence $x(4x - 1) < 0$.
- For $x = 0$ or $x = \frac{1}{4}$, $x(4x - 1) = 0$.
5. Conclusion:
- The inequality $x(4x - 1) > 0$ holds true when $x > \frac{1}{4}$.
Thus, the solution to the inequality $\sqrt{x} < 2x$ is $x > \frac{1}{4}$.
$\boxed{(A) \ x > \frac{1}{4}}$
|
\frac{1}{4}
|
deepscale
| 2,423
| |
The sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.
$(1) q > 1$ $(2) T_{2013} > 1$ $(3) S_{2012}a_{2013} < S_{2013}a_{2012}$ $(4)$ The smallest natural number $n$ for which $T_n > 1$ is $4025$ $(5) \min \left( T_n \right) = T_{2012}$
|
(1)(3)(4)(5)
|
deepscale
| 27,258
| ||
What is the greatest possible value of $x+y$ such that $x^{2} + y^{2} =90$ and $xy=27$?
|
12
|
deepscale
| 33,103
| ||
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers with $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
|
59
|
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| 36,035
| ||
Given the expansion of $\left(x-\frac{a}{x}\right)^{5}$, find the maximum value among the coefficients in the expansion.
|
10
|
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| 27,925
| ||
There are 23 socks in a drawer: 8 white and 15 black. Every minute, Marina goes to the drawer and pulls out a sock. If at any moment Marina has pulled out more black socks than white ones, she exclaims, "Finally!" and stops the process.
What is the maximum number of socks Marina can pull out before she exclaims, "Finally!"? The last sock Marina pulled out is included in the count.
|
17
|
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| 8,440
| ||
Let \( g(x) = x^2 - 4x \). How many distinct real numbers \( c \) satisfy \( g(g(g(g(c)))) = 2 \)?
|
16
|
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| 21,312
| ||
Convert the binary number $111011001001_{(2)}$ to its corresponding decimal number.
|
3785
|
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| 23,962
| ||
The base of the quadrilateral pyramid \( S A B C D \) is a rhombus \( A B C D \) with an acute angle at vertex \( A \). The height of the rhombus is 4, and the point of intersection of its diagonals is the orthogonal projection of vertex \( S \) onto the plane of the base. A sphere with radius 2 touches the planes of all the faces of the pyramid. Find the volume of the pyramid, given that the distance from the center of the sphere to the line \( A C \) is \( \frac{2 \sqrt{2}}{3} A B \).
|
8\sqrt{2}
|
deepscale
| 30,423
| ||
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
$\mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(E) \frac{1}{4} }$
|
\frac{1}{6}
|
deepscale
| 36,055
| ||
Find all values of \( x \) for which the smaller of the numbers \( \frac{1}{x} \) and \( \sin x \) is greater than \( \frac{1}{2} \). In the answer, provide the total length of the resulting intervals on the number line, rounding to the nearest hundredth if necessary.
|
2.09
|
deepscale
| 13,829
| ||
An integer greater than 9 and less than 100 is randomly chosen. What is the probability that its digits are different?
|
\frac{9}{10}
|
deepscale
| 35,195
| ||
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $I_{A}, I_{B}, I_{C}$ be the $A, B, C$ excenters of this triangle, and let $O$ be the circumcenter of the triangle. Let $\gamma_{A}, \gamma_{B}, \gamma_{C}$ be the corresponding excircles and $\omega$ be the circumcircle. $X$ is one of the intersections between $\gamma_{A}$ and $\omega$. Likewise, $Y$ is an intersection of $\gamma_{B}$ and $\omega$, and $Z$ is an intersection of $\gamma_{C}$ and $\omega$. Compute $$\cos \angle O X I_{A}+\cos \angle O Y I_{B}+\cos \angle O Z I_{C}$$
|
Let $r_{A}, r_{B}, r_{C}$ be the exradii. Using $O X=R, X I_{A}=r_{A}, O I_{A}=\sqrt{R\left(R+2 r_{A}\right)}$ (Euler's theorem for excircles), and the Law of Cosines, we obtain $$\cos \angle O X I_{A}=\frac{R^{2}+r_{A}^{2}-R\left(R+2 r_{A}\right)}{2 R r_{A}}=\frac{r_{A}}{2 R}-1$$ Therefore it suffices to compute $\frac{r_{A}+r_{B}+r_{C}}{2 R}-3$. Since $r_{A}+r_{B}+r_{C}-r=2 K\left(\frac{1}{-a+b+c}+\frac{1}{a-b+c}+\frac{1}{a+b-c}-\frac{1}{a+b+c}\right)=2 K \frac{8 a b c}{(4 K)^{2}}=\frac{a b c}{K}=4 R$ where $K=[A B C]$, this desired quantity the same as $\frac{r}{2 R}-1$. For this triangle, $r=4$ and $R=\frac{65}{8}$, so the answer is $\frac{4}{65 / 4}-1=-\frac{49}{65}$.
|
-\frac{49}{65}
|
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| 3,727
| |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$.
$(1)$ Write the equation of $C$;
$(2)$ Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. For what value of $k$ is $\overrightarrow{OA} \bot \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?
|
\frac{4\sqrt{65}}{17}
|
deepscale
| 8,029
| ||
From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board?
|
23
|
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| 13,470
| ||
Recall that the sum of the angles of a triangle is 180 degrees. In triangle $ABC$, angle $A$ is a right angle. Let $BM$ be the median of the triangle and $D$ be the midpoint of $BM$. It turns out that $\angle ABD = \angle ACD$. What are the measures of these angles?
|
30
|
deepscale
| 13,154
| ||
A trirectangular tetrahedron $M-ABC$ has three pairs of adjacent edges that are perpendicular, and a point $N$ inside the base triangle $ABC$ is at distances of $2\sqrt{2}$, $4$, and $5$ from the three faces respectively. Find the surface area of the smallest sphere that passes through both points $M$ and $N$.
|
49\pi
|
deepscale
| 24,344
| ||
The legs of a right triangle have lengths $\log_4 27$ and $\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$
|
243
|
deepscale
| 36,823
| ||
Calculate the difference between the sum of the first 2023 odd counting numbers and the sum of the first 2023 even counting numbers, where even numbers start from 2, and then multiply this difference by 3.
|
9063
|
deepscale
| 24,908
| ||
In $\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA}$.
|
- \dfrac {62}{3}
|
deepscale
| 16,542
| ||
Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle.
|
\sqrt{73}
|
deepscale
| 18,233
| ||
The nine points of this grid are equally spaced horizontally and vertically. The distance between two neighboring points is 1 unit. What is the area, in square units, of the region where the two triangles overlap?
[asy]
size(80);
dot((0,0)); dot((0,1));dot((0,2));dot((1,0));dot((1,1));dot((1,2));dot((2,0));dot((2,1));dot((2,2));
draw((0,0)--(2,1)--(1,2)--cycle, linewidth(0.6));
draw((2,2)--(0,1)--(1,0)--cycle, linewidth(0.6));
[/asy]
|
1
|
deepscale
| 36,130
| ||
In an experiment, a certain constant \( c \) is measured to be 2.43865 with an error range of \(\pm 0.00312\). The experimenter wants to publish the value of \( c \), with each digit being significant. This means that regardless of how large \( c \) is, the announced value of \( c \) (with \( n \) digits) must match the first \( n \) digits of the true value of \( c \). What is the most precise value of \( c \) that the experimenter can publish?
|
2.44
|
deepscale
| 17,807
| ||
At the beginning of the first day, a box contains 1 black ball, 1 gold ball, and no other balls. At the end of each day, for each gold ball in the box, 2 black balls and 1 gold ball are added to the box. If no balls are removed from the box, how many balls are in the box at the end of the seventh day?
|
At the beginning of the first day, the box contains 1 black ball and 1 gold ball. At the end of the first day, 2 black balls and 1 gold ball are added, so the box contains 3 black balls and 2 gold balls. At the end of the second day, $2 \times 2=4$ black balls and $2 \times 1=2$ gold balls are added, so the box contains 7 black balls and 4 gold balls. Continuing in this way, we find the following numbers of balls: End of Day 2: 7 black balls, 4 gold balls; End of Day 3: 15 black balls, 8 gold balls; End of Day 4: 31 black balls, 16 gold balls; End of Day 5: 63 black balls, 32 gold balls; End of Day 6: 127 black balls, 64 gold balls; End of Day 7: 255 black balls, 128 gold balls. At the end of the 7th day, there are thus 255+128=383 balls in the box.
|
383
|
deepscale
| 5,365
| |
A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number and less than 6? Express your answer as a common fraction.
|
\frac{3}{10}
|
deepscale
| 17,776
| ||
On the Cartesian plane in which each unit is one foot, a dog is tied to a post on the point $(4,3)$ by a $10$ foot rope. What is the greatest distance the dog can be from the origin?
|
15
|
deepscale
| 34,588
| ||
Circle $\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB+AC$ given that $BC=7$.
[asy]
unitsize(0.1 inch);
draw(circle((0,0),5));
dot((-13,0));
label("$A$",(-13,0),S);
draw((-14,-0.4)--(0,5.5));
draw((-14,0.4)--(0,-5.5));
draw((-3.3,5.5)--(-7.3,-5.5));
dot((0,0));
label("$O$",(0,0),SE);
dot((-4.8,1.5));
label("$T_3$",(-4.8,1.5),E);
dot((-1.7,4.7));
label("$T_1$",(-1.7,4.7),SE);
dot((-1.7,-4.7));
label("$T_2$",(-1.7,-4.7),SW);
dot((-3.9,3.9));
label("$B$",(-3.9,3.9),NW);
dot((-6.3,-2.8));
label("$C$",(-6.3,-2.8),SW);
[/asy]
|
17
|
deepscale
| 35,942
| ||
There exist unique nonnegative integers $A, B$ between 0 and 9, inclusive, such that $(1001 \cdot A+110 \cdot B)^{2}=57,108,249$. Find $10 \cdot A+B$.
|
We only need to bound for $A B 00$; in other words, $A B^{2} \leq 5710$ but $(A B+1)^{2} \geq 5710$. A quick check gives $A B=75$. (Lots of ways to get this...)
|
75
|
deepscale
| 4,818
| |
Suppose $a$, $b$, $c$ are positive integers such that $a+b+c=23$ and $\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.$ What is the sum of all possible distinct values of $a^2+b^2+c^2$?
|
We are given that $a+b+c=23$ and $\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9$, where $a$, $b$, and $c$ are positive integers. We need to find the sum of all possible distinct values of $a^2+b^2+c^2$.
#### Case Analysis:
Since $a+b+c=23$ is odd, the integers $a$, $b$, and $c$ cannot all be even. They must be either one odd and two evens or all odd.
**Case 1**: One odd and two evens.
- Without loss of generality, assume $a$ is odd, and $b$ and $c$ are even.
- $\gcd(a,b)$ and $\gcd(a,c)$ are odd (since $\gcd$ of an odd and even number is 1), and $\gcd(b,c)$ is even.
- Thus, $\gcd(a,b) + \gcd(b,c) + \gcd(c,a)$ would be even, contradicting the given condition that it equals 9 (odd).
- Therefore, this case is not possible.
**Case 2**: All three numbers are odd.
- All gcd values $\gcd(a,b)$, $\gcd(b,c)$, and $\gcd(c,a)$ are odd.
We consider subcases based on the values of the gcds:
**Case 2.1**: $\gcd(a,b) = 1$, $\gcd(b,c) = 1$, $\gcd(c,a) = 7$.
- The only solution that satisfies $a+b+c=23$ is $(a, b, c) = (7, 9, 7)$.
- Calculate $a^2+b^2+c^2 = 7^2 + 9^2 + 7^2 = 49 + 81 + 49 = 179$.
**Case 2.2**: $\gcd(a,b) = 1$, $\gcd(b,c) = 3$, $\gcd(c,a) = 5$.
- The only solution that satisfies $a+b+c=23$ is $(a, b, c) = (5, 3, 15)$.
- Calculate $a^2+b^2+c^2 = 5^2 + 3^2 + 15^2 = 25 + 9 + 225 = 259$.
**Case 2.3**: $\gcd(a,b) = 3$, $\gcd(b,c) = 3$, $\gcd(c,a) = 3$.
- There is no solution that satisfies $a+b+c=23$ with these gcd conditions.
#### Conclusion:
The possible distinct values of $a^2+b^2+c^2$ are 179 and 259. Adding these gives $179 + 259 = 438$.
Thus, the sum of all possible distinct values of $a^2+b^2+c^2$ is $\boxed{438}$.
|
438
|
deepscale
| 2,404
| |
A coin is tossed. If heads appear, point \( P \) moves +1 on the number line; if tails appear, point \( P \) does not move. The coin is tossed no more than 12 times, and if point \( P \) reaches coordinate +10, the coin is no longer tossed. In how many different ways can point \( P \) reach coordinate +10?
|
66
|
deepscale
| 15,058
| ||
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\frac{S}{b^{2}+2c^{2}}$.
|
\frac{\sqrt{14}}{24}
|
deepscale
| 12,864
| ||
Let $m,n$ be natural numbers such that $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ .
|
70
|
deepscale
| 11,603
| ||
Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty?
|
126
|
deepscale
| 25,633
| ||
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$.
\\
\\
Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.
|
Let \( n \geq 3 \) be an odd number and suppose that each square in an \( n \times n \) chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex. Two squares \( a \) and \( b \) are considered connected if there exists a sequence of squares \( c_1, \ldots, c_k \) with \( c_1 = a \) and \( c_k = b \) such that \( c_i \) and \( c_{i+1} \) are adjacent for \( i = 1, 2, \ldots, k-1 \).
We aim to find the maximal number \( M \) such that there exists a coloring admitting \( M \) pairwise disconnected squares.
To solve this problem, we need to consider the structure of the chessboard and the properties of the coloring. The key insight is to analyze the number of disjoint maximal monochromatic components in the board.
For a general \( (2m+1) \times (2n+1) \) board, we can prove that the maximal number of disjoint components is given by:
\[
M = (m+1)(n+1) + 1.
\]
This result can be established through induction and careful analysis of the board's configuration. The proof involves considering different types of configurations and using combinatorial arguments to bound the number of components.
Hence, the maximal number \( M \) of pairwise disconnected squares in an \( n \times n \) chessboard, where \( n \) is an odd number, is:
\[
M = \left(\frac{n+1}{2}\right)^2 + 1.
\]
The answer is: \boxed{\left(\frac{n+1}{2}\right)^2 + 1}.
|
\left(\frac{n+1}{2}\right)^2 + 1
|
deepscale
| 2,982
| |
Given the parabola $y^{2}=4x$, and the line $l$: $y=- \frac {1}{2}x+b$ intersects the parabola at points $A$ and $B$.
(I) If the $x$-axis is tangent to the circle with $AB$ as its diameter, find the equation of the circle;
(II) If the line $l$ intersects the negative semi-axis of $y$, find the maximum area of $\triangle AOB$.
|
\frac {32 \sqrt {3}}{9}
|
deepscale
| 19,043
| ||
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon.
|
745
|
deepscale
| 35,748
| ||
A company has 45 male employees and 15 female employees. A 4-person research and development team was formed using stratified sampling.
(1) Calculate the probability of an employee being selected and the number of male and female employees in the research and development team;
(2) After a month of learning and discussion, the research team decided to select two employees for an experiment. The method is to first select one employee from the team to conduct the experiment, and after that, select another employee from the remaining team members to conduct the experiment. Calculate the probability that exactly one female employee is among the two selected employees;
(3) After the experiment, the first employee to conduct the experiment obtained the data 68, 70, 71, 72, 74, and the second employee obtained the data 69, 70, 70, 72, 74. Which employee's experiment is more stable? Explain your reasoning.
|
\frac {1}{2}
|
deepscale
| 27,746
| ||
20. Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ passes through point $M\left( 1,\frac{3}{2} \right)$, $F_1$ and $F_2$ are the two foci of ellipse $C$, and $\left| MF_1 \right|+\left| MF_2 \right|=4$, $O$ is the center of ellipse $C$.
(1) Find the equation of ellipse $C$;
(2) Suppose $P,Q$ are two different points on ellipse $C$, and $O$ is the centroid of $\Delta MPQ$, find the area of $\Delta MPQ$.
|
\frac{9}{2}
|
deepscale
| 27,738
| ||
Find all real numbers $x$ satisfying $$x^{9}+\frac{9}{8} x^{6}+\frac{27}{64} x^{3}-x+\frac{219}{512}=0$$
|
Note that we can re-write the given equation as $$\sqrt[3]{x-\frac{3}{8}}=x^{3}+\frac{3}{8}$$ Furthermore, the functions of $x$ on either side, we see, are inverses of each other and increasing. Let $f(x)=\sqrt[3]{x-\frac{3}{8}}$. Suppose that $f(x)=y=f^{-1}(x)$. Then, $f(y)=x$. However, if $x<y$, we have $f(x)>f(y)$, contradicting the fact that $f$ is increasing, and similarly, if $y<x$, we have $f(x)<f(y)$, again a contradiction. Therefore, if $f(x)=f^{-1}(x)$ and both are increasing functions in $x$, we require $f(x)=x$. This gives the cubic $$x^{3}-x+\frac{3}{8}=0 \rightarrow\left(x-\frac{1}{2}\right)\left(x^{2}+\frac{1}{2} x-\frac{3}{4}\right)=0$$ giving $x=\frac{1}{2}, \frac{-1 \pm \sqrt{13}}{4}$.
|
$\frac{1}{2}, \frac{-1 \pm \sqrt{13}}{4}$
|
deepscale
| 4,560
| |
Using the digits 0 to 9, how many three-digit even numbers can be formed without repeating any digits?
|
288
|
deepscale
| 26,818
| ||
Given that $\tan \alpha = 2$, find the value of $\frac{2 \sin \alpha - \cos \alpha}{\sin \alpha + 2 \cos \alpha}$.
|
\frac{3}{4}
|
deepscale
| 8,989
| ||
Let $a_1=24$ and form the sequence $a_n$ , $n\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ?
|
88
|
deepscale
| 18,250
| ||
What is the largest prime factor of the sum of $1579$ and $5464$?
|
7043
|
deepscale
| 26,386
| ||
Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $A B, A C, A D, B C, B D, C D$. If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$.
|
This problem has far more information than necessary: $Q R$ and $U V$ are both parallel to $B C$, and $Q U$ and $R V$ are both parallel to $A D$. Hence, $Q U V R$ is a parallelogram, and the desired distance is simply the same as the side length $Q R$, namely 2001.
|
2001
|
deepscale
| 3,214
| |
Someone collected data relating the average temperature x (℃) during the Spring Festival to the sales y (ten thousand yuan) of a certain heating product. The data pairs (x, y) are as follows: (-2, 20), (-3, 23), (-5, 27), (-6, 30). Based on the data, using linear regression, the linear regression equation between sales y and average temperature x is found to be $y=bx+a$ with a coefficient $b=-2.4$. Predict the sales amount when the average temperature is -8℃.
|
34.4
|
deepscale
| 20,018
| ||
During the past summer, 100 graduates from the city of $N$ applied to 5 different universities in our country. It turned out that during the first and second waves, each university was unable to reach exactly half of the applicants to that university. In addition, representatives of at least three universities were unable to reach those graduates. What is the maximum number of graduates from the city of $N$ who could have been of interest to the military recruitment office?
|
83
|
deepscale
| 28,768
| ||
In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 \qquad \textbf{(E)}\ 7:1$
|
3:1
|
deepscale
| 36,082
| ||
Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?
|
1. **Identify the properties of the quadrilateral**: Given that $ABCD$ is a convex quadrilateral with $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. The lengths of the sides form an arithmetic progression, and $AB$ is the longest side.
2. **Introduce a point $E$ on $\overline{AB}$**: Construct point $E$ such that $BCDE$ is a parallelogram. This implies $BC = ED = b$, $CD = BE = c$, and $DA = d$. Also, $AE = 18 - c$.
3. **Apply the Law of Cosines in $\triangle ADE$**:
\[
DE^2 = AD^2 + AE^2 - 2 \cdot AD \cdot AE \cdot \cos(60^\circ)
\]
Substituting the known values and expressions, we get:
\[
b^2 = d^2 + (18-c)^2 - d(18-c)
\]
Rearranging terms, we obtain:
\[
(18-c)(18-c-d) = (b+d)(b-d)
\]
This equation will be referred to as $(\bigstar)$.
4. **Define the common difference $k$**: Let $k$ be the common difference of the arithmetic progression. The possible values for $b, c, d$ are $18-k, 18-2k, 18-3k$ in some order. The condition $0 \leq k < 6$ ensures $AB$ remains the longest side.
5. **Analyze cases based on the order of $b, c, d$**:
- **Case 1**: $(b, c, d) = (18-k, 18-2k, 18-3k)$
- Substituting into $(\bigstar)$, we find $k = 6$, which is invalid as $k < 6$.
- **Case 2**: $(b, c, d) = (18-k, 18-3k, 18-2k)$
- Substituting into $(\bigstar)$, we find $k = 5$, yielding $(b, c, d) = (13, 3, 8)$.
- **Case 3**: $(b, c, d) = (18-2k, 18-k, 18-3k)$
- Substituting into $(\bigstar)$, we find $k = 6$, which is invalid.
- **Case 4**: $(b, c, d) = (18-2k, 18-3k, 18-k)$
- Substituting into $(\bigstar)$, we find $k = 2$, yielding $(b, c, d) = (14, 12, 16)$.
- **Case 5**: $(b, c, d) = (18-3k, 18-k, 18-2k)$
- Substituting into $(\bigstar)$, we find $k = 9$, which is invalid.
- **Case 6**: $(b, c, d) = (18-3k, 18-2k, 18-k)$
- Substituting into $(\bigstar)$, we find $k = 18$, which is invalid.
6. **Sum all valid values of $a$**: From the valid cases, the values of $a$ are $13, 3, 8, 14, 12, 16$. Adding these, we get:
\[
13 + 3 + 8 + 14 + 12 + 16 = 66
\]
Thus, the sum of all possible values of $a$ is $\boxed{\textbf{(D) } 66}$.
|
66
|
deepscale
| 2,663
| |
Find the minimum possible value of
\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4},\]
given that $a,b,c,d,$ are nonnegative real numbers such that $a+b+c+d=4$ .
|
See here: https://artofproblemsolving.com/community/c5t211539f5h1434574_looks_like_mount_inequality_erupted_
or:
https://www.youtube.com/watch?v=LSYP_KMbBNc
|
\[\frac{1}{2}\]
|
deepscale
| 3,183
| |
There is a moving point \( P \) on the \( x \)-axis. Given the fixed points \( A(0,2) \) and \( B(0,4) \), what is the maximum value of \( \sin \angle APB \) when \( P \) moves along the entire \( x \)-axis?
|
\frac{1}{3}
|
deepscale
| 23,664
| ||
Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
|
1. **Identify the total number of marbles**: Initially, Tyrone has 97 marbles and Eric has 11 marbles. The total number of marbles is:
\[
97 + 11 = 108
\]
2. **Set up the equation after redistribution**: Let's denote the number of marbles Tyrone gives to Eric as $x$. After giving $x$ marbles to Eric, Tyrone has $97 - x$ marbles and Eric has $11 + x$ marbles. According to the problem, Tyrone ends up with twice as many marbles as Eric. Therefore, we can set up the equation:
\[
97 - x = 2(11 + x)
\]
3. **Solve the equation**:
\[
97 - x = 22 + 2x
\]
\[
97 - 22 = 2x + x
\]
\[
75 = 3x
\]
\[
x = \frac{75}{3} = 25
\]
4. **Verify the solution**: After Tyrone gives 25 marbles to Eric:
- Tyrone's marbles: $97 - 25 = 72$
- Eric's marbles: $11 + 25 = 36$
Check the ratio:
\[
72 = 2 \times 36
\]
This confirms that Tyrone has twice as many marbles as Eric after giving 25 marbles.
5. **Conclusion**: Tyrone gave 25 marbles to Eric. The correct answer is $\boxed{D = 25}$.
|
18
|
deepscale
| 1,090
| |
Two boards, one 5 inches wide and the other 7 inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are later separated, what is the area of the unpainted region on the five-inch board? Assume the holes caused by the nails are negligible.
|
35\sqrt{2}
|
deepscale
| 26,634
| ||
A factory produces a certain product with an annual fixed cost of $250 million. When producing $x$ million units, an additional cost of $C(x)$ million is required. When the annual production volume is less than 80 million units, $C(x)= \frac {1}{3}x^{2}+10x$; when the annual production volume is not less than 80 million units, $C(x)=51x+ \frac {10000}{x}-1450$. Assume that each unit of the product is sold for $50 million and all produced units can be sold that year.
1. Write out the annual profit $L(x)$ (million) as a function of the annual production volume $x$ (million units).
2. What is the production volume when the factory's profit from producing this product is the highest? What is the maximum profit?
|
1000
|
deepscale
| 26,741
| ||
The letters $A$, $B$, $C$ and $D$ represent digits. If $\begin{array}{ccc}&A&B\\ +&C&A\\ \hline &D&A\end{array}$and $\begin{array}{ccc}&A&B\\ -&C&A\\ \hline &&A\end{array}$,what digit does $D$ represent?
|
1. **Analyze the first equation:**
Given the addition problem:
\[
\begin{array}{cc}
& A\ B \\
+ & C\ A \\
\hline
& D\ A \\
\end{array}
\]
From the units column, we have $B + A = A$ or $B + A = A + 10$ (considering a possible carry from the tens column).
2. **Determine the value of $B$:**
Since $B + A = A$, it implies $B = 0$ (as there is no carry from the tens column that would make $B + A = A + 10$ valid).
3. **Analyze the second equation:**
Given the subtraction problem:
\[
\begin{array}{cc}
& A\ B \\
- & C\ A \\
\hline
& \ \ A \\
\end{array}
\]
From the units column, we have $B - A = 0$. Since $B = 0$, this implies $0 - A = 0$, which is only possible if $A = 0$. However, this contradicts the fact that $A$ must be a non-zero digit (as it appears in the tens place in the result). Therefore, we must consider the possibility of borrowing, which gives $10 + B - A = A$. Since $B = 0$, we have $10 - A = A$, leading to $2A = 10$ and thus $A = 5$.
4. **Substitute $A = 5$ and $B = 0$ into the equations:**
The addition problem becomes:
\[
\begin{array}{cc}
& 5\ 0 \\
+ & C\ 5 \\
\hline
& D\ 5 \\
\end{array}
\]
From the tens column, $5 + C = D$.
5. **Analyze the subtraction problem with $A = 5$ and $B = 0$:**
\[
\begin{array}{cc}
& 5\ 0 \\
- & C\ 5 \\
\hline
& \ \ 5 \\
\end{array}
\]
From the tens column, $5 - C = 0$ (no borrowing needed), so $C = 4$.
6. **Calculate $D$:**
Substitute $C = 4$ into $5 + C = D$, we get $D = 5 + 4 = 9$.
Thus, the digit $D$ represents is $\boxed{\textbf{(E)}\ 9}$.
|
9
|
deepscale
| 1,513
| |
An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length 1 unit either up or to the right. How many up-right paths from $(0,0)$ to $(7,7)$, when drawn in the plane with the line $y=x-2.021$, enclose exactly one bounded region below that line?
|
We will make use of a sort of bijection which is typically used to prove the closed form for the Catalan numbers. We will count these paths with complementary counting. Since both the starting and ending points are above the line $x-2.021$, any path which traverses below this line (and hence includes a point on the line $y=x-3$ ) will enclose at least one region. In any such path, we can reflect the portion of the path after the first visit to the line $y=x-3$ over that line to get a path from $(0,0)$ to $(10,4)$. This process is reversible for any path to $(10,4)$, so the number of paths enclosing at least one region is $\binom{14}{4}$. More difficult is to count the paths that enclose at least two regions. For any such path, consider the first and final times it intersects the line $y=x-3$. Since at least two regions are enclosed, there must be some point on the intermediate portion of the path on the line $y=x-2$. Then we can reflect only this portion of the path over the line $y=x-3$ to get a new path containing a point on the line $y=x-4$. We can then do a similar reflection starting from the first such point to get a path from $(0,0)$ to $(11,3)$. This process is reversible, so the number of paths which enclose at least two regions is $\binom{14}{3}$. Then the desired answer is just $\binom{14}{4}-\binom{14}{3}=637$.
|
637
|
deepscale
| 3,926
| |
A cashier from Aeroflot has to deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba", "Rossiya", and "Minsk". The fourth group's address will be given by tourists from "Rossiya", and the fifth group's address will be given by tourists from "Minsk". In how many ways can the cashier choose the order of visiting the hotels to deliver the tickets?
|
30
|
deepscale
| 14,577
| ||
Find the remainder when $109876543210$ is divided by $180$.
|
10
|
deepscale
| 37,686
| ||
We define a number as an ultimate mountain number if it is a 4-digit number and the third digit is larger than the second and fourth digit but not necessarily the first digit. For example, 3516 is an ultimate mountain number. How many 4-digit ultimate mountain numbers are there?
|
204
|
deepscale
| 25,146
| ||
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents?
|
To solve this problem, we need to calculate the probability that the value of the coins drawn is at least 50 cents. We will first determine the total number of ways to draw 6 coins from the 12 available, and then find the number of successful outcomes where the total value is at least 50 cents.
1. **Total Outcomes**:
The total number of ways to choose 6 coins out of 12 is given by the binomial coefficient:
\[
\binom{12}{6} = 924
\]
This represents the total possible outcomes.
2. **Successful Outcomes**:
We consider different combinations of coins that sum to at least 50 cents:
- **Case 1: 1 penny and 5 dimes**:
- Ways to choose 1 penny from 2: $\binom{2}{1} = 2$
- Ways to choose 5 dimes from 6: $\binom{6}{5} = 6$
- Total for this case: $2 \times 6 = 12$
- **Case 2: 2 nickels and 4 dimes**:
- Ways to choose 2 nickels from 4: $\binom{4}{2} = 6$
- Ways to choose 4 dimes from 6: $\binom{6}{4} = 15$
- Total for this case: $6 \times 15 = 90$
- **Case 3: 1 nickel and 5 dimes**:
- Ways to choose 1 nickel from 4: $\binom{4}{1} = 4$
- Ways to choose 5 dimes from 6: $\binom{6}{5} = 6$
- Total for this case: $4 \times 6 = 24$
- **Case 4: 6 dimes**:
- Only one way to choose all 6 dimes: $\binom{6}{6} = 1$
- Total for this case: $1$
Summing all successful outcomes:
\[
12 + 90 + 24 + 1 = 127
\]
3. **Probability Calculation**:
The probability of drawing coins worth at least 50 cents is the ratio of successful outcomes to total outcomes:
\[
\frac{127}{924}
\]
Thus, the probability that the value of the coins drawn is at least 50 cents is $\boxed{\text{C}} \ \frac{127}{924}$.
|
\frac{127}{924}
|
deepscale
| 1,698
| |
There are ten numbers \( x_1, x_2, \cdots, x_{10} \), where the maximum number is 10 and the minimum number is 2. Given that \( \sum_{i=1}^{10} x_i = 70 \), find the maximum value of \( \sum_{i=1}^{10} x_i^2 \).
|
628
|
deepscale
| 15,074
| ||
Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0<r$ and $0\leq \theta <360$. Find $\theta$.
|
\begin{eqnarray*} 0 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \frac{z^5-1}{z-1}\\ 0 &=& \frac{(z^5 - 1)(z(z-1)+1)}{z-1} = \frac{(z^2-z+1)(z^5-1)}{z-1} \end{eqnarray*}
Thus $z^5 = 1, z \neq 1 \Longrightarrow z = \mathrm{cis}\ 72, 144, 216, 288$,
or $z^2 - z + 1 = 0 \Longrightarrow z = \frac{1 \pm \sqrt{-3}}{2} = \mathrm{cis}\ 60, 300$
(see cis).
Discarding the roots with negative imaginary parts (leaving us with $\mathrm{cis} \theta,\ 0 < \theta < 180$), we are left with $\mathrm{cis}\ 60, 72, 144$; their product is $P = \mathrm{cis} (60 + 72 + 144) = \mathrm{cis} \boxed{276}$.
|
276
|
deepscale
| 6,621
| |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$?
|
\frac{1}{8}
|
deepscale
| 35,118
| ||
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$.
|
-64
|
deepscale
| 33,270
| ||
Given that $\alpha \in \left(\frac{\pi}{2}, \pi\right)$ and $\sin \alpha = \frac{4}{5}$, calculate the value of $\sin 2\alpha$.
|
-\frac{24}{25}
|
deepscale
| 17,090
| ||
If a certain number of cats ate a total of 999,919 mice, and all cats ate the same number of mice, how many cats were there in total? Additionally, each cat ate more mice than there were cats.
|
991
|
deepscale
| 7,547
| ||
Let $g(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $g(2)=0$, $90<g(9)<100$, $120<g(10)<130$, $7000k<g(150)<7000(k+1)$ for some integer $k$. What is $k$?
|
k=6
|
deepscale
| 10,317
| ||
The organizers of a ping-pong tournament have only one table. They call two participants to play, who have not yet played against each other. If after the game the losing participant suffers their second defeat, they are eliminated from the tournament (since there are no ties in tennis). After 29 games, it turned out that all participants were eliminated except for two. How many participants were there in the tournament?
|
16
|
deepscale
| 7,578
| ||
A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$?
|
15
|
deepscale
| 33,420
| ||
Given \( m > n \geqslant 1 \), find the smallest value of \( m + n \) such that
\[ 1000 \mid 1978^{m} - 1978^{n} . \
|
106
|
deepscale
| 7,610
| ||
Given a regular quadrilateral pyramid $S-ABCD$ with side edges of length $4$ and $\angle ASB = 30^\circ$, a plane passing through point $A$ intersects the side edges $SB$, $SC$, and $SD$ at points $E$, $F$, and $G$ respectively. Find the minimum perimeter of the cross-section $AEFG$.
|
4\sqrt{3}
|
deepscale
| 14,783
| ||
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$ , $x_2$ , $\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .
*Proposed by Le Duc Minh*
|
64
|
deepscale
| 24,176
| ||
What is the product of all real numbers that are tripled when added to their reciprocals?
|
-\frac{1}{2}
|
deepscale
| 23,575
| ||
Given that $a+b=3$ and $a^3+b^3=81$, find $ab$.
|
-6
|
deepscale
| 37,366
| ||
Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \_\_\_\_\_ kg.
|
70
|
deepscale
| 28,241
| ||
For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b.$
|
51
|
deepscale
| 36,540
| ||
In $\triangle ABC$, $B(-\sqrt{5}, 0)$, $C(\sqrt{5}, 0)$, and the sum of the lengths of the medians on sides $AB$ and $AC$ is $9$.
(Ⅰ) Find the equation of the trajectory of the centroid $G$ of $\triangle ABC$.
(Ⅱ) Let $P$ be any point on the trajectory found in (Ⅰ), find the minimum value of $\cos\angle BPC$.
|
-\frac{1}{9}
|
deepscale
| 24,540
| ||
The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\%$. By what percent must the larger measure be decreased so that the two angles remain complementary?
|
8\%
|
deepscale
| 38,652
| ||
Given that the coefficients $p$ and $q$ are integers and the roots $\alpha_{1}$ and $\alpha_{2}$ are irrational, a quadratic trinomial $x^{2} + px + q$ is called an irrational quadratic trinomial. Determine the minimum sum of the absolute values of the roots among all irrational quadratic trinomials.
|
\sqrt{5}
|
deepscale
| 8,645
| ||
Emma has $100$ fair coins. She flips all the coins once. Any coin that lands on tails is tossed again, and the process continues up to four times for coins that sequentially land on tails. Calculate the expected number of coins that finally show heads.
|
93.75
|
deepscale
| 23,210
| ||
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 4. The point $K$ is defined such that $\overrightarrow{L K}=3 \overrightarrow{F A}-\overrightarrow{F B}$. Determine whether $K$ lies inside, on the boundary, or outside of $A B C D E F$, and find the length of the segment $K A$.
|
\frac{4 \sqrt{3}}{3}
|
deepscale
| 9,284
| ||
Given the function $f(x)=2\sqrt{3}\sin^2{x}-\sin\left(2x-\frac{\pi}{3}\right)$,
(Ⅰ) Find the smallest positive period of the function $f(x)$ and the intervals where $f(x)$ is monotonically increasing;
(Ⅱ) Suppose $\alpha\in(0,\pi)$ and $f\left(\frac{\alpha}{2}\right)=\frac{1}{2}+\sqrt{3}$, find the value of $\sin{\alpha}$;
(Ⅲ) If $x\in\left[-\frac{\pi}{2},0\right]$, find the maximum value of the function $f(x)$.
|
\sqrt{3}+1
|
deepscale
| 30,129
| ||
There are 294 distinct cards with numbers \(7, 11, 7^{2}, 11^{2}, \ldots, 7^{147}, 11^{147}\) (each card has exactly one number, and each number appears exactly once). How many ways can two cards be selected so that the product of the numbers on the selected cards is a perfect square?
|
15987
|
deepscale
| 15,981
| ||
Consider the function
\[f(x) = \max \{-11x - 37, x - 1, 9x + 3\}\]defined for all real $x.$ Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $x_1,$ $x_2,$ $x_3.$ Find $x_1 + x_2 + x_3.$
|
-\frac{11}{2}
|
deepscale
| 37,462
| ||
A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+2k-99\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
|
\frac{2}{3}
|
deepscale
| 13,159
| ||
What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$
|
20
|
deepscale
| 37,855
| ||
Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$, find the maximum possible value of $a$.
|
\frac{50}{99}
|
deepscale
| 29,957
| ||
What is the least common multiple of 220 and 504?
|
27720
|
deepscale
| 7,606
| ||
The value of \( 333 + 33 + 3 \) is:
|
369
|
deepscale
| 22,801
| ||
Calculate: $99\times 101=\_\_\_\_\_\_$.
|
9999
|
deepscale
| 9,035
| ||
How many subsets of the set $\{1,2,3,4,5\}$ contain the number 5?
|
16
|
deepscale
| 35,172
| ||
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$
|
227.052227052227
|
deepscale
| 25,736
| ||
Given a right prism with all vertices on the same sphere, with a height of $4$ and a volume of $32$, the surface area of this sphere is ______.
|
32\pi
|
deepscale
| 19,767
| ||
Express $326_{13} + 4C9_{14}$ as a base 10 integer, where $C = 12$ in base 14.
|
1500
|
deepscale
| 22,721
| ||
How many positive two-digit integers are multiples of 5 and of 7?
|
2
|
deepscale
| 39,186
| ||
The segment connecting the centers of two intersecting circles is divided by their common chord into segments of 4 and 1. Find the length of the common chord, given that the radii of the circles are in the ratio $3:2$.
|
2 \sqrt{11}
|
deepscale
| 13,035
| ||
Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$
|
\frac{5}{2}
|
deepscale
| 18,694
| ||
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
|
Let $R$ denote the action where George selects a red ball and $B$ denote the action where he selects a blue one. To achieve a final state of three red balls and three blue balls, George must select two additional red balls and two additional blue balls during the four operations.
The possible sequences of operations that result in three red and three blue balls are: $RRBB$, $RBRB$, $RBBR$, $BBRR$, $BRBR$, and $BRRB$. The number of such sequences is given by $\binom{4}{2} = 6$, which corresponds to choosing two of the four operations to be red (the rest will be blue).
#### Case Analysis:
Each sequence has a symmetric counterpart (e.g., $RRBB$ is symmetric to $BBRR$), and we can calculate the probability for one sequence in each pair and then double it.
**Case 1: $RRBB$ and $BBRR$**
- Probability of $RRBB$:
1. First ball is red: $\frac{1}{2}$
2. Second ball is red (2 reds, 1 blue): $\frac{2}{3}$
3. Third ball is blue (3 reds, 1 blue): $\frac{1}{4}$
4. Fourth ball is blue (3 reds, 2 blues): $\frac{2}{5}$
Total probability for $RRBB$:
\[
\frac{1}{2} \times \frac{2}{3} \times \frac{1}{4} \times \frac{2}{5} = \frac{1}{30}
\]
By symmetry, the probability for $BBRR$ is the same, so the combined probability for this case is:
\[
2 \times \frac{1}{30} = \frac{1}{15}
\]
**Case 2: $RBRB$ and $BRBR$**
- Probability of $RBRB$:
1. First ball is red: $\frac{1}{2}$
2. Second ball is blue (2 reds, 1 blue): $\frac{1}{3}$
3. Third ball is red (2 reds, 2 blues): $\frac{1}{2}$
4. Fourth ball is blue (3 reds, 2 blues): $\frac{2}{5}$
Total probability for $RBRB$:
\[
\frac{1}{2} \times \frac{1}{3} \times \frac{1}{2} \times \frac{2}{5} = \frac{1}{30}
\]
By symmetry, the probability for $BRBR$ is the same, so the combined probability for this case is:
\[
2 \times \frac{1}{30} = \frac{1}{15}
\]
**Case 3: $RBBR$ and $BRRB$**
- Probability of $RBBR$:
1. First ball is red: $\frac{1}{2}$
2. Second ball is blue (2 reds, 1 blue): $\frac{1}{3}$
3. Third ball is blue (2 reds, 2 blues): $\frac{1}{2}$
4. Fourth ball is red (2 reds, 3 blues): $\frac{2}{5}$
Total probability for $RBBR$:
\[
\frac{1}{2} \times \frac{1}{3} \times \frac{1}{2} \times \frac{2}{5} = \frac{1}{30}
\]
By symmetry, the probability for $BRRB$ is the same, so the combined probability for this case is:
\[
2 \times \frac{1}{30} = \frac{1}{15}
\]
Adding up the probabilities of all cases:
\[
\frac{1}{15} + \frac{1}{15} + \frac{1}{15} = \frac{3}{15} = \frac{1}{5}
\]
Thus, the probability that the urn contains three balls of each color is $\boxed{\textbf{(B) }\frac15}$.
|
\frac{1}{5}
|
deepscale
| 1,722
| |
How long should a covered lip pipe be to produce the fundamental pitch provided by a standard tuning fork?
|
0.189
|
deepscale
| 12,630
|
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