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29,700 | A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median? | 6.5 | 27.34375 |
29,701 | Jo and Blair take turns counting numbers, starting with Jo who says "1". Blair follows by saying the next number in the sequence, which is normally the last number said by Jo plus one. However, every third turn, the speaker will say the next number plus two instead of one. Express the $53^{\text{rd}}$ number said in this new sequence. | 71 | 0 |
29,702 | The sum of seven consecutive even numbers is 686. What is the smallest of these seven numbers? Additionally, calculate the median and mean of this sequence. | 98 | 66.40625 |
29,703 | In the Cartesian coordinate plane $(xOy)$, the curve $y=x^{2}-6x+1$ intersects the coordinate axes at points that lie on circle $C$.
(1) Find the equation of circle $C$;
(2) Given point $A(3,0)$, and point $B$ is a moving point on circle $C$, find the maximum value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$, and find the length of the chord cut by line $OB$ on circle $C$ at this time. | \frac{36\sqrt{37}}{37} | 0.78125 |
29,704 | Let \( n \) be a fixed integer, \( n \geqslant 2 \).
(a) Determine the minimal constant \( c \) such that the inequality
$$
\sum_{1 \leqslant i < j \leqslant n} x_i x_j \left(x_i^2 + x_j^2\right) \leqslant c \left( \sum_{1 \leqslant i \leqslant n} x_i \right)^4
$$
holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \geqslant 0 \).
(b) For this constant \( c \), determine the necessary and sufficient conditions for equality to hold. | \frac{1}{8} | 65.625 |
29,705 | In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid. | 56 | 0 |
29,706 | Let $g_0(x) = x + |x - 150| - |x + 150|$, and for $n \geq 1$, let $g_n(x) = |g_{n-1}(x)| - 2$. For how many values of $x$ is $g_{100}(x) = 0$? | 299 | 0 |
29,707 | $\triangle PQR$ is inscribed inside $\triangle XYZ$ such that $P, Q, R$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle PYZ, \triangle QXR, \triangle RQP$ have centers $O_4, O_5, O_6$, respectively. Also, $XY = 29, YZ = 35, XZ=28$, and $\stackrel{\frown}{YR} = \stackrel{\frown}{QZ},\ \stackrel{\frown}{XR} = \stackrel{\frown}{PY},\ \stackrel{\frown}{XP} = \stackrel{\frown}{QY}$. The length of $QY$ can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$. | 31 | 1.5625 |
29,708 | A certain school club has 10 members, and two of them are put on duty each day from Monday to Friday. Given that members A and B must be scheduled on the same day, and members C and D cannot be scheduled together, the total number of different possible schedules is (▲). Choices:
A) 21600
B) 10800
C) 7200
D) 5400 | 5400 | 9.375 |
29,709 | In parallelogram ABCD, $\angle BAD=60^\circ$, $AB=1$, $AD=\sqrt{2}$, and P is a point inside the parallelogram such that $AP=\frac{\sqrt{2}}{2}$. If $\overrightarrow{AP}=\lambda\overrightarrow{AB}+\mu\overrightarrow{AD}$ ($\lambda,\mu\in\mathbb{R}$), then the maximum value of $\lambda+\sqrt{2}\mu$ is \_\_\_\_\_\_. | \frac{\sqrt{6}}{3} | 5.46875 |
29,710 | In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon. | 128 | 0 |
29,711 | Given the graph of the function $y=\cos (x+\frac{4\pi }{3})$ is translated $\theta (\theta > 0)$ units to the right, and the resulting graph is symmetrical about the $y$-axis, determine the smallest possible value of $\theta$. | \frac{\pi }{3} | 75 |
29,712 | Among the 95 numbers $1^2, 2^2, 3^2, \ldots, 95^2$, how many of them have an odd digit in the tens place? | 19 | 75 |
29,713 | Given that $O$ is the circumcenter of $\triangle ABC$, $AC \perp BC$, $AC = 3$, and $\angle ABC = \frac{\pi}{6}$, find the dot product of $\overrightarrow{OC}$ and $\overrightarrow{AB}$. | -9 | 32.8125 |
29,714 | How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$ , where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive. | 3432 | 19.53125 |
29,715 | In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. Find the value of \( x \). | 80 | 11.71875 |
29,716 | Jenny and Jack run on a circular track. Jenny runs counterclockwise and completes a lap every 75 seconds, while Jack runs clockwise and completes a lap every 70 seconds. They start at the same place and at the same time. Between 15 minutes and 16 minutes from the start, a photographer standing outside the track takes a picture that shows one-third of the track, centered on the starting line. What is the probability that both Jenny and Jack are in the picture?
A) $\frac{23}{60}$
B) $\frac{12}{60}$
C) $\frac{13}{60}$
D) $\frac{46}{60}$
E) $\frac{120}{60}$ | \frac{23}{60} | 10.9375 |
29,717 | Calculate the product of the base nine numbers $35_9$ and $47_9$, express it in base nine, and find the base nine sum of the digits of this product. Additionally, subtract $2_9$ from the sum of the digits. | 22_9 | 0 |
29,718 | Given that Jo climbs a flight of 8 stairs, and Jo can take the stairs 1, 2, 3, or 4 at a time, or a combination of these steps that does not exceed 4 with a friend, determine the number of ways Jo and the friend can climb the stairs together, including instances where either Jo or the friend climbs alone or together. | 108 | 0.78125 |
29,719 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | \dfrac{3}{4} | 0 |
29,720 | If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 16 | 20.3125 |
29,721 | A regular dodecagon \(Q_1 Q_2 \dotsb Q_{12}\) is drawn in the coordinate plane with \(Q_1\) at \((1,0)\) and \(Q_7\) at \((3,0)\). If \(Q_n\) is the point \((x_n,y_n),\) compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).\] | 4095 | 20.3125 |
29,722 | The condition for three line segments to form a triangle is: the sum of the lengths of any two line segments is greater than the length of the third line segment. Now, there is a wire 144cm long, and it needs to be cut into $n$ small segments ($n>2$), each segment being no less than 1cm in length. If any three of these segments cannot form a triangle, then the maximum value of $n$ is ____. | 10 | 85.15625 |
29,723 | A circle with a radius of 2 units rolls around the inside of a triangle with sides 9, 12, and 15 units. The circle is always tangent to at least one side of the triangle. Calculate the total distance traveled by the center of the circle when it returns to its starting position. | 24 | 80.46875 |
29,724 | If the coefficient of the $x^2$ term in the expansion of $(1-ax)(1+2x)^4$ is $4$, then $\int_{\frac{e}{2}}^{a}{\frac{1}{x}}dx =$ . | \ln(5) - 1 | 78.90625 |
29,725 | Given $\lg 2=0.3010$ and $\lg 3=0.4771$, at which decimal place does the first non-zero digit of $\left(\frac{6}{25}\right)^{100}$ occur?
(Shanghai Middle School Mathematics Competition, 1984) | 62 | 76.5625 |
29,726 | The coefficient of $\frac{1}{x}$ in the expansion of $(1-x^2)^4\left(\frac{x+1}{x}\right)^5$ needs to be determined. | -29 | 15.625 |
29,727 | The math scores of a high school's college entrance examination, denoted as $\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\xi < 110) = 0.98$, find the value of $P(90 < \xi < 100)$. | 0.48 | 85.15625 |
29,728 | There is a garden with 3 rows and 2 columns of rectangular flower beds, each measuring 6 feet long and 2 feet wide. Between the flower beds, as well as around the garden, there is a 1-foot wide path. What is the total area \( S \) of the path in square feet? | 78 | 0 |
29,729 | On three faces of a cube, diagonals are drawn such that a triangle is formed. Find the angles of this triangle. | 60 | 30.46875 |
29,730 | Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | 37 | 13.28125 |
29,731 | Given that events A and B are independent, and both are mutually exclusive with event C. It is known that $P(A) = 0.2$, $P(B) = 0.6$, and $P(C) = 0.14$. Find the probability that at least one of A, B, or C occurs, denoted as $P(A+B+C)$. | 0.82 | 53.90625 |
29,732 | Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$ . | \sqrt{3} | 0.78125 |
29,733 | Given that the height of a cylinder is $2$, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of $2\sqrt{6}$, calculate the surface area of the cylinder. | (10+4\sqrt{5})\pi | 0 |
29,734 | Find one third of 7.2, expressed as a simplified fraction or a mixed number. | 2 \frac{2}{5} | 88.28125 |
29,735 | Given vectors $\overrightarrow{a}=(\cos 25^{\circ},\sin 25^{\circ})$ and $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, let $t$ be a real number and $\overrightarrow{u}=\overrightarrow{a}+t\overrightarrow{b}$. Determine the minimum value of $|\overrightarrow{u}|$. | \frac{\sqrt{2}}{2} | 92.1875 |
29,736 | Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | 1236 | 0.78125 |
29,737 | Team X and team Y play a series where the first team to win four games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team Y wins the third game and team X wins the series, what is the probability that team Y wins the first game?
**A) $\frac{5}{12}$**
**B) $\frac{1}{2}$**
**C) $\frac{1}{3}$**
**D) $\frac{1}{4}$**
**E) $\frac{2}{3}$** | \frac{5}{12} | 14.84375 |
29,738 | Suppose $a$, $b$, and $c$ are positive integers such that $a \geq b \geq c$ and $a+b+c=2010$. Furthermore, $a!b!c! = m \cdot 10^n$, where $m$ and $n$ are integers, and $m$ is not divisible by $10$. What is the smallest possible value of $n$?
A) 499
B) 500
C) 502
D) 504
E) 506 | 500 | 31.25 |
29,739 | Given $x^{3}=4$, solve for $x$. | \sqrt[3]{4} | 10.9375 |
29,740 | Given the line $y=kx+1$, the maximum length of the chord intercepted by the ellipse $\frac{x^{2}}{4}+y^{2}=1$ as $k$ varies, determine the maximum length of the chord. | \frac{4\sqrt{3}}{3} | 12.5 |
29,741 | Given $f(x)=\frac{1}{2}\cos^{2}x-\frac{1}{2}\sin^{2}x+1-\sqrt{3}\sin x \cos x$.
$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing.
$(2)$ Find the minimum value of $f(x)$ on $[0,\frac{\pi}{2}]$ and the corresponding set of independent variables. | \left\{\frac{\pi}{3}\right\} | 31.25 |
29,742 | The Chinese government actively responds to changes in gas emissions and has set a goal to reduce carbon emission intensity by 40% by 2020 compared to 2005. It is known that in 2005, China's carbon emission intensity was about 3 tons per 10,000 yuan, and each year thereafter, the carbon emission intensity decreases by 0.08 tons per 10,000 yuan.
(1) Can the emission reduction target be achieved by 2020? Explain your reasoning;
(2) If the GDP of China in 2005 was a million yuan, and thereafter increases by 8% annually, from which year will the carbon dioxide emissions start to decrease?
(Note: "Carbon emission intensity" refers to the amount of carbon dioxide emissions per 10,000 yuan of GDP) | 2030 | 10.9375 |
29,743 | Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 54$, $b_1 = 81$, $b_n = 0$, and $$ b_{k+1} = b_{k-1} - \frac{4.5}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$. | 972 | 0 |
29,744 | We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible? | 2886 | 7.8125 |
29,745 | On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has an area of $\frac{12}{13}.$ If the real part of $z$ is positive, compute the smallest possible value of $\left| z + \frac{1}{z} \right|^2.$ | \frac{36}{13} | 1.5625 |
29,746 | A middle school cafeteria regularly purchases rice from a grain store at a price of 1500 yuan per ton. Each time rice is purchased, a transportation fee of 100 yuan is required. The cafeteria needs 1 ton of rice per day, and the storage cost for rice is 2 yuan per ton per day (less than one day is counted as one day). Assuming the cafeteria purchases rice on the day it runs out.
(1) How often should the cafeteria purchase rice to minimize the total daily cost?
(2) The grain store offers a discount: if the purchase quantity is not less than 20 tons at a time, the price of rice can enjoy a 5% discount (i.e., 95% of the original price). Can the cafeteria accept this discount condition? Please explain your reason. | 10 | 22.65625 |
29,747 | 22. In the polar coordinate system, the polar equation of circle $C$ is $\rho =4\cos \theta$. Taking the pole as the origin and the direction of the polar axis as the positive direction of the $x$-axis, a Cartesian coordinate system is established using the same unit length. The parametric equation of line $l$ is $\begin{cases} & x=\dfrac{1}{2}+\dfrac{\sqrt{2}}{2}t \\ & y=\dfrac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is the parameter).
(Ⅰ) Write the Cartesian coordinate equation of circle $C$ and the general equation of line $l$;
(Ⅱ) Given point $M\left( \dfrac{1}{2},0 \right)$, line $l$ intersects circle $C$ at points $A$ and $B$. Find the value of $\left| \left| MA \right|-\left| MB \right| \right|$. | \dfrac{\sqrt{46}}{2} | 0.78125 |
29,748 | On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$ , consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. | 67 | 13.28125 |
29,749 | A three-digit number is formed by the digits $0$, $1$, $2$, $3$, $4$, $5$, with exactly two digits being the same. There are a total of \_\_\_\_\_ such numbers. | 75 | 7.8125 |
29,750 | How many ways can a schedule of 4 mathematics courses - algebra, geometry, number theory, and calculus - be created in an 8-period day if exactly one pair of these courses can be taken in consecutive periods, and the other courses must not be consecutive? | 1680 | 10.9375 |
29,751 | Yan is somewhere between his office and a concert hall. To get to the concert hall, he can either walk directly there, or walk to his office and then take a scooter to the concert hall. He rides 5 times as fast as he walks, and both choices take the same amount of time. What is the ratio of Yan's distance from his office to his distance from the concert hall? | \frac{2}{3} | 18.75 |
29,752 | In rectangle $WXYZ$, $P$ is a point on $WY$ such that $\angle WPZ=90^{\circ}$. $UV$ is perpendicular to $WY$ with $WU=UP$, as shown. $PZ$ intersects $UV$ at $Q$. Point $R$ is on $YZ$ such that $WR$ passes through $Q$. In $\triangle PQW$, $PW=15$, $WQ=20$ and $QP=25$. Find $VZ$. (Express your answer as a common fraction.) | \dfrac{20}{3} | 0 |
29,753 | A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuckoo strike during this period? | 43 | 1.5625 |
29,754 | Suppose three hoses, X, Y, and Z, are used to fill a pool. Hoses X and Y together take 3 hours, while hose Y working alone takes 9 hours. Hoses X and Z together take 4 hours to fill the same pool. All three hoses working together can fill the pool in 2.5 hours. How long does it take for hose Z working alone to fill the pool? | 15 | 41.40625 |
29,755 | How many of the 2401 smallest positive integers written in base 7 use 3 or 6 (or both) as a digit? | 1776 | 20.3125 |
29,756 | Consider a circle of radius $4$ with center $O_1$ , a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$ , and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$ . The centers of the circle are collinear in the order $O_1$ , $O_2$ , $O_3$ . Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$ . Compute the length of $AB$ . | \sqrt{6} | 39.0625 |
29,757 | In a regular hexagon $ABCDEF$, points $P$, $Q$, $R$, and $S$ are chosen on sides $\overline{AB}$, $\overline{CD}$, $\overline{DE}$, and $\overline{FA}$ respectively, so that lines $PC$ and $RA$, as well as $QS$ and $EB$, are parallel. Moreover, the distances between these parallel lines are equal and constitute half of the altitude of the triangles formed by drawing diagonals from the vertices $B$ and $D$ to the opposite side. Calculate the ratio of the area of hexagon $APQRSC$ to the area of hexagon $ABCDEF$. | \frac{3}{4} | 10.9375 |
29,758 | Given the system of equations
\begin{align*}
4x+2y &= c, \\
6y - 12x &= d,
\end{align*}
where \(d \neq 0\), find the value of \(\frac{c}{d}\). | -\frac{1}{3} | 17.1875 |
29,759 | Given the function $f(x)=x^{2}\cos \frac {πx}{2}$, the sequence {a<sub>n</sub>} is defined as a<sub>n</sub> = f(n) + f(n+1) (n ∈ N*), find the sum of the first 40 terms of the sequence {a<sub>n</sub>}, denoted as S<sub>40</sub>. | 1680 | 3.90625 |
29,760 | Given a set of data: 10, 10, x, 8, where the median is equal to the mean, find the median of this data set. | 10 | 37.5 |
29,761 | The angle of inclination of the line $$\begin{cases} \left.\begin{matrix}x=3- \frac { \sqrt {2}}{2}t \\ y= \sqrt {5}- \frac { \sqrt {2}}{2}t\end{matrix}\right.\end{cases}$$ is ______. | \frac{\pi}{4} | 95.3125 |
29,762 | It is known that solution A has a salinity of $8\%$, and solution B has a salinity of $5\%$. After mixing both solutions, the resulting salinity is $6.2\%$. What will be the concentration when a quarter of solution A is mixed with a sixth of solution B? | $6.5\%$ | 0 |
29,763 | In the expansion of the polynomial $$(x+ \frac {1}{ \sqrt {x}})^{6}( \sqrt {x}-1)^{10}$$, the constant term is \_\_\_\_\_\_. | -495 | 21.875 |
29,764 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Calculate the value of $p + q + r + s.$ | 2028 | 37.5 |
29,765 | Read the material: Calculate $\frac{1}{30}÷(\frac{2}{3}-\frac{1}{10}+\frac{1}{6}-\frac{2}{5})$. Analysis: It is very cumbersome to calculate the result of $\frac{2}{3}-\frac{1}{10}+\frac{1}{6}-\frac{2}{5}$ using a common denominator. The following method can be used for calculation. Solution: The reciprocal of the original expression $=(\frac{2}{3}-\frac{1}{10}+\frac{1}{6}-\frac{2}{5})÷\frac{1}{30}$ $=(\frac{2}{3}-\frac{1}{10}+\frac{1}{6}-\frac{2}{5})×30$ $=\frac{2}{3}×30-\frac{1}{10}×30+\frac{1}{6}×30-\frac{2}{5}×30$ $=10$. Therefore, the original expression $=\frac{1}{10}$. Please choose the appropriate method to calculate $\frac{1}{24}÷(\frac{1}{12}-\frac{5}{16}+\frac{7}{24}-\frac{2}{3})$ based on your understanding of the material. | -\frac{2}{29} | 30.46875 |
29,766 | Without using any measuring tools or other auxiliary means, how can you cut exactly half a meter from a piece of cloth that is $\frac{8}{15}$ meters long? | 1/2 | 32.03125 |
29,767 | Into how many regions do the x-axis and the graphs of \( y = 2 - x^2 \) and \( y = x^2 - 1 \) split the plane? | 10 | 2.34375 |
29,768 | A digital watch displays time in a 24-hour format, showing hours and minutes. Calculate the largest possible sum of the digits in this display. | 24 | 0.78125 |
29,769 | Given the function $g(x)=\ln x+\frac{1}{2}x^{2}-(b-1)x$.
(1) If the function $g(x)$ has a monotonically decreasing interval, find the range of values for the real number $b$;
(2) Let $x_{1}$ and $x_{2}$ ($x_{1} < x_{2}$) be the two extreme points of the function $g(x)$. If $b\geqslant \frac{7}{2}$, find the minimum value of $g(x_{1})-g(x_{2})$. | \frac{15}{8}-2\ln 2 | 7.03125 |
29,770 | Given $A(a, 1-a^{2})$, $B(b, 1-b^{2})$, where $ab \lt 0$, tangents are drawn at points $A$ and $B$ to the parabola $y=1-x^{2}$. The minimum area enclosed by the two tangents and the $x$-axis is ____. | \frac{8\sqrt{3}}{9} | 3.90625 |
29,771 | An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of the smaller triangle? | 25\% | 77.34375 |
29,772 | Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations:
1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon.
2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to $n$ .
What is the sum of all possible values of $n$ ? | 469 | 0 |
29,773 | Given that Ron mistakenly reversed the digits of the two-digit number $a$, and the product of $a$ and $b$ was mistakenly calculated as $221$, determine the correct product of $a$ and $b$. | 527 | 27.34375 |
29,774 | Let $a>0$ and $b>0,$ and define two operations:
$$a \nabla b = \dfrac{a + b}{1 + ab}$$
$$a \Delta b = \dfrac{a - b}{1 - ab}$$
Calculate $3 \nabla 4$ and $3 \Delta 4$. | \frac{1}{11} | 97.65625 |
29,775 | In a circle with center $O$, the measure of $\angle BAC$ is $45^\circ$, and the radius of the circle $OA=15$ cm. Also, $\angle BAC$ subtends another arc $BC$ which does not include point $A$. Compute the length of arc $BC$ in terms of $\pi$. [asy]
draw(circle((0,0),1));
draw((0,0)--(sqrt(2)/2,sqrt(2)/2)--(-sqrt(2)/2,sqrt(2)/2)--(0,0));
label("$O$", (0,0), SW); label("$A$", (sqrt(2)/2,sqrt(2)/2), NE); label("$B$", (-sqrt(2)/2,sqrt(2)/2), NW);
[/asy] | 22.5\pi | 0 |
29,776 | In an extended game, each of 6 players, including Hugo, rolls a standard 8-sided die. The winner is the one who rolls the highest number. In the case of a tie for the highest roll, the tied players will re-roll until a single winner emerges. What is the probability that Hugo's first roll was a 7, given that he won the game?
A) $\frac{2772}{8192}$
B) $\frac{8856}{32768}$
C) $\frac{16056}{65536}$
D) $\frac{11028}{49152}$
E) $\frac{4428}{16384}$ | \frac{8856}{32768} | 2.34375 |
29,777 | The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$ . Compute $P(5)$ . | -24 | 59.375 |
29,778 | Given vectors satisfying $\overrightarrow{a}\cdot (\overrightarrow{a}-2\overrightarrow{b})=3$ and $|\overrightarrow{a}|=1$, with $\overrightarrow{b}=(1,1)$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac {3\pi}{4} | 95.3125 |
29,779 | Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids. | 134 | 1.5625 |
29,780 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.230 | 0 |
29,781 | There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$ , there is either a one way road from $A$ to $B$ , or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$ . Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country. | 3/2 | 8.59375 |
29,782 | Let $S$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $S$ has a product divisible by $5$. What is the maximum number of elements in $S$? | 80 | 66.40625 |
29,783 | A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week. | 36 | 23.4375 |
29,784 | Let $N = 123456789101112\dots505152$ be the number obtained by writing out the integers from 1 to 52 consecutively. Compute the remainder when $N$ is divided by 45. | 37 | 50 |
29,785 | Evaluate the expression: $\left(2\left(3\left(2\left(3\left(2\left(3 \times (2+1) \times 2\right)+2\right)\times 2\right)+2\right)\times 2\right)+2\right)$. | 5498 | 11.71875 |
29,786 | In isosceles $\triangle ABC$ where $AB = AC = 2$ and $BC = 1$, equilateral triangles $ABD$, $BCE$, and $CAF$ are constructed outside the triangle. Calculate the area of polygon $DEF$.
A) $3\sqrt{3} - \sqrt{3.75}$
B) $3\sqrt{3} + \sqrt{3.75}$
C) $2\sqrt{3} - \sqrt{3.75}$
D) $2\sqrt{3} + \sqrt{3.75}$ | 3\sqrt{3} - \sqrt{3.75} | 30.46875 |
29,787 | Given that $\sin\alpha=\frac{\sqrt{5}}{5}$ and $\sin\beta=\frac{\sqrt{10}}{10}$, where both $\alpha$ and $\beta$ are acute angles, find the value of $\alpha+\beta$. | \frac{\pi}{4} | 97.65625 |
29,788 |
A group of schoolchildren, heading to a school camp, was planned to be seated in buses so that there would be an equal number of passengers in each bus. Initially, 22 people were seated in each bus, but it turned out that three schoolchildren could not be seated. However, when one bus left empty, all the remaining schoolchildren seated equally in the other buses. How many schoolchildren were in the group, given that no more than 18 buses were provided for transporting the schoolchildren, and each bus can hold no more than 36 people? Give the answer as a number without indicating the units. | 135 | 38.28125 |
29,789 | The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers. | 3 : 2 | 9.375 |
29,790 | Dots are placed two units apart both horizontally and vertically on a coordinate grid. Calculate the number of square units enclosed by the polygon formed by connecting these dots:
[asy]
size(90);
pair a=(0,0), b=(20,0), c=(20,20), d=(40,20), e=(40,40), f=(20,40), g=(0,40), h=(0,20);
dot(a);
dot(b);
dot(c);
dot(d);
dot(e);
dot(f);
dot(g);
dot(h);
draw(a--b--c--d--e--f--g--h--cycle);
[/asy] | 12 | 0 |
29,791 | Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 15$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral? | \frac{1600}{9} | 4.6875 |
29,792 | What integer should 999,999,999 be multiplied by to get a number consisting of only ones? | 111111111 | 24.21875 |
29,793 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 10.15625 |
29,794 | Given the parabola $y^{2}=4x$ and the line $l: x+2y-2b=0$ intersecting the parabola at points $A$ and $B$.
(Ⅰ) If the circle with diameter $AB$ is tangent to the $x$-axis, find the equation of the circle;
(Ⅱ) If the line $l$ intersects the negative half of the $y$-axis, find the maximum area of $\triangle AOB$ ($O$ is the origin). | \frac{32 \sqrt{3}}{9} | 5.46875 |
29,795 | Let's define a calendar week as even or odd according to whether the sum of the day numbers within the month in that week is even or odd. Out of the 52 consecutive weeks starting from the first Monday of January, how many can be even? | 30 | 0 |
29,796 | Given real numbers $x$ and $y$ satisfy the equation $x^{2}+y^{2}-4x+1=0$.
$(1)$ Find the maximum and minimum values of $\dfrac {y}{x}$;
$(2)$ Find the maximum and minimum values of $y-x$;
$(3)$ Find the maximum and minimum values of $x^{2}+y^{2}$. | 7-4 \sqrt {3} | 0 |
29,797 | A point $(x, y)$ is randomly selected such that $0 \leq x \leq 3$ and $0 \leq y \leq 3$. What is the probability that $x + 2y \leq 6$? Express your answer as a common fraction. | \frac{1}{4} | 2.34375 |
29,798 | How many of the integers between 1 and 1500, inclusive, can be expressed as the difference of the squares of two positive integers? | 1125 | 71.875 |
29,799 | What is the least natural number that can be added to 250,000 to create a palindrome? | 52 | 0 |
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