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27,200 | There are three islands A, B, and C at sea. It is measured that the distance between islands A and B is 10n miles, $\angle BAC=60^\circ$, and $\angle ABC=75^\circ$. The distance between islands B and C is \_\_\_\_\_\_ n miles. | 5\sqrt{6} | 84.375 |
27,201 | Given that $α \in (0,π)$, and $\sin α= \frac {3}{5}$, find the value of $\tan (α- \frac {π}{4})$. | -7 | 9.375 |
27,202 | Given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(3,4)$, find the projection of vector $\overrightarrow{a}$ onto the direction of vector $\overrightarrow{b}$. | -1 | 0 |
27,203 | 29 boys and 15 girls attended a ball. Some boys danced with some of the girls (no more than once with each pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? | 29 | 2.34375 |
27,204 | A positive integer $n\geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$ | 404 | 45.3125 |
27,205 | In parallelogram $ABCD$, points $P$, $Q$, $R$, and $S$ are the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively, and point $T$ is the midpoint of segment $SR$. Given that the area of parallelogram $ABCD$ is 120 square centimeters, what is the area of $\triangle PQT$ in square centimeters? | 15 | 84.375 |
27,206 | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 40 | 0 |
27,207 | How many squares can be formed by joining four of the twelve points marked on a rectangular grid? | 11 | 1.5625 |
27,208 | Consider a sequence $\{a_n\}$ with property P: if $a_p = a_q$ for $p, q \in \mathbb{N}^{*}$, then it must hold that $a_{p+1} = a_{q+1}$. Suppose the sequence $\{a_n\}$ has property P, and it is given that $a_1=1$, $a_2=2$, $a_3=3$, $a_5=2$, and $a_6+a_7+a_8=21$. Determine the value of $a_{2017}$. | 16 | 10.9375 |
27,209 | Arrange 2002 students numbered from 1 to 2002 in a row from left to right. Counting from left to right from 1 to 11, the student who is counted as 11 remains in place, while the others leave the row. Then, the remaining students count from left to right from 1 to 11 again, and the student who is counted as 11 remains, while the others leave the row. Finally, the remaining students count from left to right from 1 to 11, and the student who is counted as 11 remains, while the others leave the row. How many students remain in the end? What are their numbers? | 1331 | 32.03125 |
27,210 | Find the smallest positive integer $n$ that satisfies the following two properties:
1. $n$ has exactly 144 distinct positive divisors.
2. Among the positive divisors of $n$, there are ten consecutive integers. | 110880 | 55.46875 |
27,211 | Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\triangle DEF$ to the area of quadrilateral $ABEF$ is: | 1: 5 | 0 |
27,212 | A rectangular swimming pool is 20 meters long, 15 meters wide, and 3 meters deep. Calculate its surface area. | 300 | 0 |
27,213 | A store decides to discount a certain type of clothing due to the change of season. If each piece of clothing is sold at 50% of the marked price, there will be a loss of 20 yuan per piece. However, if sold at 80% of the marked price, there will be a profit of 40 yuan per piece.
Question: (1) What is the marked price of each piece of clothing? (2) What is the cost of each piece of clothing? | 120 | 54.6875 |
27,214 | Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 5.46875 |
27,215 | Given a regular 14-gon, where each vertex is connected to every other vertex by a segment, three distinct segments are chosen at random. What is the probability that the lengths of these three segments refer to a triangle with positive area?
A) $\frac{73}{91}$
B) $\frac{74}{91}$
C) $\frac{75}{91}$
D) $\frac{76}{91}$
E) $\frac{77}{91}$ | \frac{77}{91} | 0 |
27,216 | Ryan is learning number theory. He reads about the *Möbius function* $\mu : \mathbb N \to \mathbb Z$ , defined by $\mu(1)=1$ and
\[ \mu(n) = -\sum_{\substack{d\mid n d \neq n}} \mu(d) \]
for $n>1$ (here $\mathbb N$ is the set of positive integers).
However, Ryan doesn't like negative numbers, so he invents his own function: the *dubious function* $\delta : \mathbb N \to \mathbb N$ , defined by the relations $\delta(1)=1$ and
\[ \delta(n) = \sum_{\substack{d\mid n d \neq n}} \delta(d) \]
for $n > 1$ . Help Ryan determine the value of $1000p+q$ , where $p,q$ are relatively prime positive integers satisfying
\[ \frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}. \]
*Proposed by Michael Kural* | 14013 | 1.5625 |
27,217 | Let $F$ , $D$ , and $E$ be points on the sides $[AB]$ , $[BC]$ , and $[CA]$ of $\triangle ABC$ , respectively, such that $\triangle DEF$ is an isosceles right triangle with hypotenuse $[EF]$ . The altitude of $\triangle ABC$ passing through $A$ is $10$ cm. If $|BC|=30$ cm, and $EF \parallel BC$ , calculate the perimeter of $\triangle DEF$ . | 12\sqrt{2} + 12 | 0.78125 |
27,218 | From the digits $1$, $2$, $3$, $4$, form a four-digit number with the first digit being $1$, and having exactly two identical digits in the number. How many such four-digit numbers are there? | 36 | 10.9375 |
27,219 | Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\Delta APF$ is minimized, the area of this triangle is \_\_\_. | \dfrac{36}{7} | 17.1875 |
27,220 | The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? | 1728 | 0 |
27,221 | Round $3.1415926$ to the nearest thousandth using the rounding rule, and determine the precision of the approximate number $3.0 \times 10^{6}$ up to which place. | 3.142 | 0.78125 |
27,222 | Jancsi usually waits for Juliska at the metro station in the afternoons. Once, he waited for 12 minutes, during which 5 trains arrived at the station. On another occasion, he waited for 20 minutes, and Juliska arrived on the seventh train. Yesterday, Jancsi waited for 30 minutes. How many trains could have arrived in the meantime, given that the time between the arrival of two trains is always the same? | 10 | 8.59375 |
27,223 | Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, determine the total area in square centimeters of the football-shaped area of regions II and III combined. Express your answer as a decimal to the nearest tenth. | 9.1 | 39.0625 |
27,224 | In square ABCD, point E is on AB and point F is on CD such that AE = 3EB and CF = 3FD. | \frac{3}{32} | 0.78125 |
27,225 | Write the number 2013 several times in a row so that the resulting number is divisible by 9. Explain the answer. | 201320132013 | 16.40625 |
27,226 | Given the function $f\left(x\right)=\cos 2x+\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\left(x\right)$, then $\cos (x_{1}+x_{2})=$____. | \frac{1}{4} | 6.25 |
27,227 | In the diagram, \(A B C D\) is a rectangle, \(P\) is on \(B C\), \(Q\) is on \(C D\), and \(R\) is inside \(A B C D\). Also, \(\angle P R Q = 30^\circ\), \(\angle R Q D = w^\circ\), \(\angle P Q C = x^\circ\), \(\angle C P Q = y^\circ\), and \(\angle B P R = z^\circ\). What is the value of \(w + x + y + z\)? | 210 | 9.375 |
27,228 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 120 | 92.96875 |
27,229 | How many distinct arrangements of the letters in the word "example" are there? | 5040 | 17.96875 |
27,230 | Let \( A_0 = (0,0) \). Points \( A_1, A_2, \dots \) lie on the \( x \)-axis, and distinct points \( B_1, B_2, \dots \) lie on the graph of \( y = x^2 \). For every positive integer \( n \), \( A_{n-1}B_nA_n \) is an equilateral triangle. What is the least \( n \) for which the length \( A_0A_n \geq 100 \)? | 10 | 0.78125 |
27,231 | A group of 11 people, including Ivanov and Petrov, are seated in a random order around a circular table. Find the probability that there will be exactly 3 people sitting between Ivanov and Petrov. | 1/10 | 1.5625 |
27,232 | If you write down twice in a row the grade I received in school for Latin, you will get my grandmother's age. What will you get if you divide this age by the number of my kittens? Imagine - you will get my morning grade, increased by fourteen-thirds.
How old is my grandmother? | 77 | 6.25 |
27,233 | Simplify $15\cdot\frac{16}{9}\cdot\frac{-45}{32}$. | -\frac{25}{6} | 0 |
27,234 | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 40 | 0.78125 |
27,235 | In the ancient Chinese mathematical masterpiece "The Mathematical Classic of Sunzi" Volume $26$, the $26$th question is: "There is an unknown quantity, when divided by $3$, the remainder is $2$; when divided by $5$, the remainder is $3; when divided by $7$, the remainder is $2$. What is the quantity?" The mathematical meaning of this question is: find a positive integer that leaves a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $2$ when divided by $7$. This is the famous "Chinese Remainder Theorem". If we arrange all positive integers that leave a remainder of $2$ when divided by $3$ and all positive integers that leave a remainder of $2$ when divided by $7$ in ascending order to form sequences $\{a_{n}\}$ and $\{b_{n}\}$ respectively, and take out the common terms from sequences $\{a_{n}\}$ and $\{b_{n}\}$ to form sequence $\{c_{n}\}$, then $c_{n}=$____; if sequence $\{d_{n}\}$ satisfies $d_{n}=c_{n}-20n+20$, and the sum of the first $n$ terms of the sequence $\left\{\frac{1}{d_{n}d_{n+1}}\right\}$ is $S_{n}$, then $S_{2023}=$____. | \frac{2023}{4050} | 10.9375 |
27,236 | What is the greatest prime factor of $15! + 18!$? | 17 | 1.5625 |
27,237 | Given $a^2 = 16$, $|b| = 3$, $ab < 0$, find the value of $(a - b)^2 + ab^2$. | 13 | 55.46875 |
27,238 | A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 5$ passes through no lattice point with $0 < x \leq 150$ for all $m$ such that $0 < m < b$. What is the maximum possible value of $b$?
**A)** $\frac{1}{150}$
**B)** $\frac{1}{151}$
**C)** $\frac{1}{152}$
**D)** $\frac{1}{153}$
**E)** $\frac{1}{154}$ | \frac{1}{151} | 91.40625 |
27,239 | Given the function $y=a^{2x}+2a^{x}-1 (a > 0$ and $a \neq 1)$, find the value of $a$ when the maximum value of the function is $14$ for the domain $-1 \leq x \leq 1$. | \frac{1}{3} | 5.46875 |
27,240 | Which number appears most frequently in the second position when listing the winning numbers of a lottery draw in ascending order? | 23 | 0 |
27,241 | Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. Then it's again Betül's turn. And the game continues until two stones remain. When two stones remain, Ali confesses the marked stone. At least in how many moves can Betül guarantee to find out the marked stone? | 11 | 6.25 |
27,242 | How many multiples of 4 are between 100 and 350? | 62 | 0.78125 |
27,243 | Calculate the area bounded by the graph of $y = \arcsin(\cos x)$ and the $x$-axis over the interval $0 \leq x \leq 2\pi$. | \pi^2 | 21.09375 |
27,244 | Given the value \(\left(\frac{11}{12}\right)^{2}\), determine the interval in which this value lies. | \frac{1}{2} | 0 |
27,245 | There are three candidates standing for one position as student president and 130 students are voting. Sally has 24 votes so far, while Katie has 29 and Alan has 37. How many more votes does Alan need to be certain he will finish with the most votes? | 17 | 3.125 |
27,246 | Given the line $x+2y=a$ intersects the circle $x^2+y^2=4$ at points $A$ and $B$, and $|\vec{OA}+ \vec{OB}|=|\vec{OA}- \vec{OB}|$, where $O$ is the origin, determine the value of the real number $a$. | -\sqrt{10} | 0 |
27,247 | A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\pi$). | 144\pi | 31.25 |
27,248 | Find the maximum value of
\[
\frac{3x + 4y + 6}{\sqrt{x^2 + 4y^2 + 4}}
\]
over all real numbers \(x\) and \(y\). | \sqrt{61} | 67.96875 |
27,249 | In triangle $XYZ$, $XY = 540$ and $YZ = 360$. Points $N$ and $O$ are located on $\overline{XY}$ and $\overline{XZ}$ respectively, such that $XN = NY$, and $\overline{ZO}$ is the angle bisector of angle $Z$. Let $Q$ be the point of intersection of $\overline{YN}$ and $\overline{ZO}$, and let $R$ be the point on line $YN$ for which $N$ is the midpoint of $\overline{RQ}$. If $XR = 216$, find $OQ$. | 216 | 3.90625 |
27,250 | If \( x = 3 \) and \( y = 7 \), then what is the value of \( \frac{x^5 + 3y^3}{9} \)? | 141 | 1.5625 |
27,251 | A student rolls two dice simultaneously, and the scores obtained are a and b respectively. The eccentricity e of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ (a > b > 0) is greater than $\frac{\sqrt{3}}{2}$. What is the probability of this happening? | \frac{1}{6} | 34.375 |
27,252 | How many four-digit positive integers are multiples of 7? | 1286 | 100 |
27,253 | The sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.
$(1) q > 1$ $(2) T_{2013} > 1$ $(3) S_{2012}a_{2013} < S_{2013}a_{2012}$ $(4)$ The smallest natural number $n$ for which $T_n > 1$ is $4025$ $(5) \min \left( T_n \right) = T_{2012}$ | (1)(3)(4)(5) | 0 |
27,254 | Calculate \[\left|\left(2 + 2i\right)^6 + 3\right|\] | 515 | 2.34375 |
27,255 | A train takes 60 seconds to pass through a 1260-meter-long bridge and 90 seconds to pass through a 2010-meter-long tunnel. What is the speed of the train in meters per second, and what is the length of the train? | 240 | 17.96875 |
27,256 | Let $c_i$ denote the $i$ th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute
\[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\]
(Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$ ) | \frac{12}{\pi^2} | 2.34375 |
27,257 | Let \( x \) be a non-zero real number such that
\[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \].
Find the product of all possible values of \( x \). | -5 | 11.71875 |
27,258 | Little Tiger places chess pieces on the grid points of a 19 × 19 Go board, forming a solid rectangular dot matrix. Then, by adding 45 more chess pieces, he transforms it into a larger solid rectangular dot matrix with one side unchanged. What is the maximum number of chess pieces that Little Tiger originally used? | 285 | 0.78125 |
27,259 | Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.\] | -7 | 15.625 |
27,260 | A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ} = \frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$. | \sqrt{5 - \sqrt{5}} | 3.125 |
27,261 | Evaluate the infinite geometric series: $$\frac{4}{3} - \frac{3}{4} + \frac{9}{16} - \frac{27}{64} + \dots$$ | \frac{64}{75} | 40.625 |
27,262 | Points $M$ , $N$ , $P$ are selected on sides $\overline{AB}$ , $\overline{AC}$ , $\overline{BC}$ , respectively, of triangle $ABC$ . Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$ .
*Proposed by Evan Chen* | 150 | 5.46875 |
27,263 | Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies? | 81 | 2.34375 |
27,264 | If \( x \) is positive, find the minimum value of \(\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\). | \sqrt{10} | 52.34375 |
27,265 | Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8$. Find the value of $r$. | \frac{\sqrt{6}}{2} | 0 |
27,266 | Given 6 people $A$, $B$, $C$, $D$, $E$, $F$, they are randomly arranged in a line. The probability of the event "$A$ is adjacent to $B$ and $A$ is not adjacent to $C$" is $\_\_\_\_\_\_$. | \frac{4}{15} | 28.125 |
27,267 | How many four-digit positive integers are multiples of 7? | 1286 | 99.21875 |
27,268 | Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.
 | 12 | 4.6875 |
27,269 | Let $a_n = n(2n+1)$ . Evaluate
\[
\biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |.
\] | 18 | 82.03125 |
27,270 | Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$. | \sqrt{2} | 10.9375 |
27,271 | How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | 42 | 0 |
27,272 | Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the length of the real axis of the hyperbola. Find the eccentricity of this hyperbola. | \frac{5}{3} | 0.78125 |
27,273 | Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\cos C=-\frac{1}{4}$.
$(1)$ If $\sin A=2\sin B$, find $b$ and $c$;
$(2)$ If $\cos (A-\frac{π}{4})=\frac{4}{5}$, find $c$. | \frac{5\sqrt{30}}{2} | 15.625 |
27,274 | Solve the following equations using appropriate methods:
$(1)\left(3x-1\right)^{2}=9$.
$(2)x\left(2x-4\right)=\left(2-x\right)^{2}$. | -2 | 0.78125 |
27,275 | A right rectangular prism has edge lengths $\log_{5}x, \log_{8}x,$ and $\log_{10}x.$ Given that the sum of its surface area and volume is twice its volume, find the value of $x$.
A) $1,000,000$
B) $10,000,000$
C) $100,000,000$
D) $1,000,000,000$
E) $10,000,000,000$ | 100,000,000 | 7.03125 |
27,276 | What is the maximum number of diagonals of a regular $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$ -gon at right angles are still considered perpendicular.
*2018 CCA Math Bonanza Tiebreaker Round #1* | 24 | 2.34375 |
27,277 | An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction. | \frac{4}{3} | 25 |
27,278 | The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 300 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? | 120 | 0 |
27,279 | What is the smallest positive integer with exactly 12 positive integer divisors? | 288 | 0 |
27,280 | In the animal kingdom, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 of each of these three types of animals, divided into 100 groups. Each group has exactly 3 animals, with exactly 2 animals of one type and 1 animal of another type.
After the groups were formed, Kung Fu Panda asked each animal, "Is there a tiger in your group?" and 138 animals responded "yes." Kung Fu Panda then asked each animal, "Is there a fox in your group?" and 188 animals responded "yes."
How many monkeys told the truth both times? | 76 | 0 |
27,281 | Given a sequence $\{a_n\}$, the sum of the first $n$ terms $S_n$ satisfies $a_{n+1}=2S_n+6$, and $a_1=6$.
(Ⅰ) Find the general formula for the sequence $\{a_n\}$;
(Ⅱ) Let $b_n=\frac{a_n}{(a_n-2)(a_{n+1}-2)}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Is there a maximum integer $m$ such that for any $n\in \mathbb{N}^*$, $T_n > \frac{m}{16}$ holds? If it exists, find $m$; if not, explain why. | m=1 | 42.96875 |
27,282 | Three rays emanate from a single point and form pairs of angles of $60^{\circ}$. A sphere with a radius of one unit touches all three rays. Calculate the distance from the center of the sphere to the initial point of the rays. | \sqrt{3} | 7.03125 |
27,283 | In a right triangular prism $\mathrm{ABC}-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$, the lengths of the base edges and the lateral edges are all 2. If $\mathrm{E}$ is the midpoint of $\mathrm{CC}_{1}$, what is the distance from $\mathrm{C}_{1}$ to the plane $\mathrm{AB} \mathrm{B}_{1} \mathrm{E}$? | \frac{\sqrt{2}}{2} | 0 |
27,284 | Given the sequence $1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2, \cdots$ where the number of 2s between consecutive 1s increases by 1 each time, calculate the sum of the first 1234 terms. | 2419 | 86.71875 |
27,285 | Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$ | 11 | 1.5625 |
27,286 | It is known that there exists a natural number \( N \) such that \( (\sqrt{3}-1)^{N} = 4817152 - 2781184 \cdot \sqrt{3} \). Find \( N \). | 16 | 25 |
27,287 | Let $(b_1, b_2, b_3, \ldots, b_{10})$ be a permutation of $(1, 2, 3, \ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations. | 126 | 4.6875 |
27,288 | In the Sweet Tooth store, they are thinking about what promotion to announce before March 8. Manager Vasya suggests reducing the price of a box of candies by $20\%$ and hopes to sell twice as many goods as usual because of this. Meanwhile, Deputy Director Kolya says it would be more profitable to raise the price of the same box of candies by one third and announce a promotion: "the third box of candies as a gift," in which case sales will remain the same (excluding the gifts). In whose version of the promotion will the revenue be higher? In your answer, specify how much greater the revenue will be if the usual revenue from selling boxes of candies is 10,000 units. | 6000 | 17.96875 |
27,289 | Given $f(x)=6-12x+x\,^{3},x\in\left[-\frac{1}{3},1\right]$, find the maximum and minimum values of the function. | -5 | 10.9375 |
27,290 | If the graph of the function $f(x) = (4-x^2)(ax^2+bx+5)$ is symmetric about the line $x=-\frac{3}{2}$, then the maximum value of $f(x)$ is ______. | 36 | 22.65625 |
27,291 | On a quadrilateral piece of paper, there are a total of 10 points, and if the vertices of the quadrilateral are included, there are a total of 14 points. It is known that any three of these points are not collinear. According to the following rules, cut this piece of paper into some triangles:
(1) Each triangle's vertices are any 3 of the 14 points;
(2) Each triangle does not contain any other points inside it.
How many triangles can this quadrilateral paper be cut into, at most? | 22 | 7.03125 |
27,292 | Given an ellipse C: $$\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \quad (a>b>0)$$ which passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its foci denoted as $F_1$ and $F_2$. The circle $x^2+y^2=2$ intersects the line $x+y+b=0$ forming a chord of length 2.
(I) Determine the standard equation of ellipse C;
(II) Let Q be a moving point on ellipse C that is not on the x-axis, with the origin O. Draw a parallel line to OQ through point $F_2$ intersecting ellipse C at two distinct points M and N.
(1) Investigate whether $\frac{|MN|}{|OQ|^2}$ is a constant value. If so, find this constant; if not, please explain why.
(2) Denote the area of $\triangle QF_2M$ as $S_1$ and the area of $\triangle OF_2N$ as $S_2$, and let $S = S_1 + S_2$. Find the maximum value of $S$. | \frac{2\sqrt{3}}{3} | 5.46875 |
27,293 | In the XY-plane, mark all the lattice points $(x, y)$ where $0 \leq y \leq 10$. For an integer polynomial of degree 20, what is the maximum number of these marked lattice points that can lie on the polynomial? | 20 | 0.78125 |
27,294 | The base of the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\angle A D C=60^{\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.
a) Find the area of the circle obtained in the cross-section of the sphere by the plane passing through the points $A_{1}, C_{1}$, and $D_{1}$.
b) Find the angle $B_{1} C_{1} A$.
c) It is additionally known that the radius of the sphere is 2. Find the volume of the prism. | 3\sqrt{3} | 0 |
27,295 | An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.) | 33 | 52.34375 |
27,296 | Triangles $\triangle ABC$ and $\triangle DEC$ share side $BC$. Given that $AB = 7\ \text{cm}$, $AC = 15\ \text{cm}$, $EC = 9\ \text{cm}$, and $BD = 26\ \text{cm}$, what is the least possible integral number of centimeters in $BC$? | 17 | 0.78125 |
27,297 | Provide a negative integer solution that satisfies the inequality $3x + 13 \geq 0$. | -1 | 0 |
27,298 | Given ellipse $C_1$ and parabola $C_2$ whose foci are both on the x-axis, the center of $C_1$ and the vertex of $C_2$ are both at the origin $O$. Two points are taken from each curve, and their coordinates are recorded in the table. Calculate the distance between the left focus of $C_1$ and the directrix of $C_2$. | \sqrt {3}-1 | 0 |
27,299 | Find the value of $y$ if $y$ is positive and $y \cdot \lfloor y \rfloor = 132$. Express your answer as a decimal. | 12 | 10.15625 |
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