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In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maintenance cost for one orphan is 200 florins per year? | 1896 | 1/8 |
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 73 | 4/8 |
Let set $P=\{0, 2, 4, 6, 8\}$, and set $Q=\{m | m=100a_1+10a_2+a_3, a_1, a_2, a_3 \in P\}$. Determine the 68th term of the increasing sequence of elements in set $Q$. | 464 | 6/8 |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $x^2 + \frac{y^2}{b^2} = 1 (0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the sequence $|AF_2|, |AB|, |BF_2|$ forms an arithmetic progression, then:
(1) Find $|AB|$;
(2) If the slope of line $l$ is $1$, find the value of $b$. | \frac{\sqrt{2}}{2} | 6/8 |
If the maximum value of the function $f(x)=a^{x} (a > 0, a \neq 1)$ on $[-2,1]$ is $4$, and the minimum value is $m$, what is the value of $m$? | \frac{1}{2} | 7/8 |
Prove that if \( a, b, c \) are the lengths of the sides of a triangle with a perimeter of 2, then \( a^{2}+b^{2}+c^{2}<2(1-abc) \). | ^{2}+b^{2}+^{2}<2(1-abc) | 7/8 |
Juca has fewer than 800 marbles. He likes to separate the marbles into groups of the same size. He noticed that if he forms groups of 3 marbles each, exactly 2 marbles are left over. If he forms groups of 4 marbles, 3 marbles are left over. If he forms groups of 5 marbles, 4 marbles are left over. Finally, if he forms groups of 7 marbles each, 6 marbles are left over.
(a) If Juca formed groups of 20 marbles each, how many marbles would be left over?
(b) How many marbles does Juca have? | 419 | 7/8 |
How many kilometers will a traveler cover in 17 days, spending 10 hours a day on this, if he has already covered 112 kilometers in 29 days, traveling 7 hours each day? | 93.79 | 2/8 |
A square has sides of length 8, and a circle centered at one of its vertices has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$. | 64 + 108\pi | 1/8 |
Given the function $$f(x)= \begin{cases} a^{x}, x<0 \\ ( \frac {1}{4}-a)x+2a, x\geq0\end{cases}$$ such that for any $x\_1 \neq x\_2$, the inequality $$\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ holds true. Determine the range of values for the real number $a$. | \frac{1}{2} | 1/8 |
A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of reading each disc will contain. | 50.5 | 7/8 |
Three trucks need to transport \(k\) full barrels, \(k\) half-full barrels, and \(k\) empty barrels such that each truck is equally loaded and each truck carries the same number of barrels. In how many ways can this be done for \(k=7\), considering the trucks and barrels with the same fullness indistinguishable? | 2 | 1/8 |
The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that from each island, there are no more than 5 bridges, and among any 7 islands, there are always two islands connected by a bridge. What is the largest possible value of $N$? | 36 | 7/8 |
The sequence $\{F_n\}$ is defined as follows: $F_1 = 1$, $F_2 = 2$, and $F_{n+2} = F_{n+1} + F_n$ for $n = 1, 2, 3, \ldots$. Prove that for any natural number $n$, $\sqrt[n]{F_{n+1}} \geq 1 + \frac{1}{\sqrt[n]{F_n}}$. | \sqrt[n]{F_{n+1}}\ge1+\frac{1}{\sqrt[n]{F_n}} | 1/8 |
Given 6 people $A$, $B$, $C$, $D$, $E$, $F$, they are randomly arranged in a line. The probability of the event "$A$ is adjacent to $B$ and $A$ is not adjacent to $C$" is $\_\_\_\_\_\_$. | \frac{4}{15} | 5/8 |
A natural number \( k \) has the property: if \( n \) is divisible by \( k \), then the number formed by writing the digits of \( n \) in reverse order is also divisible by \( k \). Prove that \( k \) is a factor of 99. | 99 | 4/8 |
Let \( S = \{1, 2, \ldots, n\} \), and let \( T \) be the set consisting of all nonempty subsets of \( S \). The function \( f: T \rightarrow S \) is "garish" if there do not exist sets \( A, B \in T \) such that \( A \) is a proper subset of \( B \) and \( f(A) = f(B) \). Determine, with proof, how many garish functions exist. | n! | 2/8 |
Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend? | 12 | 7/8 |
If \( p \) and \( q \) are positive integers, \(\max (p, q)\) is the maximum of \( p \) and \( q \) and \(\min (p, q)\) is the minimum of \( p \) and \( q \). For example, \(\max (30,40)=40\) and \(\min (30,40)=30\). Also, \(\max (30,30)=30\) and \(\min (30,30)=30\).
Determine the number of ordered pairs \((x, y)\) that satisfy the equation
$$
\max (60, \min (x, y))=\min (\max (60, x), y)
$$
where \(x\) and \(y\) are positive integers with \(x \leq 100\) and \(y \leq 100\). | 4100 | 7/8 |
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $ . Here $[x]$ denotes the greatest integer less than or equal to $x.$ | 2997 | 5/8 |
An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$ . Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$ | 10^{10} | 1/8 |
Find all natural numbers \( N \) such that the remainder when 2017 is divided by \( N \) is 17. Indicate the number of such \( N \). | 13 | 7/8 |
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$ . For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$ .
*Proposed by **Th3Numb3rThr33*** | 687 | 1/8 |
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere. The base of the prism is \( A B C \), and the lateral edges are \( A A_{1}, B B_{1}, C C_{1} \). The segment \( C D \) is a diameter of this sphere, and the point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism, given that \( C K = 2 \sqrt{6} \) and \( D K = 4 \). | 36 | 6/8 |
Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.) | 9608 | 2/8 |
Sasha chose five numbers from 1, 2, 3, 4, 5, 6, and 7 and informed Anya of their product. Based on this information, Anya realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha inform Anya of? | 420 | 4/8 |
Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not available.
- On Thursday, neither those involved earlier in the week nor Wednesday's helpers are willing to participate, but four new individuals are ready to help.
- On Friday, Tim decides he could either do it himself or get help from one previous assistant who has volunteered again.
How many different combinations of people could be involved in the prank over the week? | 48 | 2/8 |
$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals: | 1000 | 1/8 |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? | \frac{113}{648} | 4/8 |
Xiaoming and Xiaojun start simultaneously from locations A and B, heading towards each other. If both proceed at their original speeds, they meet after 5 hours. If both increase their speeds by 2 km/h, they meet after 3 hours. The distance between locations A and B is 30 km. | 30 | 2/8 |
Arrange 6 volunteers for 3 different tasks, each task requires 2 people. Due to the work requirements, A and B must work on the same task, and C and D cannot work on the same task. How many different arrangements are there? | 12 | 7/8 |
A certain product costs $6$ per unit, sells for $x$ per unit $(x > 6)$, and has an annual sales volume of $u$ ten thousand units. It is known that $\frac{585}{8} - u$ is directly proportional to $(x - \frac{21}{4})^2$, and when the selling price is $10$ dollars, the annual sales volume is $28$ ten thousand units.
(1) Find the relationship between the annual sales profit $y$ and the selling price $x$.
(2) Find the selling price that maximizes the annual profit and determine the maximum annual profit. | 135 | 2/8 |
Let \mathcal{V} be the volume enclosed by the graph $x^{2016}+y^{2016}+z^{2}=2016$. Find \mathcal{V} rounded to the nearest multiple of ten. | 360 | 1/8 |
Given the curve E with the polar coordinate equation 4(ρ^2^-4)sin^2^θ=(16-ρ^2)cos^2^θ, establish a rectangular coordinate system with the non-negative semi-axis of the polar axis as the x-axis and the pole O as the coordinate origin.
(1) Write the rectangular coordinate equation of the curve E;
(2) If point P is a moving point on curve E, point M is the midpoint of segment OP, and the parameter equation of line l is $$\begin{cases} x=- \sqrt {2}+ \frac {2 \sqrt {5}}{5}t \\ y= \sqrt {2}+ \frac { \sqrt {5}}{5}t\end{cases}$$ (t is the parameter), find the maximum value of the distance from point M to line l. | \sqrt{10} | 7/8 |
Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when $$ \sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n) $$ is divided by 50 . | 12 | 3/8 |
In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the two edge cards. What is the probability that, after 2004 moves, the center card is the queen? | \frac{1}{3}+\frac{1}{3\cdot2^{2003}} | 2/8 |
A polynomial \( p(x) \) of degree 2000 with distinct real coefficients satisfies condition \( n \) if:
1. \( p(n) = 0 \), and
2. If \( q(x) \) is obtained from \( p(x) \) by permuting its coefficients, then either \( q(n) = 0 \), or we can obtain a polynomial \( r(x) \) by transposing two coefficients of \( q(x) \) such that \( r(n) = 0 \).
Find all integers \( n \) for which there is a polynomial satisfying condition \( n \). | 01 | 6/8 |
What is the largest integer for which each pair of consecutive digits is a square? | 81649 | 5/8 |
Suppose you have 3 red candies, 2 green candies, and 4 blue candies. A flavor is defined by the ratio of red to green to blue candies, and different flavors are identified by distinct ratios (flavors are considered the same if they can be reduced to the same ratio). How many different flavors can be created if a flavor must include at least one candy of each color? | 21 | 1/8 |
The sum of the first three terms of an arithmetic progression, as well as the sum of the first six terms, are natural numbers. Additionally, its first term \( d_{1} \) satisfies the inequality \( d_{1} \geqslant \frac{1}{2} \). What is the smallest possible value that \( d_{1} \) can take? | 5/9 | 7/8 |
It is known that \(\sin \alpha - \cos \alpha = n\). Find \(\sin^{3} \alpha - \cos^{3} \alpha\). | \frac{3n-n^3}{2} | 7/8 |
We color some unit squares in a $ 99\times 99 $ square grid with one of $ 5 $ given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares. | 1900 | 1/8 |
Karim has 23 candies. He eats $n$ candies and divides the remaining candies equally among his three children so that each child gets an integer number of candies. Which of the following is not a possible value of $n$? | 9 | 1/8 |
In the diagram, what is the measure of $\angle ACB$ in degrees? [asy]
size(250);
draw((-60,0)--(0,0));
draw((0,0)--(64.3,76.6)--(166,0)--cycle);
label("$A$",(64.3,76.6),N);
label("$93^\circ$",(64.3,73),S);
label("$130^\circ$",(0,0),NW);
label("$B$",(0,0),S);
label("$D$",(-60,0),S);
label("$C$",(166,0),S);
[/asy] | 37^\circ | 6/8 |
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 | 6/8 |
Given the equation \(a \cos x + b \sin x + c = 0\) \((a^2 + b^2 \neq 0)\) has two distinct real roots \(\alpha\) and \(\beta\) in \([0, \pi]\), find the value of \(\sin (\alpha + \beta)\). | \frac{2ab}{^2+b^2} | 5/8 |
Evaluate the first three digits to the right of the decimal point in the decimal representation of $\left(10^{987} + 1\right)^{8/3}$ using the Binomial Expansion. | 666 | 1/8 |
Calculate the degree of ionization using the formula:
$$
\alpha=\sqrt{ } K_{\mathrm{HCN}} \mathrm{C}
$$
Given values:
$$
\alpha_{\text {ion }}=\sqrt{ }\left(7,2 \cdot 10^{-10}\right) / 0,1=\sqrt{ } 7,2 \cdot 10^{-9}=8,5 \cdot 10^{-5}, \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$
Alternatively, if the concentration of ions is known, you can calculate $\alpha$ as:
$$
\mathrm{C} \cdot \alpha=[\mathrm{H}^{+}]=[\mathrm{CN}^{-}], [\mathrm{H}^{+}]=[\mathrm{CN}^{-}]=8,5 \cdot 10^{-6} \text{ mol/L}
$$
Then:
$$
\alpha_{\text{ion }}=8,5 \cdot 10^{-6}, 0,1=8,5 \cdot 10^{-5} \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$ | 0.0085 | 6/8 |
In a plane, there are 4000 points, none of which are collinear. Prove that there exist 1000 pairwise non-intersecting quadrilaterals (which may not necessarily be convex) with vertices at these points. | 1000 | 4/8 |
The sum of the integer parts of all positive real solutions \( x \) that satisfy \( x^{4}-x^{3}-2 \sqrt{5} x^{2}-7 x^{2}+\sqrt{5} x+3 x+7 \sqrt{5}+17=0 \) is | 5 | 2/8 |
Point P moves on the parabola $y^2=4x$, and point Q moves on the line $x-y+5=0$. Find the minimum value of the sum of the distance $d$ from point P to the directrix of the parabola and the distance $|PQ|$ between points P and Q. | 3\sqrt{2} | 7/8 |
Solve the system of equations:
\begin{cases}
\frac{m}{3} + \frac{n}{2} = 1 \\
m - 2n = 2
\end{cases} | \frac{2}{7} | 7/8 |
In a tournament with 2017 participating teams, each round consists of three randomly chosen teams competing, with exactly one team surviving from each round. If only two teams remain, a one-on-one battle determines the winner. How many battles must take place to declare a champion? | 1008 | 5/8 |
$M$ is a subset of $\{1,2,3, \ldots, 15\}$ such that the product of any three distinct elements of $M$ is not a square. Determine the maximum number of elements in $M$. | 10 | 1/8 |
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \). | 1 | 4/8 |
We have 30 piggy banks, each with a unique key that does not open any of the other piggy banks. Someone randomly places the keys into the locked piggy banks, one per piggy bank. We then break open two piggy banks. What is the probability that we can open all the remaining piggy banks without breaking any more piggy banks? | \frac{1}{15} | 1/8 |
Given that point $P$ is an intersection point of the ellipse $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyperbola, $e_{1}$, $e_{2}$ are the eccentricities of the ellipse and hyperbola respectively, and $\angle F_{1}PF_{2} = \frac{2\pi}{3}$, find the maximum value of $\frac{1}{e_{1}} + \frac{1}{e_{2}}$. | \frac{4 \sqrt{3}}{3} | 3/8 |
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis is | 32\% | 1/8 |
If I roll 7 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number and the sum of the numbers rolled is divisible by 3? | \frac{1}{3} | 1/8 |
Given a runner who is 30 years old, and the maximum heart rate is found by subtracting the runner's age from 220, determine the adjusted target heart rate by calculating 70% of the maximum heart rate and then applying a 10% increase. | 146 | 4/8 |
A guard has detained an outsider and wants to expel him. The outsider then stated that he made a bet with his friends for 100 coins that the guard would not expel him (if the guard expels him, the outsider pays his friends 100 coins; otherwise, his friends pay him 100 coins). The outsider, deciding to buy off the guard, offered him to name a sum. What is the maximum number of coins the guard can demand so that the outsider, guided only by his own benefit, will definitely pay the guard? | 199 | 1/8 |
Let $f(x)=x^{2021}+15x^{2020}+8x+9$ have roots $a_i$ where $i=1,2,\cdots , 2021$ . Let $p(x)$ be a polynomial of the sam degree such that $p \left(a_i + \frac{1}{a_i}+1 \right)=0$ for every $1\leq i \leq 2021$ . If $\frac{3p(0)}{4p(1)}=\frac{m}{n}$ where $m,n \in \mathbb{Z}$ , $n>0$ and $\gcd(m,n)=1$ . Then find $m+n$ . | 104 | 1/8 |
Given \(\arcsin x < \arccos x < \operatorname{arccot} x\), the range of real number \(x\) is \(\quad\) | (0,\frac{\sqrt{2}}{2}) | 2/8 |
The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 300 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? | 120 | 1/8 |
What is \( \frac{1}{4} \) more than 32.5? | 32.75 | 4/8 |
A bag contains four balls, each labeled with one of the characters "美", "丽", "惠", "州". Balls are drawn with replacement until both "惠" and "州" are drawn, at which point the drawing stops. Use a random simulation method to estimate the probability that the drawing stops exactly on the third draw. Use a computer to randomly generate integer values between 0 and 3, with 0, 1, 2, and 3 representing "惠", "州", "美", and "丽" respectively. Each group of three random numbers represents the result of three draws. The following 16 groups of random numbers were generated:
232 321 230 023 123 021 132 220
231 130 133 231 331 320 122 233
Estimate the probability that the drawing stops exactly on the third draw. | \frac{1}{8} | 7/8 |
In triangle \(ABC\), point \(D\) is given on the extension of side \(CA\) beyond point \(A\), and point \(E\) is given on the extension of side \(CB\) beyond point \(B\) such that \(AB = AD = BE\). The angle bisectors of triangle \(ABC\) from vertices \(A\) and \(B\) intersect the opposite sides at points \(A_1\) and \(B_1\) respectively. What is the area of triangle \(ABC\) if the area of triangle \(DCE\) is 9 units and the area of triangle \(A_1CB_1\) is 4 units? | 6 | 3/8 |
Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart lands within the center hexagon. | \frac{1}{4} | 1/8 |
To find the equations of the tangent and normal lines to the curve at the point corresponding to the parameter value \( t = t_{0} \).
Given:
\[
\begin{cases}
x = 2t - t^2 \\
y = 3t - t^3
\end{cases}
\]
where \( t_{0} = 1 \). | -\frac{1}{3}x+\frac{7}{3} | 7/8 |
Given two concentric circles with radii \( r \) and \( R \) (\( r < R \)). A line is drawn through a point \( P \) on the smaller circle, intersecting the larger circle at points \( B \) and \( C \). The perpendicular to \( BC \) at point \( P \) intersects the smaller circle at point \( A \). Find \( |PA|^2 + |PB|^2 + |PC|^2 \). | 2(R^2+r^2) | 3/8 |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is to the left, right, up, or down, all four equally likely. Let $q$ be the probability that the object reaches $(3,3)$ in eight or fewer steps. Write $q$ in the form $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b.$ | 4151 | 1/8 |
A regular hexagon with side length \(a\) is rotated around one of the tangents to the circumscribed circle. What is the surface area described by the hexagon?
Also,
Given a circle rotated around a straight line in its plane which does not intersect it, resulting in a surface known as a torus.
Knowing the radius \(r\) of the rotating circle and the distance \(R\) from its center to the axis of rotation (\(R > r\)), find the surface area \(S\) of the torus. | 12\pi^2 | 1/8 |
Find all triples of positive real numbers \((x, y, z)\) which satisfy the system
\[
\left\{\begin{array}{l}
\sqrt[3]{x}-\sqrt[3]{y}-\sqrt[3]{z}=64 \\
\sqrt[4]{x}-\sqrt[4]{y}-\sqrt[4]{z}=32 \\
\sqrt[6]{x}-\sqrt[6]{y}-\sqrt[6]{z}=8
\end{array}\right.
\] | Nosolution | 1/8 |
Two people simultaneously step onto an escalator from opposite ends. The escalator is moving down at a speed of \( u = 1.5 \text{ m/s} \). The person moving down has a speed of \( v = 3 \text{ m/s} \) relative to the escalator, and the person moving up has a speed of \( \frac{2v}{3} \) relative to the escalator. At what distance from the lower end of the escalator will they meet? The length of the escalator is \( l = 100 \text{ m} \). | 10\, | 1/8 |
Can we find a line normal to the curves \( y = \cosh x \) and \( y = \sinh x \)? | No | 5/8 |
Given that \(a, b, c, d\) are within the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin a + \sin b + \sin c + \sin d = 1\), and \(\cos 2a + \cos 2b + \cos 2c + \cos 2d \geq \frac{10}{3}\), what is the maximum value of \(a\)? \(\quad\). | \frac{\pi}{6} | 7/8 |
A regular quadrilateral pyramid is inscribed in a sphere with radius \(R\). What is the maximum possible volume of this pyramid? | \frac{64}{81}R^3 | 6/8 |
Count the number of sequences $1 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{5}$ of integers with $a_{i} \leq i$ for all $i$. | 42 | 3/8 |
As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot Y moves clockwise, and robots J and B move counterclockwise. After 12 seconds, robot J reaches point B. After another 9 seconds, when robot J first catches up with robot B, it also meets robot Y for the first time. How many seconds after B first reaches point A will Y first reach point B? | 56 | 4/8 |
Given that 8 first-year high school students are divided evenly between two companies, A and B, with the condition that two students with excellent English grades cannot be assigned to the same company and three students with computer skills cannot be assigned to the same company, determine the number of different distribution schemes. | 36 | 7/8 |
Determine the smallest natural number $n > 2$ , or show that no such natural numbers $n$ exists, that satisfy the following condition: There exists natural numbers $a_1, a_2, \dots, a_n$ such that
\[ \gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}} \] | 4 | 2/8 |
From the hot faucet, the bathtub fills in 23 minutes, and from the cold faucet, it fills in 17 minutes. Petya first opened the hot faucet. After how many minutes should he open the cold faucet so that by the time the bathtub is filled, there will be 1.5 times more hot water than cold water? | 7 | 7/8 |
Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference between the maximum and minimum possible values for $y$ can be written as $a+b \sqrt{c}$, where $a, b$, and $c$ are integers and $c$ is positive and square-free. Find $100 a+10 b+c$. | 472 | 1/8 |
A certain fruit store deals with two types of fruits, A and B. The situation of purchasing fruits twice is shown in the table below:
| Purchase Batch | Quantity of Type A Fruit ($\text{kg}$) | Quantity of Type B Fruit ($\text{kg}$) | Total Cost ($\text{元}$) |
|----------------|---------------------------------------|---------------------------------------|------------------------|
| First | $60$ | $40$ | $1520$ |
| Second | $30$ | $50$ | $1360$ |
$(1)$ Find the purchase prices of type A and type B fruits.
$(2)$ After selling all the fruits purchased in the first two batches, the fruit store decides to reward customers by launching a promotion. In the third purchase, a total of $200$ $\text{kg}$ of type A and type B fruits are bought, and the capital invested does not exceed $3360$ $\text{元}$. Of these, $m$ $\text{kg}$ of type A fruit and $3m$ $\text{kg}$ of type B fruit are sold at the purchase price, while the remaining type A fruit is sold at $17$ $\text{元}$ per $\text{kg}$ and type B fruit is sold at $30$ $\text{元}$ per $\text{kg}$. If all $200$ $\text{kg}$ of fruits purchased in the third batch are sold, and the maximum profit obtained is not less than $800$ $\text{元}$, find the maximum value of the positive integer $m$. | 22 | 7/8 |
The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. | 98 | 5/8 |
Around a circle, 130 trees are planted: birches and lindens (both types are present). Each tree has a sign that reads: "Next to it, two different types of trees are growing." It is known that among all the trees, this statement is false for all lindens and exactly one birch. How many birches could there be? List all possible options. | 87 | 1/8 |
A group with 7 young men and 7 young women was divided into pairs randomly. Find the probability that at least one pair consists of two women. Round the answer to two decimal places. | 0.96 | 7/8 |
A point \( P \) lies at the center of square \( ABCD \). A sequence of points \( \{P_n\} \) is determined by \( P_0 = P \), and given point \( P_i \), point \( P_{i+1} \) is obtained by reflecting \( P_i \) over one of the four lines \( AB, BC, CD, DA \), chosen uniformly at random and independently for each \( i \). What is the probability that \( P_8 = P \)? | \frac{1225}{16384} | 4/8 |
The smallest variance. In a set of $n$ numbers, where one of the numbers is 0 and another is 1, what is the smallest possible variance for such a set of numbers? What should the set be to achieve this? | \frac{1}{2n} | 7/8 |
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ . | 90 | 1/8 |
The school is organizing a trip for 482 people and is renting buses with 42 seats and minibuses with 20 seats. Each person must have one seat and there must be one person per seat. What are the possible rental options for the buses and minibuses? | 2 | 1/8 |
The diagram shows two unshaded circles which touch each other and also touch a larger circle. Chord \( PQ \) of the larger circle is a tangent to both unshaded circles. The length of \( PQ \) is 6 units. What is the area, in square units, of the shaded region? | \frac{9\pi}{2} | 5/8 |
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 45 | 1/8 |
(6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by | \div | 7/8 |
Let \( p \) be the semiperimeter of an acute-angled triangle, \( R \) and \( r \) be the radii of the circumscribed and inscribed circles, respectively, and \( q \) be the semiperimeter of the triangle formed by the feet of the altitudes of the given triangle. Prove that \( \frac{R}{r} = \frac{p}{q} \). | \frac{R}{r}=\frac{p}{q} | 5/8 |
Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School?
$\text{(A)}\ 20 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 40 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 52$ | (B)\32 | 1/8 |
Two math students play a game with $k$ sticks. Alternating turns, each one chooses a number from the set $\{1,3,4\}$ and removes exactly that number of sticks from the pile (so if the pile only has $2$ sticks remaining the next player must take $1$ ). The winner is the player who takes the last stick. For $1\leq k\leq100$ , determine the number of cases in which the first player can guarantee that he will win. | 71 | 7/8 |
ABC is a triangle with circumradius \( R \) and inradius \( r \). If \( p \) is the inradius of the orthic triangle, show that
\[ \frac{p}{R} \leq 1 - \frac{1}{3} \left(1 + \frac{r}{R}\right)^2. \]
[The orthic triangle has as vertices the feet of the altitudes of \( \triangle ABC \).] | \frac{p}{R}\le1-\frac{1}{3}(1+\frac{r}{R})^2 | 1/8 |
A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .
*Proposed by Michael Ren* | 1720 | 1/8 |
From the natural numbers 1 to 2008, the maximum number of numbers that can be selected such that the sum of any two selected numbers is not divisible by 3 is ____. | 671 | 7/8 |
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