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23,916 | In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), two identical pieces are placed, with at most one piece per cell. There are ___ distinct ways to arrange the pieces. (If two arrangements can be made to coincide by rotation, they are considered the same arrangement). | 10 |
23,925 | Find the smallest number composed exclusively of ones that is divisible by 333...33 (where there are 100 threes in the number). | 300 |
23,936 | Given a moving large circle $\odot O$ tangent externally to a fixed small circle $\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\odot O_{1}$ at point $C$ and intersects $\odot O$ at points $D$ and $E$. Find $C D \cdot C E = \quad$. | 36 |
23,937 | Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3}+y^{2}=1 \). A line \( l \) with a positive slope passes through point \( F \) and intersects \( E \) at points \( A \) and \( B \). Through points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn such that \( AM \perp l \) and \( BN \perp l \), each intersecting the x-axis at points \( M \) and \( N \), respectively. Find the minimum value of \( |MN| \). | \sqrt{6} |
23,942 | Someone wrote down two numbers $5^{2020}$ and $2^{2020}$ consecutively. How many digits will the resulting number contain? | 2021 |
23,954 | In trapezoid \(ABCD\), \(AB\) is parallel to \(DC\) and \(\angle DAF = 90^\circ\). Point \(E\) is on \(DC\) such that \(EB = BC = CE\). Point \(F\) is on \(AB\) such that \(DF\) is parallel to \(EB\). In degrees, what is the measure of \(\angle FDA\)? | 30 |
23,956 | Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves
$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$ | \pi + \dfrac{3\sqrt{3}}{8} |
23,972 | The target below is made up of concentric circles with diameters $4$ , $8$ , $12$ , $16$ , and $20$ . The area of the dark region is $n\pi$ . Find $n$ .
[asy]
size(150);
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int i;
for(i=5;i>=1;i=i-1)
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if (floor(i/2)==i/2)
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filldraw(circle(origin,4*i),white);
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else
{
filldraw(circle(origin,4*i),red);
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[/asy] | 60 |
23,978 | David and Evan each repeatedly flip a fair coin. David will stop when he flips a tail, and Evan will stop once he flips 2 consecutive tails. Find the probability that David flips more total heads than Evan. | \dfrac{1}{5} |
23,989 | The diameter \( AB \) and the chord \( CD \) intersect at point \( M \). Given that \( \angle CMB = 73^\circ \) and the angular measure of arc \( BC \) is \( 110^\circ \). Find the measure of arc \( BD \). | 144 |
23,994 |
In a box, there are 3 red, 4 gold, and 5 silver stars. Stars are randomly drawn one by one from the box and placed on a Christmas tree. What is the probability that a red star is placed on the top of the tree, no more red stars are on the tree, and there are exactly 3 gold stars on the tree, if a total of 6 stars are drawn from the box? | \dfrac{5}{231} |
23,995 | Integer \( n \) such that the polynomial \( f(x) = 3x^3 - nx - n - 2 \) can be factored into a product of two non-constant polynomials with integer coefficients. Find the sum of all possible values of \( n \). | 192 |
23,998 | In $\triangle ABC$, $AB=\sqrt{5}$, $BC=1$, and $AC=2$. $I$ is the incenter of $\triangle ABC$ and the circumcircle of $\triangle IBC$ intersects $AB$ at $P$. Find $BP$. | \sqrt{5} - 2 |
24,005 | A rectangular table of dimensions \( x \) cm \(\times 80\) cm is covered with identical sheets of paper of size \( 5 \) cm \(\times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters? | 77 |
24,014 | In an isosceles trapezoid with bases \(a = 21\), \(b = 9\) and height \(h = 8\), find the radius of the circumscribed circle. | \dfrac{85}{8} |
24,021 | Given that \(\log_8 2 = 0.2525\) in base 8 (to 4 decimal places), find \(\log_8 4\) in base 8 (to 4 decimal places). | 0.5050 |
24,032 | It is known that the complex number \( z \) satisfies \( |z| = 1 \). Find the maximum value of \( u = \left| z^3 - 3z + 2 \right| \). | 3\sqrt{3} |
24,051 | In a square $\mathrm{ABCD}$, point $\mathrm{E}$ is on $\mathrm{BC}$ with $\mathrm{BE} = 2$ and $\mathrm{CE} = 1$. Point $\mathrm{P}$ moves along $\mathrm{BD}$. What is the minimum value of $\mathrm{PE} + \mathrm{PC}$? | \sqrt{13} |
24,058 | At what angle to the x-axis is the tangent to the graph of the function \( g(x) = x^2 \ln x \) inclined at the point \( x_0 = 1 \)? | \dfrac{\pi}{4} |
24,066 | In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as its diameter, intersects side \( BC \) at point \( K \). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \( BOC \). | 21 |
24,071 | Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$ .
*Proposed by David Tang* | 10 |
24,077 | A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM? | \dfrac{5}{16} |
24,079 | How many positive perfect cubes are divisors of the product \(1! \cdot 2! \cdot 3! \cdots 10!\)? | 468 |
24,080 | On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red points? | 13 |
24,087 | The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years. | \dfrac{4}{5} |
24,107 | The car engine operates with a power of \( P = 60 \text{ kW} \). Determine the car's speed \( v_0 \) if, after turning off the engine, it stops after traveling a distance of \( s = 450 \text{ m} \). The force resisting the car's motion is proportional to its speed. The mass of the car is \( m = 1000 \text{ kg} \). | 30 |
24,110 | A geometric sequence $(a_n)$ has $a_1=\sin x$ $a_2=\cos x$ , and $a_3= \tan x$ for some real number $x$ . For what value of $n$ does $a_n=1+\cos x$ | 8 |
24,139 | In the trapezoid \( ABCD \), angles \( A \) and \( D \) are right angles, \( AB = 1 \), \( CD = 4 \), \( AD = 5 \). Point \( M \) is taken on side \( AD \) such that \(\angle CMD = 2 \angle BMA \).
In what ratio does point \( M \) divide side \( AD \)? | \dfrac{2}{3} |
24,142 | In triangle \(ABC\), the bisector \(BD\) is drawn, and in triangles \(ABD\) and \(CBD\), the bisectors \(DE\) and \(DF\) are drawn, respectively. It turns out that \(EF \parallel AC\). Find the angle \(\angle DEF\). | 45^\circ |
24,153 | Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$ . Find $p+q$ .
*2022 CCA Math Bonanza Individual Round #14* | 251 |
24,154 | Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions. | 185 |
24,170 | In the plane rectangular coordinate system $xOy$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following requirement: Through the point $(a, 0)$, draw any two mutually perpendicular lines $l_{1}$ and $l_{2}$. If $l_{1}$ intersects the hyperbola $C$ at points $P$ and $Q$, and $l_{2}$ intersects $C$ at points $R$ and $S$, then $|PQ| = |RS|$ always holds. | \sqrt{2} |
24,171 | In triangle $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$ | 463 |
24,193 | A weightless pulley has a rope with masses of 3 kg and 6 kg. Neglecting friction, find the force exerted by the pulley on its axis. Consider the acceleration due to gravity to be $10 \, \mathrm{m/s}^2$. Give the answer in newtons, rounding it to the nearest whole number if necessary. | 80 |
24,218 | A and B each independently toss a fair coin. A tosses the coin 10 times, and B tosses the coin 11 times. What is the probability that the number of heads B gets is greater than the number of heads A gets? | \dfrac{1}{2} |
24,222 | What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$ $0\le v\le1,$ and $0\le w\le1$ | 16 |
24,232 | 30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem? | 26 |
24,244 | In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$. | 504 |
24,264 | A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length of the lateral edge of the pyramid. | \dfrac{\sqrt{6}}{2} |
24,276 | The minimum value of the function \( y = |\cos x| + |\cos 2x| \) (for \( x \in \mathbf{R} \)) is ______. | \dfrac{\sqrt{2}}{2} |
24,283 |
Given a triangle \(ABC\) where \(AB = AC\) and \(\angle A = 80^\circ\). Inside triangle \(ABC\) is a point \(M\) such that \(\angle MBC = 30^\circ\) and \(\angle MCB = 10^\circ\). Find \(\angle AMC\). | 70^\circ |
24,304 | We defined an operation denoted by $*$ on the integers, which satisfies the following conditions:
1) $x * 0 = x$ for every integer $x$;
2) $0 * y = -y$ for every integer $y$;
3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.
Determine the result of the operation $19 * 90$. | 1639 |
24,310 | Find the degree measure of the angle
$$
\delta=\arccos \left(\left(\sin 2541^{\circ}+\sin 2542^{\circ}+\cdots+\sin 6141^{\circ}\right)^{\cos 2520^{\circ}}+\cos 2521^{\circ}+\cdots+\cos 6120^{\circ}\right)
$$ | 69 |
24,312 | Kolya, after walking one-fourth of the way from home to school, realized that he forgot his problem book. If he does not go back for it, he will arrive at school 5 minutes before the bell rings, but if he goes back, he will be 1 minute late. How long (in minutes) does it take to get to school? | 12 |
24,314 | In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take? | 234 |
24,323 | Given a right triangle \(ABC\) with a right angle at \(A\). On the leg \(AC\), a point \(D\) is marked such that \(AD:DC = 1:3\). Circles \(\Gamma_1\) and \(\Gamma_2\) are then drawn with centers at \(A\) and \(C\) respectively, both passing through point \(D\). \(\Gamma_2\) intersects the hypotenuse at point \(E\). Another circle \(\Gamma_3\) with center at \(B\) and radius \(BE\) intersects \(\Gamma_1\) inside the triangle at a point \(F\) such that \(\angle AFB\) is a right angle. Find \(BC\), given that \(AB = 5\). | 13 |
24,332 | $ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| 30 |
24,342 | In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $r.$ | \dfrac{7}{9} |
24,350 |
A man and his faithful dog simultaneously started moving along the perimeter of a block from point \( A \) at time \( t_{0} = 0 \) minutes. The man moved at a constant speed clockwise, while the dog ran at a constant speed counterclockwise. It is known that the first time they met was \( t_{1} = 2 \) minutes after starting, and this meeting occurred at point \( B \). Given that they continued moving after this, each in their original direction and at their original speed, determine the next time they will both be at point \( B \) simultaneously. Note that \( A B = C D = 100 \) meters and \( B C = A D = 200 \) meters. | 14 |
24,395 | A regular octahedron has a side length of 1. What is the distance between two opposite faces? | \dfrac{\sqrt{6}}{3} |
24,415 | Given that \(a_{1}, a_{2}, \cdots, a_{n}\) are \(n\) people corresponding to \(A_{1}, A_{2}, \cdots, A_{n}\) cards (\(n \geq 2\), \(a_{i}\) corresponds to \(A_{i}\)). Now \(a_{1}\) picks a card from the deck randomly, and then each person in sequence picks a card. If their corresponding card is still in the deck, they take it, otherwise they pick a card randomly. What is the probability that \(a_{n}\) gets \(A_{n}\)? | \dfrac{1}{2} |
24,456 | The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid. | 96 |
24,458 | In the sum shown, each of the letters \( D, O, G, C, A \), and \( T \) represents a different digit.
$$
\begin{array}{r}
D O G \\
+C A T \\
\hline 1000
\end{array}
$$
What is the value of \( D + O + G + C + A + T \)? | 28 |
24,471 | Given the function \( f(x) \) defined on \(\mathbf{R} \) (the set of real numbers), it satisfies the following conditions:
1. \( f(1) = 1 \)
2. When \( 0 < x < 1 \), \( f(x) > 0 \)
3. For any real numbers \( x \) and \( y \), the equation \( f(x+y) - f(x-y) = 2 f(1-x) f(y) \) holds
Find \( f\left(\frac{1}{3}\right) \). | \dfrac{1}{2} |
24,473 | Two right triangles \( \triangle AXY \) and \( \triangle BXY \) have a common hypotenuse \( XY \) and side lengths (in units) \( AX=5 \), \( AY=10 \), and \( BY=2 \). Sides \( AY \) and \( BX \) intersect at \( P \). Determine the area (in square units) of \( \triangle PXY \). | \dfrac{25}{3} |
24,478 | Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$ , the decimal representation of the number $c^n+2014$ has digits all less than $5$ .
*Proposed by Evan Chen* | 10 |
24,479 | Let $P(n)$ represent the product of all non-zero digits of a positive integer $n$. For example: $P(123) = 1 \times 2 \times 3 = 6$ and $P(206) = 2 \times 6 = 12$. Find the value of $P(1) + P(2) + \cdots + P(999)$. | 97335 |
24,481 | In rectangle \(ABCD\), \(AB = 20 \, \text{cm}\) and \(BC = 10 \, \text{cm}\). Points \(M\) and \(N\) are taken on \(AC\) and \(AB\), respectively, such that the value of \(BM + MN\) is minimized. Find this minimum value. | 16 |
24,499 | To celebrate her birthday, Ana is going to prepare pear and apple pies. In the market, an apple weighs $300 \text{ g}$ and a pear weighs $200 \text{ g}$. Ana's bag can hold a maximum weight of $7 \text{ kg}$. What is the maximum number of fruits she can buy to make pies with both types of fruits? | 34 |
24,504 | Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | \dfrac{\pi}{4} |
24,505 | One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:
(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching either end.
(2) A bus departs from both Station $A$ and Station $B$ every 6 minutes. Determine the minimum number of buses required for this bus route. | 20 |
24,519 | Calculate the result of the expression \( 2015 \frac{1999}{2015} \times \frac{1}{4} - \frac{2011}{2015} \). | 503 |
24,521 | Let \( x, y \in \mathbf{R}^{+} \), and \(\frac{19}{x}+\frac{98}{y}=1\). Find the minimum value of \( x + y \). | 117 + 14\sqrt{38} |
24,532 | Let \( a, b, c, d \) be real numbers defined by
$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$
Calculate their product. | 11 |
24,545 | If the equation \( x^{2} - a|x| + a^{2} - 3 = 0 \) has a unique real solution, then \( a = \) ______. | -\sqrt{3} |
24,557 | In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$ , and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament? | 12 |
24,565 | A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid? | \dfrac{\sqrt{6} - \sqrt{2}}{4} |
24,575 | Given a triangular prism \( S-ABC \) with a base that is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = AB = 2 \). If the points \( S, A, B, C \) all lie on the surface of a sphere centered at \( O \), what is the surface area of this sphere? | \dfrac{16}{3}\pi |
24,586 | Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. | 400 |
24,587 | The set \( S \) is given by \( S = \{1, 2, 3, 4, 5, 6\} \). A non-empty subset \( T \) of \( S \) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \( T \)? | 27 |
24,591 | <u>Set 4</u> **p10.** Eve has nine letter tiles: three $C$ ’s, three $M$ ’s, and three $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$ 4 $ per package, and $ D $ batteries for $ \ $7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$ 2021 $, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell?**p12.** A deck of cards consists of $ 30 $ cards labeled with the integers $ 1 $ to $ 30 $, inclusive. The cards numbered $ 1 $ through $ 15 $ are purple, and the cards numbered $ 16 $ through $ 30 $ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $ 32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.
PS. You should use hide for answers. | \dfrac{1}{28} |
24,601 | Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively in an acute triangle \( \triangle ABC \), and it is given that \( f(A) = 1 \), \( b + c = 5 + 3 \sqrt{2} \), and \( a = \sqrt{13} \), find the area \( S \) of \( \triangle ABC \). | \dfrac{15}{2} |
24,604 | On a 10-ring target, the probabilities of hitting scores 10, 9, 8, 7, and 6 are $\frac{1}{5}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8},$ and $\frac{1}{10}$ respectively. The probability of hitting any other score (from 5 to 1) is $\frac{1}{12}$. $A$ pays $B$ the score amount in forints for any hit that is at least 6, and 1.7 forints for any other hit. How much should $B$ pay in case of a miss so that the bet is fair?
| 96 |
24,605 | In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)? | 15 |
24,608 | In the trapezium \(ABCD\), the lines \(AB\) and \(DC\) are parallel, \(BC = AD\), \(DC = 2 \times AD\), and \(AB = 3 \times AD\). The angle bisectors of \(\angle DAB\) and \(\angle CBA\) intersect at the point \(E\). What fraction of the area of the trapezium \(ABCD\) is the area of the triangle \(ABE\)? | \dfrac{3}{5} |
24,626 | Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\). | \dfrac{24}{5} |
24,635 | The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle. | 6\sqrt[4]{3} |
24,637 | On the base \(AC\) of an isosceles triangle \(ABC\), a point \(E\) is taken, and on the sides \(AB\) and \(BC\), points \(K\) and \(M\) are taken such that \(KE \parallel BC\) and \(EM \parallel AB\). What fraction of the area of triangle \(\mathrm{ABC}\) is occupied by the area of triangle \(KEM\) if \(BM:EM = 2:3\)? | \dfrac{6}{25} |
24,657 | Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths. | 0.19 |
24,672 | Seven dwarfs stood at the corners of their garden, each at one corner, and stretched a rope around the entire garden. Snow White started from Doc and walked along the rope. First, she walked four meters to the east where she met Prof. From there, she continued two meters north before reaching Grumpy. From Grumpy, she walked west and after two meters met Bashful. Continuing three meters north, she reached Happy. She then walked west and after four meters met Sneezy, from where she had three meters south to Sleepy. Finally, she followed the rope by the shortest path back to Doc, thus walking around the entire garden.
How many square meters is the entire garden?
Hint: Draw the shape of the garden, preferably on graph paper. | 22 |
24,681 | Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car? | \dfrac{5}{3} |
24,694 | Given that \( x, y, z \in \mathbf{R}_{+} \) and \( x^{2} + y^{2} + z^{2} = 1 \), find the value of \( z \) when \(\frac{(z+1)^{2}}{x y z} \) reaches its minimum. | \sqrt{2} - 1 |
24,739 | In a trapezoid $ABCD$ with $\angle A = \angle B = 90^{\circ}$, $|AB| = 5 \text{cm}$, $|BC| = 1 \text{cm}$, and $|AD| = 4 \text{cm}$, point $M$ is taken on side $AB$ such that $2 \angle BMC = \angle AMD$. Find the ratio $|AM| : |BM|$. | \dfrac{3}{2} |
24,768 | In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = 2 \) and \( BC = 5 \), then \( BX \) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \). | 8041 |
24,775 | On the sides \( BC \) and \( AC \) of triangle \( ABC \), points \( M \) and \( N \) are taken respectively such that \( CM:MB = 1:3 \) and \( AN:NC = 3:2 \). Segments \( AM \) and \( BN \) intersect at point \( K \). Find the area of quadrilateral \( CMKN \), given that the area of triangle \( ABC \) is 1. | \dfrac{3}{20} |
24,838 | A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy? | 90 |
24,843 | How many integer solutions does the inequality
$$
|x| + |y| < 1000
$$
have, where \( x \) and \( y \) are integers? | 1998001 |
24,855 | For a positive integer $n$ , let $v(n)$ denote the largest integer $j$ such that $n$ is divisible by $2^j$ . Let $a$ and $b$ be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that $v(a) > v(b)$ ? | \dfrac{341}{1024} |
24,859 | Find the smallest natural number $n$ such that $\sin n^{\circ} = \sin (2016n^{\circ})$. | 72 |
24,869 | The median \(AD\) of an acute-angled triangle \(ABC\) is 5. The orthogonal projections of this median onto the sides \(AB\) and \(AC\) are 4 and \(2\sqrt{5}\), respectively. Find the side \(BC\). | 2\sqrt{10} |
24,874 | In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth $1$ , $5$ , $25$ , and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
*Proposed by Aaron Lin* | 82 |
24,929 | \(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of the area of quadrilateral \(EFPH\) to the area of parallelogram \(ABCD\). | \dfrac{37}{72} |
24,932 | Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed? | 576 |
24,947 | Calculate: \( 4\left(\sin ^{3} \frac{49 \pi}{48} \cos \frac{49 \pi}{16} + \cos ^{3} \frac{49 \pi}{48} \sin \frac{49 \pi}{16}\right) \cos \frac{49 \pi}{12} \). | \dfrac{3}{4} |
24,953 | The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)? | 1200 |
24,967 | There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? | 512 |
24,971 | Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finished? | 120 |
24,993 | Evaluate the integral \(\int_{0}^{1} \ln x \ln (1-x) \, dx\). | 2 - \dfrac{\pi^2}{6} |
24,994 | Given the sequence \( S_{1} = 1, S_{2} = 1 - 2, S_{3} = 1 - 2 + 3, S_{4} = 1 - 2 + 3 - 4, S_{5} = 1 - 2 + 3 - 4 + 5, \cdots \), find the value of \( S_{1} + S_{2} + S_{3} + \cdots + S_{299} \). | 150 |
24,996 |
Let $x, y, z$ be positive numbers satisfying the following system of equations:
$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$
Find the value of the expression $xy + yz + xz$. | 12 |
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