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29,300 | On the lateral side $CD$ of trapezoid $ABCD (AD \parallel BC)$, a point $M$ is marked. From vertex $A$, a perpendicular $AH$ is dropped to the segment $BM$. It is found that $AD = HD$. Find the length of segment $AD$ if it is known that $BC = 16$, $CM = 8$, $MD = 9$. | 18 | 8.59375 |
29,301 | Point $Q$ lies on the diagonal $AC$ of square $EFGH$ with $EQ > GQ$. Let $R_{1}$ and $R_{2}$ be the circumcenters of triangles $EFQ$ and $GHQ$ respectively. Given that $EF = 8$ and $\angle R_{1}QR_{2} = 90^{\circ}$, find the length $EQ$ in the form $\sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c+d$. | 40 | 3.125 |
29,302 | Let $ABCD$ be a cyclic quadrilateral where the side lengths are distinct primes less than $20$ with $AB\cdot BC + CD\cdot DA = 280$. What is the largest possible value of $BD$?
A) 17
B) $\sqrt{310}$
C) $\sqrt{302.76}$
D) $\sqrt{285}$ | \sqrt{302.76} | 13.28125 |
29,303 | Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points) | \frac{9\sqrt{3}}{4} | 35.9375 |
29,304 | An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{3\sqrt{3}}{4\pi} | 28.125 |
29,305 | In a simulation experiment, 20 groups of random numbers were generated: 6830, 3013, 7055, 7430, 7740, 4422, 7884, 2604, 3346, 0952, 6807, 9706, 5774, 5725, 6576, 5929, 9768, 6071, 9138, 6754. If the numbers 1, 2, 3, 4, 5, 6 each appear exactly three times among these, it represents hitting the target exactly three times. What is the approximate probability of hitting the target exactly three times in four shots? | 25\% | 0 |
29,306 | Let $N = 99999$. Then $N^3 = \ $ | 999970000299999 | 96.875 |
29,307 | Let $T$ be the set of all ordered triples of integers $(b_1, b_2, b_3)$ with $1 \leq b_1, b_2, b_3 \leq 20$. Each ordered triple in $T$ generates a sequence according to the rule $b_n = b_{n-1} \cdot |b_{n-2} - b_{n-3}|$ for all $n \geq 4$. Find the number of such sequences for which $b_n = 0$ for some $n$. | 780 | 4.6875 |
29,308 | Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$ has an extremum of $10$ at $x=1$, find the value of $f(2)$. | 18 | 60.15625 |
29,309 | A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x^2 < y$? | \frac{\sqrt{2}}{18} | 7.8125 |
29,310 | From the set $\{1,2,3, \cdots, 20\}$, choose 4 different numbers such that these 4 numbers form an arithmetic sequence. Determine the number of such arithmetic sequences. | 114 | 53.90625 |
29,311 | Given a circle $C$ with the equation $(x-a)^2 + (y-2)^2 = 4$ ($a > 0$) and a line $l$ with the equation $x - y + 3 = 0$, which intersects the circle at points $A$ and $B$:
1. When the chord AB cut by line $l$ on circle $C$ has a length of $2\sqrt{2}$, find the value of $a$.
2. If there exists a point $P$ on the circle such that $\overrightarrow{CA} + \overrightarrow{CB} = \overrightarrow{CP}$, find the value of $a$. | 2\sqrt{2} - 1 | 10.9375 |
29,312 | Let $q(x) = x^{2007} + x^{2006} + \cdots + x + 1$, and let $s(x)$ be the polynomial remainder when $q(x)$ is divided by $x^3 + 2x^2 + x + 1$. Find the remainder when $|s(2007)|$ is divided by 1000. | 49 | 1.5625 |
29,313 | Use the Horner's method to write out the process of calculating the value of $f(x) = 1 + x + 0.5x^2 + 0.16667x^3 + 0.04167x^4 + 0.00833x^5$ at $x = -0.2$. | 0.81873 | 0 |
29,314 | In civil engineering, the stability of cylindrical columns under a load is often examined using the formula \( L = \frac{50T^3}{SH^2} \), where \( L \) is the load in newtons, \( T \) is the thickness of the column in centimeters, \( H \) is the height of the column in meters, and \( S \) is a safety factor. Given that \( T = 5 \) cm, \( H = 10 \) m, and \( S = 2 \), calculate the value of \( L \). | 31.25 | 10.15625 |
29,315 | The bug Josefína landed in the middle of a square grid composed of 81 smaller squares. She decided not to crawl away directly but to follow a specific pattern: first moving one square south, then one square east, followed by two squares north, then two squares west, and repeating the pattern of one square south, one square east, two squares north, and two squares west. On which square was she just before she left the grid? How many squares did she crawl through on this grid? | 20 | 2.34375 |
29,316 | Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\{1, 2, 3, \dots, 12\}$. What is the probability that $xyz - x - y - z$ is even? | \frac{1}{11} | 0 |
29,317 | If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0.78125 |
29,318 | In the "Lucky Sum" lottery, there are a total of $N$ balls numbered from 1 to $N$. During the main draw, 10 balls are randomly selected. During the additional draw, 8 balls are randomly selected from the same set of balls. The sum of the numbers on the selected balls in each draw is announced as the "lucky sum," and players who predicted this sum win a prize.
Can it be that the events $A$ "the lucky sum in the main draw is 63" and $B$ "the lucky sum in the additional draw is 44" are equally likely? If so, under what condition? | 18 | 4.6875 |
29,319 | Using the digits $0$, $1$, $2$, $3$, $4$ to form a five-digit number without repeating any digit, the probability that the number is even and the digits $1$, $2$ are adjacent is ______. | 0.25 | 0.78125 |
29,320 | Let \( S = \{1, 2, \cdots, 2016\} \). For any non-empty finite sets of real numbers \( A \) and \( B \), find the minimum value of
\[ f = |A \Delta S| + |B \Delta S| + |C \Delta S| \]
where
\[ X \Delta Y = \{a \in X \mid a \notin Y\} \cup \{a \in Y \mid a \notin X\} \]
is the symmetric difference between sets \( X \) and \( Y \), and
\[ C = \{a + b \mid a \in A, b \in B\} .\] | 2017 | 9.375 |
29,321 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. If $(\sqrt{3}b-c)\cos A=a\cos C$, then $\cos A=$_______. | \frac{\sqrt{3}}{3} | 76.5625 |
29,322 | Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods. | 18 | 7.03125 |
29,323 | In a New Year's cultural evening of a senior high school class, there was a game involving a box containing 6 cards of the same size, each with a different idiom written on it. The idioms were: 意气风发 (full of vigor), 风平浪静 (calm and peaceful), 心猿意马 (restless), 信马由缰 (let things take their own course), 气壮山河 (majestic), 信口开河 (speak without thinking). If two cards drawn randomly from the box contain the same character, then it's a win. The probability of winning this game is ____. | \dfrac{2}{5} | 1.5625 |
29,324 | Given a square grid of size $2023 \times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six cells. What is the smallest $n$ for which this configuration is possible? | 338 | 11.71875 |
29,325 | Given a quadratic polynomial $q(x) = x^2 - px + q$ known to be "mischievous" if the equation $q(q(x)) = 0$ is satisfied by exactly three different real numbers, determine the value of $q(2)$ for the unique polynomial $q(x)$ for which the product of its roots is minimized. | -1 | 3.125 |
29,326 | Given the digits 1, 2, 3, 4, 5, 6 to form a six-digit number (without repeating any digit), requiring that any two adjacent digits have different parity, and 1 and 2 are adjacent, determine the number of such six-digit numbers. | 40 | 67.96875 |
29,327 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sin ^{2}A+\cos ^{2}B+\cos ^{2}C=2+\sin B\sin C$.<br/>$(1)$ Find the measure of angle $A$;<br/>$(2)$ If $a=3$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$, find the maximum length of segment $AD$. | \frac{\sqrt{3}}{2} | 6.25 |
29,328 | Increase Grisha's yield by $40\%$ and Vasya's yield by $20\%$.
Grisha, the cleverest among them, calculated that their combined yield in each of the cases would increase by 5 kg, 10 kg, and 9 kg, respectively. What was the total yield of the friends (in kilograms) before meeting Hottabych? | 40 | 6.25 |
29,329 | The line joining $(4,3)$ and $(7,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Assume the square has vertices at $(4,0)$, $(7,0)$, $(7,3)$, and $(4,3)$. | \frac{5}{6} | 8.59375 |
29,330 | A point $(x, y)$ is randomly selected from inside the rectangle with vertices $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(0, 3)$. What is the probability that both $x < y$ and $x + y < 5$? | \frac{3}{8} | 6.25 |
29,331 | Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon. | 1506 | 4.6875 |
29,332 | At an exchange point, there are two types of transactions:
1) Give 2 euros - receive 3 dollars and a candy as a gift.
2) Give 5 dollars - receive 3 euros and a candy as a gift.
When the wealthy Buratino came to the exchange point, he only had dollars. When he left, he had fewer dollars, he did not get any euros, but he received 50 candies. How many dollars did Buratino spend for such a "gift"? | 10 | 0.78125 |
29,333 | There are three positive integers: large, medium, and small. The sum of the large and medium numbers equals 2003, and the difference between the medium and small numbers equals 1000. What is the sum of these three positive integers? | 2004 | 68.75 |
29,334 | Evaluate the expression $$\frac{\sin 10°}{1 - \sqrt{3}\tan 10°}.$$ | \frac{1}{2} | 4.6875 |
29,335 | Let equilateral triangle $ABC$ have side length $7$. There are three distinct triangles $AD_1E_1$, $AD_1E_2$, and $AD_2E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{21}$. Find $\sum_{k=1}^3 (CE_k)^2$. | 294 | 9.375 |
29,336 | In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac{EQ}{EF} = \frac{1}{8}$, and point $R$ is on $\overline{EH}$ such that $\frac{ER}{EH} = \frac{1}{9}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find the ratio $\frac{ES}{EG}$. | \frac{1}{9} | 1.5625 |
29,337 | Let $P$ be a moving point on curve $C_1$, and $Q$ be a moving point on curve $C_2$. The minimum value of $|PQ|$ is called the distance between curves $C_1$ and $C_2$, denoted as $d(C_1,C_2)$. If $C_1: x^{2}+y^{2}=2$, $C_2: (x-3)^{2}+(y-3)^{2}=2$, then $d(C_1,C_2)=$ \_\_\_\_\_\_ ; if $C_3: e^{x}-2y=0$, $C_4: \ln x+\ln 2=y$, then $d(C_3,C_4)=$ \_\_\_\_\_\_ . | \sqrt{2} | 10.9375 |
29,338 | Jessica's six assignment scores are 87, 94, 85, 92, 90, and 88. What is the arithmetic mean of these six scores? | 89.3 | 0 |
29,339 | Arrange the sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cycle: $(3)$, $(5, 7)$, $(9, 11, 13)$, $(15, 17, 19, 21)$, $(23)$, $(25, 27)$, $(29, 31, 33)$, $(35, 37, 39, 41)$, $(43)$, ..., then 2013 is the number in the $\boxed{\text{nth}}$ bracket. | 403 | 13.28125 |
29,340 | In a rectangular parallelepiped with dimensions AB = 4, BC = 2, and CG = 5, point M is the midpoint of EF, calculate the volume of the rectangular pyramid with base BDFE and apex M. | \frac{40}{3} | 24.21875 |
29,341 | Find the number of permutations of the 6 characters $a, b, c, d, e, f$ such that the subsequences $a c e$ and $d f$ do not appear. | 582 | 67.1875 |
29,342 | In $\triangle ABC$, if $\angle B=30^{\circ}$, $AB=2 \sqrt {3}$, $AC=2$, find the area of $\triangle ABC$. | 2 \sqrt {3} | 0 |
29,343 | One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube. | 64 | 9.375 |
29,344 | How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 | 0 |
29,345 | A rectangular grid consists of 5 rows and 6 columns with equal square blocks. How many different squares can be traced using the lines in the grid? | 70 | 35.9375 |
29,346 | In a right-angled triangle, the lengths of the two legs are 12 and 5, respectively. Find the length of the hypotenuse and the height from the right angle to the hypotenuse. | \frac{60}{13} | 42.96875 |
29,347 | Given the enclosure dimensions are 15 feet long, 8 feet wide, and 7 feet tall, with each wall and floor being 1 foot thick, determine the total number of one-foot cubical blocks used to create the enclosure. | 372 | 16.40625 |
29,348 | Find the smallest positive integer \( n \) that is not less than 9, such that for any \( n \) integers (which can be the same) \( a_{1}, a_{2}, \cdots, a_{n} \), there always exist 9 numbers \( a_{i_{1}}, a_{i_{2}}, \cdots, a_{i_{9}} \) (where \(1 \leq i_{1} < i_{2} < \cdots < i_{9} \leq n \)) and \( b_{i} \in \{4,7\} \) (for \(i=1,2,\cdots,9\)), such that \( b_{1} a_{i_{1}} + b_{2} a_{i_{2}} + \cdots + b_{9} a_{i_{9}} \) is divisible by 9. | 13 | 37.5 |
29,349 | In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is $\begin{cases}x=1+t\cdot\cos \alpha \\ y=2+t\cdot\sin \alpha \end{cases} (t \text{ is a parameter})$, and in the polar coordinate system (with the same length unit as the rectangular coordinate system $(xOy)$, and with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis), the equation of curve $C$ is $\rho=6\sin \theta$.
(1) Find the rectangular coordinate equation of curve $C$ and the ordinary equation of line $l$ when $\alpha=\frac{\pi}{2}$;
(2) Suppose curve $C$ intersects line $l$ at points $A$ and $B$, and point $P$ has coordinates $(1,2)$. Find the minimum value of $|PA|+|PB|$. | 2\sqrt{7} | 48.4375 |
29,350 | Given that it is currently between 4:00 and 5:00 o'clock, and eight minutes from now, the minute hand of a clock will be exactly opposite to the position where the hour hand was six minutes ago, determine the exact time now. | 4:45\frac{3}{11} | 3.90625 |
29,351 | The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$. | 13 | 27.34375 |
29,352 | The total number of matches played in the 2006 World Cup competition can be calculated by summing the number of matches determined at each stage of the competition. | 64 | 43.75 |
29,353 | If 640 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? | 24 | 8.59375 |
29,354 | Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\). | 39 | 20.3125 |
29,355 | A certain chemical reaction requires a catalyst to accelerate the reaction, but using too much of this catalyst affects the purity of the product. If the amount of this catalyst added is between 500g and 1500g, and the 0.618 method is used to arrange the experiment, then the amount of catalyst added for the second time is \_\_\_\_\_\_ g. | 882 | 11.71875 |
29,356 | Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{7}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\] | -\frac{4}{3} | 3.90625 |
29,357 | A student reads a book, reading 35 pages on the first day and then 5 more pages each subsequent day, until only 35 pages are left on the last day. The second time he reads it, he reads 45 pages on the first day and then 5 more pages each subsequent day, until only 40 pages are left on the last day. How many pages does the book have in total? | 385 | 2.34375 |
29,358 | (The full score for this question is 8 points) There are 4 red cards labeled with the numbers 1, 2, 3, 4, and 2 blue cards labeled with the numbers 1, 2. Four different cards are drawn from these 6 cards.
(1) If it is required that at least one blue card is drawn, how many different ways are there to draw the cards?
(2) If the sum of the numbers on the four drawn cards equals 10, and they are arranged in a row, how many different arrangements are there? | 96 | 2.34375 |
29,359 | For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$ . (Once a sapling is turned into a perfectly-shaped apple tree, it will stay a perfectly-shaped apple tree.) The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Find $100m+n$ .
*Proposed by Yannick Yao* | 4910 | 4.6875 |
29,360 | In the quadrilateral $ABCD$ , the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$ , as well with side $AD$ an angle of $30^o$ . Find the acute angle between the diagonals $AC$ and $BD$ . | 80 | 0.78125 |
29,361 | The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM? | 30 | 13.28125 |
29,362 | Given that each side of a large square is divided into four equal parts, a smaller square is inscribed in such a way that its corners are at the division points one-fourth and three-fourths along each side of the large square, calculate the ratio of the area of this inscribed square to the area of the large square. | \frac{1}{4} | 32.8125 |
29,363 | Five candidates are to be selected to perform four different jobs, where one candidate can only work as a driver and the other four can do all the jobs. Determine the number of different selection schemes. | 48 | 1.5625 |
29,364 | Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum? | 10 | 25 |
29,365 | Given positive integers \( N \) and \( k \), we counted how many different ways the number \( N \) can be written in the form \( a + b + c \), where \( 1 \leq a, b, c \leq k \), and the order of the summands matters. Could the result be 2007? | 2007 | 0 |
29,366 | Gulliver arrives in the land of the Lilliputians with 7,000,000 rubles. He uses all the money to buy kefir at a price of 7 rubles per bottle (an empty bottle costs 1 ruble at that time). After drinking all the kefir, he returns the bottles and uses the refunded money to buy more kefir. During this process, he notices that the cost of both the kefir and the empty bottle doubles each visit to the store. He continues this cycle: drinking kefir, returning bottles, and buying more kefir, with prices doubling between each store visit. How many bottles of kefir did Gulliver drink? | 1166666 | 0 |
29,367 | Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} | 0 |
29,368 | Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number. | 7200 | 10.9375 |
29,369 | Given $e^{i \theta} = \frac{3 + i \sqrt{8}}{5}$, find $\cos 4 \theta$. | -\frac{287}{625} | 3.90625 |
29,370 | Given the ellipse $$E: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$$ has an eccentricity of $$\frac{\sqrt{2}}{2}$$, and the point $$A(1, \sqrt{2})$$ is on the ellipse.
(1) Find the equation of ellipse E;
(2) If a line $l$ with a slope of $$\sqrt{2}$$ intersects the ellipse E at two distinct points B and C, find the maximum area of triangle ABC. | \sqrt{2} | 15.625 |
29,371 | Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 73 | 0.78125 |
29,372 | Given function $f(x)$ defined on $\mathbb{R}$ satisfies: (1) The graph of $y = f(x - 1)$ is symmetric about the point $(1, 0)$; (2) For all $x \in \mathbb{R}$, $$f\left( \frac {3}{4}-x\right) = f\left( \frac {3}{4}+x\right)$$ holds; (3) When $x \in \left(- \frac {3}{2}, - \frac {3}{4}\right]$, $f(x) = \log_{2}(-3x + 1)$, then find $f(2011)$. | -2 | 14.84375 |
29,373 | What is the smallest positive integer with exactly 12 positive integer divisors? | 108 | 0 |
29,374 | Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9? | 32 | 0 |
29,375 | Find the smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,
\[\sqrt{\frac{x}{y + z + w}} + \sqrt{\frac{y}{x + z + w}} + \sqrt{\frac{z}{x + y + w}} + \sqrt{\frac{w}{x + y + z}} < M.\] | \frac{4}{\sqrt{3}} | 33.59375 |
29,376 | In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has three sides each of length 5. If the length of the chord is $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers, find the value of $m+n$. | 409 | 0 |
29,377 | Determine the sum of all real numbers \(x\) satisfying
\[
(x^2 - 5x + 3)^{x^2 - 6x + 3} = 1.
\] | 16 | 3.90625 |
29,378 | When the radius $r$ of a circle is increased by $5$, the area is quadrupled. What was the original radius $r$? Additionally, find the new perimeter of the circle after this radius increase.
A) Original radius: 4, New perimeter: $18\pi$
B) Original radius: 5, New perimeter: $20\pi$
C) Original radius: 6, New perimeter: $22\pi$
D) Original radius: 5, New perimeter: $19\pi$ | 20\pi | 2.34375 |
29,379 | Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 9x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 38 | 0.78125 |
29,380 | Given the function $f(x)= \sqrt {3}\cos ( \frac {π}{2}+x)\cdot \cos x+\sin ^{2}x$, where $x\in R$.
(I) Find the interval where $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, angles $A$, $B$, and $C$ have corresponding opposite sides $a$, $b$, and $c$. If $B= \frac {π}{4}$, $a=2$, and angle $A$ satisfies $f(A)=0$, find the area of $\triangle ABC$. | \frac {3+ \sqrt {3}}{3} | 0 |
29,381 | If the integers \( a, b, \) and \( c \) satisfy:
\[
a + b + c = 3, \quad a^3 + b^3 + c^3 = 3,
\]
then what is the maximum value of \( a^2 + b^2 + c^2 \)? | 57 | 61.71875 |
29,382 | Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} | 0.78125 |
29,383 | A partition of a number \( n \) is a sequence of positive integers, arranged in descending order, whose sum is \( n \). For example, \( n=4 \) has 5 partitions: \( 1+1+1+1=2+1+1=2+2=3+1=4 \). Given two different partitions of the same number, \( n=a_{1}+a_{2}+\cdots+a_{k}=b_{1}+b_{2}+\cdots+b_{l} \), where \( k \leq l \), the first partition is said to dominate the second if all of the following inequalities hold:
\[
\begin{aligned}
a_{1} & \geq b_{1} ; \\
a_{1}+a_{2} & \geq b_{1}+b_{2} ; \\
a_{1}+a_{2}+a_{3} & \geq b_{1}+b_{2}+b_{3} ; \\
& \vdots \\
a_{1}+a_{2}+\cdots+a_{k} & \geq b_{1}+b_{2}+\cdots+b_{k} .
\end{aligned}
\]
Find as many partitions of the number \( n=20 \) as possible such that none of the partitions dominates any other. Your score will be the number of partitions you find. If you make a mistake and one of your partitions does dominate another, your score is the largest \( m \) such that the first \( m \) partitions you list constitute a valid answer. | 20 | 0.78125 |
29,384 | Given an acute angle \( \theta \), the equation \( x^{2} + 4x \cos \theta + \cot \theta = 0 \) has a double root. Find the radian measure of \( \theta \). | \frac{5\pi}{12} | 0 |
29,385 | If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 1.5625 |
29,386 | Evaluate the sum
\[
\sum_{k=1}^{50} (-1)^k \cdot \frac{k^3 + k^2 + 1}{(k+1)!}.
\]
Determine the form of the answer as a difference between two terms, where each term is a fraction involving factorials. | \frac{126001}{51!} | 0 |
29,387 | Given that $\alpha$ and $\beta$ are acute angles, $\cos\alpha=\frac{{\sqrt{5}}}{5}$, $\cos({\alpha-\beta})=\frac{3\sqrt{10}}{10}$, find the value of $\cos \beta$. | \frac{\sqrt{2}}{10} | 9.375 |
29,388 | Given complex numbers \( z \) and \( \omega \) that satisfy the following two conditions:
1. \( z + \omega + 3 = 0 \);
2. \( | z |, 2, | \omega | \) form an arithmetic sequence.
Does \( \cos (\arg z - \arg \omega) \) have a maximum value? If it does, find the maximum value. | \frac{1}{8} | 0 |
29,389 | For the largest possible $n$, can you create two bi-infinite sequences $A$ and $B$ such that any segment of length $n$ from sequence $B$ is contained in $A$, where $A$ has a period of 1995, but $B$ does not have this property (it is either not periodic or has a different period)?
Comment: The sequences can consist of arbitrary symbols. The problem refers to the minimal period. | 1995 | 42.96875 |
29,390 | How many "plane-line pairs" are formed by a line and a plane parallel to each other in a rectangular box, where the line is determined by two vertices and the plane contains four vertices? | 48 | 8.59375 |
29,391 | Define a "spacy" set of integers such that it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 181 | 0 |
29,392 | Four cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a 2, the third card is a 3, and the fourth card is a 4? Assume that the dealing is done without replacement. | \frac{16}{405525} | 0 |
29,393 | What is the smallest positive integer with exactly 12 positive integer divisors? | 288 | 0 |
29,394 | $-14-(-2)^{3}\times \dfrac{1}{4}-16\times \left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{3}{8}\right)$. | -22 | 55.46875 |
29,395 | An eight-sided die (numbered 1 through 8) is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? | 960 | 1.5625 |
29,396 | A cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}, \overline{P_1P_3}, \overline{P_1P_4}, \overline{P_1'P_2'}, \overline{P_1'P_3'},$ and $\overline{P_1'P_4'}$ with each vertex positioned $\frac{2}{3}$ of the distance from $P_1$ or $P_1'$. | \frac{4\sqrt{2}}{3} | 9.375 |
29,397 | Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ *Gheorghe Iurea* | 15 | 0 |
29,398 | Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt 3$ . We construct four semicircles $\omega_1$ , $\omega_2$ , $\omega_3$ , $\omega_4$ whose diameters are the segments $AB$ , $BC$ , $CD$ , $DA$ . It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$ ). Compute the square of the area of $X_1X_2X_3X_4$ .
*Proposed by Evan Chen* | 243 | 7.8125 |
29,399 | On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. | 21 | 2.34375 |
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