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stringlengths 10
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stringclasses 890
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stringlengths 0
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|---|---|---|---|---|---|
In $\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos B = \frac{4}{5}$ and $b = 2$.
1. Find the value of $a$ when $A = \frac{\pi}{6}$.
2. Find the value of $a + c$ when the area of $\triangle ABC$ is $3$.
|
2\sqrt{10}
|
deepscale
| 23,064
| ||
Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$ . Let
\[AO = 5, BO =6, CO = 7, DO = 8.\]
If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ , determine $\frac{OM}{ON}$ .
|
35/48
|
deepscale
| 28,734
| ||
An omino is a 1-by-1 square or a 1-by-2 horizontal rectangle. An omino tiling of a region of the plane is a way of covering it (and only it) by ominoes. How many omino tilings are there of a 2-by-10 horizontal rectangle?
|
There are exactly as many omino tilings of a 1-by-$n$ rectangle as there are domino tilings of a 2-by-$n$ rectangle. Since the rows don't interact at all, the number of omino tilings of an $m$-by-$n$ rectangle is the number of omino tilings of a 1-by-$n$ rectangle raised to the $m$ th power, $F_{n}^{m}$. The answer is $89^{2}=7921$.
|
7921
|
deepscale
| 3,434
| |
Let \( A, B, C, D, E, F, G, H \) be distinct digits from 0 to 7 that satisfy the following equation:
\[ \overrightarrow{A B C} + \overline{D E} = \overline{F G H} \]
Find \(\overline{D E}\) if \(\overline{A B C} = 146\).
(Note: \(\overline{A B C}\) denotes a three-digit number composed of digits \( A, B, \) and \( C \), similarly, \(\overline{F G H}\) and \(\overline{D E}\) are constructed.)
|
57
|
deepscale
| 8,889
| ||
The graph of the parabola $x = 2y^2 - 6y + 3$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.
|
6
|
deepscale
| 34,118
| ||
Round to the nearest hundredth: 18.4851
|
18.49
|
deepscale
| 39,176
| ||
Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located.
|
\frac{\pi}{4}
|
deepscale
| 30,211
| ||
A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is found that if the price of each shirt is reduced by $1, the store can sell an additional 2 shirts per day on average. If the store makes an average daily profit of $2000, what should be the selling price of each shirt?
|
120
|
deepscale
| 28,920
| ||
Find the product of all constants $t$ such that the quadratic $x^2 + tx + 6$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.
|
1225
|
deepscale
| 17,256
| ||
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 35, what is the probability that this number will be divisible by 11?
A) $\frac{1}{4}$
B) $\frac{1}{8}$
C) $\frac{1}{5}$
D) $\frac{1}{10}$
E) $\frac{1}{15}$
|
\frac{1}{8}
|
deepscale
| 31,265
| ||
Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Calculate the sum $S$ of this series.
|
\frac{5}{26}
|
deepscale
| 14,533
| ||
$(1)$ Calculate: $2^{-1}+|\sqrt{6}-3|+2\sqrt{3}\sin 45^{\circ}-\left(-2\right)^{2023}\cdot (\frac{1}{2})^{2023}$.
$(2)$ Simplify and then evaluate: $\left(\frac{3}{a+1}-a+1\right) \div \frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-1$, $2$, $3$ for evaluation.
|
-4
|
deepscale
| 22,991
| ||
In Perfectville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Jane runs around the block on the $500$-foot side of the street, while John runs on the opposite side of the street. How many more feet than Jane does John run for every lap around the block?
|
240
|
deepscale
| 18,929
| ||
Given that the terminal side of angle $\alpha$ ($0 < \alpha < \frac{\pi}{2}$) passes through the point $(\cos 2\beta, 1+\sin 3\beta \cos \beta - \cos 3\beta \sin \beta)$, where $\frac{\pi}{2} < \beta < \pi$ and $\beta \neq \frac{3\pi}{4}$, calculate $\alpha - \beta$.
|
-\frac{3\pi}{4}
|
deepscale
| 29,143
| ||
A square $WXYZ$ with side length 8 units is divided into four smaller squares by drawing lines from the midpoints of one side to the midpoints of the opposite sides. The top right square of each iteration is shaded. If this dividing and shading process is done 100 times, what is the total area of the shaded squares?
A) 18
B) 21
C) $\frac{64}{3}$
D) 25
E) 28
|
\frac{64}{3}
|
deepscale
| 8,895
| ||
Given that $\sin \alpha = 3 \sin \left(\alpha + \frac{\pi}{6}\right)$, find the value of $\tan \left(\alpha + \frac{\pi}{12}\right)$.
|
2 \sqrt{3} - 4
|
deepscale
| 24,671
| ||
Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed?
|
The problem is equivalent to finding the number of ways to partition 10 into a sum of three (unordered) positive integers. These can be computed by hand to be $(1,1,8),(1,2,7),(1,3,6),(1,4,5),(2,2,6),(2,3,5),(2,4,4),(3,3,4)$
|
8
|
deepscale
| 3,377
| |
Given that $\operatorname{tg} \theta$ and $\operatorname{ctg} \theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\alpha < \theta < \frac{5 \pi}{4}$, find the value of $\cos \theta - \sin \theta$.
|
-\sqrt{\frac{5 - 2\sqrt{5}}{5}}
|
deepscale
| 14,520
| ||
1990-1980+1970-1960+\cdots -20+10 =
|
1. **Identify the pattern and the number of terms**: The sequence given is $1990 - 1980 + 1970 - 1960 + \cdots - 20 + 10$. We observe that the sequence alternates between addition and subtraction, starting with a subtraction. The sequence starts at $1990$ and ends at $10$, decreasing by $10$ each step.
2. **Calculate the total number of terms**: The sequence decreases by $10$ each step from $1990$ to $10$. The number of terms can be calculated by finding how many steps it takes to go from $1990$ to $10$:
\[
\frac{1990 - 10}{10} + 1 = 198 + 1 = 199 \text{ terms}
\]
3. **Group the terms**: We notice that every pair of terms (starting from the second term) forms a group that sums to $10$. For example, $(-1980 + 1970) = -10$, $(+1960 - 1950) = +10$, and so on. Each of these pairs sums to $10$.
4. **Count the number of pairs**: Since there are $199$ terms, the first $198$ terms can be grouped into $99$ pairs (as each pair consists of two terms):
\[
\frac{198}{2} = 99 \text{ pairs}
\]
5. **Sum of all pairs**: Each pair sums to $10$, so the sum of all $99$ pairs is:
\[
99 \times 10 = 990
\]
6. **Add the last term**: The last term, which is $10$, was not included in any pair. Therefore, we add this term to the sum of all pairs:
\[
990 + 10 = 1000
\]
7. **Conclusion**: The sum of the entire sequence is $1000$. Therefore, the answer is $\boxed{\text{D}}$.
|
1000
|
deepscale
| 2,583
| |
Find the positive integer $n\,$ for which \[\lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994\] (For real $x\,$, $\lfloor x\rfloor\,$ is the greatest integer $\le x.\,$)
|
Note that if $2^x \le a<2^{x+1}$ for some $x\in\mathbb{Z}$, then $\lfloor\log_2{a}\rfloor=\log_2{2^{x}}=x$.
Thus, there are $2^{x+1}-2^{x}=2^{x}$ integers $a$ such that $\lfloor\log_2{a}\rfloor=x$. So the sum of $\lfloor\log_2{a}\rfloor$ for all such $a$ is $x\cdot2^x$.
Let $k$ be the integer such that $2^k \le n<2^{k+1}$. So for each integer $j<k$, there are $2^j$ integers $a\le n$ such that $\lfloor\log_2{a}\rfloor=j$, and there are $n-2^k+1$ such integers such that $\lfloor\log_2{a}\rfloor=k$.
Therefore, $\lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor= \sum_{j=0}^{k-1}(j\cdot2^j) + k(n-2^k+1) = 1994$.
Through computation: $\sum_{j=0}^{7}(j\cdot2^j)=1538<1994$ and $\sum_{j=0}^{8}(j\cdot2^j)=3586>1994$. Thus, $k=8$.
So, $\sum_{j=0}^{k-1}(j\cdot2^j) + k(n-2^k+1) = 1538+8(n-2^8+1)=1994 \Rightarrow n = \boxed{312}$.
Alternatively, one could notice this is an arithmetico-geometric series and avoid a lot of computation.
|
312
|
deepscale
| 6,584
| |
In rectangle $ABCD$, $AB=5$ and $BC =3$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF = 1$ and $GC=2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$. Express your answer as a common fraction. [asy]
pair A,B,C,D,I,F,G;
A=(0,0);
B=(5,0);
C=(5,3);
D=(0,3);
F=(1,3);
G=(3,3);
I=(1.67,5);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--B--I--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$F$",F,SE);
label("$G$",G,SW);
label("$E$",I,N);
label("1",(0.5,3),N);
label("2",(4,3),N);
label("3",(0,1.5),W);
label("3",(5,1.5),E);
label("5",(2.5,0),S);
[/asy]
|
\frac{25}{2}
|
deepscale
| 35,851
| ||
Given the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) with its right focus at $F(c, 0)$. A circle centered at the origin $O$ with radius $c$ intersects the hyperbola in the first quadrant at point $A$. The tangent to the circle at point $A$ has a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola.
|
\sqrt{2}
|
deepscale
| 7,448
| ||
For each interger $n\geq 4$, we consider the $m$ subsets $A_1, A_2,\dots, A_m$ of $\{1, 2, 3,\dots, n\}$, such that
$A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. Find the maximum value of $m$.
|
To find the maximum value of \( m \), we need to evaluate the constraints given by the problem, specifically that none of the subsets \( A_1, A_2, \ldots, A_m \) is contained in any other subset. Each subset \( A_i \) has exactly \( i \) elements.
Let's consider the key points to derive the maximum value of \( m \):
1. **Condition on Subsets:**
- There are \( n \) distinct elements in total.
- Subset \( A_1 \) contains exactly 1 element, \( A_2 \) contains exactly 2, and so on, up to \( A_m \), which contains exactly \( m \) elements.
- No subset \( A_i \) is contained within another subset \( A_j \).
2. **Binomial Coefficient Representation:**
- The total number of ways to choose subsets of varying sizes from a set with \( n \) elements is given by the binomial coefficients.
- Therefore, for subset \( A_i \), a key constraint is that \(\binom{n}{i}\) must account for at least \( i \) elements being distinct in the choice so that none of the subsets are contained within another.
3. **Deriving the Maximum \( m \):**
- To adhere to the condition that none of these subsets is contained in any other subset, the sum of sizes of all subsets cannot exceed \( n \), the total number of distinct elements:
\[
1 + 2 + \dots + m \leq n
\]
- The left hand side is the sum of the first \( m \) positive integers, which is \(\frac{m(m + 1)}{2}\).
- So, we form the inequality:
\[
\frac{m(m + 1)}{2} \leq n
\]
4. **Solving the Inequality:**
- Multiply both sides by 2 to clear the fraction:
\[
m(m + 1) \leq 2n
\]
- This quadratic inequality can be solved for \( m \) by rearranging and setting up the quadratic equation:
\[
m^2 + m - 2n = 0
\]
- Using the quadratic formula:
\[
m = \frac{-1 \pm \sqrt{1 + 8n}}{2}
\]
- Since \( m \) has to be a positive integer and we want the maximum \( m \), we use the floor function:
\[
m = \left\lfloor \frac{-1 + \sqrt{1 + 8n}}{2} \right\rfloor
\]
Thus, the maximum value of \( m \) is:
\[
\boxed{\left\lfloor \frac{-1 + \sqrt{1 + 8n}}{2} \right\rfloor}
\]
|
\left\lfloor \frac{-1 + \sqrt{1 + 8n}}{2} \right\rfloor
|
deepscale
| 6,401
| |
Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \max\{f(x),g(x),h(x)\},$$$$k(x) = \min\{f(x),g(x),h(x)\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greatest; similarly, $k(x)$ is the least of these three values.
Shown below is the graph of $y=j(x)$ for $-3.5\le x\le 3.5$.
Let $\ell$ be the length of the graph of $y=k(x)$ for $-3.5\le x\le 3.5$. What is the value of $\ell^2$?
[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(-5,5,-5,5);
draw((-3.5,5)--(-2,2)--(2,2)--(3.5,5),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy]
|
245
|
deepscale
| 33,940
| ||
Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$ .
|
79
|
deepscale
| 30,119
| ||
Independent trials are conducted, in each of which event \( A \) can occur with a probability of 0.001. What is the probability that in 2000 trials, event \( A \) will occur at least two and at most four times?
|
0.541
|
deepscale
| 13,722
| ||
What is the remainder of $5^{2010}$ when it is divided by 7?
|
1
|
deepscale
| 37,764
| ||
Simplify
\[\left( \frac{1 + i}{1 - i} \right)^{1000}.\]
|
1
|
deepscale
| 36,394
| ||
A circle with a radius of 2 is inscribed in triangle \(ABC\) and touches side \(BC\) at point \(D\). Another circle with a radius of 4 touches the extensions of sides \(AB\) and \(AC\), as well as side \(BC\) at point \(E\). Find the area of triangle \(ABC\) if the measure of angle \(\angle ACB\) is \(120^{\circ}\).
|
\frac{56}{\sqrt{3}}
|
deepscale
| 8,118
| ||
How many positive integers \( n \) are there such that \( n \) is a multiple of 4, and the least common multiple of \( 4! \) and \( n \) equals 4 times the greatest common divisor of \( 8! \) and \( n \)?
|
12
|
deepscale
| 25,104
| ||
Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$.
|
\frac {175}{264}
|
deepscale
| 18,970
| ||
Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$
|
Dividing both sides by $n!^{2}$, we obtain $$\begin{aligned} \frac{1!2!\ldots(n-3)!(n-2)!(n-1)!}{[n(n-1)!][n(n-1)(n-2)!]} & >1 \\ \frac{1!2!\ldots(n-3)!}{n^{2}(n-1)} & >1 \\ 1!2!\ldots(n-3)! & >n^{2}(n-1) \end{aligned}$$ Factorials are small at first, so we can rule out some small cases: when $n=6$, the left hand side is $1!2!3!=12$, which is much smaller than $6^{2} \cdot 5$. (Similar calculations show that $n=1$ through $n=5$ do not work. either.) Setting $n=7$, the left-hand side is 288 , which is still smaller than $7^{2} \cdot 6$. However, $n=8$ gives $34560>448$, so 8 is the smallest integer for which the inequality holds.
|
8
|
deepscale
| 4,454
| |
A certain scenic area has two attractions that require tickets for visiting. The three ticket purchase options presented at the ticket office are as follows:
Option 1: Visit attraction A only, $30$ yuan per person;
Option 2: Visit attraction B only, $50$ yuan per person;
Option 3: Combined ticket for attractions A and B, $70$ yuan per person.
It is predicted that in April, $20,000$ people will choose option 1, $10,000$ people will choose option 2, and $10,000$ people will choose option 3. In order to increase revenue, the ticket prices are adjusted. It is found that when the prices of options 1 and 2 remain unchanged, for every $1$ yuan decrease in the price of the combined ticket (option 3), $400$ people who originally planned to buy tickets for attraction A only and $600$ people who originally planned to buy tickets for attraction B only will switch to buying the combined ticket.
$(1)$ If the price of the combined ticket decreases by $5$ yuan, the number of people buying tickets for option 1 will be _______ thousand people, the number of people buying tickets for option 2 will be _______ thousand people, the number of people buying tickets for option 3 will be _______ thousand people; and calculate how many tens of thousands of yuan the total ticket revenue will be?
$(2)$ When the price of the combined ticket decreases by $x$ (yuan), find the functional relationship between the total ticket revenue $w$ (in tens of thousands of yuan) in April and $x$ (yuan), and determine at what price the combined ticket should be to maximize the total ticket revenue in April. What is the maximum value in tens of thousands of yuan?
|
188.1
|
deepscale
| 27,807
| ||
In the diagram, \(AB\) is a diameter of a circle with center \(O\). \(C\) and \(D\) are points on the circle. \(OD\) intersects \(AC\) at \(P\), \(OC\) intersects \(BD\) at \(Q\), and \(AC\) intersects \(BD\) at \(R\). If \(\angle BOQ = 60^{\circ}\) and \(\angle APO = 100^{\circ}\), calculate the measure of \(\angle BQO\).
|
95
|
deepscale
| 25,470
| ||
The union of sets $A$ and $B$ is $A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\}$. When $A \neq B$, the pairs $(A, B)$ and $(B, A)$ are considered different. How many such pairs $(A, B)$ are there?
|
27
|
deepscale
| 20,740
| ||
Two positive integers \( x \) and \( y \) have \( xy=24 \) and \( x-y=5 \). What is the value of \( x+y \)?
|
The positive integer divisors of 24 are \( 1,2,3,4,6,8,12,24 \). The pairs of divisors that give a product of 24 are \( 24 \times 1,12 \times 2,8 \times 3 \), and \( 6 \times 4 \). We want to find two positive integers \( x \) and \( y \) whose product is 24 and whose difference is 5. Since \( 8 \times 3=24 \) and \( 8-3=5 \), then \( x=8 \) and \( y=3 \) are the required integers. Here, \( x+y=8+3=11 \).
|
11
|
deepscale
| 5,320
| |
Find the smallest positive prime that divides \( n^2 + 5n + 23 \) for some integer \( n \).
|
17
|
deepscale
| 7,821
| ||
For each real number $x$, let $\lfloor x \rfloor$ denote the greatest integer that does not exceed x. For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_{2} n \rfloor$ is a positive even integer?
|
For integers $k$, we want $\lfloor \log_2 n\rfloor = 2k$, or $2k \le \log_2 n < 2k+1 \Longrightarrow 2^{2k} \le n < 2^{2k+1}$. Thus, $n$ must satisfy these inequalities (since $n < 1000$):
$4\leq n <8$
$16\leq n<32$
$64\leq n<128$
$256\leq n<512$
There are $4$ for the first inequality, $16$ for the second, $64$ for the third, and $256$ for the fourth, so the answer is $4+16+64+256=\boxed{340}$.
|
340
|
deepscale
| 6,612
| |
In triangle $XYZ$, the medians $\overline{XT}$ and $\overline{YS}$ are perpendicular. If $XT = 15$ and $YS = 20$, find the length of side $XZ$.
|
\frac{50}{3}
|
deepscale
| 32,084
| ||
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $|a_n-a_{n-1}|= \frac {1}{2^n}$ $(n\geqslant 2,n\in\mathbb{N})$, and the subsequence $\{a_{2n-1}\}$ is decreasing, while $\{a_{2n}\}$ is increasing, find the value of $5-6a_{10}$.
|
\frac {1}{512}
|
deepscale
| 12,997
| ||
The three row sums and the three column sums of the array \[
\left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right]
\]
are the same. What is the least number of entries that must be altered to make all six sums different from one another?
|
1. **Identify the initial sums**: First, we calculate the row sums and column sums of the given matrix:
\[
\begin{matrix}
4 & 9 & 2\\
8 & 1 & 6\\
3 & 5 & 7
\end{matrix}
\]
- Row sums: $4+9+2=15$, $8+1+6=15$, $3+5+7=15$
- Column sums: $4+8+3=15$, $9+1+5=15$, $2+6+7=15$
All sums are equal to 15.
2. **Consider changing 3 entries**: We analyze the effect of changing 3 entries in the matrix. Two possible configurations are:
- Changing one entry in each row and column (sudoku-style):
\[
\begin{matrix}
* & 9 & 2\\
8 & * & 6\\
3 & 5 & *
\end{matrix}
\]
- Leaving at least one row and one column unchanged:
\[
\begin{matrix}
* & 9 & 2\\
* & * & 6\\
3 & 5 & 7
\end{matrix}
\]
3. **Analyze the effect of changes**:
- In the sudoku-style change, altering one entry in each row and column affects two sums (one row and one column) for each change. This configuration does not guarantee all sums will be different because each altered entry affects exactly two sums, potentially keeping some sums equal.
- In the second configuration, where one row and one column remain unchanged, those unchanged sums remain equal to each other, thus not all sums can be made different.
4. **Determine if 4 changes are sufficient**: Building on the second configuration, we consider changing one more entry in either the untouched row or column:
\[
\begin{matrix}
* & 9 & 2\\
* & * & 6\\
3 & * & 7
\end{matrix}
\]
By choosing appropriate values for the changed entries (e.g., setting them to zero), we can achieve different sums for all rows and columns.
5. **Conclusion**: Changing 4 entries allows us to make all row and column sums different. Therefore, the least number of entries that must be altered to make all six sums different from one another is $\boxed{4}$.
|
4
|
deepscale
| 2,022
| |
In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. Calculate the length of side $AB$.
|
\sqrt{17}
|
deepscale
| 18,229
| ||
Consider the sum $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ .
|
3031
|
deepscale
| 31,454
| ||
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $\sqrt{3}c\sin A = a\cos C$.
$(I)$ Find the value of $C$;
$(II)$ If $c=2a$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$.
|
\frac{\sqrt{15} - \sqrt{3}}{2}
|
deepscale
| 19,801
| ||
When the number "POTOP" was added together 99,999 times, the resulting number had the last three digits of 285. What number is represented by the word "POTOP"? (Identical letters represent identical digits.)
|
51715
|
deepscale
| 13,900
| ||
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.
|
-2
|
deepscale
| 23,909
| ||
In $\triangle ABC$, $A=\frac{\pi}{4}, B=\frac{\pi}{3}, BC=2$.
(I) Find the length of $AC$;
(II) Find the length of $AB$.
|
1+ \sqrt{3}
|
deepscale
| 22,304
| ||
In the triangle shown, for $\angle A$ to be the largest angle of the triangle, it must be that $m<x<n$. What is the least possible value of $n-m$, expressed as a common fraction? [asy]
draw((0,0)--(1,0)--(.4,.5)--cycle);
label("$A$",(.4,.5),N); label("$B$",(1,0),SE); label("$C$",(0,0),SW);
label("$x+9$",(.5,0),S); label("$x+4$",(.7,.25),NE); label("$3x$",(.2,.25),NW);
[/asy]
|
\frac{17}{6}
|
deepscale
| 36,267
| ||
Let $T = (2+i)^{20} - (2-i)^{20}$, where $i = \sqrt{-1}$. Find $|T|$.
|
19531250
|
deepscale
| 16,601
| ||
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today's date.
What number does Caitlin wear?
|
Let's denote the uniform numbers of Ashley, Bethany, and Caitlin as $a$, $b$, and $c$ respectively. According to the problem, all these numbers are 2-digit primes. The smallest 2-digit primes are $11, 13, 17, 19, 23, \ldots$.
From the conversation:
- Bethany's birthday earlier this month corresponds to the sum of Ashley's and Caitlin's numbers: $a + c$.
- Caitlin's birthday later this month corresponds to the sum of Ashley's and Bethany's numbers: $a + b$.
- Today's date corresponds to the sum of Bethany's and Caitlin's numbers: $b + c$.
Given that all dates are valid days of a month, the sums $a+c$, $a+b$, and $b+c$ must all be between $1$ and $31$ (inclusive).
From the solution provided:
- $a + c = 24$ (Bethany's birthday)
- $a + b = 30$ (Caitlin's birthday)
- $b + c = 28$ (Today's date)
We need to find values of $a$, $b$, and $c$ that are 2-digit primes and satisfy these equations. Let's solve these equations step-by-step:
1. **Solve for $a$:**
\[
a = 30 - b \quad \text{(from } a + b = 30\text{)}
\]
2. **Substitute $a$ in the equation for today's date:**
\[
(30 - b) + c = 28 \implies c = 28 - 30 + b = b - 2
\]
3. **Substitute $a$ and $c$ in the equation for Bethany's birthday:**
\[
(30 - b) + (b - 2) = 24 \implies 30 - 2 = 24 \implies 28 = 24 \quad \text{(contradiction)}
\]
This contradiction suggests an error in the initial assumptions or calculations. Let's recheck the possible values for $a$, $b$, and $c$:
- If $a = 11$, then $b + c = 28$, $a + b = 30 \implies b = 19$, $c = 9$ (not a prime).
- If $a = 13$, then $b + c = 28$, $a + b = 30 \implies b = 17$, $c = 11$ (both primes).
Checking the sums:
- $a + c = 13 + 11 = 24$ (Bethany's birthday)
- $a + b = 13 + 17 = 30$ (Caitlin's birthday)
- $b + c = 17 + 11 = 28$ (Today's date)
All conditions are satisfied with $a = 13$, $b = 17$, and $c = 11$. Therefore, Caitlin wears the number $\boxed{11}$.
|
11
|
deepscale
| 2,019
| |
A club has 10 members, 5 boys and 5 girls. Two of the members are chosen at random. What is the probability that they are both girls?
|
\dfrac{2}{9}
|
deepscale
| 34,807
| ||
If we exchange a 10-dollar bill into dimes and quarters, what is the total number \( n \) of different ways to have two types of coins?
|
20
|
deepscale
| 27,788
| ||
In triangle $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $C = \frac{2\pi}{3}$ and $a = 6$:
(Ⅰ) If $c = 14$, find the value of $\sin A$;
(Ⅱ) If the area of $\triangle ABC$ is $3\sqrt{3}$, find the value of $c$.
|
2\sqrt{13}
|
deepscale
| 19,279
| ||
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
|
1. **Understanding the discount offer**: The store offers one free window for every four purchased. This effectively means a discount of $100$ dollars for every five windows bought, as the fifth window is free.
2. **Calculating individual purchases**:
- **Dave's purchase**: Dave needs 7 windows. According to the store's offer, he would get one free window for buying five, so he pays for 6 windows:
\[
\text{Cost for Dave} = 6 \times 100 = 600 \text{ dollars}
\]
- **Doug's purchase**: Doug needs 8 windows. He would get one free window for buying five, and pays for the remaining three:
\[
\text{Cost for Doug} = (5 + 3) \times 100 - 100 = 700 \text{ dollars}
\]
- **Total cost if purchased separately**:
\[
\text{Total separate cost} = 600 + 700 = 1300 \text{ dollars}
\]
3. **Calculating joint purchase**:
- **Joint purchase**: Together, Dave and Doug need 15 windows. For every five windows, they get one free. Thus, for 15 windows, they get three free:
\[
\text{Cost for joint purchase} = (15 - 3) \times 100 = 1200 \text{ dollars}
\]
4. **Calculating savings**:
- **Savings**: The savings when purchasing together compared to purchasing separately is:
\[
\text{Savings} = \text{Total separate cost} - \text{Cost for joint purchase} = 1300 - 1200 = 100 \text{ dollars}
\]
5. **Conclusion**: Dave and Doug will save $100$ dollars if they purchase the windows together rather than separately.
Thus, the answer is $\boxed{100\ \mathrm{(A)}}$.
|
100
|
deepscale
| 2,381
| |
In $\triangle ABC$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, and angles $A, B, C$ form an arithmetic sequence.
(1) Find the measure of angle $B$.
(2) If $a=4$ and the area of $\triangle ABC$ is $S=5\sqrt{3}$, find the value of $b$.
|
\sqrt{21}
|
deepscale
| 18,258
| ||
A line \( y = -\frac{2}{3}x + 6 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r,s) \) is on the line segment \( PQ \). If the area of \( \triangle POQ \) is four times the area of \( \triangle TOP \), what is the value of \( r+s \)?
|
8.25
|
deepscale
| 20,766
| ||
Let the function $f(x)= \begin{cases} \log_{2}(1-x) & (x < 0) \\ g(x)+1 & (x > 0) \end{cases}$. If $f(x)$ is an odd function, determine the value of $g(3)$.
|
-3
|
deepscale
| 30,347
| ||
An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times.
|
\frac{11}{2048}
|
deepscale
| 24,837
| ||
The altitude of an equilateral triangle is $\sqrt{12}$ units. What is the area and the perimeter of the triangle, expressed in simplest radical form?
|
12
|
deepscale
| 32,465
| ||
Given the coordinates of points $A(3, 0)$, $B(0, -3)$, and $C(\cos\alpha, \sin\alpha)$, where $\alpha \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$. If $\overrightarrow{OC}$ is parallel to $\overrightarrow{AB}$ and $O$ is the origin, find the value of $\alpha$.
|
\frac{3\pi}{4}
|
deepscale
| 12,524
| ||
Given the function $f(x)=4\cos (3x+φ)(|φ| < \dfrac{π}{2})$, its graph is symmetric about the line $x=\dfrac{11π}{12}$. When $x\_1$, $x\_2∈(−\dfrac{7π}{12},−\dfrac{π}{12})$, $x\_1≠x\_2$, and $f(x\_1)=f(x\_2)$, determine the value of $f(x\_1+x\_2)$.
|
2\sqrt{2}
|
deepscale
| 17,693
| ||
Given $x \gt 0$, $y \gt 0$, $x+2y=1$, calculate the minimum value of $\frac{{(x+1)(y+1)}}{{xy}}$.
|
8+4\sqrt{3}
|
deepscale
| 14,243
| ||
In the row of Pascal's triangle that starts with 1 and then 12, what is the fourth number?
|
220
|
deepscale
| 34,913
| ||
In the geometric sequence ${a_{n}}$, $a_{n}=9$, $a_{5}=243$, find the sum of the first 4 terms.
|
120
|
deepscale
| 8,252
| ||
Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\] for some real number $x.$
|
854
|
deepscale
| 25,121
| ||
Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$ ?
|
3267
|
deepscale
| 25,103
| ||
Four mice: White, Gray, Fat, and Thin were dividing a cheese wheel. They cut it into 4 apparently identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's slice, and White's slice weighed 8 grams less than Gray's slice. However, White wasn't upset because his slice weighed exactly one-quarter of the total cheese weight.
Gray cut 8 grams from his piece, and Fat cut 20 grams from his piece. How should the 28 grams of removed cheese be divided so that all mice end up with equal amounts of cheese? Don't forget to explain your answer.
|
14
|
deepscale
| 8,272
| ||
A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \_\_\_\_\_\_\_\_.
|
2.376
|
deepscale
| 7,852
| ||
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$.
|
1+3\sqrt{2}
|
deepscale
| 21,532
| ||
Find \( g\left(\frac{1}{1996}\right) + g\left(\frac{2}{1996}\right) + g\left(\frac{3}{1996}\right) + \cdots + g\left(\frac{1995}{1996}\right) \) where \( g(x) = \frac{9x}{3 + 9x} \).
|
997
|
deepscale
| 31,766
| ||
In a company, employees have a combined monthly salary of $10,000. A kind manager proposes to triple the salaries of those earning up to $500, and to increase the salaries of others by $1,000, resulting in a total salary of $24,000. A strict manager proposes to reduce the salaries of those earning more than $500 to $500, while leaving others' salaries unchanged. What will the total salary be in this case?
|
7000
|
deepscale
| 10,090
| ||
This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours I could have exercised this month?
|
22
|
deepscale
| 15,247
| ||
A regular dodecagon \(Q_1 Q_2 \ldots Q_{12}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_7\) at \((4,0)\). If \(Q_n\) is the point \((x_n, y_n)\), compute the numerical value of the product:
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \ldots (x_{12} + y_{12} i).
\]
|
531440
|
deepscale
| 31,478
| ||
In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written?
|
19
|
deepscale
| 14,353
| ||
The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression.
|
247
|
deepscale
| 15,346
| ||
A and B start from points A and B simultaneously, moving towards each other and meet at point C. If A starts 2 minutes earlier, then their meeting point is 42 meters away from point C. Given that A's speed is \( a \) meters per minute, B's speed is \( b \) meters per minute, where \( a \) and \( b \) are integers, \( a > b \), and \( b \) is not a factor of \( a \). What is the value of \( a \)?
|
21
|
deepscale
| 23,045
| ||
The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\overrightarrow{MF}⋅\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find the area of $\triangle PMN$.
|
108
|
deepscale
| 29,936
| ||
Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\circ}$ and $85^{\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\circ}$ more than twice $C$. Find the measure of the largest angle.
|
205^\circ
|
deepscale
| 39,483
| ||
Given the ellipse E: $\\frac{x^{2}}{4} + \\frac{y^{2}}{2} = 1$, O is the coordinate origin, and a line with slope k intersects ellipse E at points A and B. The midpoint of segment AB is M, and the angle between line OM and AB is θ, with tanθ = 2 √2. Find the value of k.
|
\frac{\sqrt{2}}{2}
|
deepscale
| 18,023
| ||
Given the function $f(x) = 2\cos(x)(\sin(x) + \cos(x))$ where $x \in \mathbb{R}$,
(Ⅰ) Find the smallest positive period of function $f(x)$;
(Ⅱ) Find the intervals on which function $f(x)$ is monotonically increasing;
(Ⅲ) Find the minimum and maximum values of function $f(x)$ on the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.
|
\sqrt{2} + 1
|
deepscale
| 29,223
| ||
What is the coefficient of $x^5$ in the expansion of $(2x+3)^7$?
|
6048
|
deepscale
| 35,185
| ||
Provided $x$ is a multiple of $27720$, determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+7)(3x+8)$ and $x$.
|
168
|
deepscale
| 31,246
| ||
On a highway, there are checkpoints D, A, C, and B arranged in sequence. A motorcyclist and a cyclist started simultaneously from A and B heading towards C and D, respectively. After meeting at point E, they exchanged vehicles and each continued to their destinations. As a result, the first person spent 6 hours traveling from A to C, and the second person spent 12 hours traveling from B to D. Determine the distance of path AB, given that the speed of anyone riding a motorcycle is 60 km/h, and the speed on a bicycle is 25 km/h. Additionally, the average speed of the first person on the path AC equals the average speed of the second person on the path BD.
|
340
|
deepscale
| 15,486
| ||
Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist?
|
15
|
deepscale
| 15,889
| ||
The perimeter of triangle \( \mathrm{ABC} \) is 1. A circle touches side \( \mathrm{AB} \) at point \( P \) and the extension of side \( \mathrm{AC} \) at point \( Q \). A line passing through the midpoints of sides \( \mathrm{AB} \) and \( \mathrm{AC} \) intersects the circumcircle of triangle \( \mathrm{APQ} \) at points \( X \) and \( Y \). Find the length of segment \( X Y \).
|
\frac{1}{2}
|
deepscale
| 18,032
| ||
Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \, \text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both boys spent 10 minutes at the respective relative's house. On their way back, they met again $400 \, \text{m}$ away from Károly's house. How far apart do the two families live? What assumptions can we make to answer this question?
|
1760
|
deepscale
| 13,843
| ||
Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number).
$261 \times 345$
$234 \times 345$
$256 \times 345$
$562 \times 345$
|
$234 \times 345$
|
deepscale
| 32,332
| ||
Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Diagram
[asy] size(250); pair A,B,C,D,E,F,G,H,M,N,O,O2,P,W,X,Y,Z; A=(-76.537,184.776); B=(76.537,184.776); C=(184.776,76.537); D=(184.776,-76.537); E=(76.537,-184.776); F=(-76.537,-184.776); G=(-184.776,-76.537); H=(-184.776,76.537); M=(A+B)/2; N=(C+D)/2; O=(E+F)/2; O2=(A+E)/2; P=(G+H)/2; W=(100,-41.421); X=(-41.421,-100); Y=(-100,41.421); Z=(41.421,100); draw(A--B--C--D--E--F--G--H--A); label("$A_1$",A,dir(112.5)); label("$A_2$",B,dir(67.5)); label("$\textcolor{blue}{A_3}$",C,dir(22.5)); label("$A_4$",D,dir(337.5)); label("$A_5$",E,dir(292.5)); label("$A_6$",F,dir(247.5)); label("$A_7$",G,dir(202.5)); label("$A_8$",H,dir(152.5)); label("$M_1$",M,dir(90)); label("$\textcolor{blue}{M_3}$",N,dir(0)); label("$M_5$",O,dir(270)); label("$M_7$",P,dir(180)); label("$O$",O2,dir(152.5)); draw(M--W,red); draw(N--X,red); draw(O--Y,red); draw(P--Z,red); draw(O2--(W+X)/2,red); draw(O2--N,red); label("$\textcolor{blue}{B_1}$",W,dir(292.5)); label("$B_2$",(W+X)/2,dir(292.5)); label("$B_3$",X,dir(202.5)); label("$B_5$",Y,dir(112.5)); label("$B_7$",Z,dir(22.5)); [/asy] All distances are to scale.
|
Let $A_1A_2 = 2$. Then $B_1$ and $B_3$ are the projections of $M_1$ and $M_5$ onto the line $B_1B_3$, so $2=B_1B_3=-M_1M_5\cos x$, where $x = \angle A_3M_3B_1$. Then since $M_1M_5 = 2+2\sqrt{2}, \cos x = \dfrac{-2}{2+2\sqrt{2}}= 1-\sqrt{2}$, $\cos 2x = 2\cos^2 x -1 = 5 - 4\sqrt{2} = 5-\sqrt{32}$, and $m+n=\boxed{037}$.
|
37
|
deepscale
| 7,014
| |
Given $\cos\left(\alpha- \frac{\beta}{2}\right) = -\frac{1}{9}$ and $\sin\left(\frac{\alpha}{2} - \beta\right) = \frac{2}{3}$, with $\frac{\pi}{2} < \alpha < \pi$ and $0 < \beta < \frac{\pi}{2}$, find $\cos(\alpha + \beta)$.
|
-\frac{239}{729}
|
deepscale
| 26,953
| ||
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$.
Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.
i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
|
Let \( A_1B_1C_1D_1 \) be an arbitrary convex quadrilateral. \( P \) is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point \( P \) is acute. We recursively define points \( A_k, B_k, C_k, D_k \) symmetric to \( P \) with respect to lines \( A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1}, D_{k-1}A_{k-1} \) respectively for \( k \ge 2 \). Consider the sequence of quadrilaterals \( A_iB_iC_iD_i \).
i) Among the first 12 quadrilaterals, the ones that are similar to the 1997th quadrilateral are the 1st, 5th, and 9th quadrilaterals.
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, the ones that are cyclic are the 1st, 3rd, 5th, 7th, 9th, and 11th quadrilaterals.
The answer is:
\[
\begin{aligned}
&\text{1. } \boxed{1, 5, 9} \\
&\text{2. } \boxed{1, 3, 5, 7, 9, 11}
\end{aligned}
\]
|
1, 5, 9
|
deepscale
| 5,775
| |
If the universal set $U = \{-1, 0, 1, 2\}$, and $P = \{x \in \mathbb{Z} \,|\, -\sqrt{2} < x < \sqrt{2}\}$, determine the complement of $P$ in $U$.
|
\{2\}
|
deepscale
| 22,828
| ||
How many distinct triangles can be constructed by connecting three different vertices of a cube? (Two triangles are distinct if they have different locations in space.)
|
56
|
deepscale
| 34,971
| ||
Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords?
|
\sqrt{2-\sqrt{2}}
|
deepscale
| 29,657
| ||
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$
|
505
|
deepscale
| 35,937
| ||
Find the minimum value of the maximum of \( |x^2 - 2xy| \) over \( 0 \leq x \leq 1 \) for \( y \) in \( \mathbb{R} \).
|
3 - 2\sqrt{2}
|
deepscale
| 7,953
| ||
Calculate the result of the expression \((19 \times 19 - 12 \times 12) \div \left(\frac{19}{12} - \frac{12}{19}\right)\).
|
228
|
deepscale
| 8,890
| ||
Karen has seven envelopes and seven letters of congratulations to various HMMT coaches. If she places the letters in the envelopes at random with each possible configuration having an equal probability, what is the probability that exactly six of the letters are in the correct envelopes?
|
0, since if six letters are in their correct envelopes the seventh is as well.
|
0
|
deepscale
| 3,155
| |
Given points P and Q are on a circle of radius 7 and the length of chord PQ is 8. Point R is the midpoint of the minor arc PQ. Find the length of the line segment PR.
|
\sqrt{98 - 14\sqrt{33}}
|
deepscale
| 30,810
| ||
A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers?
|
6
|
deepscale
| 39,396
| ||
In square ABCD, point E is on AB and point F is on CD such that AE = 3EB and CF = 3FD.
|
\frac{3}{32}
|
deepscale
| 27,229
|
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