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stringlengths 10
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27,500 | Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | \frac{180}{37} | 0.78125 |
27,501 | Two circles, circle $A$ with radius 2 and circle $B$ with radius 1.5, are to be constructed with the following process: The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ to $(3,0)$. The center of circle $B$ is chosen uniformly and at random, and independently from the first choice, from the line segment joining $(1,2)$ to $(4,2)$. What is the probability that circles $A$ and $B$ intersect?
A) 0.90
B) 0.95
C) 0.96
D) 1.00 | 0.96 | 59.375 |
27,502 | Identical matches of length 1 are used to arrange the following pattern. If \( c \) denotes the total length of matches used, find \( c \). | 700 | 0 |
27,503 | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 98.4375 |
27,504 | If set $A=\{x\in N\left|\right.-1 \lt x\leqslant 2\}$, $B=\{x\left|\right.x=ab,a,b\in A\}$, then the number of non-empty proper subsets of set $B$ is ______. | 14 | 7.03125 |
27,505 | An entrepreneur invested $\$20,\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $\$22,\!446.40$. If the annual interest rate of the second term deposit is $s\%$, what is $s?$ | 7.840 | 0 |
27,506 | Given that the sum of three numbers, all equally likely to be $1$, $2$, $3$, or $4$, drawn from an urn with replacement, is $9$, calculate the probability that the number $3$ was drawn each time. | \frac{1}{13} | 0.78125 |
27,507 |
Monsieur Dupont remembered that today is their wedding anniversary and invited his wife to dine at a fine restaurant. Upon leaving the restaurant, he noticed that he had only one fifth of the money he initially took with him. He found that the centimes he had left were equal to the francs he initially had (1 franc = 100 centimes), while the francs he had left were five times less than the initial centimes he had.
How much did Monsieur Dupont spend at the restaurant? | 7996 | 11.71875 |
27,508 | Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$. | 11 | 0 |
27,509 | A four-digit number satisfies the following conditions:
(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;
(2) When divided by 9, the remainder is 8.
Find the smallest odd four-digit number that satisfies these conditions.
(Shandong Province Mathematics Competition, 1979) | 1979 | 17.1875 |
27,510 | Consider the function $y=a\sqrt{1-x^2} + \sqrt{1+x} + \sqrt{1-x}$ ($a\in\mathbb{R}$), and let $t= \sqrt{1+x} + \sqrt{1-x}$ ($\sqrt{2} \leq t \leq 2$).
(1) Express $y$ as a function of $t$, denoted as $m(t)$.
(2) Let the maximum value of the function $m(t)$ be $g(a)$. Find $g(a)$.
(3) For $a \geq -\sqrt{2}$, find all real values of $a$ that satisfy $g(a) = g\left(\frac{1}{a}\right)$. | a = 1 | 28.90625 |
27,511 | There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes?
(Once a safe is opened, the key inside the safe can be used to open another safe.) | 1/47 | 1.5625 |
27,512 | Let $A$, $B$, $R$, $M$, and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AB) + \log_{10} (AM) &= 2, \\
\log_{10} (ML) + \log_{10} (MR) &= 3, \\
\log_{10} (RA) + \log_{10} (RB) &= 5.
\end{align*}
Compute the value of the product $ABRML$. | 100 | 1.5625 |
27,513 | Determine the smallest integer $B$ such that there exist several consecutive integers, including $B$, that add up to 2024. | -2023 | 13.28125 |
27,514 | For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | 47 | 2.34375 |
27,515 | In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is | 67.5 | 1.5625 |
27,516 | Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$ , $|BD|=6$ , and $|AD|\cdot|CE|=|DC|\cdot|AE|$ , find the area of the quadrilateral $ABCD$ . | 9\sqrt{2} | 2.34375 |
27,517 | John earned scores of 92, 85, and 91 on his first three physics examinations. If John receives a score of 95 on his fourth exam, then by how much will his average increase? | 1.42 | 20.3125 |
27,518 | If the direction vector of line $l$ is $\overrightarrow{e}=(-1,\sqrt{3})$, calculate the inclination angle of line $l$. | \frac{2\pi}{3} | 57.03125 |
27,519 | For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ . | 1680 | 61.71875 |
27,520 | The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$? | 12 | 6.25 |
27,521 | Given the sequence \(\left\{a_{n}\right\}\) that satisfies
\[ a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad \text{for} \quad n \in \mathbf{Z}_{+}, \]
where \([x]\) denotes the greatest integer less than or equal to the real number \(x\). Let \(C\) be a real number such that for any positive integer \(n\),
\[ \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C. \]
Find the minimum value of \(C\). | \frac{\sqrt{5} - 2}{4} | 0 |
27,522 | Given that in the rectangular coordinate system $(xOy)$, the origin is the pole and the positive semi-axis of $x$ is the polar axis to establish a polar coordinate system, the polar coordinate equation of the conic section $(C)$ is $p^{2}= \frac {12}{3+\sin ^{2}\theta }$, the fixed point $A(0,- \sqrt {3})$, $F\_{1}$, $F\_{2}$ are the left and right foci of the conic section $(C)$, and the line $l$ passes through point $F\_{1}$ and is parallel to the line $AF\_{2}$.
(I) Find the rectangular coordinate equation of conic section $(C)$ and the parametric equation of line $l$;
(II) If line $l$ intersects conic section $(C)$ at points $M$ and $N$, find $|F\_{1}M|⋅|F\_{1}N|$. | \frac {12}{5} | 92.96875 |
27,523 | Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.
1. Determine whether the function $f(x) = x^2$ belongs to set $M$ and explain your reasoning.
2. If $f(x) = \sin x \in M$, find all companion pairs $(a,k)$ for the function $f(x)$.
3. If $(1,1)$ and $(2,-1)$ are both companion pairs of the function $f(x)$, where $f(x) = \cos(\frac{\pi}{2}x)$ for $1 \leq x < 2$ and $f(x) = 0$ for $x=2$. Find all zeros of the function $y=f(x)$ when $2014 \leq x \leq 2016$. | 2016 | 0 |
27,524 | Given the function $f\left( x \right)=2\sin (\omega x+\varphi )\left( \omega \gt 0,\left| \varphi \right|\lt \frac{\pi }{2} \right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\left( \frac{\pi }{18},\frac{\pi }{3} \right)$. The graph of $f\left( x \right)$ is shifted to the left by $\pi$ units and coincides with the original graph. When ${x}_{1}$, ${x}_{2} \in \left( -\frac{17\pi }{12},-\frac{2\pi }{3} \right)$ and ${x}_{1} \ne {x}_{2}$, if $f\left( {x}_{1} \right)=f\left( {x}_{2} \right)$, find $f({x}_{1}+{x}_{2})$. | -1 | 44.53125 |
27,525 | A circle with radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares it can pass through? | 800 | 3.90625 |
27,526 | In a circle with a radius of 5 units, \( CD \) and \( AB \) are mutually perpendicular diameters. A chord \( CH \) intersects \( AB \) at \( K \) and has a length of 8 units, calculate the lengths of the two segments into which \( AB \) is divided. | 8.75 | 0 |
27,527 | Real numbers \(a, b, c\) and a positive number \(\lambda\) satisfy \(f(x) = x^3 + a x^2 + b x + c\), which has 3 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 |
27,528 | You are given the numbers $0$, $2$, $3$, $4$, $6$. Use these numbers to form different combinations and calculate the following:
$(1)$ How many unique three-digit numbers can be formed?
$(2)$ How many unique three-digit numbers that can be divided by $3$ can be formed? (Note: Write the result of each part in data form) | 20 | 40.625 |
27,529 | I live on a very short street with 14 small family houses. The odd-numbered houses from 1 are on one side of the street, and the even-numbered houses from 2 are on the opposite side (e.g., 1 and 2 are opposite each other).
On one side of the street, all families have surnames that are colors, and on the other side, the surnames indicate professions.
Szabó and Fazekas live opposite to Zöld and Fehér, respectively, who are both neighbors of Fekete.
Kovács is the father-in-law of Lakatos.
Lakatos lives in a higher-numbered house than Barna. The sum of the house numbers of Lakatos and Barna is equal to the sum of the house numbers of Fehér and Fazekas. Kádárné's house number is twice the house number of her sister, Kalaposné.
Sárga lives opposite Pék.
If Bordóné's house number is two-digit and she lives opposite her sister, Kádárné, what is the house number of Mr. Szürke? | 13 | 10.15625 |
27,530 | How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | 41 | 0.78125 |
27,531 | In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 1500$ and $PS = 3000$. Let $\angle P = 37^\circ$, $\angle S = 53^\circ$, and $X$ and $Y$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $XY$. | 750 | 12.5 |
27,532 | In the textbook, students were once asked to explore the coordinates of the midpoint of a line segment: In a plane Cartesian coordinate system, given two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, the midpoint of the line segment $AB$ is $M$, then the coordinates of $M$ are ($\frac{{x}_{1}+{x}_{2}}{2}$, $\frac{{y}_{1}+{y}_{2}}{2}$). For example, if point $A(1,2)$ and point $B(3,6)$, then the coordinates of the midpoint $M$ of line segment $AB$ are ($\frac{1+3}{2}$, $\frac{2+6}{2}$), which is $M(2,4)$. Using the above conclusion to solve the problem: In a plane Cartesian coordinate system, if $E(a-1,a)$, $F(b,a-b)$, the midpoint $G$ of the line segment $EF$ is exactly on the $y$-axis, and the distance to the $x$-axis is $1$, then the value of $4a+b$ is ____. | 4 \text{ or } 0 | 39.84375 |
27,533 | Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
\[f(x) f(-x) = f(x^3).\] | 12 | 0 |
27,534 | Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 7.03125 |
27,535 | Given the function $f(x)=\begin{cases} x+2 & (x\leqslant -1) \\ x^{2} & (-1< x < 2) \\ 2x & (x\geqslant 2) \end{cases}$
$(1)$ Find $f(2)$, $f\left(\dfrac{1}{2}\right)$, $f[f(-1)]$;
$(2)$ If $f(a)=3$, find the value of $a$. | \sqrt {3} | 0 |
27,536 | What is the greatest prime factor of $15! + 18!$? | 17 | 0.78125 |
27,537 | In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\triangle ABC$. | 2\sqrt{3} | 7.03125 |
27,538 | Given an ellipse $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, whose left and right foci are $F_{1}$ and $F_{2}$ respectively, and the top vertex is $B$. If the perimeter of $\triangle BF_{1}F_{2}$ is $6$, and the distance from point $F_{1}$ to the line $BF_{2}$ is $b$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ Let $A_{1}, A_{2}$ be the two endpoints of the major axis of ellipse $C$, and point $P$ is any point on ellipse $C$ other than $A_{1}, A_{2}$. The line $A_{1}P$ intersects the line $x = m$ at point $M$. If the circle with diameter $MP$ passes through point $A_{2}$, find the value of the real number $m$. | 14 | 0.78125 |
27,539 | How many of the natural numbers from 1 to 700, inclusive, contain the digit 3 at least once? | 214 | 0 |
27,540 | The function $y=f(x)$ is an even function with the smallest positive period of $4$, and when $x \in [-2,0]$, $f(x)=2x+1$. If there exist $x\_1$, $x\_2$, $…x\_n$ satisfying $0 \leqslant x\_1 < x\_2 < … < x\_n$, and $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_1)|+…+|f(x\_{n-1}-f(x\_n))|=2016$, then the minimum value of $n+x\_n$ is \_\_\_\_\_\_. | 1513 | 1.5625 |
27,541 | How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | 511 | 3.125 |
27,542 | The solution to the inequality
$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$
is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.
Recall that $[x]$ denotes the greatest integer less than or equal to $x$. | 78.00 | 4.6875 |
27,543 | What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,8,Ticks(1.0,NoZero));
yaxis(0,8,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,7)--(0,6)--(6,6)--cycle);
label("$E$",(0,0),SW);
label("$F$",(0,7),N);
label("$G$",(7,7),NE);
label("$H$",(7,0),E);
[/asy] | 67\% | 0 |
27,544 | What is the smallest five-digit positive integer congruent to $2 \pmod{17}$? | 10013 | 2.34375 |
27,545 | A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated contents of the bag? | \frac{91}{169} | 0 |
27,546 | Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7. | 60718293 | 0 |
27,547 | Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(10,2)$, respectively. Calculate its area. | 52\sqrt{3} | 10.9375 |
27,548 | A certain intelligence station has four different passwords A, B, C, and D. Each week, one of the passwords is used, and the password for each week is equally likely to be randomly selected from the three passwords not used in the previous week. If password A is used in the first week, what is the probability that password A is also used in the seventh week? (Express your answer as a simplest fraction.) | 1/3 | 0.78125 |
27,549 | How many positive integers \( n \) satisfy \[ (n + 9)(n - 4)(n - 13) < 0 \]? | 11 | 0 |
27,550 | Jia participates in a shooting practice with 4 bullets, among which one is a blank (a "blank" means a bullet without a projectile).
(1) If Jia shoots only once, calculate the probability of the shot being a blank;
(2) If Jia shoots a total of 3 times, calculate the probability of a blank appearing in these three shots;
(3) If an equilateral triangle PQR with a side length of 10 is drawn on the target, and Jia, using live rounds, aims and randomly shoots at the area of triangle PQR, calculate the probability that all bullet holes are more than 1 unit away from the vertices of △PQR (ignoring the size of the bullet holes). | 1 - \frac{\sqrt{3}\pi}{150} | 1.5625 |
27,551 |
A random point \(N\) on a line has coordinates \((t, -2-t)\), where \(t \in \mathbb{R}\). A random point \(M\) on a parabola has coordinates \(\left( x, x^2 - 4x + 5 \right)\), where \(x \in \mathbb{R}\). The square of the distance between points \(M\) and \(N\) is given by \(\rho^2(x, t) = (x - t)^2 + \left( x^2 - 4x + 7 + t \right)^2\). Find the coordinates of points \(M\) and \(N\) that minimize \(\rho^2\).
When the point \(M\) is fixed, \(\rho^2(t)\) depends on \(t\), and at the point of minimum, its derivative with respect to \(t\) is zero: \[-2(x - t) + 2\left( x^2 - 4x + 7 + t \right) = 0.\]
When point \(N\) is fixed, the function \(\rho^2(x)\) depends on \(x\), and at the point of minimum, its derivative with respect to \(x\) is zero:
\[2(x - t) + 2\left( x^2 - 4x + 7 + t \right)(2x - 4) = 0.\]
We solve the system:
\[
\begin{cases}
2 \left( x^2 - 4x + 7 + t \right) (2x - 3) = 0 \\
4(x - t) + 2 \left( x^2 - 4x + 7 + t \right) (2x - 5) = 0
\end{cases}
\]
**Case 1: \(x = \frac{3}{2}\)**
Substituting \(x = \frac{3}{2}\) into the second equation, we get \(t = -\frac{7}{8}\). Critical points are \(N^* \left( -\frac{7}{8}, -\frac{9}{8} \right)\) and \(M^* \left( \frac{3}{2}, \frac{5}{4} \right)\). The distance between points \(M^*\) and \(N^*\) is \(\frac{19\sqrt{2}}{8}\). If \(2r \leq \frac{19\sqrt{2}}{8}\), then the circle does not intersect the parabola (it touches at points \(M^*\) and \(N^*\)).
**Case 2: \(x \neq \frac{3}{2}\)**
Then \(\left( x^2 - 4x + 7 + t \right) = 0\) and from the second equation \(x = t\). Substituting \(x = t\) into the equation \(\left( x^2 - 4x + 7 + t \right) = 0\), we get the condition for \(t\), i.e. \(t^2 - 3t + 7 = 0\). Since this equation has no solutions, case 2 is not realizable. The maximum radius \(r_{\max} = \frac{19\sqrt{2}}{16}\). | \frac{19\sqrt{2}}{8} | 6.25 |
27,552 | If \(A\ \clubsuit\ B\) is defined as \(A\ \clubsuit\ B = 3A^2 + 2B + 7\), what is the value of \(A\) for which \(A\ \clubsuit\ 7 = 61\)? | \frac{2\sqrt{30}}{3} | 21.875 |
27,553 | Given the arithmetic sequence $\{a_n\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms. | 180 | 0 |
27,554 | Given a regular 2017-sided polygon \( A_{1} A_{2} \cdots A_{2017} \) inscribed in a unit circle \(\odot O\), choose any two distinct vertices \( A_{i} \) and \( A_{j} \). What is the probability that \( \overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2} \)? | 2/3 | 7.8125 |
27,555 | Evaluate the absolute value of the expression $|7 - \sqrt{53}|$.
A) $7 - \sqrt{53}$
B) $\sqrt{53} - 7$
C) $0.28$
D) $\sqrt{53} + 7$
E) $-\sqrt{53} + 7$ | \sqrt{53} - 7 | 60.15625 |
27,556 | Given $x+x^{-1}=3$, calculate the value of $x^{ \frac {3}{2}}+x^{- \frac {3}{2}}$. | \sqrt{5} | 0 |
27,557 | Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group. | 63 | 89.0625 |
27,558 | Determine the smallest positive integer $n$, different from 2004, such that there exists a polynomial $f(x)$ with integer coefficients for which the equation $f(x) = 2004$ has at least one integer solution and the equation $f(x) = n$ has at least 2004 different integer solutions. | (1002!)^2 + 2004 | 0 |
27,559 | Let $f(n)=1 \times 3 \times 5 \times \cdots \times (2n-1)$ . Compute the remainder when $f(1)+f(2)+f(3)+\cdots +f(2016)$ is divided by $100.$ *Proposed by James Lin* | 74 | 0 |
27,560 | Find the minimum value of the function \( f(x) = \tan^2 x - 4 \tan x - 8 \cot x + 4 \cot^2 x + 5 \) on the interval \( \left( \frac{\pi}{2}, \pi \right) \). | 9 - 8\sqrt{2} | 0 |
27,561 | All the complex roots of $(z + 2)^6 = 64z^6$, when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{\sqrt{3}} | 1.5625 |
27,562 | How many ways are there to arrange the letters of the word $\text{CA}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{T}_1\text{T}_2$, where there are three A's, two N's, and two T's, with each A, N, and T considered distinct? | 5040 | 0.78125 |
27,563 | Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 1.5625 |
27,564 | Biejia and Vasha are playing a game. Biejia selects 100 non-negative numbers \(x_1, x_2, \cdots, x_{100}\) (they can be the same), whose sum equals 1. Vasha then pairs these numbers into 50 pairs in any way he chooses, computes the product of the two numbers in each pair, and writes the largest product on the blackboard. Biejia wants the number written on the blackboard to be as large as possible, while Vasha wants it to be as small as possible. What will be the number written on the blackboard under optimal strategy? | 1/396 | 9.375 |
27,565 | In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, $BD = 17^3$, and $\cos B = m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 162 | 2.34375 |
27,566 | The surface of a 3x3x3 Rubik's cube consists of 54 cells. What is the maximum number of cells that can be marked such that no two marked cells share a common vertex? | 14 | 24.21875 |
27,567 | Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[\frac{(a - b)(c - d)}{(b - c)(d - a)} = \frac{3}{7}.\] Find the product of all possible values of
\[\frac{(a - c)(b - d)}{(a - b)(c - d)}.\] | -\frac{4}{3} | 0.78125 |
27,568 | The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1, calculate the sum of these numbers. | 4436 | 1.5625 |
27,569 | A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$ . An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ?
[center][/center] | 18 | 16.40625 |
27,570 | A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$ . We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$ . Determine the greatest value of $n$ . | 6050 | 15.625 |
27,571 | In the rectangular coordinate system, an ellipse C passes through points A $(\sqrt{3}, 0)$ and B $(0, 2)$.
(I) Find the equation of ellipse C;
(II) Let P be any point on the ellipse, find the maximum area of triangle ABP, and find the coordinates of point P when the area of triangle ABP is maximum. | \sqrt{6} + \sqrt{3} | 0 |
27,572 | Given $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\cos 2C+2c\cos A\cos C+a+b=0$.
$(1)$ Find the size of angle $C$;
$(2)$ If $b=4\sin B$, find the maximum value of the area $S$ of $\triangle ABC$. | \sqrt{3} | 28.90625 |
27,573 | How many four-digit positive integers are multiples of 7? | 1286 | 100 |
27,574 | In acute triangle $ABC$ , points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$ . Compute $EC - EB$ . | 54 | 25.78125 |
27,575 | Let $x = \frac{\sum\limits_{n=1}^{30} \cos n^\circ}{\sum\limits_{n=1}^{30} \sin n^\circ}$. What is the smallest integer that does not fall below $100x$? | 360 | 0 |
27,576 | Given the function $f(x)=\frac{1}{2}x^{2}+(2a^{3}-a^{2})\ln x-(a^{2}+2a-1)x$, and $x=1$ is its extreme point, find the real number $a=$ \_\_\_\_\_\_. | -1 | 29.6875 |
27,577 | The real numbers \(x_{1}, x_{2}, \cdots, x_{2001}\) satisfy \(\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001\). Let \(y_{k}=\frac{1}{k}\left(x_{1}+ x_{2} + \cdots + x_{k}\right)\) for \(k=1, 2, \cdots, 2001\). Find the maximum possible value of \(\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|\). | 2000 | 37.5 |
27,578 |
A travel agency conducted a promotion: "Buy a trip to Egypt, bring four friends who also buy trips, and get your trip cost refunded." During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of these customers each brought exactly four new clients, while the remaining 100 brought no one. How many tourists went to the Land of the Pyramids for free? | 29 | 0 |
27,579 | Find the smallest positive integer which cannot be expressed in the form \(\frac{2^{a}-2^{b}}{2^{c}-2^{d}}\) where \(a, b, c, d\) are non-negative integers. | 11 | 85.9375 |
27,580 | A regular hexagon with side length 1 has an arbitrary interior point that is reflected over the midpoints of its six sides. Calculate the area of the hexagon formed in this way. | \frac{9\sqrt{3}}{2} | 0 |
27,581 | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____. | \frac{1}{3}\pi | 0 |
27,582 | In the equation "中环杯是 + 最棒的 = 2013", different Chinese characters represent different digits. What is the possible value of "中 + 环 + 杯 + 是 + 最 + 棒 + 的"? (If there are multiple solutions, list them all). | 1250 + 763 | 0 |
27,583 | Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\in\mathbb{N}^*$), and that $\{a_{2n-1}\}$ is an increasing sequence, $\{a_{2n}\}$ is a decreasing sequence, find the limit $$\lim_{n\to\infty} \frac{a_{2n-1}}{a_{2n}} = \_\_\_\_\_\_.$$ | -\frac{1}{2} | 56.25 |
27,584 | A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$ . If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$ , find the diameter of $\omega_{1998}$ . | 3995 | 0.78125 |
27,585 | In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$. | \frac{1}{2} | 5.46875 |
27,586 | Find the sum of all integer values of \( c \) for which the equation \( 15|p-1| + |3p - |p + c|| = 4 \) has at least one solution for \( p \). | -2 | 13.28125 |
27,587 | On a circular keychain, I need to arrange six keys. Two specific keys, my house key and my car key, must always be together. How many different ways can I arrange these keys considering that arrangements can be rotated or reflected? | 48 | 0.78125 |
27,588 | (1) Let the function $f(x) = |x - 2| + |x + a|$. If the inequality $f(x) \geq 3$ always holds for all $x \in \mathbb{R}$, find the range of values for the real number $a$.
(2) Given positive numbers $x, y, z$ that satisfy $x + 2y + 3z = 1$, find the minimum value of $\frac {3}{x} + \frac {2}{y} + \frac {1}{z}$. | 16 + 8\sqrt{3} | 9.375 |
27,589 | A hexagonal lattice now consists of 13 points where each point is one unit distance from its nearest neighbors. Points are arranged with one center point, six points forming a hexagon around the center, and an additional outer layer of six points forming a larger hexagon. How many equilateral triangles have all three vertices in this new expanded lattice? | 18 | 1.5625 |
27,590 | Let \( z \) be a complex number such that \( |z| = 2 \). Find the maximum value of
\[
|(z - 2)(z + 2)^2|.
\] | 16 \sqrt{2} | 4.6875 |
27,591 | What is the least positive integer with exactly $12$ positive factors? | 72 | 0 |
27,592 | Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the absolute value of the slope of this line? | \frac{4}{3} | 0 |
27,593 | In triangle $\triangle ABC$, point $G$ satisfies $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\overrightarrow{AM}=m\overrightarrow{AB}$ $(m>0)$ and $\overrightarrow{AN}=n\overrightarrow{AC}$ $(n>0)$, then the minimum value of $3m+n$ is ______. | \frac{4}{3}+\frac{2\sqrt{3}}{3} | 0 |
27,594 | A regular tetrahedron has the numbers $1,2,3,4$ written on its four faces. Four such identical regular tetrahedrons are thrown simultaneously onto a table. What is the probability that the product of the numbers on the four faces in contact with the table is divisible by $4$? | $\frac{13}{16}$ | 0 |
27,595 | Sarah multiplied an integer by itself. Which of the following could be the result? | 36 | 0.78125 |
27,596 | Given a circle $C: (x-3)^2 + (y-4)^2 = 25$, the shortest distance from a point on circle $C$ to line $l: 3x + 4y + m = 0 (m < 0)$ is $1$. If point $N(a, b)$ is located on the part of line $l$ in the first quadrant, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{7 + 4\sqrt{3}}{55} | 1.5625 |
27,597 | David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
*Ray Li* | 30 | 0.78125 |
27,598 | $ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ? | 1/3 | 13.28125 |
27,599 | Suppose there is an octahedral die with the numbers 1, 2, 3, 4, 5, 6, 7, and 8 written on its eight faces. Each time the die is rolled, the chance of any of these numbers appearing is the same. If the die is rolled three times, and the numbers appearing on the top face are recorded in sequence, let the largest number be represented by $m$ and the smallest by $n$.
(1) Let $t = m - n$, find the range of values for $t$;
(2) Find the probability that $t = 3$. | \frac{45}{256} | 3.125 |
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