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Calculate the areas of the figures bounded by the curves:
a)
\[
\left\{
\begin{array}{l}
y = x \sqrt{9 - x^{2}} \\
y = 0 \quad (0 \leq x \leq 3)
\end{array}
\right.
\]
b)
\[
\left\{
\begin{array}{l}
y = 2x - x^{2} + 3 \\
y = x^{2} - 4x + 3
\end{array}
\right.
\] | 9 | 0.75 |
There are 101 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers? | 34 | 0.375 |
Given that \(a, b, c, d\) are positive numbers and
\[ a + 20b = c + 20d = 2, \]
find the minimum value of \(\frac{1}{a} + \frac{1}{bcd}\). | \frac{441}{2} | 0.125 |
Little children were eating candies. Each ate 7 fewer candies than all the others combined, but still more than one candy. How many candies were eaten in total? | 21 | 0.875 |
Find all positive integers \( M \) less than 10 such that 5 divides \( 1989^M + M^{1889} \). | 1 \text{ and } 4 | 0.75 |
For which real \( a \) are there distinct reals \( x \) and \( y \) such that \( x = a - y^2 \) and \( y = a - x^2 \) ? | a > \frac{3}{4} | 0.75 |
If a die is rolled 500 times, what is the most probable number of times that the face showing 1 dot will appear? | 83 | 0.75 |
Given the sequences $\{a_n\}$ and $\{b_n\}$ such that $a_1 = -1$, $b_1 = 2$, and the recurrence relations $a_{n+1} = -b_n$, $b_{n+1} = 2a_n - 3b_n$ for $n \in \mathbf{N}^*$, find the value of $b_{2015} + b_{2016}$. | -3 \cdot 2^{2015} | 0.25 |
Let \( a \) and \( b \) be positive real numbers such that \(\frac{a}{1+a}+\frac{b}{1+b}=1\). Show that
\[ \frac{a}{1+b^{2}}-\frac{b}{1+a^{2}}=a-b \] | a - b | 0.875 |
The shorter side of a rectangle is equal to 1, and the acute angle between the diagonals is $60^\circ$. Find the radius of the circle circumscribed around the rectangle. | 1 | 0.75 |
In the figure, segment \(DE\) divides square sheet \(ABCD\) into \(\triangle ADE\) and quadrilateral \(EDCB\). Given that the area ratio \(S_{\triangle ADE} : S_{EDCB} = 5 : 19\), determine the ratio of the perimeter of \(\triangle ADE\) to the perimeter of quadrilateral \(EDCB\). | \frac{15}{22} | 0.125 |
If the real numbers \( x \) and \( y \) satisfy \( x^{2} + y^{2} - 2x + 4y = 0 \), find the maximum value of \( x - 2y \). | 10 | 0.75 |
Given ten points on the boundary line of a half-plane, how many ways can the points be paired such that the points paired together can be connected by non-intersecting lines within the half-plane? | 42 | 0.625 |
In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that no sum of two different numbers of the same color equals a power of two? | 256 | 0.25 |
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 | 0.75 |
Given the sets \( A = \{(x, y) \mid y^2 - x - 1 = 0\} \), \( B = \{(x, y) \mid 4x^2 + 2x - 2y + 5 = 0\} \), and \( C = \{(x, y) \mid y = kx + b\} \), find all non-negative integers \( k \) and \( b \) such that \( (A \cup B) \cap C = \emptyset \). | (k, b) = (1, 2) | 0.625 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{5}-8}-n \sqrt{n\left(n^{2}+5\right)}}{\sqrt{n}}$$ | -\frac{5}{2} | 0.75 |
The sum of the first three terms of an increasing arithmetic progression is 15. If 1 is subtracted from each of the first two terms and 1 is added to the third term, the resulting three numbers form a geometric progression. Find the sum of the first ten terms of the arithmetic progression. | 120 | 0.875 |
The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \( F \) is greater than \( \frac{1}{12} \), the probability of obtaining the face opposite \( F \) is less than \( \frac{1}{12} \) while the probability of obtaining any one of the other ten faces is \( \frac{1}{12} \).
When two such dice are rolled, the probability of obtaining a sum of 13 is \( \frac{29}{384} \).
What is the probability of obtaining face \( F \)? | \frac{7}{48} | 0.75 |
The sides of rectangle $ABCD$ are $AB=3$ and $BC=2$. Point $P$ is on side $AB$ such that line $PD$ touches the circle with diameter $BC$ at point $E$. The line passing through the center of the circle and point $E$ intersects side $AB$ at point $Q$. What is the area of triangle $PQE$? | \frac{1}{24} | 0.75 |
If the maximum value and the minimum value of the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \) sum up to 4, then find \( a + b \). | 3 | 0.625 |
n people each have exactly one unique secret. How many phone calls are needed so that each person knows all n secrets? You should assume that in each phone call the caller tells the other person every secret he knows, but learns nothing from the person he calls. | 2n-2 | 0.125 |
The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 2202 | 0.375 |
Given a non-empty subset family \( U \) of \( S = \{a_1, a_2, \ldots, a_n\} \) that satisfies the property: if \( A \in U \) and \( A \subseteq B \), then \( B \in U \); and a non-empty subset family \( V \) of \( S \) that satisfies the property: if \( A \in V \) and \( A \supseteq B \), then \( B \in V \). Find the maximum possible value of \( \frac{|U \cap V|}{|U| \cdot |V|} \). | \frac{1}{2^n} | 0.125 |
Find all integers \( k \geq 1 \) such that there exists a pair of integers \( (n, m) \) for which \( 9n^6 = 2^k + 5m^2 + 2 \). | 1 | 0.625 |
Let \( d \) be a randomly chosen divisor of 2016. Find the expected value of
\[
\frac{d^{2}}{d^{2}+2016}
\] | \frac{1}{2} | 0.75 |
Given the sequence \( a_{n}=\left\lfloor(\sqrt{2}+1)^{n}+\left(\frac{1}{2}\right)^{n}\right\rfloor \) for \( n \geq 0 \), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \), compute the sum:
\[ \sum_{n=1}^{\infty} \frac{1}{a_{n-1} a_{n+1}} \] | \frac{1}{8} | 0.5 |
Someone wrote down two numbers $5^{2020}$ and $2^{2020}$ consecutively. How many digits will the resulting number contain? | 2021 | 0.375 |
For any positive integer \( n \), let \( a_{n} \) denote the number of triangles with integer side lengths whose longest side is \( 2n \).
(1) Find an expression for \( a_{n} \) in terms of \( n \);
(2) Given the sequence \( \{b_{n}\} \) satisfies
$$
\sum_{k=1}^{n}(-1)^{n-k} \binom{n}{k} b_{k}=a_{n}\quad (n \in \mathbf{Z}_{+}),
$$
find the number of positive integers \( n \) such that \( b_{n} \leq 2019 a_{n} \). | 12 | 0.125 |
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Find the number of possible routes the ant has taken. | 546 | 0.25 |
Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C$ for the first time, and at the same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. How long after the race starts will $C$ and $D$ meet for the first time? | 53 | 0.375 |
\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\] | -1 | 0.875 |
In triangle \(ABC\), the sides are \(AB = 10\), \(AC = 24\), and \(BC = 26\). The medians \(AM\) and \(CN\) are drawn, where points \(M\) and \(N\) are the midpoints of sides \(BC\) and \(AB\), respectively. Point \(I\) lies on side \(AC\), and \(BI\) is the angle bisector. Find the area of triangle \(MNI\). | 30 | 0.5 |
At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian again at 13:00. How many minutes did the repair take? (The pedestrian's speed is constant, and he moved without stopping; the cyclist's speed is also constant except for the repair interval.) | 100 \text{ minutes} | 0.625 |
\(3.420 \sin 10^{\circ} \cdot \sin 20^{\circ} \cdot \sin 30^{\circ} \cdot \sin 40^{\circ} \cdot \sin 50^{\circ} \cdot \sin 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 80^{\circ} = \frac{3}{256} \cdot\) | \frac{3}{256} | 0.125 |
Determine the number of permutations that can be made using 3 green balls, 2 red balls, 2 white balls, and 3 yellow balls such that no two yellow balls are adjacent. | 11760 | 0.875 |
A regular triangular prism $ABC A_1B_1C_1$ is inscribed in a sphere, with base $ABC$ and lateral edges $AA_1, BB_1, CC_1$. Segment $CD$ is a diameter of this sphere, and point $K$ is the midpoint of edge $AA_1$. Find the volume of the prism if $CK = 2 \sqrt{6}$ and $DK = 4$. | 36 | 0.25 |
When \( s \) and \( t \) range over all real numbers, what is the minimum value of \( (s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2} \)? | 2 | 0.625 |
Several students decided to buy a tape recorder priced between 170 and 195 dollars. However, at the last moment, two students decided not to participate in the purchase, so each of the remaining students had to contribute 1 dollar more. How much did the tape recorder cost? | 180 | 0.875 |
Calculate the area of the region bounded by the graphs of the functions:
$$
x=\sqrt{e^{y}-1}, x=0, y=\ln 2
$$ | 2 - \frac{\pi}{2} | 0.875 |
Given the function
\[ f(x) = \sqrt{2x^{2} + 2x + 41} - \sqrt{2x^{2} + 4x + 4} \quad (x \in \mathbb{R}), \]
determine the maximum value of \( f(x) \). | 5 | 0.5 |
Lele's family raises some chickens and ducks. Grabbing any 6 of them, there are at least 2 that are not ducks; Grabbing any 9 of them, there is at least 1 that is a duck. What is the maximum number of chickens and ducks in Lele's family? | 12 | 0.75 |
The product of the digits of any multi-digit number is always less than this number. If we calculate the product of the digits of a given multi-digit number, then the product of the digits of this product, and so on, we will necessarily reach a single-digit number after some number of steps. This number of steps is called the persistence of the number. For example, the number 723 has a persistence of 2 because $7 \cdot 2 \cdot 3 = 42$ (1st step) and $4 \cdot 2 = 8$ (2nd step).
1. Find the largest odd number with distinct digits that has a persistence of 1.
2. Find the largest even number with distinct nonzero digits that has a persistence of 1.
3. Find the smallest natural number that has a persistence of 3. | 39 | 0.125 |
Given that the equation \(|x| - \frac{4}{x} = \frac{3|x|}{x}\) has \(k\) distinct real root(s), find the value of \(k\). | 1 | 0.75 |
Luis wrote the sequence of natural numbers starting from 1:
$$
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \cdots
$$
When did he write the digit 3 for the 25th time? | 131 | 0.125 |
Show that if \(a, b, c,\) and \(d\) are four positive real numbers such that \( abcd = 1 \), then
\[
a^{2}+b^{2}+c^{2}+d^{2}+ab+ac+ad+bc+bd+cd \geq 10
\] | 10 | 0.875 |
A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots? | \frac{3}{4} | 0.25 |
Place 6 points inside a rectangle with dimensions $4 \times 3$. Show that we can find two points whose distance is less than or equal to $\sqrt{5}$. | \sqrt{5} | 0.5 |
Given the sequence $\left\{a_{n}\right\}$ such that:
$$
a_{1}=-2,
$$
and $S_{n}=\frac{3}{2} a_{n}+n$ (where $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$). Let $f(x)$ be an odd function defined on $\mathbf{R}$, which satisfies:
$$
f(2-x)=f(x).
$$
Find $f\left(a_{2021}\right)=$ $\qquad$ | 0 | 0.625 |
If the 200th day of some year is a Sunday and the 100th day of the following year is also a Sunday, what day of the week was the 300th day of the previous year? Provide the answer as the number of the day of the week (if Monday, then 1; if Tuesday, then 2, etc.). | 1 | 0.125 |
From the eight natural numbers 1 to 8, how many ways are there to choose three numbers such that no two numbers are consecutive? | 20 | 0.625 |
The solutions to the equation \( x^3 - 4 \lfloor x \rfloor = 5 \), where \( x \) is a real number, are denoted by \( x_1, x_2, x_3, \ldots, x_k \) for some positive integer \( k \). Find \( \sum_{i=1}^{k} x_{i}^{3} \). | 10 | 0.625 |
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 | 0.5 |
Several points, including points \(A\) and \(B\), are marked on a line. All possible segments with endpoints at the marked points are considered. Vasya counted that point \(A\) is inside 50 of these segments, and point \(B\) is inside 56 of these segments. How many points were marked? (The endpoints of a segment are not considered its internal points.) | 16 | 0.75 |
If a two-digit number is divided by a certain integer, the quotient is 3 and the remainder is 8. If the digits of the dividend are swapped and the divisor remains the same, the quotient is 2 and the remainder is 5. Find the original value of the dividend. | 53 | 0.625 |
Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \). | 1003 | 0.375 |
There are two types of tables in a restaurant: a square table can seat 4 people, and a round table can seat 9 people. The restaurant manager calls a number a "wealth number" if the total number of diners can exactly fill a certain number of tables. How many "wealth numbers" are there among the numbers from 1 to 100? | 88 | 0.375 |
The function \( f \) maps the set of positive integers onto itself and satisfies the equation
\[
f(f(n)) + f(n) = 2n + 6
\]
What could this function be? | f(n) = n + 2 | 0.125 |
Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black? | \frac{7}{15} | 0.625 |
Given that the unit digit of \(1+2+3+\ldots+1997+1998+1999+1998+1997+\ldots+3+2+1\) is \(P\), find the value of \(P\). | 1 | 0.625 |
Vitya has five math lessons per week, one on each day from Monday to Friday. He knows that there is a probability of \( \frac{1}{2} \) that the teacher will not check his homework at all during the week, and a probability of \( \frac{1}{2} \) that the teacher will check it, but only on one of the math lessons, with equal chances on any lesson.
At the end of his math lesson on Thursday, Vitya realized that the teacher has not yet checked his homework this week. What is the probability that the homework will be checked on Friday? | \frac{1}{6} | 0.75 |
Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? | 33 | 0.375 |
Given the sequence \( \left\{a_{n}\right\} \) with the sum of its first \( n \) terms defined by \( S_{n}=2a_{n}-1 \) for \( n=1,2, \cdots \), and the sequence \( \left\{b_{n}\right\} \) that satisfies \( b_{1}=3 \) and \( b_{k+1}=a_{k}+b_{k} \) for \( k= 1,2, \cdots \), find the sum of the first \( n \) terms of the sequence \( \left\{b_{n}\right\} \). | 2^n + 2n - 1 | 0.375 |
Determine the largest prime factor of the sum \(\sum_{k=1}^{11} k^{5}\). | 263 | 0.375 |
The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the lesson. What fraction of the way to school had he covered when he remembered about the pen? | \frac{1}{4} | 0.5 |
Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from twelve consecutive natural numbers to a certain number \(a\) is 358, and the sum of the distances from these same twelve numbers to another number \(b\) is 212. Find all possible values of \(a\), given that \(a + b = 114.5\). | \frac{190}{3} | 0.125 |
Given a triangular pyramid \( S-ABC \) with vertex \( S \). The projection of \( S \) onto the base \( \triangle ABC \) is the orthocenter \( H \) of \( \triangle ABC \). Additionally, \( BC = 2 \), \( SB = SC \), and the dihedral angle between the face \( SBC \) and the base is \( 60^\circ \). Determine the volume of the pyramid. | \frac{\sqrt{3}}{3} | 0.625 |
On the island of Misfortune, there live knights who always tell the truth and liars who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle and each said, "Both of my neighbors are liars". How many different values can $N$ take?
| 337 | 0.375 |
There are large, medium, and small cubic pools, with inner edge lengths of 6 meters, 3 meters, and 2 meters, respectively. When two piles of gravel are submerged in the medium and small pools, the water levels in the respective pools rise by 6 centimeters and 4 centimeters. If these two piles of gravel are both submerged in the large pool, by how many centimeters will the water level rise? | \frac{35}{18} | 0.5 |
Determine the minimum value of \( \sum_{k=1}^{50} x_{k} \), where the summation is done over all possible positive numbers \( x_{1}, \ldots, x_{50} \) satisfying \( \sum_{k=1}^{50} \frac{1}{x_{k}} = 1 \). | 2500 | 0.875 |
Find the largest real number \( k \) such that there exists a sequence of positive real numbers \(\{a_i\}\) for which \(\sum_{n=1}^{\infty} a_n\) converges but \(\sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{n^k}\) does not. | \frac{1}{2} | 0.625 |
Let \( ABC \) be a right triangle at \( C \). On the line \( (AC) \), place a point \( D \) such that \( CD = BC \), with \( C \) positioned between \( A \) and \( D \). The perpendicular to \( (AB) \) passing through \( D \) intersects \( (BC) \) at \( E \). Show that \( AC = CE \). | AC = CE | 0.75 |
A flock of geese was flying. At each lake, half of the geese and half a goose landed. The rest continued flying. All the geese landed on $n$ lakes.
How many geese were there in the flock in total? | 2^n - 1 | 0.5 |
To determine the roots of the following system of equations with an error less than $0.01$, how many decimal places must we consider for the value of $\sqrt{2}$?
$$
\begin{aligned}
& \sqrt{2} x + 8.59 y = 9.98 \\
& 1.41 x + 8.59 y = 10
\end{aligned}
$$ | 5 | 0.625 |
Given acute angles \(A\) and \(B\) such that \(\tan (A+B) = 2 \tan A\), what is the maximum value of \(\tan B\)? | \frac{\sqrt{2}}{4} | 0.875 |
The sum of a set of numbers is the sum of all its elements. Let \( S \) be a set consisting of positive integers not exceeding 15, where the sums of any two disjoint subsets of \( S \) are not equal. Among all sets with this property, find the \( S \) with the maximum sum, and determine the sum of this set. | 61 | 0.375 |
The triangle \(ABC\) is isosceles with \(AB = BC\). The point \(D\) is a point on \(BC\), between \(B\) and \(C\), such that \(AC = AD = BD\). What is the size of angle \(ABC\)? | 36^\circ | 0.625 |
A plane divides the medians of the faces $ABC$, $ACD$, and $ABD$ of a tetrahedron $ABCD$ emanating from vertex $A$ in the ratios $1:2$, $1:1$, and $1:2$ respectively, considering from point $A$. Find the ratio of the volumes of the parts into which this plane divides the tetrahedron. | \frac{1}{15} | 0.25 |
The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 cm and 40 cm. Find the length of the arc of the semicircle that lies between the points where it touches the legs of the triangle. | 12\pi | 0.875 |
Given the polynomial
$$
\begin{aligned}
P(x)= & x^{15}-2008 x^{14}+2008 x^{13}-2008 x^{12}+2008 x^{11} \\
& -\cdots+2008 x^{3}-2008 x^{2}+2008 x,
\end{aligned}
$$
determine \( P(2007) \). | 2007 | 0.5 |
Given that \( a, b, c \) are all positive numbers, find the maximum value of \( y = \frac{ab + 2bc}{a^2 + b^2 + c^2} \). | \frac{\sqrt{5}}{2} | 0.875 |
On the radius \( AO \) of a circle with center \( O \), a point \( M \) was chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) were chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \), given that the radius of the circle is \( 14 \) and \( \sin \alpha = \frac{\sqrt{33}}{7} \). | 16 | 0.25 |
Let \( x, y \in \mathbf{R} \). Denote the minimum value among \( 2^{-x}, 2^{x-y}, 2^{y-1} \) as \( P \). When \( 0 < x < 1 \) and \( 0 < y < 1 \), what is the maximum value of \( P \)? | 2^{-\frac{1}{3}} | 0.625 |
In a chess tournament, \( n \) women and \( 2n \) men participated. Everyone played exactly one game with each other. There were no draws, and the number of games won by the women is in the ratio of \( 7: 5 \) to the number of games won by the men. What is \( n \)? | 3 | 0.625 |
What is the largest prime divisor of a number in the form $\overline{xyxyxy}$ in the decimal system? | 97 | 0.375 |
In the tetrahedron \( P-ABC \), \( PA=BC=\sqrt{6} \), \( PB=AC=\sqrt{8} \), and \( PC=AB=\sqrt{10} \). Find the radius of the circumscribed sphere of the tetrahedron. | \sqrt{3} | 0.375 |
\( 1111111 \times 1111111 = \) | 1234567654321 | 0.75 |
Consider a large cube of dimensions \(4 \times 4 \times 4\) composed of 64 unit cubes. Select 16 of these unit cubes and color them red, ensuring that within every \(1 \times 1 \times 4\) rectangular prism formed by 4 unit cubes, exactly 1 unit cube is colored red. How many different ways can the 16 unit cubes be colored red? Provide a justification for your answer. | 576 | 0.25 |
In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that no sum of two different numbers of the same color equals a power of two? | 256 | 0.25 |
Determine the mass of a sphere with radius \( r \), given that the density at each point is proportional to its distance from the center of the sphere. | k \pi r^4 | 0.125 |
How many integers at a minimum must be selected from the set $\{1,2, \ldots, 20\}$ to ensure that this selection includes two integers \( a \) and \( b \) such that \( a - b = 2 \)? | 11 | 0.875 |
Show that if I take 3 integers, I can find 2 of them whose sum is even. Then, if I take 5 integers, I can find 3 of them whose sum is divisible by 3. Then show that if I take 11 integers, I can find 6 of them whose sum is divisible by 6. Find the smallest integer \( n \) such that if I take \( n \) integers, I can always find 18 of them whose sum is divisible by 18. | 35 | 0.875 |
Does there exist a positive integer $m$ such that the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{a b c} = \frac{m}{a + b + c}$ has infinitely many solutions in positive integers? | 12 | 0.375 |
Person A and Person B decided to go to a restaurant. Due to high demand, Person A arrived first and took a waiting number, while waiting for Person B. After a while, Person B arrived but did not see Person A, so he also took a waiting number. While waiting, Person B saw Person A, and they compared their waiting numbers. They found that the digits of these two numbers are two-digit numbers in reverse order, and the sum of the digits of both numbers is 8. Additionally, Person B's number is 18 greater than Person A's. What is Person A's number? $\qquad$ | 35 | 0.25 |
Consider a square of side length 1. Draw four lines that each connect a midpoint of a side with a corner not on that side, such that each midpoint and each corner is touched by only one line. Find the area of the region completely bounded by these lines. | \frac{1}{5} | 0.375 |
Two squares are arranged as shown in the figure. If the overlapping part is subtracted from the smaller square, $52\%$ of its area remains. If the overlapping part is subtracted from the larger square, $73\%$ of its area remains. Find the ratio of the side of the smaller square to the side of the larger square. | 0.75 | 0.125 |
A castle has a number of halls and \( n \) doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he just used to enter that hall. Find a strategy allowing the knight to get outside after visiting no more than \( 2n \) halls (a hall is counted each time it is entered). | 2n | 0.625 |
The base and one side of a triangle are 30 and 14, respectively. Find the area of this triangle if the median drawn to the base is 13. | 168 | 0.75 |
From a three-digit number \( A \), which does not contain zeroes in its digits, a two-digit number \( B \) is obtained by replacing the first two digits with their sum (for example, the number 243 becomes 63). Find \( A \) if it is known that \( A = 3B \). | 135 | 0.75 |
The sports event lasted for \( n \) days, during which \( m \) medals were distributed. On the first day, 1 medal was given out, plus \(\frac{1}{7}\) of the remaining \((m-1)\) medals. On the second day, 2 medals were given out, plus \(\frac{1}{7}\) of the remaining medals after that. This pattern continued for subsequent days. On the last day, the \( n \)-th day, the remaining \( n \) medals were given out. How many days did the event last, and how many medals were distributed in total? | n = 6, \, m = 36 | 0.125 |
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