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31,500 | If $a$ and $b$ are additive inverses, $c$ and $d$ are multiplicative inverses, and the absolute value of $m$ is 1, find $(a+b)cd-2009m=$ \_\_\_\_\_\_. | 2009 | 33.59375 |
31,501 | A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is fi ve tiles wide. That red border is surrounded by a white border which is fi ve tiles wide. Finally, the white border is surrounded by a red border which is five tiles wide. The resulting pattern is pictured below. In all, $2900$ red tiles are used to tile the large rectangle. Find the perimeter of the large rectangle.
[asy]
import graph;
size(5cm);
fill((-5,-5)--(0,-5)--(0,35)--(-5,35)--cycle^^(50,-5)--(55,-5)--(55,35)--(50,35)--cycle,red);
fill((0,30)--(0,35)--(50,35)--(50,30)--cycle^^(0,-5)--(0,0)--(50,0)--(50,-5)--cycle,red);
fill((-15,-15)--(-10,-15)--(-10,45)--(-15,45)--cycle^^(60,-15)--(65,-15)--(65,45)--(60,45)--cycle,red);
fill((-10,40)--(-10,45)--(60,45)--(60,40)--cycle^^(-10,-15)--(-10,-10)--(60,-10)--(60,-15)--cycle,red);
fill((-10,-10)--(-5,-10)--(-5,40)--(-10,40)--cycle^^(55,-10)--(60,-10)--(60,40)--(55,40)--cycle,white);
fill((-5,35)--(-5,40)--(55,40)--(55,35)--cycle^^(-5,-10)--(-5,-5)--(55,-5)--(55,-10)--cycle,white);
for(int i=0;i<16;++i){
draw((-i,-i)--(50+i,-i)--(50+i,30+i)--(-i,30+i)--cycle,linewidth(.5));
}
[/asy] | 350 | 11.71875 |
31,502 | As shown in the diagram, the area of parallelogram \(ABCD\) is 60. The ratio of the areas of \(\triangle ADE\) and \(\triangle AEB\) is 2:3. Find the area of \(\triangle BEF\). | 12 | 18.75 |
31,503 | Given a sequence $\{a_n\}$ where the first term is 1 and the common difference is 2,
(1) Find the general formula for $\{a_n\}$;
(2) Let $b_n=\frac{1}{a_n \cdot a_{n-1}}$, and the sum of the first n terms of the sequence $\{b_n\}$ is $T_n$. Find the minimum value of $T_n$. | \frac{1}{3} | 1.5625 |
31,504 | Given vectors $m=(2\sin \omega x, \cos^2 \omega x - \sin^2 \omega x)$ and $n=(\sqrt{3}\cos \omega x, 1)$, where $\omega > 0$ and $x \in R$. If the smallest positive period of the function $f(x)=m \cdot n$ is $\pi$.
(1) Find the value of $\omega$;
(2) In $\triangle ABC$, if $f(B)=-2$, $BC= \sqrt{3}$, $\sin B= \sqrt{3}\sin A$, find the value of $\overrightarrow{BA} \cdot \overrightarrow{BC}$. | -\frac{3}{2} | 22.65625 |
31,505 | The number of distinct even numbers that can be formed using the digits 0, 1, 2, and 3. | 10 | 36.71875 |
31,506 | In the Cartesian coordinate system $xOy$, given the circle $O: x^2 + y^2 = 1$ and the circle $C: (x-4)^2 + y^2 = 4$, a moving point $P$ is located between two points $E$ and $F$ on the line $x + \sqrt{3}y - 2 = 0$. Tangents to circles $O$ and $C$ are drawn from point $P$, with the points of tangency being $A$ and $B$, respectively. If it satisfies $PB \geqslant 2PA$, then the length of the segment $EF$ is. | \frac{2 \sqrt{39}}{3} | 11.71875 |
31,507 | What is the median of the following list of $4100$ numbers?
\[1, 2, 3, \ldots, 2050, 1^2, 2^2, 3^2, \ldots, 2050^2\]
A) $1977.5$
B) $2004.5$
C) $2005.5$
D) $2006.5$
E) $2025.5$ | 2005.5 | 23.4375 |
31,508 | Consider a fictional country where license plates are made using an extended Rotokas alphabet with fourteen letters: A, E, G, I, K, M, O, P, R, S, T, U, V, Z. License plates are six letters long and must start with G, end with M, and not contain the letters S, T, or Z. Additionally, no letters may repeat. How many such license plates are possible? | 3024 | 57.8125 |
31,509 | Given vector $\vec{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$, and $\vec{a}\cdot \vec{b}=\frac{1}{2}$, calculate the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$. | \frac{1}{2} | 0 |
31,510 | When $0.76\overline{204}$ is expressed as a fraction in the form $\frac{x}{999000}$, what is the value of $x$? | 761280 | 15.625 |
31,511 | $ABCD$ is a rectangle with $AB = CD = 2$ . A circle centered at $O$ is tangent to $BC$ , $CD$ , and $AD$ (and hence has radius $1$ ). Another circle, centered at $P$ , is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$ . If line $AT$ is tangent to both circles at $T$ , find the radius of circle $P$ . | 3 - 2\sqrt{2} | 75 |
31,512 | In recent years, handicrafts made by a certain handicraft village have been very popular overseas. The villagers of the village have established a cooperative for exporting handicrafts. In order to strictly control the quality, the cooperative invites 3 experts to inspect each handicraft made by the villagers. The quality control process is as follows: $(i)$ If all 3 experts consider a handicraft to be of good quality, then the quality of the handicraft is rated as grade $A$; $(ii)$ If only 1 expert considers the quality to be unsatisfactory, then the other 2 experts will conduct a second quality check. If both experts in the second check consider the quality to be good, then the handicraft is rated as grade $B$. If one or both of the experts in the second check consider the quality to be unsatisfactory, then the handicraft is rated as grade $C$; $(iii)$ If 2 or all 3 experts consider the quality to be unsatisfactory, then the handicraft is rated as grade $D$. It is known that the probability of 1 handicraft being considered unsatisfactory by 1 expert in each quality check is $\frac{1}{3}$, and the quality of each handicraft is independent of each other. Find: $(1)$ the probability that 1 handicraft is rated as grade $B$; $(2)$ if handicrafts rated as grade $A$, $B$, and $C$ can be exported with profits of $900$ yuan, $600$ yuan, and $300$ yuan respectively, while grade $D$ cannot be exported and has a profit of $100$ yuan. Find: $①$ the most likely number of handicrafts out of 10 that cannot be exported; $②$ if the profit of 1 handicraft is $X$ yuan, find the distribution and mean of $X$. | \frac{13100}{27} | 6.25 |
31,513 | Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\). | 2 - \sqrt{2} | 0.78125 |
31,514 | Given \( P = 3659893456789325678 \) and \( 342973489379256 \), the product \( P \) is calculated. The number of digits in \( P \) is: | 34 | 90.625 |
31,515 |
A circle $\omega$ is circumscribed around the triangle $ABC$. A circle $\omega_{1}$ is tangent to the line $AB$ at point $A$ and passes through point $C$, and a circle $\omega_{2}$ is tangent to the line $AC$ at point $A$ and passes through point $B$. A tangent to circle $\omega$ at point $A$ intersects circle $\omega_{1}$ again at point $X$ and intersects circle $\omega_{2}$ again at point $Y$. Find the ratio $\frac{AX}{XY}$. | \frac{1}{2} | 68.75 |
31,516 | If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 | 3.125 |
31,517 | Given that a basketball player has a 40% chance of scoring with each shot, estimate the probability that the player makes exactly two out of three shots using a random simulation method. The simulation uses a calculator to generate a random integer between 0 and 9, with 1, 2, 3, and 4 representing a scored shot, and 5, 6, 7, 8, 9, 0 representing a missed shot. Three random numbers are grouped to represent the results of three shots. After simulating, the following 20 groups of random numbers were generated:
```
907 966 191 925 271 932 812 458 569 683
431 257 393 027 556 488 730 113 537 989
```
Based on this, estimate the probability that the athlete scores twice out of three shots. | \frac {1}{4} | 0 |
31,518 | A solid cube of side length $2$ is removed from each corner of a larger solid cube of side length $4$. Find the number of edges of the remaining solid. | 36 | 60.15625 |
31,519 | For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative? | 12 | 38.28125 |
31,520 | Let X be a set containing 10 elements, and A, B be two disjoint subsets of X, containing 3 and 4 elements respectively. Calculate the number of subsets of X that contain neither A nor B. | 840 | 0 |
31,521 | My five friends and I play doubles badminton every weekend. Each weekend, two of us play as a team against another two, while the remaining two rest. How many different ways are there for us to choose the two teams and the resting pair? | 45 | 32.03125 |
31,522 | What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003}+1)^{11/7}$? | 571 | 21.875 |
31,523 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, and $\overrightarrow{a}\cdot \overrightarrow{b}=-\sqrt{2}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{3\pi}{4} | 94.53125 |
31,524 | A deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.) | 210 | 6.25 |
31,525 | Given the ellipse $C:\dfrac{x^2}{m^2}+y^2=1$ (where $m > 1$ is a constant), $P$ is a moving point on curve $C$, and $M$ is the right vertex of curve $C$. The fixed point $A$ has coordinates $(2,0)$.
$(1)$ If $M$ coincides with $A$, find the coordinates of the foci of curve $C$;
$(2)$ If $m=3$, find the maximum and minimum values of $|PA|$. | \dfrac{\sqrt{2}}{2} | 57.03125 |
31,526 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent piece. What is the probability that at least 40 cents worth of coins come up heads? | \frac{9}{16} | 0.78125 |
31,527 | Consider all angles measured in degrees. Calculate the product $\prod_{k=1}^{22} \csc^2(4k-3)^\circ = m^n$, where $m$ and $n$ are integers greater than 1. Find the value of $m+n$. | 23 | 0.78125 |
31,528 | Given a point P on the curve $y = x^2 - \ln x$, find the minimum distance from point P to the line $y = x + 2$. | \sqrt{2} | 83.59375 |
31,529 | For Beatrix's latest art installation, she has fixed a $2 \times 2$ square sheet of steel to a wall. She has two $1 \times 2$ magnetic tiles, both of which she attaches to the steel sheet, in any orientation, so that none of the sheet is visible and the line separating the two tiles cannot be seen. One tile has one black cell and one grey cell; the other tile has one black cell and one spotted cell. How many different looking $2 \times 2$ installations can Beatrix obtain?
A 4
B 8
C 12
D 14
E 24 | 12 | 48.4375 |
31,530 | Given that the random variable $X$ follows a two-point distribution with $E(X) = 0.7$, determine its success probability.
A) $0$
B) $1$
C) $0.3$
D) $0.7$ | 0.7 | 98.4375 |
31,531 | How many $3$-digit positive integers have digits whose product equals $36$? | 21 | 37.5 |
31,532 | Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$ , and let $M = \left( \sqrt{3}+1 \right)x + 2y$ . Determine the maximum possible value of $M^2$ .
*Proposed by Varun Mohan* | 16104 | 56.25 |
31,533 | What is the number of radians in the smaller angle formed by the hour and minute hands of a clock at 3:40? Express your answer as a decimal rounded to three decimal places. | 2.278 | 0 |
31,534 | Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 3 and 9, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Calculate the length of the chord expressed in the form $\frac{m\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, and provide $m+n+p.$ | 22 | 7.03125 |
31,535 | In a square with a side length of 12 cm, the midpoints of its adjacent sides are connected to each other and to the opposite side of the square. Find the radius of the circle inscribed in the resulting triangle. | 2\sqrt{5} - \sqrt{2} | 0.78125 |
31,536 | A group of adventurers is showing their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. Additionally, it is known that:
- If an adventurer has diamonds, then they have either emeralds or sapphires (but not both simultaneously);
- If an adventurer has emeralds, then they have either rubies or diamonds (but not both simultaneously).
What is the minimum number of adventurers that can be in such a group? | 16 | 20.3125 |
31,537 | A factory produces a certain type of component, and the inspector randomly selects 16 of these components from the production line each day to measure their dimensions (in cm). The dimensions of the 16 components selected in one day are as follows:
10.12, 9.97, 10.01, 9.95, 10.02, 9.98, 9.21, 10.03, 10.04, 9.99, 9.98, 9.97, 10.01, 9.97, 10.03, 10.11
The mean ($\bar{x}$) and standard deviation ($s$) are calculated as follows:
$\bar{x} \approx 9.96$, $s \approx 0.20$
(I) If there is a component with a dimension outside the range of ($\bar{x} - 3s$, $\bar{x} + 3s$), it is considered that an abnormal situation has occurred in the production process of that day, and the production process of that day needs to be inspected. Based on the inspection results of that day, is it necessary to inspect the production process of that day? Please explain the reason.
(II) Among the 16 different components inspected that day, two components are randomly selected from those with dimensions in the range of (10, 10.1). Calculate the probability that the dimensions of both components are greater than 10.02. | \frac{1}{5} | 44.53125 |
31,538 | The product of two consecutive integers is $20{,}412$. What is the sum of these two integers? | 287 | 6.25 |
31,539 | A line with slope $k$ intersects the parabola $y^{2}=4x$ at points $A$ and $B$, and is tangent to the circle $(x-5)^{2}+y^{2}=9$ at point $M$, where $M$ is the midpoint of segment $AB$. Find the value of $k$. | \frac{2\sqrt{5}}{5} | 1.5625 |
31,540 | The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$. If the terms of the sequence $\{a_n\}$ are arranged according to the following rule: $$\frac {1}{2}, \frac {1}{3}, \frac {2}{3}, \frac {1}{4}, \frac {2}{4}, \frac {3}{4}, \frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \ldots, \frac {1}{n}, \frac {2}{n}, \ldots, \frac {n-1}{n}, \ldots$$ If there exists a positive integer k such that $S_{k-1} < 10$ and $S_k > 10$, then $a_k = \_\_\_\_\_\_$. | \frac{6}{7} | 9.375 |
31,541 | The value of \( a \) is chosen such that the number of roots of the first equation \( 4^{x} - 4^{-x} = 2 \cos a x \) is 2007. How many roots does the second equation \( 4^{x} + 4^{-x} = 2 \cos a x + 4 \) have for the same \( a \)? | 4014 | 17.1875 |
31,542 | The horse walks a distance that is half of the previous day's distance each day, and after walking for 7 days, it has covered a total distance of 700 li. Calculate the total distance it will cover from the 8th day to the 14th day. | \frac {175}{32} | 16.40625 |
31,543 | The complex number $2+i$ and the complex number $\frac{10}{3+i}$ correspond to points $A$ and $B$ on the complex plane, calculate the angle $\angle AOB$. | \frac{\pi}{4} | 63.28125 |
31,544 | Consider numbers of the form $1a1$ , where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$ , $91719$ .* | 55 | 28.125 |
31,545 | How many natural numbers between 200 and 400 are divisible by 8? | 24 | 0.78125 |
31,546 | Given the function $f(x) = \sqrt{ax^2 + bx + c}$ $(a < 0)$, and all points $\left(\begin{matrix}s, f(t) \end{matrix}\right)$ (where $s, t \in D$) form a square region, determine the value of $a$. | -4 | 12.5 |
31,547 | The coefficient of \\(x^4\\) in the expansion of \\((1+x+x^2)(1-x)^{10}\\). | 135 | 96.875 |
31,548 | How many kings can be placed on an $8 \times 8$ chessboard without any of them being in check? | 16 | 6.25 |
31,549 | A regular hexagon $ABCDEF$ has sides of length 2. Find the area of $\bigtriangleup ADF$. Express your answer in simplest radical form. | 4\sqrt{3} | 1.5625 |
31,550 | Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \cdots - 59 + 63 - 67 + 71$. | -36 | 7.03125 |
31,551 | Given the sequence $\{a\_n\}$ that satisfies the condition: when $n \geqslant 2$ and $n \in \mathbb{N}^+$, we have $a\_n + a\_{n-1} = (-1)^n \times 3$. Calculate the sum of the first 200 terms of the sequence $\{a\_n\}$. | 300 | 35.9375 |
31,552 | If Greg rolls six fair six-sided dice, what is the probability that he rolls more 2's than 5's? | \dfrac{16710}{46656} | 0 |
31,553 | In 2000, there were 60,000 cases of a disease reported in a country. By 2020, there were only 300 cases reported. Assume the number of cases decreased exponentially rather than linearly. Determine how many cases would have been reported in 2010. | 4243 | 96.09375 |
31,554 | The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? | \frac{53}{3} | 5.46875 |
31,555 | Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one tie such that the shirt and hat are never of the same color, and the tie must match the color of the hat? | 18900 | 17.96875 |
31,556 | 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. | 0.472 | 0 |
31,557 | (1) Given that $\tan \alpha = -2$, calculate the value of $\dfrac {3\sin \alpha + 2\cos \alpha}{5\cos \alpha - \sin \alpha}$.
(2) Given that $\sin \alpha = \dfrac {2\sqrt{5}}{5}$, calculate the value of $\tan (\alpha + \pi) + \dfrac {\sin \left( \dfrac {5\pi}{2} + \alpha \right)}{\cos \left( \dfrac {5\pi}{2} - \alpha \right)}$. | -\dfrac{5}{2} | 20.3125 |
31,558 |
Anton makes custom watches for a jewelry store. Each watch consists of a bracelet, a gemstone, and a clasp.
Bracelets are available in silver, gold, and steel. Anton has precious stones: cubic zirconia, emerald, quartz, diamond, and agate - and clasps: classic, butterfly, buckle. Anton is happy only when three watches are displayed in a row from left to right according to the following rules:
- There must be a steel watch with a classic clasp and cubic zirconia stone.
- Watches with a classic clasp must be flanked by gold and silver watches.
- The three watches in a row must have different bracelets, gemstones, and clasps.
How many ways are there to make Anton happy? | 72 | 0.78125 |
31,559 | If the positive numbers $a$ and $b$ satisfy the equation $2+\log_{2}a=3+\log_{3}b=\log_{6}(a+b)$, find the value of $\frac{1}{a}+\frac{1}{b}$. | 108 | 0 |
31,560 | The sequence is formed by taking all positive multiples of 4 that contain at least one digit that is either a 2 or a 3. What is the $30^\text{th}$ term of this sequence? | 132 | 14.84375 |
31,561 | Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $ABC$, and $\frac{a-c}{b-\sqrt{2}c}=\frac{sin(A+C)}{sinA+sinC}$.
$(Ⅰ)$ Find the measure of angle $A$;
$(Ⅱ)$ If $a=\sqrt{2}$, $O$ is the circumcenter of triangle $ABC$, find the minimum value of $|3\overrightarrow{OA}+2\overrightarrow{OB}+\overrightarrow{OC}|$;
$(Ⅲ)$ Under the condition of $(Ⅱ)$, $P$ is a moving point on the circumcircle of triangle $ABC$, find the maximum value of $\overrightarrow{PB}•\overrightarrow{PC}$. | \sqrt{2} + 1 | 0 |
31,562 | Given that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $c\cos A=5$, $a\sin C=4$.
(1) Find the length of side $c$;
(2) If the area of $\triangle ABC$, $S=16$, find the perimeter of $\triangle ABC$. | 13+ \sqrt {41} | 0 |
31,563 | Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. | -10 | 24.21875 |
31,564 | Given complex numbers $z_1 = \cos\theta - i$ and $z_2 = \sin\theta + i$, the maximum value of the real part of $z_1 \cdot z_2$ is \_\_\_\_\_\_, and the maximum value of the imaginary part is \_\_\_\_\_\_. | \sqrt{2} | 98.4375 |
31,565 | There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? | 512 | 2.34375 |
31,566 | Given that $\overrightarrow{OA}=(1,0)$, $\overrightarrow{OB}=(1,1)$, and $(x,y)=λ \overrightarrow{OA}+μ \overrightarrow{OB}$, if $0\leqslant λ\leqslant 1\leqslant μ\leqslant 2$, then the maximum value of $z= \frac {x}{m}+ \frac{y}{n}(m > 0,n > 0)$ is $2$. Find the minimum value of $m+n$. | \frac{5}{2}+ \sqrt{6} | 3.90625 |
31,567 | Given an isosceles right triangle ∆PQR with hypotenuse PQ = 4√2, let S be the midpoint of PR. On segment QR, point T divides it so that QT:TR = 2:1. Calculate the area of ∆PST. | \frac{8}{3} | 21.09375 |
31,568 | Triangle PQR is a right triangle with PQ = 6, QR = 8, and PR = 10. Point S is on PR, and QS bisects the right angle at Q. The inscribed circles of triangles PQS and QRS have radii rp and rq, respectively. Find rp/rq. | \frac{3}{28}\left(10-\sqrt{2}\right) | 0 |
31,569 | Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints. | 9\pi | 37.5 |
31,570 | Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .
*2021 CCA Math Bonanza Lightning Round #3.4* | 164 | 60.9375 |
31,571 | Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x-1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{4} | 90.625 |
31,572 | Begin by adding 78.652 to 24.3981. After adding, subtract 0.025 from the result. Finally, round the answer to the nearest hundredth. | 103.03 | 82.03125 |
31,573 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $C= \dfrac {\pi}{3}$, $b=8$. The area of $\triangle ABC$ is $10 \sqrt {3}$.
(I) Find the value of $c$;
(II) Find the value of $\cos (B-C)$. | \dfrac {13}{14} | 79.6875 |
31,574 | If \(\sin A = \frac{p}{5}\) and \(\frac{\cos A}{\tan A} = \frac{q}{15}\), find \(q\). | 16 | 0 |
31,575 | $\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously? | 1440 | 1.5625 |
31,576 | You have $7$ red shirts, $8$ green shirts, $10$ pairs of pants, $10$ blue hats, $10$ red hats, and $5$ scarves (each distinct). How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one scarf, without having the same color of shirts and hats? | 7500 | 15.625 |
31,577 | Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from twelve consecutive natural numbers to a certain number \(a\) is 358, and the sum of the distances from these same twelve numbers to another number \(b\) is 212. Find all possible values of \(a\), given that \(a + b = 114.5\). | \frac{190}{3} | 0 |
31,578 | A four digit number is called *stutterer* if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers. | 7744 | 100 |
31,579 | Let the domain of the function $f(x)$ be $R$. $f(x+1)$ is an odd function, and $f(x+2)$ is an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, calculate $f(\frac{9}{2})$. | \frac{5}{2} | 3.125 |
31,580 | We call a four-digit number with the following property a "centered four-digit number": Arrange the four digits of this four-digit number in any order, and when all the resulting four-digit numbers (at least 2) are sorted from smallest to largest, the original four-digit number is exactly in the middle position. For example, 2021 is a "centered four-digit number". How many "centered four-digit numbers" are there, including 2021? | 90 | 2.34375 |
31,581 | In the Cartesian coordinate system, the coordinates of point P(a,b) satisfy $a \neq b$, and both $a$ and $b$ are elements of the set $\{1,2,3,4,5,6\}$. Additionally, the distance from point P to the origin, $|OP|$, is greater than or equal to 5. The number of such points P is ______. | 20 | 57.8125 |
31,582 | In triangle $∆ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$ respectively. The vector $\overrightarrow{m}\left(a, \sqrt{3b}\right)$ is parallel to $\overrightarrow{n}=\left(\cos A,\sin B\right)$.
(1) Find $A$;
(2) If $a= \sqrt{7},b=2$, find the area of $∆ABC$. | \dfrac{3 \sqrt{3}}{2} | 86.71875 |
31,583 | Given the sequence $103, 1003, 10003, 100003, \dots$, find how many of the first 2048 numbers in this sequence are divisible by 101. | 20 | 0.78125 |
31,584 | 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. | 0.293 | 0 |
31,585 | In triangle $ABC$, $AB = 5$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD$. | \sqrt{0.45} | 0 |
31,586 | What is the smallest positive value of $m$ so that the equation $15x^2 - mx + 315 = 0$ has integral solutions? | 150 | 14.0625 |
31,587 | Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have. | 10 | 28.125 |
31,588 | Given that the area of obtuse triangle $\triangle ABC$ is $2 \sqrt {3}$, $AB=2$, $BC=4$, find the radius of the circumcircle. | \dfrac {2 \sqrt {21}}{3} | 0 |
31,589 | A fly trapped inside a rectangular prism with dimensions $1$ meter, $2$ meters, and $3$ meters decides to tour the corners of the box. It starts from the corner $(0,0,0)$ and ends at the corner $(0,0,3)$, visiting each of the other corners exactly once. Determine the maximum possible length, in meters, of its path assuming that it moves in straight lines.
A) $\sqrt{14} + 4 + \sqrt{13} + \sqrt{5}$
B) $\sqrt{14} + 4 + \sqrt{5} + \sqrt{10}$
C) $2\sqrt{14} + 4 + 2$
D) $\sqrt{14} + 6 + \sqrt{13} + 1$
E) $\sqrt{14} + 6 + \sqrt{13} + \sqrt{5}$ | \sqrt{14} + 6 + \sqrt{13} + \sqrt{5} | 26.5625 |
31,590 |
Xiaoming takes 100 RMB to the store to buy stationery. After returning, he counts the money he received in change and finds he has 4 banknotes of different denominations and 4 coins of different denominations. The banknotes have denominations greater than 1 yuan, and the coins have denominations less than 1 yuan. Furthermore, the total value of the banknotes in units of "yuan" must be divisible by 3, and the total value of the coins in units of "fen" must be divisible by 7. What is the maximum amount of money Xiaoming could have spent?
(Note: The store gives change in denominations of 100 yuan, 50 yuan, 20 yuan, 10 yuan, 5 yuan, and 1 yuan banknotes, and coins with values of 5 jiao, 1 jiao, 5 fen, 2 fen, and 1 fen.) | 63.37 | 0 |
31,591 | Let $a,$ $b,$ $c,$ $d$ be distinct integers, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ Find the smallest possible value of
\[|a + b \omega + c \omega^2 + d \omega^3|.\] | \sqrt{4.5} | 0 |
31,592 | As shown in the diagram, points \(E\), \(F\), \(G\), and \(H\) are located on the four sides of the square \(ABCD\) as the trisection points closer to \(B\), \(C\), \(D\), and \(A\) respectively. If the side length of the square is 6, what is the area of the shaded region? | 18 | 7.8125 |
31,593 | In triangle $PQR,$ $PQ = 24,$ $QR = 25,$ $PR = 7,$ and point $H$ is the intersection of the altitudes (the orthocenter). Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles $PQR$ and $P'Q'R'$? | 84 | 3.125 |
31,594 | Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$ . | 505 | 18.75 |
31,595 | Determine how many five-letter words can be formed such that they begin and end with the same letter and the third letter is always a vowel. | 87880 | 50 |
31,596 | The function \[f(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x} & \quad \text{ if } x \ge 5 \end{aligned} \right.\] has an inverse $f^{-1}.$ Find the value of $f^{-1}(0) + f^{-1}(1) + \dots + f^{-1}(9).$ | 291 | 0.78125 |
31,597 | Evaluate the expression: \[ M = \frac{\sqrt{\sqrt{7} + 3} + \sqrt{\sqrt{7} - 3}}{\sqrt{\sqrt{7} + 2}} + \sqrt{5 - 2\sqrt{6}}, \]
and determine the value of $M$.
A) $\sqrt{9}$
B) $\sqrt{2} - 1$
C) $2\sqrt{2} - 1$
D) $\sqrt{\frac{7}{2}}$
E) $1$ | 2\sqrt{2} - 1 | 3.90625 |
31,598 | Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win of 30 rubles, and there are no other prizes. The Scientist decided to take a risk since he has nothing to lose. Find the probability that the Absent-Minded Scientist will be able to win enough money to buy a bus ticket. | 0.19 | 27.34375 |
31,599 | Let sets $X$ and $Y$ have $30$ and $25$ elements, respectively, and there are at least $10$ elements in both sets. Find the smallest possible number of elements in $X \cup Y$. | 45 | 57.03125 |
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