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28,300 | Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $360$. | 360 | 0 |
28,301 | In $\Delta XYZ$, $\overline{MN} \parallel \overline{XY}$, $XM = 5$ cm, $MY = 8$ cm, and $NZ = 9$ cm. What is the length of $\overline{YZ}$? | 23.4 | 31.25 |
28,302 | Shirley has a magical machine. If she inputs a positive even integer $n$ , the machine will output $n/2$ , but if she inputs a positive odd integer $m$ , the machine will output $m+3$ . The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$ . What number should she input? | 67 | 91.40625 |
28,303 | If two lines $l' $ and $m'$ have equations $y = -3x + 9$, and $y = -5x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l'$ will fall between $l'$ and $m'$? Express your answer as a decimal to the nearest hundredth. | 0.4 | 0 |
28,304 | Determine $\sqrt[5]{102030201}$ without a calculator. | 101 | 17.96875 |
28,305 | Find all odd natural numbers greater than 500 but less than 1000, each of which has the sum of the last digits of all its divisors (including 1 and the number itself) equal to 33. | 729 | 96.875 |
28,306 | Consider a sequence $F_0=2$ , $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$ . If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$ , what is the positive difference between $r_1$ and $r_2$ ?
| \frac{\sqrt{17}}{2} | 3.125 |
28,307 | Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right)\]
for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.$ | \frac{4047}{2} | 34.375 |
28,308 | The symphony orchestra has more than 200 members but fewer than 300 members. When they line up in rows of 6, there are two extra members; when they line up in rows of 8, there are three extra members; and when they line up in rows of 9, there are four extra members. How many members are in the symphony orchestra? | 260 | 2.34375 |
28,309 | The table shows the vertical drops of six roller coasters in Fibonacci Fun Park:
\begin{tabular}{|l|c|}
\hline
Speed Demon & 150 feet \\
\hline
Looper & 230 feet \\
\hline
Dare Devil & 160 feet \\
\hline
Giant Drop & 190 feet \\
\hline
Sky Scream & 210 feet \\
\hline
Hell Spiral & 180 feet \\
\hline
\end{tabular}
What is the positive difference between the mean and the median of these values? | 1.67 | 13.28125 |
28,310 | The number 119 has the following property:
- Division by 2 leaves a remainder of 1;
- Division by 3 leaves a remainder of 2;
- Division by 4 leaves a remainder of 3;
- Division by 5 leaves a remainder of 4;
- Division by 6 leaves a remainder of 5.
How many positive integers less than 2007 satisfy this property? | 32 | 0 |
28,311 | Find the area bounded by the graph of \( y = \arcsin(\cos x) \) and the \( x \)-axis on the interval \( \left[0, 2\pi\right] \). | \frac{\pi^2}{2} | 12.5 |
28,312 | Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer \(k\), denote by \(S_{k}\) the set of all visible lattice points \((x, y)\) such that \(x^{2}+y^{2}=k^{2}\). Let \(D\) denote the set of all positive divisors of \(2021 \cdot 2025\). Compute the sum
$$
\sum_{d \in D}\left|S_{d}\right|
$$
Here, a lattice point is a point \((x, y)\) on the plane where both \(x\) and \(y\) are integers, and \(|A|\) denotes the number of elements of the set \(A\). | 20 | 0 |
28,313 | At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time? | 9:36 | 7.03125 |
28,314 | Consider a fictional language with ten letters in its alphabet: A, B, C, D, F, G, H, J, L, M. Suppose license plates of six letters utilize only letters from this alphabet. How many license plates of six letters are possible that begin with either B or D, end with J, cannot contain any vowels (A), and have no letters that repeat? | 1680 | 52.34375 |
28,315 | Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$ , respectively. | 18 | 42.1875 |
28,316 | The slant height of a cone forms an angle $\alpha$ with the plane of its base, where $\cos \alpha = \frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of tangency of the sphere and the lateral surface of the cone. The volume of the part of the cone enclosed between this plane and the base plane of the cone is 37. Find the volume of the remaining part of the cone. | 37 | 5.46875 |
28,317 | On November 1, 2019, at exactly noon, two students, Gavrila and Glafira, set their watches (both have standard watches where the hands make a full rotation in 12 hours) accurately. It is known that Glafira's watch gains 12 seconds per day, and Gavrila's watch loses 18 seconds per day. After how many days will their watches simultaneously show the correct time again? Give the answer as a whole number, rounding if necessary. | 1440 | 14.0625 |
28,318 | **The first term of a sequence is $2089$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $2089^{\text{th}}$ term of the sequence?** | 16 | 1.5625 |
28,319 | A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip. | 110 | 19.53125 |
28,320 | Given a sequence $\{a_n\}$ with $16$ terms, and $a_1=1, a_8=4$. Let the function related to $x$ be $f_n(x)=\frac{x^3}{3}-a_nx^2+(a_n^2-1)x$, where $n\in \mathbf{N}^*$. If $x=a_{n+1}$ ($1\leqslant n\leqslant 15$) is the extremum point of the function $f_n(x)$, and the slope of the tangent line at the point $(a_{16}, f_8(a_{16}))$ on the curve $y=f_8(x)$ is $15$, then the number of sequences $\{a_n\}$ that satisfy the conditions is ______. | 1176 | 2.34375 |
28,321 | What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | 15 | 27.34375 |
28,322 | A grocery store manager decides to design a more compact pyramid-like stack of apples with a rectangular base of 4 apples by 6 apples. Each apple above the first level still rests in a pocket formed by four apples below, and the stack is completed with a double row of apples on top. Determine the total number of apples in the stack. | 53 | 8.59375 |
28,323 | A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element? | 87 | 49.21875 |
28,324 | In a square $ABCD$ with side length $4$, find the probability that $\angle AMB$ is an acute angle. | 1-\dfrac{\pi}{8} | 0 |
28,325 | Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______. | 288 | 32.03125 |
28,326 | The letter T is formed by placing a $2\:\text{inch}\!\times\!6\:\text{inch}$ rectangle vertically and a $3\:\text{inch}\!\times\!2\:\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in inches?
```
[No graphic input required]
``` | 22 | 34.375 |
28,327 | The lateral face of a regular triangular pyramid \( SABC \) is inclined to the base \( ABC \) at an angle \(\alpha = \arctan \frac{3}{4}\). Points \( M, N, K \) are the midpoints of the sides of the base \( ABC \). The triangle \( MNK \) forms the lower base of a right prism. The edges of the upper base of the prism intersect the lateral edges of the pyramid \( SABC \) at points \( F, P \), and \( R \) respectively. The total surface area of the polyhedron with vertices at points \( M, N, K, F, P, R \) is \( 53 \sqrt{3} \). Find the side length of the triangle \( ABC \). (16 points) | 16 | 4.6875 |
28,328 | John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number? | 1001 | 17.96875 |
28,329 | Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 7.8125 |
28,330 | Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$ . If $a_1=a_2=1$ , and $k=18$ , determine the number of elements of $\mathcal{A}$ . | 1597 | 3.125 |
28,331 | A boy named Vasya wrote down the nonzero coefficients of a tenth-degree polynomial \( P(x) \) in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, and continued this process until he arrived at a constant, which he also wrote down.
What is the minimum number of different numbers he could have ended up with?
Coefficients are written down with their signs, and constant terms are also recorded. If there is a term of the form \(\pm x^n\), \(\pm 1\) is written down. | 10 | 7.8125 |
28,332 | A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
*Proposed by Brandon Wang* | 2019 | 67.96875 |
28,333 | Three cards are dealt successively without replacement from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$, the second card is a King, and the third card is a $\spadesuit$? | \frac{13}{2550} | 67.1875 |
28,334 | Given that $|\vec{a}|=1$, $|\vec{b}|=2$, and $(\vec{a}+\vec{b})\cdot \vec{b}=3$, find the angle between $\vec{b}$ and $\vec{a}$. | \frac{2\pi}{3} | 93.75 |
28,335 | A right rectangular prism has integer side lengths $a$ , $b$ , and $c$ . If $\text{lcm}(a,b)=72$ , $\text{lcm}(a,c)=24$ , and $\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?
*Proposed by Deyuan Li and Andrew Milas* | 3024 | 15.625 |
28,336 | Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b > 0)$ with its right focus $F$ lying on the line $2x-y-2=0$, where $A$ and $B$ are the left and right vertices of $C$, and $|AF|=3|BF|$.<br/>$(1)$ Find the standard equation of $C$;<br/>$(2)$ A line $l$ passing through point $D(4,0)$ intersects $C$ at points $P$ and $Q$, with the midpoint of segment $PQ$ denoted as $N$. If the slope of line $AN$ is $\frac{2}{5}$, find the slope of line $l$. | -\frac{1}{4} | 33.59375 |
28,337 | A point is marked one quarter of the way along each side of a triangle. What fraction of the area of the triangle is shaded? | $\frac{5}{8}$ | 0 |
28,338 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 | 28.90625 |
28,339 | Given that in the expansion of \\((1+2x)^{n}\\), only the coefficient of the fourth term is the maximum, then the constant term in the expansion of \\((1+ \dfrac {1}{x^{2}})(1+2x)^{n}\\) is \_\_\_\_\_\_. | 61 | 57.8125 |
28,340 | Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 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 this edge. | \frac{\sqrt{3}}{4} | 5.46875 |
28,341 | A subset \( S \) of the set of integers \{ 0, 1, 2, ..., 99 \} is said to have property \( A \) if it is impossible to fill a 2x2 crossword puzzle with the numbers in \( S \) such that each number appears only once. Determine the maximal number of elements in sets \( S \) with property \( A \). | 25 | 0.78125 |
28,342 | For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)^2
\]equal to zero? | 337 | 0.78125 |
28,343 | Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits. | 2605 | 37.5 |
28,344 | A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours? | 225 | 42.96875 |
28,345 | Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$ , satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$ , where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$ . Find the least positive integer $k$ for which in every $k$ -subset of $S$ , having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1. | 1003 | 24.21875 |
28,346 | Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$ . | 1010 | 0 |
28,347 | If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$. | -16 | 7.8125 |
28,348 | Let \( a \) and \( b \) be integers such that \( ab = 72 \). Find the minimum value of \( a + b \). | -17 | 0.78125 |
28,349 | Quantities \(r\) and \( s \) vary inversely. When \( r \) is \( 1500 \), \( s \) is \( 0.4 \). Alongside, quantity \( t \) also varies inversely with \( r \) and when \( r \) is \( 1500 \), \( t \) is \( 2.5 \). What is the value of \( s \) and \( t \) when \( r \) is \( 3000 \)? Express your answer as a decimal to the nearest thousandths. | 1.25 | 3.125 |
28,350 | Compute $\sin(-30^\circ)$ and verify by finding $\cos(-30^\circ)$, noticing the relationship, and confirming with the unit circle properties. | \frac{\sqrt{3}}{2} | 78.125 |
28,351 | How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$ ? | 42 | 59.375 |
28,352 | Calculate the value of $V_3$ in Horner's method (also known as Qin Jiushao algorithm) for finding the value of the polynomial $f(x) = 4x^6 + 3x^5 + 4x^4 + 2x^3 + 5x^2 - 7x + 9$ when $x = 4$. | 80 | 0 |
28,353 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2\sqrt{3}$, and $\overrightarrow{a} \perp (\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$. | -3 | 21.875 |
28,354 | Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$ . | 45578899 | 2.34375 |
28,355 | During the flower exhibition in Zhengzhou Green Expo Garden, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that areas A and B each have one person, and the remaining two areas each have two people. Among them, Little Li and Little Wang cannot be together. Determine the total number of different arrangements possible. | 156 | 10.9375 |
28,356 | A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ Determine the length of $AC.$ | 5 + \frac{15\sqrt{2}}{2} | 2.34375 |
28,357 | In the diagram, \(\angle AFC = 90^\circ\), \(D\) is on \(AC\), \(\angle EDC = 90^\circ\), \(CF = 21\), \(AF = 20\), and \(ED = 6\). Determine the total area of quadrilateral \(AFCE\). | 297 | 7.8125 |
28,358 | Given that Ms. Demeanor's class consists of 50 students, more than half of her students bought crayons from the school bookstore, each buying the same number of crayons, with each crayon costing more than the number of crayons bought by each student, and the total cost for all crayons was $19.98, determine the cost of each crayon in cents. | 37 | 31.25 |
28,359 | In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. What is the minimum distance the ray could travel? | \frac{10 \sqrt{3}}{9} | 0 |
28,360 | One angle of a triangle is twice another, and the sides opposite these angles have lengths 12 and 18. Compute the length of the third side of the triangle. | 15 | 35.15625 |
28,361 | How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | 216 | 0 |
28,362 | Given an arithmetic sequence $\{a_n\}$ with the first term $a_1=11$ and common difference $d=2$, and $a_n=2009$, find $n$. | 1000 | 97.65625 |
28,363 | Ivan Tsarevich is fighting the Dragon Gorynych on the Kalinov Bridge. The Dragon has 198 heads. With one swing of his sword, Ivan Tsarevich can cut off five heads. However, new heads immediately grow back, the number of which is equal to the remainder when the number of heads left after Ivan's swing is divided by 9. If the remaining number of heads is divisible by 9, no new heads grow. If the Dragon has five or fewer heads before the swing, Ivan Tsarevich can kill the Dragon with one swing. How many sword swings does Ivan Tsarevich need to defeat the Dragon Gorynych? | 40 | 26.5625 |
28,364 | Given that $a$, $b$, and $c$ are integers, and $a-2b=4$, $ab+c^2-1=0$, find the value of $a+b+c$. | -3 | 5.46875 |
28,365 | On an island, there are knights who always tell the truth and liars who always lie. At the main celebration, 100 islanders sat around a large round table. Half of the attendees said the phrase: "both my neighbors are liars," while the remaining said: "among my neighbors, there is exactly one liar." What is the maximum number of knights that can sit at this table? | 67 | 22.65625 |
28,366 | How many multiples of 4 are between 100 and 350? | 62 | 0.78125 |
28,367 | Given that \( x \) and \( y \) are positive numbers, determine the minimum value of \(\left(x+\frac{1}{y}\right)^{2}+\left(y+\frac{1}{2x}\right)^{2}\). | 3 + 2 \sqrt{2} | 7.03125 |
28,368 | Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 95 | 9.375 |
28,369 | Given Allison's birthday cake is in the form of a $5 \times 5 \times 3$ inch rectangular prism with icing on the top, front, and back sides but not on the sides or bottom, calculate the number of $1 \times 1 \times 1$ inch smaller prisms that will have icing on exactly two sides. | 30 | 0.78125 |
28,370 | The base of the quadrilateral prism \( A B C D A_{1} B_{1} C_{1} D_{1} \) is a rhombus \( A B C D \) with \( B D = 12 \) and \( \angle B A C = 60^{\circ} \). A sphere passes through the vertices \( D, A, B, B_{1}, C_{1}, D_{1} \).
a) Find the area of the circle obtained in the cross section of the sphere by the plane passing through points \( A_{1}, B_{1} \), and \( C_{1} \).
b) Find the angle \( A_{1} C B \).
c) Given that the radius of the sphere is 8, find the volume of the prism. | 192\sqrt{3} | 0.78125 |
28,371 | Find the number of functions of the form \( f(x) = ax^3 + bx^2 + cx + d \) such that
\[ f(x)f(-x) = f(x^3). \] | 16 | 0.78125 |
28,372 | Pablo is solving a quadratic equation and notices that the ink has smeared over the coefficient of $x$, making it unreadable. He recalls that the equation has two distinct negative, integer solutions. He needs to calculate the sum of all possible coefficients that were under the smeared ink.
Given the equation form:
\[ x^2 + ?x + 24 = 0 \]
What is the sum of all distinct coefficients that could complete this equation? | -60 | 8.59375 |
28,373 | Given Tom paid $180, Dorothy paid $200, Sammy paid $240, and Alice paid $280, and they agreed to split the costs evenly, calculate the amount that Tom gave to Sammy minus the amount that Dorothy gave to Alice. | 20 | 36.71875 |
28,374 | An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$. | \frac{1}{13} | 1.5625 |
28,375 | Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ.$ Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ.$ Find the number of degrees in $\angle CMB.$ | 83 | 2.34375 |
28,376 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy the conditions: $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a}$ is perpendicular to $(2\overrightarrow{b} - \overrightarrow{a})$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 99.21875 |
28,377 | The perimeter of the quadrilateral formed by the four vertices of the ellipse $C: \frac {x^{2}}{4}+ \frac {y^{2}}{16}=1$ is equal to _____. | 8 \sqrt {5} | 0 |
28,378 |
The road from Petya's house to the school takes 20 minutes. One day, on his way to school, he remembered that he forgot his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home to get the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the class. What fraction of the way had he traveled when he remembered about the pen? | \frac{7}{20} | 21.875 |
28,379 | There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points. | 60 | 3.90625 |
28,380 | Given a triangle $\triangle ABC$, where the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that vector $\overrightarrow{m} = (\sin A + \sin C, \sin B - \sin A)$ and vector $\overrightarrow{n} = (\sin A - \sin C, \sin B)$ are orthogonal.
1. Find the measure of angle $C$.
2. If $a^2 = b^2 + \frac{1}{2}c^2$, find the value of $\sin(A - B)$. | \frac{\sqrt{3}}{4} | 14.0625 |
28,381 | Four identical point charges are initially placed at the corners of a square, storing a total energy of 20 Joules. Determine the total amount of energy stored if one of these charges is moved to the center of the square. | 10\sqrt{2} + 10 | 0 |
28,382 | Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters? | 28 | 21.875 |
28,383 | An equilateral triangle $ABC$ has an area of $27\sqrt{3}$. The rays trisecting $\angle BAC$ intersect side $BC$ at points $D$ and $E$. Find the area of $\triangle ADE$. | 3\sqrt{3} | 17.1875 |
28,384 | What is the greatest prime factor of $15! + 18!$? | 17 | 2.34375 |
28,385 | Given sets $M=\{1, 2, a^2 - 3a - 1 \}$ and $N=\{-1, a, 3\}$, and the intersection of $M$ and $N$ is $M \cap N = \{3\}$, find the set of all possible real values for $a$. | \{4\} | 11.71875 |
28,386 | The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$ . Find the volume of the tetrahedron. | 1/8 | 42.96875 |
28,387 | An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:
- All connections operate in both directions
- If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.
Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there? | 512 | 14.0625 |
28,388 | Tom's favorite number is between $100$ and $150$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Tom's favorite number? | 143 | 52.34375 |
28,389 | If the chord cut by the line $ax-by+2=0$ on the circle $x^{2}+y^{2}+2x-4y+1=0$ is $4$, find the minimum value of $\dfrac{2}{a}+\dfrac{3}{b}$. | 4+2\sqrt{3} | 14.84375 |
28,390 | Given that the graph of a power function passes through the points $(2, 16)$ and $\left( \frac{1}{2}, m \right)$, then $m = \_\_\_\_\_\_$. | \frac{1}{16} | 68.75 |
28,391 | Use all digits from 1 to 9 to form three three-digit numbers such that their product is:
a) the smallest;
b) the largest. | 941 \times 852 \times 763 | 0 |
28,392 |
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is marked on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\) and \(AK: KE = 3:7\). Find \(CL: AB\).
A regular pentagon is a pentagon where all sides and all angles are equal. | 0.7 | 0 |
28,393 | In a 6 by 6 grid, each of the 36 small squares measures 3 cm by 3 cm and is initially shaded. Five unshaded squares of side 1.5 cm are placed on the grid in such a way that some may overlap each other and/or the edges of the grid. If such overlap occurs, the overlapping parts are not visible as shaded. The area of the visible shaded region can be written in the form $A-B\phi$. What is the value $A+B$? | 335.25 | 9.375 |
28,394 | If: (1) \(a, b, c, d\) are all elements of the set \(\{1,2,3,4\}\); (2) \(a \neq b\), \(b \neq c\), \(c \neq d\), \(d \neq a\); (3) \(a\) is the smallest among \(a, b, c, d\). Then, how many different four-digit numbers \(\overline{abcd}\) can be formed? | 24 | 2.34375 |
28,395 | Find the largest perfect square such that when the last two digits are removed, it is still a perfect square. (It is assumed that one of the digits removed is not zero.) | 1681 | 61.71875 |
28,396 | Given the parametric equation of line $l$ as $\begin{cases}x=-\frac{1}{2}+\frac{\sqrt{2}}{2}t \\ y=\frac{1}{2}+\frac{\sqrt{2}}{2}t\end{cases}$ and the parametric equation of ellipse $C$ as $\begin{cases}x=2\cos\theta \\ y=\sqrt{3}\sin\theta\end{cases}$, in a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinates of point $A$ are $(\frac{\sqrt{2}}{2},\frac{3}{4}\pi)$.
(1) Convert the coordinates of point $A$ to rectangular coordinates and convert the parametric equation of the ellipse to a standard form.
(2) Line $l$ intersects ellipse $C$ at points $P$ and $Q$. Find the value of $|AP|\cdot|AQ|$. | \frac{41}{14} | 10.9375 |
28,397 | The function \( f(n) \) is defined on the set of natural numbers \( N \) as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{ if } n \geqslant 1000, \\
f[f(n + 7)], & \text{ if } n < 1000.
\end{cases} \]
What is the value of \( f(90) \)? | 999 | 66.40625 |
28,398 | In right triangle $ABC$ with $\angle BAC = 90^\circ$, we have $AB = 15$ and $BC = 17$. Find $\tan A$ and $\sin A$. | \frac{8}{17} | 5.46875 |
28,399 | Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ such that $WI = 9 \cdot IZ$. Determine the ratio of the area of square $IJKL$ to the area of square $WXYZ$.
A) $\frac{1}{20}$
B) $\frac{1}{50}$
C) $\frac{1}{40}$
D) $\frac{1}{64}$
E) $\frac{1}{80}$ | \frac{1}{50} | 37.5 |
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