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A cashier from Aeroflot has to deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba", "Rossiya", and "Minsk". The fourth group's address will be given by tourists from "Rossiya", and the fifth group's address will be given by tourists from "Minsk". In how many ways can the cashier choose the order of visiting the hotels to deliver the tickets? | 30 | 0.625 |
In a right triangle \(ABC\) with hypotenuse \(AC\) equal to 2, medians \(AM\) and \(CN\) are drawn. A circle can be circumscribed around quadrilateral \(ANMC\). Find the radius of this circle. | \frac{\sqrt{5}}{2} | 0.5 |
Let \( S = \{1, 2, \cdots, 2005\} \). Find the minimum value of \( n \) such that any set of \( n \) pairwise coprime elements from \( S \) contains at least one prime number. | 16 | 0.375 |
A chessboard has its rows and columns numbered as illustrated in the figure. In each cell, the number written is the sum of the numbers of the row and column of that cell. For example, in the cell located in row 4 and column 5, the number written is $4+5=9$.
a) What is the sum of the numbers written in all the cells of the chessboard?
b) Let $S_{\text {black}}$ be the sum of all the numbers written in the black cells and $S_{\text {white}}$ be the sum of all the numbers written in the white cells. What is the value of the difference $S_{\text {black}}-S_{\text {white}}$?
c) What is the value of $S_{\text {black}}$? | 288 | 0.125 |
Each of the lateral edges of a pyramid is equal to 269/32. The base of the pyramid is a triangle with sides 13, 14, 15. Find the volume of the pyramid. | \frac{483}{8} | 0.25 |
How many zeros are at the end of the number "100!" (i.e., the product of all integers from 1 to 100)? | 24 | 0.875 |
A circle is inscribed in a right triangle. One of the legs is divided by the point of tangency into segments measuring 6 and 10, starting from the vertex of the right angle. Find the area of the triangle. | 240 | 0.625 |
A binary operation $\star$ satisfies $(a \star b) \star c = a + b + c$ for all real numbers $a, b$, and $c$. Show that $\star$ is the usual addition. | a \star b = a + b | 0.25 |
In the regular quadrilateral pyramid $P-ABCD$, $G$ is the centroid of $\triangle PBC$. Find the value of $\frac{V_{G-PAD}}{V_{G-PAB}}$. | 2 | 0.625 |
If the sum of roots of \(5 x^{2} + a x - 2 = 0\) is twice the product of roots, find the value of \(a\).
Given that \(y = a x^{2} - b x - 13\) passes through \((3,8)\), find the value of \(b\).
If there are \(c\) ways of arranging \(b\) girls in a circle, find the value of \(c\).
If \(\frac{c}{4}\) straight lines and 3 circles are drawn on a paper, and \(d\) is the largest number of points of intersection, find the value of \(d\). | 57 | 0.875 |
Let \( \triangle ABC \) be an acute triangle with circumcenter \( O \) such that \( AB = 4 \), \( AC = 5 \), and \( BC = 6 \). Let \( D \) be the foot of the altitude from \( A \) to \( BC \), and \( E \) be the intersection of \( AO \) with \( BC \). Suppose that \( X \) is on \( BC \) between \( D \) and \( E \) such that there is a point \( Y \) on \( AD \) satisfying \( XY \parallel AO \) and \( YO \perp AX \). Determine the length of \( BX \). | \frac{96}{41} | 0.5 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 1} \frac{1+\cos \pi x}{\operatorname{tg}^{2} \pi x}
\] | \frac{1}{2} | 0.75 |
The MK-97 microcalculator can perform only three operations on numbers stored in its memory:
1. Check if two selected numbers are equal.
2. Add selected numbers.
3. Given selected numbers \(a\) and \(b\), find the roots of the equation \(x^2 + ax + b = 0\), and indicate if there are no roots.
All results of these actions are stored in memory. Initially, one number \(x\) is stored in memory. How can one use the MK-97 to determine if this number is equal to one? | x = 1 | 0.875 |
Given positive real numbers \( x \) and \( y \) (\( x > y \)) satisfying \( x y = 490 \) and \( (\lg x - \lg 7)(\lg y - \lg 7) = -\frac{143}{4} \), determine the number of digits in the integer part of \( x \) in decimal representation. | 8 | 0.5 |
The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.
How many fruits in total grew on the miracle tree? | 30 | 0.375 |
Consider the integer sequence \( a_{1}, a_{2}, \cdots, a_{10} \) satisfying:
\[
a_{10} = 3a_{1}, \quad a_{2} + a_{8} = 2a_{5}
\]
and \( a_{i+1} \in \{1 + a_{i}, 2 + a_{i}\} \) for \(i = 1, 2, \cdots, 9\). Find the number of such sequences. | 80 | 0.375 |
The sum and difference of 44 and 18 consist of the same digits, just in reverse order. How many pairs of two-digit numbers have this property? | 9 | 0.25 |
On side $BC$ of a triangle $ABC$ with an obtuse angle at $C$, a point $M$ is marked. A point $D$ is chosen such that triangle $BCD$ is acute, and points $A$ and $D$ lie on opposite sides of line $BC$. Circles $\omega_{B}$ and $\omega_{C}$ are circumscribed around triangles $BMD$ and $CMD$ respectively. Side $AB$ intersects circle $\omega_{B}$ again at point $P$, and ray $AC$ intersects circle $\omega_{C}$ again at point $Q$. Segment $PD$ intersects circle $\omega_{C}$ again at point $R$, and ray $QD$ intersects circle $\omega_{B}$ again at point $S$. Find the ratio of the areas of triangles $ABR$ and $ACS$. | 1 | 0.625 |
On the sides \( BC \) and \( CD \) of the square \( ABCD \), points \( M \) and \( K \) are marked respectively such that \(\angle BAM = \angle CKM = 30^{\circ}\). Find \(\angle AKD\). | 75^\circ | 0.75 |
Define the efficiency of a natural number \( n \) as the proportion of all natural numbers from 1 to \( n \) (inclusive) that share a common divisor with \( n \) greater than 1. For example, the efficiency of the number 6 is \( \frac{2}{3} \).
a) Does a number exist with efficiency greater than 80%? If so, find the smallest such number.
b) Is there a number whose efficiency is the highest possible (i.e., not less than that of any other number)? If so, find the smallest such number. | 30030 | 0.625 |
Let the function \( f(x)=\frac{x^{2}+x+16}{x} \) where \( 2 \leqslant x \leqslant a \) and the real number \( a > 2 \). If the range of \( f(x) \) is \([9, 11]\), then find the range of values for \( a \). | [4, 8] | 0.75 |
The function \( f \) is defined on the set of integers and satisfies:
\[
f(n)=\left\{
\begin{array}{ll}
n-3 & \text{if } n \geq 1000 \\
f[f(n+5)] & \text{if } n < 1000
\end{array}
\right.
\]
Find \( f(84) \). | 997 | 0.25 |
A section of a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ by a plane passing through its vertex $D$ intersects the lateral edges $A A_{1}$, $B B_{1}$, and $C C_{1}$ at points $K$, $M$, and $N$, respectively. Find the ratio of the volume of the pyramid with vertex at point $P$ and base $D K M N$ to the volume of the given parallelepiped, if it is known that point $P$ divides the edge $D D_{1}$ in the ratio $D P : P D_{1} = m : n$. | \frac{m}{3(m+n)} | 0.75 |
For which values of \( n \) is the expression \( 2^{n} + 1 \) a nontrivial power of a natural number? | 3 | 0.75 |
Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron? | \frac{1}{8} | 0.625 |
Pigsy picked a bag of wild fruits from the mountain to bring back to Tang Monk. As he walked, he got hungry and took out half of them, hesitated, put back two, and then ate the rest. This happened four times: taking out half, putting back two, and then eating. By the time he reached Tang Monk, there were 5 wild fruits left in the bag. How many wild fruits did Pigsy originally pick? | 20 | 0.875 |
The polynomial \( P(x) \) is such that \( P\left(x^{2}\right) \) has \( 2n+1 \) roots. What is the minimum number of roots that the derivative of the polynomial \( P(x) \) can have?
(In both cases, distinct roots are considered, without considering multiplicity.) | n | 0.75 |
There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number of white balls drawn? | \frac{19}{60} | 0.5 |
In an isosceles triangle \( ABC \) (\( AB = BC \)), a circle is inscribed. A line parallel to side \( BC \) and tangent to the circle intersects side \( AB \) at a point \( N \) such that \( AN = \frac{3}{8} AB \). Find the radius of the circle if the area of triangle \( ABC \) is 12. | \frac{3}{2} | 0.375 |
In a \(3 \times 3\) table, 9 numbers are arranged such that all six products of these numbers in the rows and columns of the table are different. What is the maximum number of numbers in this table that can be equal to one? | 5 | 0.75 |
Divide an equilateral triangle into $n^{2}$ equally sized smaller equilateral triangles. Number some of these smaller triangles with the numbers $1, 2, 3, \cdots, m$ such that adjacent triangles have adjacent numbers. Show that $m \leq n^{2} - n + 1$. | m \leq n^2 - n + 1 | 0.75 |
Find all triplets of positive integers \((m, n, p)\) with \(p\) being a prime number, such that \(2^{m} p^{2} + 1 = n^{5}\). | (1, 3, 11) | 0.75 |
The triangle ABC has area 1. Take points X on AB and Y on AC so that the centroid G is on the opposite side of XY to B and C. Show that the area of BXGY plus the area of CYGX is greater than or equal to 4/9. When do we have equality? | \frac{4}{9} | 0.875 |
In the right triangle \(ABC\) with the right angle at \(C\), points \(P\) and \(Q\) are the midpoints of the angle bisectors drawn from vertices \(A\) and \(B\). The circle inscribed in the triangle touches the hypotenuse at point \(H\). Find the angle \(PHQ\). | 90^\circ | 0.75 |
Let \( a_1, a_2, \ldots, a_{24} \) be integers with sum 0 and satisfying \( |a_i| \leq i \) for all \( i \). Find the greatest possible value of \( a_1 + 2a_2 + 3a_3 + \cdots + 24a_{24} \). | 1432 | 0.375 |
Find all strictly increasing functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all \( n \geq 0 \), \( f(f(n)) < n + 1 \). | f(n) = n | 0.875 |
Triangle \(ABC\) has side lengths \(AB=2\sqrt{5}\), \(BC=1\), and \(CA=5\). Point \(D\) is on side \(AC\) such that \(CD=1\), and \(F\) is a point such that \(BF=2\) and \(CF=3\). Let \(E\) be the intersection of lines \(AB\) and \(DF\). Find the area of \(CDEB\). | \frac{22}{35} | 0.5 |
Given that $\left\{a_{n}\right\}$ is a geometric sequence and $a_{1} a_{2017}=1$, and if $f(x)=\frac{2}{1+x^{2}}$, find $\sum_{i=1}^{2017} f\left(a_{i}\right)$. | 2017 | 0.75 |
Triangle \(A B C\) has side lengths \(A B = 65\), \(B C = 33\), and \(A C = 56\). Find the radius of the circle tangent to sides \(A C\) and \(B C\) and to the circumcircle of triangle \(A B C\). | 24 | 0.375 |
In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \( a \) and \( b \). What is the maximum number of marked points that can lie on a line that does not coincide with \( a \) or \( b \)? | 23 | 0.125 |
Let $S$ be a set of $n$ distinct real numbers, and let $A_{s}$ be the set of averages of all distinct pairs of elements from $S$. For a given $n \geq 2$, what is the minimum possible number of elements in $A_{s}$? | 2n - 3 | 0.875 |
Calculate the areas of the figures bounded by the lines given in polar coordinates.
$$
r=\frac{5}{2} \sin \phi, \quad r=\frac{3}{2} \sin \phi
$$ | \pi | 0.625 |
In the right triangle \(ABC\) with a right angle at \(A\), the altitude \(AH\) is drawn. A circle passing through points \(A\) and \(H\) intersects the legs \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. Find the length of segment \(AC\) given that \(AX = 5\), \(AY = 6\), and \(AB = 9\). | 13.5 | 0.25 |
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle. | 16 | 0.125 |
If \( P(x, y) \) is a point on the hyperbola \(\frac{x^{2}}{8}-\frac{y^{2}}{4}=1\), then the minimum value of \(|x-y|\) is ________. | 2 | 0.75 |
Find the smallest positive integer \( n \) such that, if there are initially \( 2n \) townspeople and 1 goon, then the probability that the townspeople win is greater than 50%. | 3 | 0.125 |
The sum of \( m \) distinct positive even numbers and \( n \) distinct positive odd numbers is 1987. Determine the maximum value of \( 3m + 4n \) for all such \( m \) and \( n \). Provide a proof for your conclusion. | 221 | 0.375 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=p, a_{2}=p+1, a_{n+2}-2 a_{n+1}+a_{n}=n-20$, where $p$ is a given real number and $n$ is a positive integer, find the value of $n$ that makes $a_{n}$ minimal. | 40 | 0.625 |
How many functions \( f: \{1, 2, \ldots, 10\} \rightarrow \{1, 2, \ldots, 10\} \) satisfy the property that \( f(i) + f(j) = 11 \) for all values of \( i \) and \( j \) such that \( i + j = 11 \)? | 100000 | 0.75 |
Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______. | \frac{\pi}{6} | 0.75 |
In preparation for an exam, three students solved 100 problems. Each student solved 60 problems, and every problem was solved by at least one student. A problem is considered difficult if it was solved by only one student. A problem is considered easy if it was solved by all three students. Are there more easy problems or difficult problems, and by how many? | 20 | 0.875 |
A $10 \times 10$ table is filled with numbers from 1 to 100: in the first row, the numbers from 1 to 10 are listed in ascending order from left to right; in the second row, the numbers from 11 to 20 are listed in the same way, and so on; in the last row, the numbers from 91 to 100 are listed from left to right. Can a segment of 7 cells of the form $\quad$ be found in this table, where the sum of the numbers is $455?$ (The segment can be rotated.) | Yes | 0.75 |
There are 70 points on a circle. Choose one point and label it 1. Move clockwise, skipping one point, and label the next point 2. Skip two points and label the next point 3. Continue this pattern, until all numbers from 1 to 2014 have been labeled on the points. Each point may have more than one number labeled on it. What is the smallest integer labeled on the point that is also labeled with 2014? | 5 | 0.5 |
How many pairs of two-digit positive integers have a difference of 50? | 40 | 0.75 |
The famous Fibonacci sequence is defined as follows:
$$
\begin{gathered}
F_{0}=0, F_{1}=1 \\
F_{n+2}=F_{n+1}+F_{n}
\end{gathered}
$$
1. Let $\varphi$ and $\varphi'$ be the two roots of the equation $x^2 - x - 1 = 0$.
$$
\varphi=\frac{1+\sqrt{5}}{2}, \varphi^{\prime}=\frac{1-\sqrt{5}}{2}
$$
Show that there exist two real numbers $\lambda$ and $\mu$ such that
$$
F_{n}=\lambda \cdot \varphi^{n}+\mu \cdot \varphi^{\prime n}
$$
Can you generalize the result to any sequences of the type $u_{n+2} = a u_{n+1} + b u_{n}$?
2. Show that
$$
\lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_{n}} = \varphi
$$
3. Show the following formulas:
$$
\begin{gathered}
\sum_{i=1}^{n} F_{i}^{2}=F_{n} F_{n+1} \\
F_{2n-1}=F_{n}^{2} + F_{n+1}^{2} \\
m \mid n \Rightarrow F_{m} \mid F_{n}
\end{gathered}
$$ | \varphi | 0.375 |
A polynomial \( P \) has four roots: \( \frac{1}{4}, \frac{1}{2}, 2, 4 \). The product of the roots is 1, and \( P(1) = 1 \). Find \( P(0) \). | \frac{8}{9} | 0.875 |
Find the number of all five-digit numbers \( \overline{abcde} \) where all digits are different, and \( a < b < c > d > e \). | 1134 | 0.125 |
In a \( 29 \times 29 \) table, the numbers \( 1, 2, 3, \ldots, 29 \) are each written 29 times. It is found that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table. | 15 | 0.625 |
The great commander, Marshal of the Soviet Union Georgy Konstantinovich Zhukov, was born in the village of Strelkovka in the Kaluga province. He lived for 78 years. In the 20th century, he lived 70 years more than in the 19th century. In what year was G.K. Zhukov born? | 1896 | 0.375 |
On the minor axis of an ellipse, at a distance $e=O F$ (where $O$ is the center of the ellipse and $F$ is one of the foci) from the center, two points are taken, and from these points, perpendiculars to an arbitrary tangent are drawn. Show that the sum of the squares of these perpendiculars is $2 a^{2}$. | 2 a^2 | 0.5 |
In triangle \( A B C \), the angle bisector \( A M \) is perpendicular to the median \( B K \). Find the ratios \( B P : P K \) and \( A P : P M \), where \( P \) is the intersection point of the angle bisector and the median. | AP : PM = 3:1 | 0.125 |
In a magical country, there are only two types of people: type A people who always tell the truth and type B people who always lie. One day, 2014 citizens of this country lined up in a row, and each person said, "There are more type B people behind me than type A people in front of me." How many type A people are there among these 2014 citizens? | 1007 | 0.875 |
A steamboat travels from point $A$ to point $B$ in $a$ hours and back in $b$ hours. How many hours will a raft take to drift from $A$ to $B$? | \frac{2ab}{b-a} | 0.75 |
Let \( A \) and \( B \) be two positive four-digit integers such that \( A \times B = 16^5 + 2^{10} \). Find the value of \( A + B \). | 2049 | 0.375 |
The transgalactic ship encountered an amazing meteor shower. One part of the meteors flies along a straight line with equal speeds, one after another, at equal distances from each other. Another part flies similarly but along another straight line, parallel to the first one, with the same speeds but in the opposite direction, at the same distance from each other. The ship flies parallel to these lines. Astronaut Gavrila recorded that the ship encounters meteors flying towards it every 7 seconds, and those flying in the same direction as the ship every 13 seconds. He wondered how often the meteors would pass by if the ship were stationary. He thought it necessary to take the arithmetic mean of the two given times. Is Gavrila right? If so, write this arithmetic mean as the answer. If not, indicate the correct time in seconds, rounded to one decimal place. | 9.1 | 0.25 |
Find the smallest number \( k \) such that \(\frac{t_{a} + t_{b}}{a + b} < k\), where \( a \) and \( b \) are two side lengths of a triangle, and \( t_{a} \) and \( t_{b} \) are the lengths of the angle bisectors corresponding to these sides, respectively. | \frac{4}{3} | 0.375 |
Suppose \( S = \{1,2, \cdots, 2005\} \). Find the minimum value of \( n \) such that every subset of \( S \) consisting of \( n \) pairwise coprime numbers contains at least one prime number. | 16 | 0.625 |
Petra had natural numbers from 1 to 9 written down. She added two of these numbers, erased them, and wrote the resulting sum in place of the erased addends. She then had eight numbers left, which she was able to divide into two groups with the same product.
Determine the largest possible product of these groups. | 504 | 0.625 |
A circle is tangent to the extensions of two sides \( AB \) and \( AD \) of the square \( ABCD \), and the point of tangency cuts off a segment of 2 cm from vertex \( A \). Two tangents are drawn from point \( C \) to this circle. Find the side of the square if the angle between the tangents is \( 30^\circ \), and it is known that \( \sin 15^\circ = \frac{\sqrt{3}-1}{2 \sqrt{2}} \). | 2 \sqrt{3} | 0.25 |
$p$ and $q$ are primes such that the numbers $p+q$ and $p+7q$ are both squares. Find the value of $p$. | 2 | 0.25 |
A group consisting of 7 young men and 7 young women is randomly paired up. Find the probability that at least one pair consists of two young women. Round your answer to the nearest hundredth. | 0.96 | 0.625 |
Through the midpoint of the altitude $BB_1$ of triangle $ABC$, a line $DE$ is drawn parallel to the base. How does the area of triangle $BDE$ compare to the area of trapezoid $ADEC$? | 1:3 | 0.125 |
There are 207 different cards with numbers $1, 2, 3, 2^{2}, 3^{2}, \ldots, 2^{103}, 3^{103}$ (each card has one number, and each number appears exactly once). How many ways can you choose 3 cards such that the product of the numbers on the selected cards is a square of an integer that is divisible by 6? | 267903 | 0.375 |
Determine the value of \( c \) such that the equation
$$
5x^{2} - 2x + c = 0
$$
has roots with a ratio of \(-\frac{3}{5}\). | -3 | 0.875 |
The diagonals of a trapezoid are 12 and 6, and the sum of the bases is 14. Find the area of the trapezoid. | 16 \sqrt{5} | 0.5 |
There are 100 pieces, and two people take turns taking pieces. Each time, one is allowed to take either 1 or 2 pieces. The person who takes the last piece wins. If you go first, how many pieces should you take in the first turn to guarantee a win? | 1 | 0.875 |
In triangle \( \triangle ABC \), the sides opposite to the angles \( A \), \( B \), and \( C \) are of lengths \( a \), \( b \), and \( c \) respectively. Point \( G \) satisfies
$$
\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = \mathbf{0}, \quad \overrightarrow{GA} \cdot \overrightarrow{GB} = 0.
$$
If \((\tan A + \tan B) \tan C = m \tan A \cdot \tan B\), then \( m = \). | \frac{1}{2} | 0.125 |
Given a constant \( a \in (0,1) \), and \( |x| + |y| \leqslant 1 \), what is the maximum value of the function \( f(x, y) = a x + y \)? | 1 | 0.625 |
Find the value of the expression \( 1! \cdot 3 - 2! \cdot 4 + 3! \cdot 5 - 4! \cdot 6 + \ldots - 2000! \cdot 2002 + 2001! \). | 1 | 0.875 |
By how much did the dollar exchange rate change during the year 2014 (from January 1, 2014, to December 31, 2014)? Give your answer in rubles, rounded to the nearest whole number. | 24 | 0.125 |
In an $n$-throw game of dice, the sum of the points from all throws must be greater than $2^n$ to pass the round.
1. How many rounds can a person pass at most?
2. What is the probability that the person passes the first three rounds in succession?
(Note: The dice is a fair six-sided cube with faces numbered 1 through 6. When the dice come to rest after being thrown, the number on the upper face is the point scored.) | \frac{100}{243} | 0.375 |
The radii of the excircles of a triangle are 3, 10, and 15 respectively. What is the perimeter of the triangle? | 30 | 0.5 |
8 people decide to hold daily meetings subject to the following rules. At least one person must attend each day. A different set of people must attend on different days. On day N, for each 1 ≤ k < N, at least one person must attend who was present on day k. How many days can the meetings be held? | 128 | 0.5 |
In triangle \(ABC\), points \(C_{1}\), \(B_{1}\), and \(A_{1}\) are located on sides \(AB\), \(AC\), and \(CB\), respectively, such that
\[ \frac{AC_{1}}{C_{1}B} = \frac{BA_{1}}{A_{1}C} = \frac{CB_{1}}{B_{1}A} = \lambda \]
Let the radii of the incircles of triangles \(AC_{1}B_{1}\), \(BC_{1}A_{1}\), \(CA_{1}B_{1}\), \(A_{1}B_{1}C_{1}\), and \(ABC\) be \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\), and \(r\) respectively. For what values of \(\lambda\) does the following equation hold?
\[ \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{1}{r_{4}} + \frac{4}{r} \] | \lambda = 1 | 0.875 |
Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles intersecting at points $\mathrm{P}$ and $\mathrm{Q}$. Let $A$ and $B$ be two points on $\Gamma_{1}$ such that $A, P, Q,$ and $B$ are concyclic in this order, and let $C$ and $D$ be the second intersections with $\Gamma_{2}$ of lines (AP) and (BQ) respectively. Show that $(AB) \parallel (CD)$. | AB \parallel CD | 0.375 |
Find the convergence region of the series:
1) \(\sum_{n=1}^{\infty} \frac{x^{n}}{n!}=\frac{x}{11}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\ldots+\frac{x^{n}}{n!}+\ldots\);
2) \(\sum_{n=1}^{\infty} n \cdot n! x^{n}=1! x+2! x^{2}+3! x^{3}+\ldots+n! x^{n}+\ldots\);
3) \(\sum_{n=1}(-1)^{n-1} \frac{x^{n}}{n^{2}}=\frac{x}{1^{2}}-\frac{x^{2}}{2^{2}}+\frac{x^{3}}{3^{2}}-\frac{x^{4}}{4^{2}}+\ldots+(-1)^{n-1} \frac{x^{n}}{n^{2}}+\ldots\);
4) \(\sum_{n=0}^{\infty} 2^{n}(x-1)^{n}=1+2(x-1)+2^{2}(x-1)^{2}+\ldots+2^{n}(x-1)^{n}+\ldots\);
5) \(\sum_{n=1}^{\infty} \frac{x^{n}}{n^{n}}\). | (-\infty, \infty) | 0.25 |
Given \(0 < x < 1, a > 0, a \neq 1\), compare the magnitudes of \(p = \left|\log_a(1 - x)\right|\) and \(q = \left|\log_a(1 + x)\right| \). | p > q | 0.75 |
Given that \( f(x) \) is an \( n \)-degree polynomial of \( x \) where \( n > 0 \), and for any real number \( x \), the equation \( 8 f\left(x^{3}\right) - x^{6} f(2x) - 2 f\left(x^{2}\right) + 12 = 0 \) holds.
(1) Find \( f(x) \). | f(x) = x^3 - 2 | 0.875 |
In quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(AC\) and is perpendicular to side \(AD\), while diagonal \(AC\) is perpendicular to side \(CD\). On side \(AD\), a point \(K\) is taken such that \(AC = AK\). The angle bisector of \(\angle ADC\) intersects \(BK\) at point \(M\). Find the angle \(\angle ACM\). | 45^\circ | 0.875 |
Solve the systems of equations:
a)
\[
\begin{cases}
x + y + xy = 5, \\
xy(x + y) = 6;
\end{cases}
\]
b)
\[
\begin{cases}
x^3 + y^3 + 2xy = 4, \\
x^2 - xy + y^2 = 1.
\end{cases}
\] | (1, 1) | 0.375 |
Let \( ABC \) be a triangle such that \(\angle CAB = 20^\circ\). Let \( D \) be the midpoint of segment \([AB]\). It is given that \(\angle CDB = 40^\circ\). What is the measure of \(\angle ABC\)? | 70^\circ | 0.875 |
Oleg has 550 rubles and wants to buy an odd number of tulips, making sure that no color is repeated. In the store where Oleg goes, one tulip costs 49 rubles, and there are eleven different shades available. How many different ways are there for Oleg to give flowers to his mother? (The answer should be a compact expression that does not contain summation signs, ellipses, etc.) | 1024 | 0.625 |
Given \(\frac{ab}{a+b}=3\), \(\frac{bc}{b+c}=6\), and \(\frac{ac}{a+c}=9\), determine the value of \(\frac{c}{ab}\). | -\frac{35}{36} | 0.875 |
Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\). | 2 | 0.125 |
In a rhombus with an acute angle of $30^{\circ}$, a circle is inscribed, and a square is inscribed in the circle. Find the ratio of the area of the rhombus to the area of the square. | 4 | 0.875 |
Let \( n \) be a fixed integer, \( n \geq 2 \).
(I) Find the minimum constant \( c \) such that the inequality
\[ \sum_{1 \leq i < j \leq n} x_i x_j \left( x_i^2 + x_j^2 \right) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]
holds.
(II) For this constant \( c \), determine the necessary and sufficient conditions for which equality holds. | \frac{1}{8} | 0.375 |
2002 is a palindromic year, meaning it reads the same backward and forward. The previous palindromic year was 11 years ago (1991). What is the maximum number of non-palindromic years that can occur consecutively (between the years 1000 and 9999)? | 109 | 0.375 |
As shown in the figure, an equilateral triangle $ABC$ with a side length of 4 is folded along a line $EF$ parallel to side $BC$ in such a way that plane $AEF$ is perpendicular to plane $BCFE$. Let $O$ be the midpoint of $EF$.
(1) Find the cosine value of the dihedral angle $F-AE-B$.
(2) If $BE$ is perpendicular to plane $AOC$, find the length of the fold $EF$. | \frac{8}{3} | 0.25 |
Given that \( 100^{2} + 1^{2} = 65^{2} + 76^{2} = pq \) for some primes \( p \) and \( q \). Find \( p + q \). | 210 | 0.5 |
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | 21 | 0.25 |
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