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25,500 | Let triangle $PQR$ be a right triangle with $\angle PRQ = 90^\circ$. A circle is tangent to the sides $PQ$ and $PR$ at points $S$ and $T$ respectively. The points on the circle diametrically opposite $S$ and $T$ both lie on side $QR$. Given that $PQ = 12$, find the area of the portion of the circle that lies outside triangle $PQR$. | 4\pi - 8 | 0 |
25,501 | Find the smallest number which when successively divided by \( 45,454,4545 \) and 45454 leaves remainders of \( 4, 45,454 \) and 4545 respectively. | 35641667749 | 0 |
25,502 | Find the smallest positive integer $N$ such that any "hydra" with 100 necks, where each neck connects two heads, can be defeated by cutting at most $N$ strikes. Here, one strike can sever all the necks connected to a particular head $A$, and immediately after, $A$ grows new necks to connect with all previously unconnected heads (each head connects to one neck). The hydra is considered defeated when it is divided into two disconnected parts. | 10 | 0 |
25,503 | Two positive integers \(m\) and \(n\) are chosen such that \(m\) is the smallest positive integer with only two positive divisors, and \(n\) is the largest integer less than 200 that has exactly four positive divisors. What is \(m+n\)? | 192 | 1.5625 |
25,504 | Given real numbers $x$, $y$, $z$ satisfying $\begin{cases} xy+2z=1 \\ x^{2}+y^{2}+z^{2}=5 \end{cases}$, the minimum value of $xyz$ is \_\_\_\_\_\_. | 9 \sqrt {11}-32 | 0 |
25,505 | Given point O is the circumcenter of ∆ABC, and |→BA|=2, |→BC|=6, calculate →BO⋅→AC. | 16 | 21.09375 |
25,506 | Let \( a \) be the number of six-digit numbers divisible by 13 but not divisible by 17, and \( b \) be the number of six-digit numbers divisible by 17 but not divisible by 13.
Find \( a - b \). | 16290 | 78.125 |
25,507 | How many distinct four letter arrangements can be formed by rearranging the letters found in the word **FLUFFY**? For example, FLYF and ULFY are two possible arrangements. | 72 | 80.46875 |
25,508 | In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ | \frac{8\sqrt{2}}{9} | 1.5625 |
25,509 | In triangle $PQR$, $PQ = 4$, $PR = 8$, and $\cos \angle P = \frac{1}{10}$. Find the length of angle bisector $\overline{PS}$. | 4.057 | 0 |
25,510 | Approximate the reading indicated by the arrow in the diagram of a measuring device. | 42.3 | 0 |
25,511 | The measure of angle $ACB$ is 30 degrees. If ray $CA$ is rotated 510 degrees about point $C$ clockwise, what will be the positive measure of the acute angle $ACB$, in degrees? | 120 | 0.78125 |
25,512 | It is known that P and Q are two points on the unit circle centered at the origin O, and they are located in the first and fourth quadrants, respectively. The x-coordinate of point P is $\frac{4}{5}$, and the x-coordinate of point Q is $\frac{5}{13}$. Then, $\cos \angle POQ = \_\_\_\_\_\_$. | \frac{56}{65} | 7.8125 |
25,513 | Two students were asked to add two positive integers. Alice subtracted the two numbers by mistake and obtained 3. Bob mistakenly multiplied the same two integers and got 63. What was the correct sum of the two integers? | 17 | 13.28125 |
25,514 |
A circle touches the extensions of two sides \(AB\) and \(AD\) of square \(ABCD\), and the point of tangency cuts off a segment of length \(2 + \sqrt{5 - \sqrt{5}}\) cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square, given that the angle between the tangents is \(72^\circ\), and it is known that \(\sin 36^\circ = \frac{\sqrt{5 - \sqrt{5}}}{2\sqrt{2}}\). | \frac{\sqrt{\sqrt{5} - 1} \cdot \sqrt[4]{125}}{5} | 0 |
25,515 |
In a department store, they received 10 suitcases and 10 keys separately in an envelope. Each key opens only one suitcase, and every suitcase can be matched with a corresponding key.
A worker in the department store, who received the suitcases, sighed:
- So much hassle with matching keys! I know how stubborn inanimate objects can be!! You start matching the key to the first suitcase, and it always turns out that only the tenth key fits. You'll try the keys ten times because of one suitcase, and because of ten - a whole hundred times!
Let’s summarize the essence briefly. A salesperson said that the number of attempts is no more than \(10+9+8+\ldots+2+1=55\), and another employee proposed to reduce the number of attempts since if the key does not fit 9 suitcases, it will definitely fit the tenth one. Thus, the number of attempts is no more than \(9+8+\ldots+1=45\). Moreover, they stated that this will only occur in the most unfortunate scenario - when each time the key matches the last suitcase. It should be expected that in reality the number of attempts will be roughly
\[\frac{1}{2} \times \text{the maximum possible number of attempts} = 22.5.\]
Igor Fedorovich Akulich from Minsk wondered why the expected number of attempts is half the number 45. After all, the last attempt is not needed only if the key does not fit any suitcase except the last one, but in all other cases, the last successful attempt also takes place. Akulich assumed that the statement about 22.5 attempts is unfounded, and in reality, it is a bit different.
**Problem:** Find the expected value of the number of attempts (all attempts to open the suitcases are counted - unsuccessful and successful, in the case where there is no clarity). | 29.62 | 0 |
25,516 | The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers? | 26 | 0 |
25,517 | What is the smallest four-digit number that is divisible by $35$? | 1015 | 87.5 |
25,518 | Let \(a_{1}, a_{2}, \cdots, a_{10}\) be any 10 distinct positive integers such that \(a_{1} + a_{2} + \cdots + a_{10} = 1995\). Find the minimum value of \(a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{10} a_{1}\). | 6044 | 0 |
25,519 | Formulas for shortened multiplication (other).
Common fractions | 198719871987 | 0 |
25,520 | Let $g: \mathbb{R} \to \mathbb{R}$ be a function such that
\[g((x + y)^2) = g(x)^2 - 2xg(y) + 2y^2\]
for all real numbers $x$ and $y.$ Find the number of possible values of $g(1)$ and the sum of all possible values of $g(1)$. | \sqrt{2} | 0 |
25,521 | Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\frac{\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajectory of the moving point $P$. | \frac{5\pi}{6} | 17.1875 |
25,522 | 20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.
Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from village $B$. The stronger fighter wins. We say that village $A$ is stronger than village $B$ if a fighter from village $A$ wins at least $k$ of the 400 fights.
It turns out that each village is stronger than its neighboring village in a clockwise direction. Determine the maximum value of $k$ so that this can be the case. | 290 | 0 |
25,523 | Distribute five students, namely A, B, C, D, and E, to Peking University, Tsinghua University, and Renmin University of China for recommendation, with the condition that each university gets at least one student and student A cannot be recommended to Peking University. How many different recommendation plans are there? (Answer with a number) | 100 | 0 |
25,524 | In the arithmetic sequence $\{a_n\}$, $a_3+a_6+a_9=54$. Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Then, determine the value of $S_{11}$. | 99 | 0 |
25,525 | Given that $\triangle ABC$ is an equilateral triangle with side length $s$, determine the value of $s$ when $AP = 2$, $BP = 2\sqrt{3}$, and $CP = 4$. | \sqrt{14} | 0.78125 |
25,526 | For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by at least $3$ from all multiples of $p$. Considering the prime numbers $5$, $7$, and $11$, find the number of positive integers less than or equal to $15,000$ which are simultaneously $5$-safe, $7$-safe, and $11$-safe. | 975 | 0 |
25,527 | Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer. | 15015 | 6.25 |
25,528 | Given the set of digits {1, 2, 3, 4, 5}, find the number of three-digit numbers that can be formed with the digits 2 and 3, where 2 is positioned before 3. | 12 | 1.5625 |
25,529 | It is known that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, the number \( \log_{m} n \) is rational, and additionally,
$$
k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}}
$$
Find the minimum possible value of the sum \( k + 5m + n \). | 278 | 12.5 |
25,530 | Given the function $f(x)=\ln x+\frac{1}{2}x^2-ax$, where $a\in \mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$.
(I) When $a=3$, find the extreme values of the function $f(x)$.
(II) If $x_2\geqslant ex_1$ ($e$ is the base of the natural logarithm), find the maximum value of $f(x_2)-f(x_1)$. | 1-\frac{e}{2}+\frac{1}{2e} | 1.5625 |
25,531 | Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determine the length of the portion of $\overline{XY}$ contained in the cube with edge length $4$. | 4\sqrt{3} | 25 |
25,532 | Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=? | 150 | 48.4375 |
25,533 | For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$. | 5120 | 28.125 |
25,534 | Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$ , where $a,\,b,\,c,\,d$ are positive integers. | 11 | 59.375 |
25,535 | Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | 15.5 | 0 |
25,536 | Determine the length of side $x$ in the following diagram:
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(2))--cycle);
draw((0,0)--(-2,0)--(0,2*sqrt(2))--cycle);
label("10",(-1, sqrt(2)),NW);
label("$x$",(sqrt(2),sqrt(2)),NE);
draw("$30^{\circ}$",(2.2,0),NW);
draw("$45^{\circ}$",(-1.8,0),NE);
draw(rightanglemark((0,2*sqrt(2)),(0,0),(2,0),4));
[/asy] | \frac{20\sqrt{3}}{3} | 0 |
25,537 | At Central Middle School, the $150$ students who participate in the Math Club plan to gather for a game night. Due to an overlapping school event, attendance is expected to drop by $40\%$. Walter and Gretel are preparing cookies using a new recipe that makes $18$ cookies per batch. The other details remain the same. How many full recipes should they prepare? | 10 | 36.71875 |
25,538 | Solve the equation $x^2 + 14x = 72$. The positive solution has the form $\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c + d$? | 128 | 0.78125 |
25,539 | A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone. | 94.5 | 0 |
25,540 | Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. Additionally, the letter $C$ must appear at least once in the word. How many six-letter good words are there? | 94 | 0 |
25,541 | Given a parabola $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two tangent lines to circle $M$ are drawn, intersecting the x-axis at points $A$ and $B$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Find the minimum value of the area of $\triangle PAB$. | \frac{25}{2} | 0.78125 |
25,542 | Let $g$ be a function taking the positive integers to the positive integers, such that:
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$),
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $g(m) = n^2$ or $g(n) = m^2.$
Find the sum of all possible values of $g(18).$ | 104976 | 2.34375 |
25,543 | 4. Find the biggest positive integer $n$ , lesser thar $2012$ , that has the following property:
If $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ . | 1944 | 3.125 |
25,544 | There are 15 rectangular sheets of paper. In each move, one of the sheets is chosen and cut with a straight line, not passing through its vertices, into two sheets. After 60 moves, it turned out that all the sheets are triangles or hexagons. How many hexagons are there? | 25 | 3.125 |
25,545 | For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 14 | 2.34375 |
25,546 | What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 45]=180$? | 292 | 0 |
25,547 | Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\] | \sqrt{2} | 7.03125 |
25,548 | Two rectangles, one measuring \(8 \times 10\) and the other \(12 \times 9\), are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction. | 65 | 0 |
25,549 | Form a five-digit number with no repeated digits using the numbers 0, 1, 2, 3, and 4, where exactly one even number is sandwiched between two odd numbers. The total number of such five-digit numbers is | 28 | 78.125 |
25,550 | Find the modular inverse of \( 31 \), modulo \( 45 \).
Express your answer as an integer from \( 0 \) to \( 44 \), inclusive. | 15 | 0 |
25,551 | Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a\cdot (a-b)=3$, find the angle between $a$ and $b$. | \frac{\pi }{3} | 97.65625 |
25,552 | How many four-digit positive integers are multiples of 7? | 1286 | 99.21875 |
25,553 | The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$? | 15 | 39.0625 |
25,554 | Find the greatest common divisor of $8!$ and $(6!)^2.$ | 2880 | 0 |
25,555 | Calculate the product of all prime numbers between 1 and 20. | 9699690 | 100 |
25,556 | Given the set $A={m+2,2m^2+m}$, find the value of $m$ if $3\in A$. | -\frac{3}{2} | 3.90625 |
25,557 | Given an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpendicular to the x-axis.
(1) Find the equation of ellipse C;
(2) Find the minimum value of the slope of line MN, where line l, tangent to the parabola $y^2=4x$ in the first quadrant, intersects ellipse C at points A and B, intersects the x-axis at point M, and the perpendicular bisector of segment AB intersects the y-axis at point N. | - \frac { \sqrt {3}}{12} | 0 |
25,558 | Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$ , find the sum of digits of $E$ | 13 | 16.40625 |
25,559 | 9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost? | 1.11 | 32.03125 |
25,560 | In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c= \sqrt {7}$, $C= \frac {\pi}{3}$.
(1) If $2\sin A=3\sin B$, find $a$ and $b$;
(2) If $\cos B= \frac {3 \sqrt {10}}{10}$, find the value of $\sin 2A$. | \frac {3-4 \sqrt {3}}{10} | 0 |
25,561 | During a year when Valentine's Day, February 14, falls on a Tuesday, what day of the week is Cinco de Mayo (May 5) and how many days are between February 14 and May 5 inclusively? | 81 | 17.1875 |
25,562 | If the point corresponding to the complex number (2-i)z is in the second quadrant of the complex plane, calculate the value of z. | -1 | 24.21875 |
25,563 | Given the binary operations $\clubsuit$ and $\spadesuit$ defined as $\clubsuit$ : $a^{\log_5(b)}$ and $\spadesuit$ : $a^{\frac{1}{\log_5(b)}}$ for real numbers $a$ and $b$ where these expressions are defined, a sequence $(b_n)$ is defined recursively as $b_4 = 4 \spadesuit 2$ and $b_n = (n \spadesuit (n-2)) \clubsuit b_{n-1}$ for all integers $n \geq 5$. Find $\log_5(b_{2023})$ to the nearest integer. | 11 | 2.34375 |
25,564 | In the polar coordinate system, the polar coordinate equation of the curve $\Gamma$ is $\rho= \frac {4\cos \theta}{\sin ^{2}\theta}$. Establish a rectangular coordinate system with the pole as the origin, the polar axis as the positive semi-axis of $x$, and the unit length unchanged. The lines $l_{1}$ and $l_{2}$ both pass through the point $F(1,0)$, and $l_{1} \perp l_{2}$. The slope angle of line $l_{1}$ is $\alpha$.
(1) Write the rectangular coordinate equation of the curve $\Gamma$; write the parameter equations of $l_{1}$ and $l_{2}$;
(2) Suppose lines $l_{1}$ and $l_{2}$ intersect curve $\Gamma$ at points $A$, $B$ and $C$, $D$ respectively. The midpoints of segments $AB$ and $CD$ are $M$ and $N$ respectively. Find the minimum value of $|MN|$. | 4 \sqrt {2} | 0 |
25,565 | The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much should each resident contribute to collect exactly the required rent? | 10.54 | 35.9375 |
25,566 | Let $k$ be the product of every third positive integer from $2$ to $2006$ , that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$ . Find the number of zeros there are at the right end of the decimal representation for $k$ . | 168 | 22.65625 |
25,567 | Xiao Wang left home at 8:30 to visit a housing exhibition and returned home at 12:00, while his alarm clock at home was showing 11:46 when he got back. Calculate the time in minutes until the alarm clock will point to 12:00 exactly. | 15 | 4.6875 |
25,568 | Given vectors $\overrightarrow{a}=m \overrightarrow{i}+5 \overrightarrow{j}- \overrightarrow{k}, \overrightarrow{b}=3 \overrightarrow{i}+ \overrightarrow{j}+r \overrightarrow{k}$, if $\overrightarrow{a}//\overrightarrow{b}$, find the values of real numbers $m=$ \_\_\_\_\_\_ and $r=$ \_\_\_\_\_\_. | -\frac{1}{5} | 69.53125 |
25,569 | Compute the product of the sums of the squares and the cubes of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 1 = 0,\] given that all roots are real and nonnegative. | 13754 | 16.40625 |
25,570 | Let $g$ be a function taking the positive integers to the positive integers, such that:
(i) $g$ is increasing (i.e., $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$,
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$,
(iv) $g(2) = 3$.
Find the sum of all possible values of $g(18)$. | 108 | 2.34375 |
25,571 | 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 |
25,572 | $ABCDEFGH$ is a cube. Find $\sin \angle BAE$, where $E$ is the top vertex directly above $A$. | \frac{1}{\sqrt{2}} | 0 |
25,573 | A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$? | \frac{1}{10} | 2.34375 |
25,574 | Tom adds up all the even integers from 2 to 600, inclusive. Lara adds up all the integers from 1 to 200, inclusive. What is Tom's sum divided by Lara's sum? | 4.5 | 20.3125 |
25,575 | Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$ , followings are satisfied.
i. $f(n+1) > f(n)$ ii. $f(f(n)) = 2n+2$ Find the value of $f(2013)$ .
(Here, $\mathbb{N}$ is the set of all positive integers.) | 4026 | 30.46875 |
25,576 | How many positive integer divisors of $1800^{1800}$ are divisible by exactly 180 positive integers? | 18 | 1.5625 |
25,577 | Given that the coordinates of a direction vector of line $l$ are $({-1,\sqrt{3}})$, the inclination angle of line $l$ is ____. | \frac{2\pi}{3} | 61.71875 |
25,578 | In triangle $XYZ$ with right angle at $Z$, $\angle XYZ < 45^\circ$ and $XY = 6$. A point $Q$ on $\overline{XY}$ is chosen such that $\angle YQZ = 3\angle XQZ$ and $QZ = 2$. Determine the ratio $\frac{XQ}{YQ}$ in simplest form. | \frac{7 + 3\sqrt{5}}{2} | 0 |
25,579 | Given integers $x$ and $y$ satisfy the equation $2xy + x + y = 83$, find the values of $x + y$. | -85 | 8.59375 |
25,580 | First, factorize 42 and 30 into prime factors, then answer the following questions:
(1) 42= , 30= .
(2) The common prime factors of 42 and 30 are .
(3) The unique prime factors of 42 and 30 are .
(4) The greatest common divisor (GCD) of 42 and 30 is .
(5) The least common multiple (LCM) of 42 and 30 is .
(6) From the answers above, you can conclude that . | 210 | 24.21875 |
25,581 | Given the function $f\left(x\right)=\cos x+\left(x+1\right)\sin x+1$ on the interval $\left[0,2\pi \right]$, find the minimum and maximum values of $f(x)$. | \frac{\pi}{2}+2 | 9.375 |
25,582 | Find the sum of the distinct prime factors of $7^7 - 7^4$. | 24 | 0 |
25,583 | A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads? | \frac{3}{128} | 10.15625 |
25,584 | The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arrange the program list? | 720 | 4.6875 |
25,585 | In trapezoid $ABCD$ , $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$ . Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$ .
*Proposed by Lewis Chen* | 260 | 1.5625 |
25,586 | Given a triangle whose side lengths are all positive integers, and only one side length is 5, which is not the shortest side, the number of such triangles is . | 14 | 0 |
25,587 | A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square? | \frac{3\pi}{32} | 16.40625 |
25,588 | There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points)
(Ⅰ) If the 3 girls must stand together, how many different arrangements are there?
(Ⅱ) If no two girls are next to each other, how many different arrangements are there?
(Ⅲ) If there are exactly three people between person A and person B, how many different arrangements are there? | 720 | 3.125 |
25,589 | The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? | $-\frac{2\sqrt{3}}{3}$ | 0 |
25,590 | Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\] | 7\sqrt{2}. | 0 |
25,591 | The numbers \( p_1, p_2, p_3, q_1, q_2, q_3, r_1, r_2, r_3 \) are equal to the numbers \( 1, 2, 3, \dots, 9 \) in some order. Find the smallest possible value of
\[
P = p_1 p_2 p_3 + q_1 q_2 q_3 + r_1 r_2 r_3.
\] | 214 | 89.84375 |
25,592 | Given that $M(m,n)$ is any point on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.
(I) Find the maximum and minimum values of $|MQ|$;
(II) Find the maximum and minimum values of $\frac{n-3}{m+2}$. | 2-\sqrt{3} | 6.25 |
25,593 | Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse? | 199 | 25.78125 |
25,594 | How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 38 | 3.90625 |
25,595 | A sphere is cut into three equal wedges. The circumference of the sphere is $18\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\pi$. | 162\pi | 71.09375 |
25,596 | A large square region is paved with $n^2$ square black tiles, where each tile measures $t$ inches on each side. Surrounding each tile is a white border that is $w$ inches wide. When $n=30$, it's given that the black tiles cover $81\%$ of the area of the large square region. Find the ratio $\frac{w}{t}$ in this scenario.
A) $\frac{1}{8}$
B) $\frac{1}{9}$
C) $\frac{2}{9}$
D) $\frac{1}{10}$
E) $\frac{1}{11}$ | \frac{1}{9} | 27.34375 |
25,597 |
a) Find all the divisors of the number 572 based on its prime factorization.
b) How many divisors does the number \(572 a^{3} b c\) have if:
I. \(a, b\), and \(c\) are prime numbers greater than 20 and different from each other?
II. \(a = 31\), \(b = 32\), and \(c = 33\)? | 384 | 87.5 |
25,598 | In a rectangular grid where grid lines are spaced $1$ unit apart, the acronym XYZ is depicted below. The X is formed by two diagonal lines crossing, the Y is represented with a 'V' shape starting from a bottom point going up to join two endpoints with horizontal lines, the Z is drawn with a top horizontal line, a diagonal from top right to bottom left and a bottom horizontal line. Calculate the sum of lengths of the line segments that form the acronym XYZ.
A) $6 + 3\sqrt{2}$
B) $4 + 5\sqrt{2}$
C) $3 + 6\sqrt{2}$
D) $5 + 4\sqrt{2}$ | 4 + 5\sqrt{2} | 56.25 |
25,599 | In the diagram, $JKLM$ and $NOPM$ are squares each of area 25. If $Q$ is the midpoint of both $KL$ and $NO$, find the total area of polygon $JMQPON$.
[asy]
unitsize(3 cm);
pair J, K, L, M, N, O, P, Q;
O = (0,0);
P = (1,0);
M = (1,1);
N = (0,1);
Q = (N + O)/2;
J = reflect(M,Q)*(P);
K = reflect(M,Q)*(O);
L = reflect(M,Q)*(N);
draw(J--K--L--M--cycle);
draw(M--N--O--P--cycle);
label("$J$", J, N);
label("$K$", K, W);
label("$L$", L, S);
label("$M$", M, NE);
label("$N$", N, NW);
label("$O$", O, SW);
label("$P$", P, SE);
label("$Q$", Q, SW);
[/asy] | 25 | 10.15625 |
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