problem stringlengths 10 5.15k | answer stringlengths 0 1.22k |
|---|---|
Let \( S = \{1, 2, \cdots, 2009\} \). \( A \) is a 3-element subset of \( S \) such that all elements in \( A \) form an arithmetic sequence. How many such 3-element subsets \( A \) are there? | 1008016 |
Two players, \(A\) and \(B\), play rock-paper-scissors continuously until player \(A\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play? | 12 |
Given \(0<\theta<\pi\), a complex number \(z_{1}=1-\cos \theta+i \sin \theta\) and \(z_{2}=a^{2}+a i\), where \(a \in \mathbb{R}\), it is known that \(z_{1} z_{2}\) is a pure imaginary number, and \(\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\). Determine the value of \(\theta\) when \(\bar{a}\) is a negative real number... | \frac{\pi}{2} |
Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are eliminated. The remaining team wins. How ma... | 3432 |
By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\). | 310 |
The numbers \(a\) and \(b\) are such that \(|a| \neq |b|\) and \(\frac{a+b}{a-b} + \frac{a-b}{a+b} = 6\). Find the value of the expression \(\frac{a^{3} + b^{3}}{a^{3} - b^{3}} + \frac{a^{3} - b^{3}}{a^{3} + b^{3}}\). | \frac{18}{7} |
A waiter at the restaurant U Šejdíře always adds the current date to the bill: he increases the total amount spent by as many crowns as the day of the month it is.
In September, a group of three friends dined at the restaurant twice. The first time, each person paid separately, and the waiter added the date to each bi... | 15 |
The base of a pyramid is a square. The height of the pyramid intersects the diagonal of the base. Find the maximum volume of such a pyramid if the perimeter of the diagonal cross-section that contains the height of the pyramid is 5. | \frac{\sqrt{5}}{3} |
$A, B, C, D$ attended a meeting, and each of them received the same positive integer. Each person made three statements about this integer, with at least one statement being true and at least one being false. Their statements are as follows:
$A:\left(A_{1}\right)$ The number is less than 12;
$\left(A_{2}\right)$ 7 can... | 89 |
Through the vertices \( A \) and \( C \) of triangle \( ABC \), lines are drawn perpendicular to the bisector of angle \( ABC \), intersecting lines \( CB \) and \( BA \) at points \( K \) and \( M \) respectively. Find \( AB \) if \( BM = 10 \) and \( KC = 2 \). | 12 |
What is the three-digit number that is one less than twice the number formed by switching its outermost digits? | 793 |
Given the fraction \(\frac{5}{1+\sqrt[3]{32 \cos ^{4} 15^{\circ}-10-8 \sqrt{3}}}\). Simplify the expression under the cubic root to a simpler form, and then reduce the fraction. | 1 - \sqrt[3]{4} + \sqrt[3]{16} |
Given a hyperbola \( x^{2} - y^{2} = t \) (where \( t > 0 \)), the right focus is \( F \). Any line passing through \( F \) intersects the right branch of the hyperbola at points \( M \) and \( N \). The perpendicular bisector of \( M N \) intersects the \( x \)-axis at point \( P \). When \( t \) is a positive real nu... | \frac{\sqrt{2}}{2} |
In a kindergarten, each child was given three cards, each of which has either "MA" or "NY" written on it. It turned out that 20 children can arrange their cards to spell the word "MAMA", 30 children can arrange their cards to spell the word "NYANYA", and 40 children can arrange their cards to spell the word "MANYA". Ho... | 10 |
A positive integer \( n \) is said to be increasing if, by reversing the digits of \( n \), we get an integer larger than \( n \). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positive integers are increasing? | 4005 |
Nine barrels. In how many ways can nine barrels be arranged in three tiers so that the numbers written on the barrels that are to the right of any barrel or below it are larger than the number written on the barrel itself? An example of a correct arrangement is having 123 in the top row, 456 in the middle row, and 789 ... | 42 |
Persons A, B, and C set out from location $A$ to location $B$ at the same time. Their speed ratio is 4: 5: 12, respectively, where A and B travel by foot, and C travels by bicycle. C can carry one person with him on the bicycle (without changing speed). In order for all three to reach $B$ at the same time in the shorte... | 7/10 |
How can you measure 15 minutes using a 7-minute hourglass and an 11-minute hourglass? | 15 |
In trapezoid \(ABCD\) with bases \(AB\) and \(CD\), it holds that \(|AD| = |CD|\), \(|AB| = 2|CD|\), \(|BC| = 24 \text{ cm}\), and \(|AC| = 10 \text{ cm}\).
Calculate the area of trapezoid \(ABCD\). | 180 |
Given the acute angle \( x \) that satisfies the equation \( \sin^3 x + \cos^3 x = \frac{\sqrt{2}}{2} \), find \( x \). | \frac{\pi}{4} |
Suppose the function \( y= \left| \log_{2} \frac{x}{2} \right| \) has a domain of \([m, n]\) and a range of \([0,2]\). What is the minimum length of the interval \([m, n]\)? | 3/2 |
Given that \(\tan \frac{\alpha+\beta}{2}=\frac{\sqrt{6}}{2}\) and \(\cot \alpha \cdot \cot \beta=\frac{7}{13}\), find the value of \(\cos (\alpha-\beta)\). | \frac{2}{3} |
Let \( f(n) = 3n^2 - 3n + 1 \). Find the last four digits of \( f(1) + f(2) + \cdots + f(2010) \). | 1000 |
Triangles \(ABC\) and \(ABD\) are inscribed in a semicircle with diameter \(AB = 5\). A perpendicular from \(D\) to \(AB\) intersects segment \(AC\) at point \(Q\), ray \(BC\) at point \(R\), and segment \(AB\) at point \(P\). It is known that \(PR = \frac{27}{10}\), and \(PQ = \frac{5}{6}\). Find the length of segment... | 1.5 |
Find the product of two approximate numbers: $0.3862 \times 0.85$. | 0.33 |
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)). There exists a point \( P \) on the right branch of the hyperbola such that \( \left( \overrightarrow{OP} + \overrightarrow{OF_{2}} \right) \cdot \overrightarro... | \sqrt{3} + 1 |
In the coordinate plane \(xOy\), given points \(A(1,3)\), \(B\left(8 \frac{1}{3}, 1 \frac{2}{3}\right)\), and \(C\left(7 \frac{1}{3}, 4 \frac{2}{3}\right)\), the extended lines \(OA\) and \(BC\) intersect at point \(D\). Points \(M\) and \(N\) are on segments \(OD\) and \(BD\) respectively, with \(OM = MN = BN\). Find ... | \frac{5 \sqrt{10}}{3} |
There is a cube of size \(10 \times 10 \times 10\) made up of small unit cubes. A grasshopper is sitting at the center \(O\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, provided that the distance to point \(O\) increases.... | \frac{27!}{(9!)^3} |
Given the parabola \( C: y^2 = 2x \) whose directrix intersects the x-axis at point \( A \), a line \( l \) passing through point \( B(-1,0) \) is tangent to the parabola \( C \) at point \( K \). A line parallel to \( l \) is drawn through point \( A \) and intersects the parabola \( C \) at points \( M \) and \( N \)... | \frac{1}{2} |
In a kindergarten's junior group, there are two (small) Christmas trees and five children. The caregivers want to divide the children into two round dances around each of the trees, with each round dance having at least one child. The caregivers distinguish between children but do not distinguish between the trees: two... | 50 |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of segment \( BM \). Given \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \). | \sqrt{3} + 1 |
The number $\overline{x y z t}$ is a perfect square such that the number $\overline{t z y x}$ is also a perfect square, and the quotient of the numbers $\overline{x y z t}$ and $\overline{t z y x}$ is also a perfect square. Determine the number $\overline{x y z t}$. (The overline indicates that the number is written in... | 9801 |
30 students from 5 grades participated in answering 40 questions. Each student answered at least 1 question. Every two students from the same grade answered the same number of questions, and students from different grades answered a different number of questions. How many students answered only 1 question? | 26 |
Let \( G \) be the centroid of \(\triangle ABC\). Given \( BG \perp CG \) and \( BC = \sqrt{2} \), find the maximum value of \( AB + AC \). | 2\sqrt{5} |
In a park, there is a row of flags arranged in the sequence of 3 yellow flags, 2 red flags, and 4 pink flags. Xiaohong sees that the row ends with a pink flag. Given that the total number of flags does not exceed 200, what is the maximum number of flags in this row? | 198 |
Severus Snape, the potions professor, prepared three potions, each in an equal volume of 400 ml. The first potion makes the drinker smarter, the second makes them more beautiful, and the third makes them stronger. To ensure the effect of any potion, it is sufficient to drink at least 30 ml of that potion. Snape intend... | 60 |
The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \). | 3 : 4 |
There are 5 integers written on the board. By summing them in pairs, the following set of 10 numbers is obtained: 2, 6, 10, 10, 12, 14, 16, 18, 20, 24. Determine which numbers are written on the board and write their product as the answer. | -3003 |
Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$
If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$
then the smallest positive period of \( f(x) \... | \pi |
In a company, some pairs of people are friends (if $A$ is friends with $B$, then $B$ is friends with $A$). It turns out that among every set of 100 people in the company, the number of pairs of friends is odd. Find the largest possible number of people in such a company. | 101 |
A subscriber forgot the last digit of a phone number and therefore dials it randomly. What is the probability that they will have to dial the number no more than three times? | 0.3 |
In how many ways can you form 5 quartets from 5 violinists, 5 violists, 5 cellists, and 5 pianists? | (5!)^3 |
Find the smallest natural number that cannot be written in the form \(\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\), where \(a\), \(b\), \(c\), and \(d\) are natural numbers. | 11 |
In the isosceles right triangle \(ABC\) with \(\angle A = 90^\circ\) and \(AB = AC = 1\), a rectangle \(EHGF\) is inscribed such that \(G\) and \(H\) lie on the side \(BC\). Find the maximum area of the rectangle \(EHGF\). | 1/4 |
A man went to a bank to cash a check. The cashier made a mistake and gave him the same number of cents instead of dollars and the same number of dollars instead of cents. The man did not count the money, put it in his pocket, and additionally dropped a 5-cent coin. When he got home, he discovered that he had exactly tw... | 31.63 |
In parallelogram \(ABCD\), point \(P\) is taken on side \(BC\) such that \(3PB = 2PC\), and point \(Q\) is taken on side \(CD\) such that \(4CQ = 5QD\). Find the ratio of the area of triangle \(APQ\) to the area of triangle \(PQC\). | 37/15 |
Two decks, each containing 36 cards, were placed on a table. The first deck was shuffled and placed on top of the second deck. For each card in the first deck, the number of cards between it and the same card in the second deck was counted. What is the sum of these 36 numbers? | 1260 |
An aluminum part and a copper part have the same volume. The density of aluminum is $\rho_{A} = 2700 \, \text{kg/m}^3$, and the density of copper is $\rho_{M} = 8900 \, \text{kg/m}^3$. Find the mass of the aluminum part, given that the masses of the parts differ by $\Delta m = 60 \, \text{g}$. | 26.13 |
Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same.
$\square + \square \square = \square \square \square$.
What is the largest possible three-digit number in the equation? | 105 |
Anya, Vanya, Dania, Sanya, and Tanya were collecting apples. It turned out that each of them collected an integer percentage of the total number of apples, and all these percentages are different and greater than zero. What is the minimum number of apples that could have been collected? | 20 |
The population of a city increases annually by $1 / 50$ of the current number of inhabitants. In how many years will the population triple? | 55 |
Let \(a, b, c, d, e\) be positive integers. Their sum is 2345. Let \(M = \max (a+b, b+c, c+d, d+e)\). Find the smallest possible value of \(M\). | 782 |
How many three-digit positive integers \( x \) are there with the property that \( x \) and \( 2x \) have only even digits? (One such number is \( x=420 \), since \( 2x=840 \) and each of \( x \) and \( 2x \) has only even digits.) | 18 |
Two people, A and B, take turns to draw candies from a bag. A starts by taking 1 candy, then B takes 2 candies, A takes 4 candies next, then B takes 8 candies, and so on. This continues. When the number of candies remaining in the bag is less than the number they are supposed to take, they take all the remaining candie... | 260 |
Given a triangle with side lengths \( l, m, n \), where \( l > m > n \) are integers, and it is known that:
\[
\left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\}
\]
where \( \{x\} = x - \lfloor x \rfloor \), and \( \lfloor x \rfloor \) denotes the greatest i... | 3003 |
There are 8 different positive integers. Among them, at least 6 are multiples of 2, at least 5 are multiples of 3, at least 3 are multiples of 5, and at least 1 is a multiple of 7. To minimize the largest number among these 8 integers, what is this largest number? | 20 |
Kolya started playing WoW when the hour and minute hands were opposite each other. He finished playing after a whole number of minutes, at which point the minute hand coincided with the hour hand. How long did he play (assuming he played for less than 12 hours)? | 360 |
If the functions \( f(x) \) and \( g(x) \) are defined for all real numbers, and they satisfy the equation \( f(x-y) = f(x) g(y) - g(x) f(y) \), with \( f(-2) = f(1) \neq 0 \), then find \( g(1) + g(-1) \). | -1 |
Consider all permutations of the numbers $1, 2, \cdots, 8$ as eight-digit numbers. How many of these eight-digit numbers are multiples of 11? | 4608 |
Given \(\alpha, \beta \in [0, \pi]\), find the maximum value of \((\sin \alpha + \sin (\alpha + \beta)) \cdot \sin \beta\). | \frac{8\sqrt{3}}{9} |
Compute
$$
\int_{1}^{2} \frac{9x+4}{x^{5}+3x^{2}+x} \, dx. | \ln \frac{80}{23} |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{... | 1078 |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in... | 3001 |
The school plans to arrange 6 leaders to be on duty from May 1st to May 3rd. Each leader must be on duty for 1 day, with 2 leaders assigned each day. If leader A cannot be on duty on the 2nd, and leader B cannot be on duty on the 3rd, how many different methods are there to arrange the duty schedule? | 42 |
It is known that
$$
\sqrt{9-8 \sin 50^{\circ}}=a+b \sin c^{\circ}
$$
for exactly one set of positive integers \((a, b, c)\), where \(0 < c < 90\). Find the value of \(\frac{b+c}{a}\). | 14 |
In an arithmetic sequence \(\{a_{n}\}\), if \(\frac{a_{11}}{a_{10}} < -1\) and the sum of the first \(n\) terms \(S_{n}\) has a maximum value, then the value of \(n\) when \(S_{n}\) attains its smallest positive value is \(\qquad\). | 19 |
In a three-dimensional Cartesian coordinate system, there is a sphere with its center at the origin and a radius of 3 units. How many lattice points lie on the surface of the sphere? | 30 |
In a physical education class, 25 students from class 5"B" stood in a line. Each student is either an honor student who always tells the truth or a troublemaker who always lies.
Honor student Vlad took the 13th position. All the students, except Vlad, said: "There are exactly 6 troublemakers between me and Vlad." How ... | 12 |
Calculate the integral
$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x
$$
with an accuracy of $\alpha=0.001$. | 0.090 |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-104)^{2}
$$
If the result is a non-integer, round it to the nearest integer. | 49608 |
Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done? | 179940 |
When programming a computer to print the first 10,000 natural numbers greater than 0: $1,2,3, \cdots, 10000$, the printer unfortunately has a malfunction. Each time it prints the digits 7 or 9, it prints $x$ instead. How many numbers are printed incorrectly? | 5904 |
The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \geq 1$. | 106 |
A regular $n$-gon is inscribed in a circle with radius $R$, and its area is equal to $3 R^{2}$. Find $n$. | 12 |
The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When should the tires be swapped so that they wear out at the same time? | 9375 |
Given 5 distinct real numbers, any two of which are summed to yield 10 sums. Among these sums, the smallest three are 32, 36, and 37, and the largest two are 48 and 51. What is the largest of these 5 numbers? | 27.5 |
An equilateral triangle is subdivided into 4 smaller equilateral triangles. Using red and yellow to paint the vertices of the triangles, each vertex must be colored and only one color can be used per vertex. If two colorings are considered the same when they can be made identical by rotation, how many different colorin... | 24 |
For non-negative integers \( x \), the function \( f(x) \) is defined as follows:
$$
f(0) = 0, \quad f(x) = f\left(\left\lfloor \frac{x}{10} \right\rfloor\right) + \left\lfloor \log_{10} \left( \frac{10}{x - 10\left\lfloor \frac{x-1}{10} \right\rfloor} \right) \right\rfloor .
$$
For \( 0 \leqslant x \leqslant 2006 \), ... | 1111 |
Petya places "+" and "-" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple times. Petya then sums all the numbers on the boar... | 32 |
Determine the following number:
\[
\frac{12346 \cdot 24689 \cdot 37033 + 12347 \cdot 37034}{12345^{2}}
\] | 74072 |
Point \( O \) is located on side \( AC \) of triangle \( ABC \) such that \( CO : CA = 2 : 3 \). When this triangle is rotated by a certain angle around point \( O \), vertex \( B \) moves to vertex \( C \), and vertex \( A \) moves to point \( D \), which lies on side \( AB \). Find the ratio of the areas of triangles... | 1/6 |
There are 294 distinct cards with numbers \(7, 11, 7^{2}, 11^{2}, \ldots, 7^{147}, 11^{147}\) (each card has exactly one number, and each number appears exactly once). How many ways can two cards be selected so that the product of the numbers on the selected cards is a perfect square? | 15987 |
In the coordinate plane, a rectangle is drawn with vertices at $(34,0),(41,0),(34,9),(41,9)$. Find the smallest value of the parameter $a$ for which the line $y=ax$ divides this rectangle into two parts such that the area of one part is twice the area of the other. If the answer is not an integer, write it as a decimal... | 0.08 |
Masha came up with the number \( A \), and Pasha came up with the number \( B \). It turned out that \( A + B = 2020 \), and the fraction \( \frac{A}{B} \) is less than \( \frac{1}{4} \). What is the maximum value that the fraction \( \frac{A}{B} \) can take? | 403/1617 |
Let \(a, b, c\) be the side lengths of a right triangle, with \(a \leqslant b < c\). Determine the maximum constant \(k\) such that the inequality \(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c\) holds for all right triangles, and specify when equality occurs. | 2 + 3\sqrt{2} |
There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switches numbered 2, 3, and 5 are flipped, how many lamps remain on? | 499 |
Find the total length of the intervals on the number line where the inequalities $x < 1$ and $\sin (\log_{2} x) < 0$ hold. | \frac{2^{\pi}}{1+2^{\pi}} |
Let's call a natural number "remarkable" if it is the smallest among natural numbers with the same sum of digits. What is the sum of the digits of the two-thousand-and-first remarkable number? | 2001 |
In a math lesson, each gnome needs to find a three-digit number without zero digits, divisible by 3, such that when 297 is added to it, the resulting number consists of the same digits but in reverse order. What is the minimum number of gnomes that must be in the lesson so that among the numbers they find, there are al... | 19 |
In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) in the following ratio: \( AM : AC = CN : CE = r \). If the points \( B, M, N \) are collinear, find the ratio \( r \). | \frac{\sqrt{3}}{3} |
What is the value of the sum $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} 2002\right]$? | 17984 |
Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the color painted by the second painter. The firs... | 49 |
Inside triangle \(ABC\), a random point \(M\) is chosen. What is the probability that the area of one of the triangles \(ABM\), \(BCM\), or \(CAM\) is greater than the sum of the areas of the other two? | 0.75 |
The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.
How many triangles can be formed with vertices at the marked points? | 1130 |
In a certain year, a specific date was never a Sunday in any month. Determine this date. | 31 |
On an island, there are 1000 villages, each with 99 inhabitants. Each inhabitant is either a knight, who always tells the truth, or a liar, who always lies. It is known that the island has exactly 54,054 knights. One day, each inhabitant was asked the question: "Are there more knights or liars in your village?" It turn... | 638 |
The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$
has the largest element as $\qquad$ . | \sqrt[3]{3} |
A target is a triangle divided by three sets of parallel lines into 100 equal equilateral triangles with unit sides. A sniper shoots at the target. He aims at a triangle and hits either it or one of the adjacent triangles sharing a side. He can see the results of his shots and can choose when to stop shooting. What is ... | 25 |
How many of the integers from \(2^{10}\) to \(2^{18}\) inclusive are divisible by \(2^{9}\)? | 511 |
The line \( l: (2m+1)x + (m+1)y - 7m - 4 = 0 \) intersects the circle \( C: (x-1)^{2} + (y-2)^{2} = 25 \) to form the shortest chord length of \(\qquad \). | 4 \sqrt{5} |
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