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40.3k
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5.15k
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1.22k
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100
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|---|---|---|---|
32,100 | Consider a series of squares where each square has a border of dots around it, each border containing dots on all four sides. The smallest square (first in the series) has a single dot at the center. Each successive square surrounds the previous one with an additional border of dots, with the number of dots on each side of the square equal to the term number. Determine how many dots are in the fourth square. | 37 | 3.125 |
32,101 | Given that $\{a_n\}$ is an arithmetic sequence with the first term $a_1 > 0$, and $a_{1007} + a_{1008} > 0$, $a_{1007} \cdot a_{1008} < 0$, determine the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$. | 2014 | 9.375 |
32,102 | Given a trapezoid \( ABCD \) with \( AD \parallel BC \). It turns out that \( \angle ABD = \angle BCD \). Find the length of segment \( BD \) if \( BC = 36 \) and \( AD = 64 \). | 48 | 64.84375 |
32,103 | Let $x_1, x_2, \ldots, x_7$ be natural numbers, and $x_1 < x_2 < \ldots < x_6 < x_7$, also $x_1 + x_2 + \ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. | 61 | 0 |
32,104 | Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.). | 205_{16} | 35.15625 |
32,105 | The sum of all of the digits of the integers from 1 to 2008 is to be calculated. | 28054 | 6.25 |
32,106 | Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$ . Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$ . | 2/3 | 1.5625 |
32,107 | A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. Calculate the measure of angle $ADB$. | 90 | 0 |
32,108 | A triangle has sides of lengths 40 units, 50 units, and 70 units. An altitude is dropped from the vertex opposite the side of length 70 units. Calculate the length of this altitude.
A) $\frac{40\sqrt{7}}{7}$ units
B) $\frac{80\sqrt{7}}{7}$ units
C) $\frac{120\sqrt{7}}{7}$ units
D) $\frac{160\sqrt{7}}{7}$ units | \frac{80\sqrt{7}}{7} | 29.6875 |
32,109 | The numbers \(a, b, c, d\) belong to the interval \([-12.5, 12.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 650 | 7.03125 |
32,110 | Find the area of the region defined by the inequality: \( |y - |x - 2| + |x|| \leq 4 \). | 32 | 38.28125 |
32,111 | In $\triangle PQR$, points $M$ and $N$ lie on $\overline{PQ}$ and $\overline{PR}$, respectively. If $\overline{PM}$ and $\overline{QN}$ intersect at point $S$ so that $PS/SM = 4$ and $QS/SN = 3$, what is $RN/NQ$? | \frac{4}{3} | 5.46875 |
32,112 | Given a set of data arranged in ascending order, which are -1, 0, 4, x, 7, 14, and the median is 5, find the variance of this set of data. | \frac{74}{3} | 89.0625 |
32,113 | Given that $\alpha \in (-\frac{\pi }{2},\frac{\pi }{2})$, $\beta \in (-\frac{\pi }{2},\frac{\pi }{2})$, and $\tan \alpha$ and $\tan \beta$ are the two real roots of the equation $x^{2}+3\sqrt{3}x+4=0$, find the value of $\alpha + \beta$ = \_\_\_\_\_\_\_\_\_\_\_\_. | - \frac{2\pi}{3} | 16.40625 |
32,114 | Given $a, b, c > 0$ and $(a+b)bc = 5$, find the minimum value of $2a+b+c$. | 2\sqrt{5} | 3.125 |
32,115 | The bank plans to invest 40% of a certain fund in project M for one year, and the remaining 60% in project N. It is estimated that project M can achieve an annual profit of 19% to 24%, while project N can achieve an annual profit of 29% to 34%. By the end of the year, the bank must recover the funds and pay a certain rebate rate to depositors. To ensure that the bank's annual profit is no less than 10% and no more than 15% of the total investment in M and N, what is the minimum rebate rate that should be given to the depositors? | 10 | 8.59375 |
32,116 | Square $EFGH$ has sides of length 4. Segments $EK$ and $EL$ divide the square's area into two equal parts. Calculate the length of segment $EK$. | 4\sqrt{2} | 36.71875 |
32,117 | In $\triangle ABC$, it is given that $BD:DC = 3:2$ and $AE:EC = 3:4$. Point $M$ is the intersection of $AD$ and $BE$. If the area of $\triangle ABC$ is 1, what is the area of $\triangle BMD$? | $\frac{4}{15}$ | 0 |
32,118 | Given $\triangle ABC$ with its three interior angles $A$, $B$, and $C$, and $2\sin^{2}(B+C)= \sqrt{3}\sin 2A$.
(Ⅰ) Find the degree of $A$;
(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\triangle ABC$. | 10\sqrt{3} | 76.5625 |
32,119 | Given that the function $y = (m^2 + 2m - 2)x^{\frac{1}{m-1}}$ is a power function, find the value of $m$. | -3 | 2.34375 |
32,120 | A survey of $150$ teachers determined the following:
- $90$ had high blood pressure
- $60$ had heart trouble
- $50$ had diabetes
- $30$ had both high blood pressure and heart trouble
- $20$ had both high blood pressure and diabetes
- $10$ had both heart trouble and diabetes
- $5$ had all three conditions
What percent of the teachers surveyed had none of the conditions? | 3.33\% | 74.21875 |
32,121 | Let $x, y, z$ be real numbers such that $x + y + z = 2$, and $x \ge -\frac{2}{3}$, $y \ge -1$, and $z \ge -2$. Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 4} + \sqrt{3z + 7}.\] | \sqrt{57} | 0 |
32,122 | In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ with an inclination angle $\alpha = 60^{\circ}$ is $\left\{\begin{array}{l}{x=2+t\cos\alpha}\\{y=t\sin\alpha}\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole, and the non-negative half-axis of the $x$-axis as the polar axis. Establish a polar coordinate system with the same unit length as the rectangular coordinate system. The polar coordinate equation of the curve $C$ in the polar coordinate system is $\rho =\rho \cos ^{2}\theta +4\cos \theta$. <br/>$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$; <br/>$(2)$ Let point $P(2,0)$. The line $l$ intersects the curve $C$ at points $A$ and $B$, and the midpoint of chord $AB$ is $D$. Find the value of $\frac{|PD|}{|PA|}+\frac{|PD|}{|PB|}$. | \frac{\sqrt{7}}{3} | 9.375 |
32,123 | When \(0 < x < \frac{\pi}{2}\), the value of the function \(y = \tan 3x \cdot \cot^3 x\) cannot take numbers within the open interval \((a, b)\). Find the value of \(a + b\). | 34 | 4.6875 |
32,124 | $\triangle PQR$ is similar to $\triangle XYZ$. What is the number of centimeters in the length of $\overline{YZ}$? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(10,-2)--(8,6)--cycle);
label("10cm",(6,3),NW);
label("7cm",(10.2,2.5),NE);
draw((15,0)--(23,-1.8)--(22,4.5)--cycle);
label("$P$",(10,-2),E);
label("4cm",(21.2,1.3),NE);
label("$Q$",(8,6),N);
label("$R$",(0,0),SW);
label("$X$",(23,-1.8),E);
label("$Y$",(22,4.5),NW);
label("$Z$",(15,0),SW);
[/asy] | 5.7 | 3.90625 |
32,125 | In the drawing, there is a grid composed of 25 small equilateral triangles.
How many rhombuses can be formed from two adjacent small triangles? | 30 | 10.15625 |
32,126 | Let $a_1 = 1$ and $a_{n+1} = a_n \cdot p_n$ for $n \geq 1$ where $p_n$ is the $n$ th prime number, starting with $p_1 = 2$ . Let $\tau(x)$ be equal to the number of divisors of $x$ . Find the remainder when $$ \sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d) $$ is divided by 91 for positive integers $d$ . Recall that $d|a_n$ denotes that $d$ divides $a_n$ .
*Proposed by Minseok Eli Park (wolfpack)* | 40 | 3.90625 |
32,127 | Three equilateral triangles $ABC$, $BCD$, and $CDE$ are positioned such that $B$, $C$, and $D$ are collinear, and $C$ is the midpoint of $BD$. Triangle $CDE$ is positioned such that $E$ is on the same side of line $BD$ as $A$. What is the value of $AE \div BC$ when expressed in simplest radical form?
[asy]
draw((0,0)--(5,8.7)--(10,0)--cycle);
draw((10,0)--(12.5,4.35)--(15,0)--cycle);
label("$A$",(0,0),SW);
label("$B$",(5,8.7),N);
label("$C$",(10,0),S);
label("$D$",(15,0),SE);
label("$E$",(12.5,4.35),N);
[/asy] | \sqrt{3} | 15.625 |
32,128 | Let $x$ be chosen randomly from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}5x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.
A) $\frac{1}{9}$
B) $\frac{1}{10}$
C) $\frac{1}{8}$
D) $\frac{1}{7}$
E) $\frac{1}{6}$ | \frac{1}{9} | 15.625 |
32,129 | A straight line $l$ passes through a vertex and a focus of an ellipse. If the distance from the center of the ellipse to $l$ is one quarter of its minor axis length, calculate the eccentricity of the ellipse. | \dfrac{1}{2} | 3.90625 |
32,130 | A school plans to purchase two brands of soccer balls, brand A and brand B. It is known that the unit price of brand A soccer balls is $30 less than the unit price of brand B soccer balls. The quantity of brand A soccer balls that can be purchased with $1000 is the same as the quantity of brand B soccer balls that can be purchased with $1600.<br/>$(1)$ Find the unit prices of brand A and brand B soccer balls.<br/>$(2)$ The school plans to purchase a total of 80 soccer balls of both brand A and brand B. Let $a$ be the number of brand A soccer balls to be purchased, where the quantity of brand A soccer balls is at least 30 but not more than 3 times the quantity of brand B soccer balls. Determine how many brand A soccer balls should be purchased to minimize the total cost $W$. | 60 | 62.5 |
32,131 | Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students. | \frac{1}{4} | 4.6875 |
32,132 | Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is along side $EG$ of the triangle, and side $AB$ is now one-third the length of side $AD$. The altitude from $F$ to side $EG$ is 12 inches, and side $EG$ is 15 inches. Determine the area of rectangle $ABCD$. | \frac{10800}{289} | 31.25 |
32,133 | Square \(ABCD\) has sides of length 14. A circle is drawn through \(A\) and \(D\) so that it is tangent to \(BC\). What is the radius of the circle? | 8.75 | 0 |
32,134 | In the third season of "China Poetry Conference", there were many highlights under the theme of "Life has its own poetry". In each of the ten competitions, there was a specially designed opening poem recited in unison by a hundred people under the coordination of lights and dances. The poems included "Changsha Spring in the Jingyuan Garden", "The Road to Shu is Difficult", "The Song of Xi'le", "The Ballad of the Wanderer", "Moon Over the Guan Mountains", and "Six Plates Mountain in the Qingping Melody". The first six competitions were arranged in a certain order, with "The Road to Shu is Difficult" before "The Ballad of the Wanderer", and neither "Changsha Spring in the Jingyuan Garden" nor "Six Plates Mountain in the Qingping Melody" were in the last competition or adjacent to each other. The number of arrangements is $\_\_\_\_\_\_$. (Answer with a number.) | 144 | 10.15625 |
32,135 | In triangle \(XYZ,\) \(XY = 5,\) \(XZ = 7,\) \(YZ = 9,\) and \(W\) lies on \(\overline{YZ}\) such that \(\overline{XW}\) bisects \(\angle YXZ.\) Find \(\cos \angle YXW.\) | \frac{3\sqrt{5}}{10} | 63.28125 |
32,136 | The perpendicular bisectors of the sides of triangle $DEF$ meet its circumcircle at points $D'$, $E'$, and $F'$, respectively. If the perimeter of triangle $DEF$ is 42 and the radius of the circumcircle is 10, find the area of hexagon $DE'F'D'E'F$. | 105 | 0.78125 |
32,137 | The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs? | 75 | 43.75 |
32,138 | Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack? | \frac{17}{442} | 0.78125 |
32,139 | Point $(x, y)$ is randomly picked from the rectangular region with vertices at $(0, 0), (3030, 0), (3030, 3031), and (0, 3031)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | \frac{505}{3031} | 46.09375 |
32,140 | If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$. | -\frac{9}{2} | 10.9375 |
32,141 | Given that the magnitude of vector $\overrightarrow {a}$ is 1, the magnitude of vector $\overrightarrow {b}$ is 2, and the magnitude of $\overrightarrow {a}+ \overrightarrow {b}$ is $\sqrt {7}$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac {\pi}{3} | 99.21875 |
32,142 | Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A. | \frac{\sqrt{6}}{3} | 21.875 |
32,143 | If the line $(m+2)x+3y+3=0$ is parallel to the line $x+(2m-1)y+m=0$, then the real number $m=$ \_\_\_\_\_\_. | -\frac{5}{2} | 28.125 |
32,144 | Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$ . Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all
numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.
*Proposed by **FedeX333X*** | 135 | 18.75 |
32,145 | In a certain high school physical examination for seniors, the heights (in centimeters) of 12 students are $173$, $174$, $166$, $172$, $170$, $165$, $165$, $168$, $164$, $173$, $175$, $178$. Find the upper quartile of this data set. | 173.5 | 29.6875 |
32,146 | Given a region bounded by a larger quarter-circle with a radius of $5$ units, centered at the origin $(0,0)$ in the first quadrant, a smaller circle with radius $2$ units, centered at $(0,4)$ that lies entirely in the first quadrant, and the line segment from $(0,0)$ to $(5,0)$, calculate the area of the region. | \frac{9\pi}{4} | 35.9375 |
32,147 | Calculate the sum of the series $1-2-3+4+5-6-7+8+9-10-11+\cdots+1998+1999-2000-2001$. | 2001 | 2.34375 |
32,148 | Neznaika is drawing closed paths inside a $5 \times 8$ rectangle, traveling along the diagonals of $1 \times 2$ rectangles. In the illustration, an example of a path passing through 12 such diagonals is shown. Help Neznaika draw the longest possible path. | 20 | 3.90625 |
32,149 | Points $P$ and $Q$ are on a circle with radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of line segment $PR$. | \sqrt{32} | 0 |
32,150 | Let $T$ be the set of ordered triples $(x,y,z)$ of real numbers where
\[\log_{10}(2x+2y) = z \text{ and } \log_{10}(x^{2}+2y^{2}) = z+2.\]
Find constants $c$ and $d$ such that for all $(x,y,z) \in T$, the expression $x^{3} + y^{3}$ equals $c \cdot 10^{3z} + d \cdot 10^{z}.$ What is the value of $c+d$?
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{5}{16}$
D) $\frac{1}{4}$
E) $\frac{1}{2}$ | \frac{5}{16} | 3.125 |
32,151 | Given the sequence $\{a\_n\}$ satisfies $a\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, then the minimum value of $\{\frac{S_{n}+4}{n}\}$ is ___. | \frac{9}{2} | 2.34375 |
32,152 | There is a group of monkeys transporting peaches from location $A$ to location $B$. Every 3 minutes a monkey departs from $A$ towards $B$, and it takes 12 minutes for a monkey to complete the journey. A rabbit runs from $B$ to $A$. When the rabbit starts, a monkey has just arrived at $B$. On the way, the rabbit encounters 5 monkeys walking towards $B$, and continues to $A$ just as another monkey leaves $A$. If the rabbit's running speed is 3 km/h, find the distance between locations $A$ and $B$. | 300 | 0 |
32,153 | Given $\left(a + \frac{1}{a}\right)^2 = 5$, find the value of $a^3 + \frac{1}{a^3}$. | 2\sqrt{5} | 30.46875 |
32,154 | Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?
[asy]
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
[/asy] | 540 | 17.1875 |
32,155 | Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. What time does Leo think it is?
A) 11:30
B) 11:45
C) 12:00
D) 12:30
E) 12:45 | 12:30 | 26.5625 |
32,156 | In the tetrahedron $P-ABC$, $\Delta ABC$ is an equilateral triangle, and $PA=PB=PC=3$, $PA \perp PB$. The volume of the circumscribed sphere of the tetrahedron $P-ABC$ is __________. | \frac{27\sqrt{3}\pi}{2} | 1.5625 |
32,157 | In rectangle $ABCD$, $\overline{CE}$ bisects angle $C$ (no trisection this time), where $E$ is on $\overline{AB}$, $F$ is still on $\overline{AD}$, but now $BE=10$, and $AF=5$. Find the area of $ABCD$. | 200 | 7.8125 |
32,158 | Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, $a_1=15$, and it satisfies $\frac{a_{n+1}}{2n-3} = \frac{a_n}{2n-5}+1$, knowing $n$, $m\in\mathbb{N}$, and $n > m$, find the minimum value of $S_n - S_m$. | -14 | 0 |
32,159 | In a bag, there are $4$ red balls, $m$ yellow balls, and $n$ green balls. Now, two balls are randomly selected from the bag. Let $\xi$ be the number of red balls selected. If the probability of selecting two red balls is $\frac{1}{6}$ and the probability of selecting one red and one yellow ball is $\frac{1}{3}$, then $m-n=$____, $E\left(\xi \right)=$____. | \frac{8}{9} | 56.25 |
32,160 | In $\triangle ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$, and $A$, $B$, $C$ form an arithmetic sequence.
(I) If $b=7$ and $a+c=13$, find the area of $\triangle ABC$.
(II) Find the maximum value of $\sqrt{3}\sin A + \sin(C - \frac{\pi}{6})$ and the size of angle $A$ when the maximum value is reached. | \frac{\pi}{3} | 74.21875 |
32,161 | Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction. | \frac{2}{3} | 10.15625 |
32,162 | A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receiving the task, the club members immediately started producing helium gas at a rate of $xL$ per day. It is known that the cost of raw materials for producing $1L$ of helium gas is $100$ yuan. If the daily production of helium gas is less than $4L$, the additional cost per day is $W_1=4x^2+16$ yuan. If the daily production of helium gas is greater than or equal to $4L$, the additional cost per day is $W_2=17x+\frac{9}{x}-3$ yuan. The production cost consists of raw material cost and additional cost.
$(1)$ Write the relationship between the total cost $W$ (in yuan) and the daily production $x$ (in $L$).
$(2)$ When the club produces how many liters of helium gas per day, the total cost is minimized? What is the minimum cost? | 4640 | 15.625 |
32,163 | Given 6 persons, with the restriction that person A and person B cannot visit Paris, calculate the total number of distinct selection plans for selecting 4 persons to visit Paris, London, Sydney, and Moscow, where each person visits only one city. | 240 | 18.75 |
32,164 | Given $α-β=\frac{π}{3}$ and $tanα-tanβ=3\sqrt{3}$, calculate the value of $\cos \left(\alpha +\beta \right)$. | -\frac{1}{6} | 14.84375 |
32,165 | A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percentage of the original list price should the marked price be? | 107\% | 4.6875 |
32,166 | In a village, a plot of land is shaped as a right triangle, where one of the legs measures 5 units and another measures 12 units. Farmer Euclid decides to leave a small unplanted square at the vertex of this right angle, and the shortest distance from this unplanted square to the hypotenuse is 3 units. Determine the fraction of the plot that is planted.
A) $\frac{412}{1000}$
B) $\frac{500}{1000}$
C) $\frac{290}{1000}$
D) $\frac{145}{1000}$
E) $\frac{873}{1000}$ | \frac{412}{1000} | 10.15625 |
32,167 | In Pascal's Triangle, we know each number is the combination of two numbers just above it. What is the sum of the middle three numbers in each of Rows 5, 6, and 7? | 157 | 1.5625 |
32,168 | In the $4 \times 5$ grid shown, six of the $1 \times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \times 10$ grid are drawn, how many of the $1 \times 1$ squares are not intersected by either diagonal? | 48 | 7.8125 |
32,169 | Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer. | 28 | 98.4375 |
32,170 | Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$ .
Find the greatest value of $M = abc$ | 6171 | 92.96875 |
32,171 | Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.
Determine the maximum possible number of positive integers among the 2018 integers. | 1009 | 25 |
32,172 | A relatively prime date is defined as a date where the day and the month number are coprime. Determine how many relatively prime dates are in the month with 31 days and the highest number of non-relatively prime dates? | 11 | 3.90625 |
32,173 | If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$? | 18 | 45.3125 |
32,174 | The inclination angle of the line $\sqrt{3}x+y-1=0$ is ____. | \frac{2\pi}{3} | 57.8125 |
32,175 | Two adjacent faces of a tetrahedron are equilateral triangles with a side length of 1 and form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that contains the given edge. | \frac{\sqrt{3}}{4} | 5.46875 |
32,176 | Given that the function $y = f(x)$ is an even function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = \log_2(x+2) - 3$. Find the values of $f(6)$ and $f(f(0))$. | -1 | 86.71875 |
32,177 | Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. | \frac{17\pi}{36} | 4.6875 |
32,178 | Simplify the expression: $({1-\frac{1}{{x+3}}})÷\frac{{{x^2}-9}}{{{x^2}+6x+9}}$, then choose a suitable number from $-3$, $2$, $3$ to substitute and evaluate. | -4 | 88.28125 |
32,179 | Let $P(x) = 3\sqrt[3]{x}$, and $Q(x) = x^3$. Determine $P(Q(P(Q(P(Q(4))))))$. | 108 | 78.125 |
32,180 | Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \). Additionally, find its height dropped from vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Vertices:
- \( A_{1}(-1, 2, 4) \)
- \( A_{2}(-1, -2, -4) \)
- \( A_{3}(3, 0, -1) \)
- \( A_{4}(7, -3, 1) \) | 24 | 0 |
32,181 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 14.84375 |
32,182 | A three-digit number has distinct digits. By arbitrarily swapping the positions of its digits, five other three-digit numbers can be obtained. If the sum of these six three-digit numbers equals 2220, then among all the numbers that meet this condition, the smallest three-digit number is ____. | 127 | 21.09375 |
32,183 | How many natural numbers between 200 and 400 are divisible by 8? | 25 | 56.25 |
32,184 | Given the function $f$ mapping from set $M$ to set $N$, where $M=\{a, b, c\}$ and $N=\{-3, -2, -1, 0, 1, 2, 3\}$, how many mappings $f$ satisfy the condition $f(a) + f(b) + f(c) = 0$? | 37 | 100 |
32,185 | **p22.** Consider the series $\{A_n\}^{\infty}_{n=0}$ , where $A_0 = 1$ and for every $n > 0$ , $$ A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]}, $$ where $[x]$ denotes the largest integer value smaller than or equal to $x$ . Find the $(2023^{3^2}+20)$ -th element of the series.**p23.** The side lengths of triangle $\vartriangle ABC$ are $5$ , $7$ and $8$ . Construct equilateral triangles $\vartriangle A_1BC$ , $\vartriangle B_1CA$ , and $\vartriangle C_1AB$ such that $A_1$ , $B_1$ , $C_1$ lie outside of $\vartriangle ABC$ . Let $A_2$ , $B_2$ , and $C_2$ be the centers of $\vartriangle A_1BC$ , $\vartriangle B_1CA$ , and $\vartriangle C_1AB$ , respectively. What is the area of $\vartriangle A_2B_2C_2$ ?**p24.**There are $20$ people participating in a random tag game around an $20$ -gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the $20$ -gon (no matter where they were at the beginning). If there are currently $10$ taggers, let $E$ be the expected number of untagged people at the end of the next round. If $E$ can be written as $\frac{a}{b}$ for $a, b$ relatively prime positive integers, compute $a + b$ .
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | 653 | 0 |
32,186 | Given the parametric equation of curve $C\_1$ is $\begin{cases} x=3\cos \alpha \ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C\_2$ is $\rho \cos \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$.
(I) Find the rectangular coordinate equation of curve $C\_2$ and the maximum value of the distance $|OP|$ between the moving point $P$ on curve $C\_1$ and the coordinate origin $O$;
(II) If curve $C\_2$ intersects with curve $C\_1$ at points $A$ and $B$, and intersects with the $x$-axis at point $E$, find the value of $|EA|+|EB|$. | \frac{6 \sqrt{3}}{5} | 0.78125 |
32,187 | What fraction of the area of an isosceles trapezoid $KLMN (KL \parallel MN)$ is the area of triangle $ABC$, where $A$ is the midpoint of base $KL$, $B$ is the midpoint of base $MN$, and $C$ is the midpoint of leg $KN$? | \frac{1}{4} | 85.15625 |
32,188 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\sin C+\cos C=1-\sin \frac{C}{2}$.
$(1)$ Find the value of $\sin C$.
$(2)$ If $a^{2}+b^{2}=4(a+b)-8$, find the value of side $c$. | 1+ \sqrt{7} | 0 |
32,189 | On the side \( BC \) of triangle \( ABC \), points \( A_1 \) and \( A_2 \) are marked such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On the side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) lie on the same line. Find \( B_1B_2 \). | 12 | 7.8125 |
32,190 | Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money. | 12.5\% | 21.875 |
32,191 | The maximum value of the function \( y = \tan x - \frac{2}{|\cos x|} \) is to be determined. | -\sqrt{3} | 17.1875 |
32,192 | Given that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation when one piece of each product is sold. | 5.92 | 0 |
32,193 | Given that the sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$ by $255$.
$(1)$ Find all the rational terms of $x$ in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$;
$(2)$ If $(2x-1)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots+a_{n}x^{n}$, find the value of $(a_{0}+a_{2}+a_{4})^{2}-(a_{1}+a_{3})^{2}$. | 81 | 80.46875 |
32,194 | Using the digits $1, 2, 3, 4$, 24 unique four-digit numbers can be formed without repeating any digit. If these 24 four-digit numbers are arranged in ascending order, find the sum of the two middle numbers. | 4844 | 0.78125 |
32,195 |
In triangle \( \triangle ABC \), \( AB = BC = 2 \) and \( AC = 3 \). Let \( O \) be the incenter of \( \triangle ABC \). If \( \overrightarrow{AO} = p \overrightarrow{AB} + q \overrightarrow{AC} \), find the value of \( \frac{p}{q} \). | 2/3 | 8.59375 |
32,196 | In an isosceles triangle $\triangle ABC$ with vertex angle $A = \frac{2\pi}{3}$ and base $BC = 2\sqrt{3}$, find the dot product $\vec{BA} \cdot \vec{AC}$. | -2 | 27.34375 |
32,197 | Let $ABCD$ be an isosceles trapezoid with $AB$ and $CD$ as parallel bases, and $AB > CD$. A point $P$ inside the trapezoid connects to the vertices $A$, $B$, $C$, $D$, creating four triangles. The areas of these triangles, starting from the triangle with base $\overline{CD}$ moving clockwise, are $3$, $4$, $6$, and $7$. Determine the ratio $\frac{AB}{CD}$.
- **A**: $2$
- **B**: $\frac{5}{2}$
- **C**: $\frac{7}{3}$
- **D**: $3$
- **E**: $\sqrt{5}$ | \frac{7}{3} | 31.25 |
32,198 | Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$. | -7 | 9.375 |
32,199 | From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. | \frac{6}{35} | 32.03125 |
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