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|---|---|---|---|
25,400 | Express the quotient and remainder of $3232_5 \div 21_5$ in base $5$. | 130_5 \, R2_5 | 0 |
25,401 | Evaluate the expression $\dfrac{13! - 12! - 2 \times 11!}{10!}$. | 1430 | 0 |
25,402 | Given the function $y=3\sin \left(2x-\frac{\pi }{8}\right)$, determine the horizontal shift required to transform the graph of the function $y=3\sin 2x$. | \frac{\pi}{8} | 0 |
25,403 | Sam and Lee run at equal and constant rates. They also cycle and skateboard at equal and constant rates. Sam covers $120$ kilometers after running for $4$ hours, cycling for $5$ hours, and skateboarding for $3$ hours while Lee covers $138$ kilometers after running for $5$ hours, skateboarding for $4$ hours, and cycling for $3$ hours. Their running, cycling, and skateboarding rates are all whole numbers of kilometers per hour. Find the sum of the squares of Sam's running, cycling, and skateboarding rates. | 436 | 0.78125 |
25,404 | Given that $F$ is the focus of the parabola $y^{2}=4x$, and a perpendicular line to the directrix is drawn from a point $M$ on the parabola, with the foot of the perpendicular being $N$. If $|MF|= \frac{4}{3}$, then $\angle NMF=$ . | \frac{2\pi}{3} | 17.96875 |
25,405 | In a right triangle \( A B C \) (with right angle at \( C \)), the medians \( A M \) and \( B N \) are drawn with lengths 19 and 22, respectively. Find the length of the hypotenuse of this triangle. | 29 | 0 |
25,406 | Assume that $y_1, y_2, ... , y_8$ are real numbers such that
\[
\begin{aligned}
y_1 + 4y_2 + 9y_3 + 16y_4 + 25y_5 + 36y_6 + 49y_7 + 64y_8 &= 3, \\
4y_1 + 9y_2 + 16y_3 + 25y_4 + 36y_5 + 49y_6 + 64y_7 + 81y_8 &= 15, \\
9y_1 + 16y_2 + 25y_3 + 36y_4 + 49y_5 + 64y_6 + 81y_7 + 100y_8 &= 140.
\end{aligned}
\]
Find the value of $16y_1 + 25y_2 + 36y_3 + 49y_4 + 64y_5 + 81y_6 + 100y_7 + 121y_8$. | 472 | 0 |
25,407 | Given a right triangle \(ABC\). On the extension of the hypotenuse \(BC\), a point \(D\) is chosen such that the line \(AD\) is tangent to the circumscribed circle \(\omega\) of triangle \(ABC\). The line \(AC\) intersects the circumscribed circle of triangle \(ABD\) at point \(E\). It turns out that the angle bisector of \(\angle ADE\) is tangent to the circle \(\omega\). In what ratio does point \(C\) divide the segment \(AE\)? | 1:2 | 28.90625 |
25,408 | Zeus starts at the origin \((0,0)\) and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. What is the smallest number of moves that he can make to get to the point \((1056,1007)\)? | 2111 | 0 |
25,409 | The side lengths \(a, b, c\) of triangle \(\triangle ABC\) satisfy the conditions:
1. \(a, b, c\) are all integers;
2. \(a, b, c\) form a geometric sequence;
3. At least one of \(a\) or \(c\) is equal to 100.
Find all possible sets of the triplet \((a, b, c)\). | 10 | 0 |
25,410 | Find the greatest common divisor of $8!$ and $(6!)^3.$ | 11520 | 0 |
25,411 | Find the area of the circle described by the equation $x^2 - 4x + y^2 - 8y + 12 = 0$ that lies above the line $y = 3$. | 4\pi | 2.34375 |
25,412 | How many positive integers less than 10,000 have at most three different digits? | 4119 | 0 |
25,413 | In the grid made up of $1 \times 1$ squares, four digits of 2015 are written in the shaded areas. The edges are either horizontal or vertical line segments, line segments connecting the midpoints of adjacent sides of $1 \times 1$ squares, or the diagonals of $1 \times 1$ squares. What is the area of the shaded portion containing the digits 2015? | $47 \frac{1}{2}$ | 0 |
25,414 | It is now 3:00:00 PM, as read on a 12-hour digital clock. In 315 hours, 58 minutes, and 16 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? | 77 | 9.375 |
25,415 | Let $g(x) = |3\{x\} - 1.5|$ where $\{x\}$ denotes the fractional part of $x$. Determine the smallest positive integer $m$ such that the equation \[m g(x g(x)) = x\] has at least $3000$ real solutions. | 23 | 0 |
25,416 | Let \( c \) be the constant term in the expansion of \( \left(2 x+\frac{b}{\sqrt{x}}\right)^{3} \). Find the value of \( c \). | \frac{3}{2} | 0 |
25,417 | The price of a stock increased by $25\%$ during January, fell by $15\%$ during February, rose by $20\%$ during March, and fell by $x\%$ during April. The price of the stock at the end of April was the same as it had been at the beginning of January. Calculate the value of $x$. | 22 | 14.0625 |
25,418 | Given an arithmetic sequence ${a_{n}}$ with the sum of its first $n$ terms denoted as $S_{n}$, if $S_{5}$, $S_{4}$, and $S_{6}$ form an arithmetic sequence, then determine the common ratio of the sequence ${a_{n}}$, denoted as $q$. | -2 | 6.25 |
25,419 | The sum of four numbers $p, q, r$, and $s$ is 100. If we increase $p$ by 10, we get the value $M$. If we decrease $q$ by 5, we get the value $M$. If we multiply $r$ by 10, we also get the value $M$. Lastly, if $s$ is divided by 2, it equals $M$. Determine the value of $M$. | 25.610 | 0 |
25,420 | In the convex quadrilateral \(ABCD\), \(\angle ABC=60^\circ\), \(\angle BAD=\angle BCD=90^\circ\), \(AB=2\), \(CD=1\), and the diagonals \(AC\) and \(BD\) intersect at point \(O\). Find \(\sin \angle AOB\). | \frac{15 + 6\sqrt{3}}{26} | 0 |
25,421 | Find the minimum value of
\[9y + \frac{1}{y^6}\] for \(y > 0\). | 10 | 0 |
25,422 | In the arithmetic sequence $\{a\_n\}$, the common difference $d=\frac{1}{2}$, and the sum of the first $100$ terms $S\_{100}=45$. Find the value of $a\_1+a\_3+a\_5+...+a\_{99}$. | -69 | 0 |
25,423 | For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + 2x + 15\] has three distinct roots, which are also roots of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 75x + c.\] Determine the value of $f(-1)$. | -2773 | 0 |
25,424 | An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. After the recruitment and departure, 7 new leaders are elected from outside the organization. How many people total will be in the organization after four years? | 172 | 0.78125 |
25,425 | A certain school organized a Chinese traditional culture activity week to promote traditional Chinese culture. During the activity period, a Chinese traditional culture knowledge competition was held, with classes participating in the competition. Each class selected 5 representatives through a Chinese traditional culture knowledge quiz to participate in the grade-level competition. The grade-level competition was divided into two stages: preliminary and final. In the preliminary stage, the questions of Chinese traditional culture were placed in two boxes labeled A and B. Box A contained 5 multiple-choice questions and 3 fill-in-the-blank questions, while box B contained 4 multiple-choice questions and 3 fill-in-the-blank questions. Each class representative team was required to randomly draw two questions to answer from either box A or box B. Each class representative team first drew one question to answer, and after answering, the question was not returned to the box. Then they drew a second question to answer. After answering the two questions, the questions were returned to their original boxes.
$(1)$ If the representative team of Class 1 drew 2 questions from box A, mistakenly placed the questions in box B after answering, and then the representative team of Class 2 answered the questions, with the first question drawn from box B. It is known that the representative team of Class 2 drew a multiple-choice question from box B. Find the probability that the representative team of Class 1 drew 2 multiple-choice questions from box A.
$(2)$ After the preliminary round, the top 6 representative teams from Class 6 and Class 18 entered the final round. The final round was conducted in the form of an idiom solitaire game, using a best-of-five format, meaning the team that won three rounds first won the match and the game ended. It is known that the probability of Class 6 winning the first round is $\frac{3}{5}$, the probability of Class 18 winning is $\frac{2}{5}$, and the probability of the winner of each round winning the next round is $\frac{2}{5}$, with each round having a definite winner. Let the random variable $X$ represent the number of rounds when the game ends. Find the expected value of the random variable $X$, denoted as $E(X)$. | \frac{537}{125} | 3.125 |
25,426 | Let $ABC$ be a triangle with $AB=5$ , $AC=12$ and incenter $I$ . Let $P$ be the intersection of $AI$ and $BC$ . Define $\omega_B$ and $\omega_C$ to be the circumcircles of $ABP$ and $ACP$ , respectively, with centers $O_B$ and $O_C$ . If the reflection of $BC$ over $AI$ intersects $\omega_B$ and $\omega_C$ at $X$ and $Y$ , respectively, then $\frac{O_BO_C}{XY}=\frac{PI}{IA}$ . Compute $BC$ .
*2016 CCA Math Bonanza Individual #15* | \sqrt{109} | 0 |
25,427 | Three different integers are randomly chosen from the set $$\{ -6, -3, 0, 2, 5, 7 \}$$. What is the probability that their sum is even? Express your answer as a common fraction. | \frac{19}{20} | 0 |
25,428 | Given that the terminal side of angle $a$ passes through point P(4, -3), find:
1. The value of $2\sin{a} - \cos{a}$
2. The coordinates of point P where the terminal side of angle $a$ intersects the unit circle. | -2 | 0 |
25,429 | Given a square pyramid \(M-ABCD\) with a square base such that \(MA = MD\), \(MA \perp AB\), and the area of \(\triangle AMD\) is 1, find the radius of the largest sphere that can fit into this square pyramid. | \sqrt{2} - 1 | 22.65625 |
25,430 | Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. | 619 | 20.3125 |
25,431 | Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$ , all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\] | 2^3 * 3 * 5 * 17 | 0 |
25,432 | On eight cards, the numbers $1, 1, 2, 2, 3, 3, 4, 4$ are written. Is it possible to arrange these cards in a row such that there is exactly one card between the ones, two cards between the twos, three cards between the threes, and four cards between the fours? | 41312432 | 0 |
25,433 | Given that $0 < a \leqslant \frac{5}{4}$, find the range of real number $b$ such that all real numbers $x$ satisfying the inequality $|x - a| < b$ also satisfy the inequality $|x - a^2| < \frac{1}{2}$. | \frac{3}{16} | 0 |
25,434 | Henry constructs a model involving four spherical clay pieces of radii 1 inch, 4 inches, 6 inches, and 3 inches. Calculate the square of the total volume of the clay used, subtracting 452 from the squared result, and express your answer in terms of $\pi$. | 168195.11\pi^2 | 0 |
25,435 | A cube is suspended in space with its top and bottom faces horizontal. The cube has one top face, one bottom face, and four side faces. Determine the number of ways to move from the top face to the bottom face, visiting each face at most once, without moving directly from the top face to the bottom face, and not moving from side faces back to the top face. | 20 | 0.78125 |
25,436 | If the shortest chord is cut by the line $y = kx + 1$ on the circle $C: x^2 + y^2 - 2x - 3 = 0$, then $k = \boxed{\_\_\_\_\_\_\_\_}$. | -1 | 33.59375 |
25,437 | If $a>0$ and $b>0,$ a new operation $\Delta$ is defined as follows: $$a \Delta b = \frac{a^2 + b^2}{1 + ab}.$$ Calculate $(2 \Delta 3) \Delta 4$. | \frac{6661}{2891} | 0 |
25,438 | The graph of the function f(x) = sin(2x) is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function g(x). Find the analytical expression for g(x). Also, find the minimum value of $|x_1 - x_2|$ for $x_1$ and $x_2$ that satisfy $|f(x_1) - g(x_2)| = 2$. | \frac{\pi}{2} | 1.5625 |
25,439 | Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:<br/>
| | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150 | 50 | 200 |
| Machine B | 120 | 80 | 200 |
| Total | 270 | 130 | 400 |
$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?<br/>
$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?<br/>
Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>
| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$ | 3.841 | 6.635 | 10.828| | 99\% | 8.59375 |
25,440 | Given a sufficiently large positive integer \( n \), which can be divided by all the integers from 1 to 250 except for two consecutive integers \( k \) and \( k+1 \), find \( k \). | 127 | 2.34375 |
25,441 | Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head. | 48 | 4.6875 |
25,442 | (a) In a tennis tournament with 64 players, how many matches are played?
(b) In a tournament with 2011 players, how many matches are played? | 113 | 0 |
25,443 | Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\]Find $x + y.$ | 30 | 87.5 |
25,444 | For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have? | 150 | 3.125 |
25,445 | (15) Given the following propositions:
(1) "If $x > 2$, then $x > 0$" - the negation of the proposition
(2) "For all $a \in (0, +\infty)$, the function $y = a^x$ is strictly increasing on its domain" - the negation
(3) "$π$ is a period of the function $y = \sin x$" or "$2π$ is a period of the function $y = \sin 2x$"
(4) "$x^2 + y^2 = 0$" is a necessary condition for "$xy = 0$"
The sequence number(s) of the true proposition(s) is/are _______. | (2)(3) | 0 |
25,446 | A natural number \( x \) in a base \( r \) system (\( r \leq 36 \)) is represented as \( \overline{ppqq} \), where \( 2q = 5p \). It turns out that the base-\( r \) representation of \( x^2 \) is a seven-digit palindrome with a middle digit of zero. (A palindrome is a number that reads the same from left to right and from right to left). Find the sum of the digits of the number \( x^2 \) in base \( r \). | 36 | 8.59375 |
25,447 | Given $f(x)=\ln x$, $g(x)= \frac {1}{3}x^{3}+ \frac {1}{2}x^{2}+mx+n$, and line $l$ is tangent to both the graphs of $f(x)$ and $g(x)$ at point $(1,0)$.
1. Find the equation of line $l$ and the expression for $g(x)$.
2. If $h(x)=f(x)-g′(x)$ (where $g′(x)$ is the derivative of $g(x)$), find the range of the function $h(x)$. | \frac {1}{4}- \ln 2 | 0 |
25,448 | There are 15 married couples among 30 people. Calculate the total number of handshakes that occurred among these people. | 301 | 0 |
25,449 | $ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are located on edges $AB$ and $CD$, respectively, such that $BE < CF$. The rectangle is folded over line $EF$ so that point $C$ maps to $C'$ on side $AD$ and point $B$ maps to $B'$ on side $AD$ such that $\angle{AB'C'} \cong \angle{B'EA}$ and $\angle{B'C'A} = 90^\circ$. If $AB' = 3$ and $BE = 12$, compute the area of rectangle $ABCD$ in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Compute $a + b + c$. | 57 | 0 |
25,450 | A fair coin is tossed 4 times. What is the probability of at least two consecutive heads? | \frac{5}{8} | 6.25 |
25,451 | What is the smallest positive integer that satisfies the congruence $5x \equiv 17 \pmod{31}$? | 26 | 5.46875 |
25,452 | Given the function $f(x)=|2x-1|$.
(1) Solve the inequality $f(x) < 2$;
(2) If the minimum value of the function $g(x)=f(x)+f(x-1)$ is $a$, and $m+n=a$ $(m > 0,n > 0)$, find the minimum value of $\frac{m^{2}+2}{m}+\frac{n^{2}+1}{n}$. | 2\sqrt{2}-2 | 0 |
25,453 | Which of the following numbers is not an integer? | $\frac{2014}{4}$ | 0 |
25,454 | Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$ . | 843 | 0 |
25,455 | If \( 6x + t = 4x - 9 \), what is the value of \( x + 4 \)? | -4 | 0 |
25,456 | Calculate the sum:
\[\sum_{N = 1}^{2048} \lfloor \log_3 N \rfloor.\] | 12049 | 0 |
25,457 | Given a geometric sequence $\{a_n\}$ with a common ratio of $2$ and the sum of the first $n$ terms denoted by $S_n$. If $a_2= \frac{1}{2}$, find the expression for $a_n$ and the value of $S_5$. | \frac{31}{16} | 0 |
25,458 | In triangle \( ABC \) with \( AB = 8 \) and \( AC = 10 \), the incenter \( I \) is reflected across side \( AB \) to point \( X \) and across side \( AC \) to point \( Y \). Given that segment \( XY \) bisects \( AI \), compute \( BC^2 \). (The incenter \( I \) is the center of the inscribed circle of triangle \( ABC \).) | 84 | 2.34375 |
25,459 | A circle is inscribed in quadrilateral $EFGH$, tangent to $\overline{EF}$ at $R$ and to $\overline{GH}$ at $S$. Given that $ER=24$, $RF=31$, $GS=40$, and $SH=29$, find the square of the radius of the circle. | 945 | 0 |
25,460 | A polyhedron has 12 faces and is such that:
(i) all faces are isosceles triangles,
(ii) all edges have length either \( x \) or \( y \),
(iii) at each vertex either 3 or 6 edges meet, and
(iv) all dihedral angles are equal.
Find the ratio \( x / y \). | 3/5 | 0 |
25,461 | A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of a square \(ABCD\), and the point of tangency cuts off a segment of length \(6 - 2\sqrt{5}\) cm from vertex \(A\). Two tangents are drawn to this circle from point \(C\). Find the side length of the square, given that the angle between the tangents is \(36^{\circ}\), and it is known that \(\sin 18^{\circ} = \frac{\sqrt{5} - 1}{4}\). | (\sqrt{5} - 1)(2\sqrt{2} - \sqrt{5} + 1) | 0 |
25,462 | Person A and Person B start simultaneously from points A and B respectively, walking towards each other. Person A starts from point A, and their speed is 4 times that of Person B. The distance between points A and B is \( S \) kilometers, where \( S \) is a positive integer with 8 factors. The first time they meet at point C, the distance \( AC \) is an integer. The second time they meet at point D, the distance \( AD \) is still an integer. After the second meeting, Person B feels too slow, so they borrow a motorbike from a nearby village near point D. By the time Person B returns to point D with the motorbike, Person A has reached point E, with the distance \( AE \) being an integer. Finally, Person B chases Person A with the motorbike, which travels at 14 times the speed of Person A. Both arrive at point A simultaneously. What is the distance between points A and B?
\[ \text{The distance between points A and B is } \qquad \text{kilometers.} \] | 105 | 3.125 |
25,463 | In the diagram, a rectangle has a perimeter of $40$, and a triangle has a height of $40$. If the rectangle and the triangle have the same area, what is the value of $x?$ Assume the length of the rectangle is twice its width. [asy]
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);
draw((4,0)--(7,0)--(7,5)--cycle);
draw((6.8,0)--(6.8,.2)--(7,.2));
label("$x$",(5.5,0),S);
label("40",(7,2.5),E);
[/asy] | 4.4445 | 0 |
25,464 | Given a cube with an unknown volume, two of its dimensions are increased by $1$ and the third is decreased by $2$, and the volume of the resulting rectangular solid is $27$ less than that of the cube. Determine the volume of the original cube. | 125 | 28.90625 |
25,465 | In the diagram, \(AB\) is a diameter of a circle with center \(O\). \(C\) and \(D\) are points on the circle. \(OD\) intersects \(AC\) at \(P\), \(OC\) intersects \(BD\) at \(Q\), and \(AC\) intersects \(BD\) at \(R\). If \(\angle BOQ = 60^{\circ}\) and \(\angle APO = 100^{\circ}\), calculate the measure of \(\angle BQO\). | 95 | 0 |
25,466 | Each number in the list $1,2,3,\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . What is $m+n$ ?
*2021 CCA Math Bonanza Team Round #9* | 455 | 0 |
25,467 | The product of positive integers $a$, $b$, and $c$ equals 3960. What is the minimum possible value of the sum $a + b + c$? | 72 | 0 |
25,468 | Integers $x$ and $y$ with $x > y > 0$ satisfy $x + y + xy = 119$. What is $x$? | 39 | 11.71875 |
25,469 | Given the function $f(x) = (\sin x + \cos x)^2 + \cos 2x - 1$.
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. | -\sqrt{2} | 34.375 |
25,470 | Real numbers \(a, b, c\) and positive number \(\lambda\) make the function \(f(x) = x^3 + ax^2 + bx + c\) have three real roots \(x_1, x_2, x_3\), such that (1) \(x_2 - x_1 = \lambda\); (2) \(x_3 > \frac{1}{2}(x_1 + x_2)\). Find the maximum value of \(\frac{2a^3 + 27c + 9ab}{\lambda^3}\). | \frac{3\sqrt{3}}{2} | 0 |
25,471 | In triangle $PQR$, $QR = 24$. An incircle of the triangle trisects the median $PS$ from $P$ to side $QR$. Given that the area of the triangle is $k \sqrt{p}$, where $k$ and $p$ are integers and $p$ is square-free, find $k+p$. | 106 | 0 |
25,472 | Evaluate the expression \[ \frac{a+3}{a+1} \cdot \frac{b-2}{b-3} \cdot \frac{c + 9}{c+7} , \] given that $c = b-11$, $b = a+3$, $a = 5$, and none of the denominators are zero. | \frac{1}{3} | 0 |
25,473 | Consider a triangle with vertices at points \( (0, 0), (30, 0), \) and \( (18, 26) \). The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the original triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | 3380 | 0 |
25,474 | Given the curves $C_{1}: \begin{cases} x=2+\cos t \\ y=\sin t-1 \end{cases}$ (with $t$ as the parameter), and $C_{2}: \begin{cases} x=4\cos \alpha \\ y=\sin \alpha \end{cases}$ (with $\alpha$ as the parameter), in the polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, there is a line $C_{3}: \theta= \frac {\pi}{4} (\rho \in \mathbb{R})$.
1. Find the standard equations of curves $C_{1}$ and $C_{2}$, and explain what curves they respectively represent;
2. If the point $P$ on $C_{2}$ corresponds to the parameter $\alpha= \frac {\pi}{2}$, and $Q$ is a point on $C_{1}$, find the minimum distance $d$ from the midpoint $M$ of $PQ$ to the line $C_{3}$. | \frac { \sqrt {2}-1}{ \sqrt {2}} | 0 |
25,475 | Let $T$ be the set of all positive integer divisors of $144,000$. Calculate the number of numbers that are the product of two distinct elements of $T$. | 451 | 0 |
25,476 | What is the largest integer that must divide the product of any 5 consecutive integers? | 60 | 1.5625 |
25,477 | What is the smallest positive integer that ends in 3 and is divisible by 11? | 113 | 0 |
25,478 | The sequence of integers $\{a_i\}_{i = 0}^{\infty}$ satisfies $a_0 = 3$ , $a_1 = 4$ , and
\[a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil\]
for $n \ge 0$ . Evaluate the sum
\[\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).\] | 0.518 | 0 |
25,479 | The number \(n\) is a three-digit positive integer and is the product of the three factors \(x\), \(y\), and \(5x+2y\), where \(x\) and \(y\) are integers less than 10 and \((5x+2y)\) is a composite number. What is the largest possible value of \(n\) given these conditions? | 336 | 0 |
25,480 | How many even divisors does \(8!\) have, and how many of those are also multiples of both 2 and 3? | 84 | 0 |
25,481 | Let \(ABCD\) be a quadrilateral inscribed in a unit circle with center \(O\). Suppose that \(\angle AOB = \angle COD = 135^\circ\), and \(BC = 1\). Let \(B'\) and \(C'\) be the reflections of \(A\) across \(BO\) and \(CO\) respectively. Let \(H_1\) and \(H_2\) be the orthocenters of \(AB'C'\) and \(BCD\), respectively. If \(M\) is the midpoint of \(OH_1\), and \(O'\) is the reflection of \(O\) about the midpoint of \(MH_2\), compute \(OO'\). | \frac{1}{4}(8-\sqrt{6}-3\sqrt{2}) | 0 |
25,482 | The numbers \(a, b,\) and \(c\) (not necessarily integers) satisfy the conditions
\[
a + b + c = 0 \quad \text{and} \quad \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 100
\]
What is the value of \(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\)? | -101 | 9.375 |
25,483 | On graph paper, a stepwise right triangle was drawn with legs equal to 6 cells each. Then, all grid lines inside the triangle were outlined. What is the maximum number of rectangles that can be found in this drawing? | 126 | 10.9375 |
25,484 | Eight distinct pieces of candy are to be distributed among three bags: red, blue, and white, with each bag receiving at least one piece of candy. Determine the total number of arrangements possible. | 846720 | 0 |
25,485 | The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges start at the vertices of a square, and this configuration stores 20 Joules of energy. How much more energy, in Joules, would be stored if one of these charges was moved to the center of the square? | 5(3\sqrt{2} - 3) | 0 |
25,486 | Given that the terminal side of angle $\alpha$ passes through point $P$((-$ sqrt{3}$, $m$)), and $\sin \alpha$ = $\frac{\sqrt{3}}{4}$ $m$($m$($m$ne 0)), determine in which quadrant angle $\alpha$ lies, and find the value of $\tan \alpha$. | \frac{\sqrt{7}}{3} | 7.8125 |
25,487 | Given integers $a$ and $b$ whose sum is $20$, determine the number of distinct integers that can be written as the sum of two, not necessarily different, fractions $\frac{a}{b}$. | 14 | 0 |
25,488 | Given a parallelogram \(A B C D\) with \(\angle B = 111^\circ\) and \(B C = B D\). On the segment \(B C\), there is a point \(H\) such that \(\angle B H D = 90^\circ\). Point \(M\) is the midpoint of side \(A B\). Find the angle \(A M H\). Provide the answer in degrees. | 132 | 0 |
25,489 | Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]
What is the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \)? | 1511 | 0 |
25,490 | Points \(P, Q, R,\) and \(S\) lie in the plane of the square \(EFGH\) such that \(EPF\), \(FQG\), \(GRH\), and \(HSE\) are equilateral triangles. If \(EFGH\) has an area of 25, find the area of quadrilateral \(PQRS\). Express your answer in simplest radical form. | 100 + 50\sqrt{3} | 3.90625 |
25,491 | Suppose that $a, b, c$ , and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$ . | 63 | 3.125 |
25,492 | The positive integers \(a\) and \(b\) are such that the numbers \(15a + 165\) and \(16a - 155\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481 | 0 |
25,493 | Compute $1-2+3-4+ \dots -100+101$. | 76 | 0 |
25,494 | Given the line $y=x+\sqrt{6}$, the circle $(O)$: $x^2+y^2=5$, and the ellipse $(E)$: $\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$ $(b > 0)$ with an eccentricity of $e=\frac{\sqrt{3}}{3}$. The length of the chord intercepted by line $(l)$ on circle $(O)$ is equal to the length of the major axis of the ellipse. Find the product of the slopes of the two tangent lines to ellipse $(E)$ passing through any point $P$ on circle $(O)$, if the tangent lines exist. | -1 | 2.34375 |
25,495 | What is the smallest number, all of whose digits are 1 or 2, and whose digits add up to $10$? | 111111112 | 12.5 |
25,496 | There are 2014 points marked on a circle. A grasshopper sits on one of these points and makes jumps either 57 divisions or 10 divisions clockwise. It is known that he visited all the marked points, making the minimum number of jumps. | 18 | 0 |
25,497 | Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\mathbf{v}$ such that
\[\begin{pmatrix} 3 & 4 \\ 6 & 3 \end{pmatrix} \mathbf{v} = k \mathbf{v}.\] | 3 - 2\sqrt{6} | 2.34375 |
25,498 | If three different numbers are selected from 2, 3, 4, 5, 6 to be $a$, $b$, $c$ such that $N = abc + ab + bc + a - b - c$ reaches its maximum value, then this maximum value is. | 167 | 0.78125 |
25,499 | Given circles $P$, $Q$, and $R$ each have radius 2, circle $P$ and $Q$ are tangent to each other, and circle $R$ is tangent to the midpoint of $\overline{PQ}$. Calculate the area inside circle $R$ but outside circle $P$ and circle $Q$. | 2\pi | 6.25 |
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