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Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral \(ABED\). | \frac{28}{3} | 0.375 |
Olya drew $N$ different lines on a plane, any two of which intersect. It turned out that among any 15 lines, there are always two that form an angle of $60^{\circ}$ between them. What is the largest possible value of $N$ for which this is possible? | 42 | 0.125 |
Shan solves the simultaneous equations
$$
xy = 15 \text { and } (2x - y)^4 = 1
$$
where \(x\) and \(y\) are real numbers. She calculates \(z\), the sum of the squares of all the \(y\)-values in her solutions. What is the value of \(z\)? | 122 | 0.75 |
On the board, the number \( x = 9999 \) is written in a numeral system with an even base \( r \). Vasya found out that the \( r \)-ary representation of \( x^2 \) is an eight-digit palindrome, in which the sum of the second and third digits is 24. (A palindrome is a number that reads the same from left to right and right to left). For which \( r \) is this possible? | r = 26 | 0.125 |
The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minutes, given that it pointed exactly to 12 o'clock at the beginning of the five-minute interval of correct movement? | 28 | 0.75 |
Given a three-digit number \( N \), the remainders when \( N \) is divided by 3, 7, and 11 are 1, 3, and 8, respectively. Find the maximum value of \( N \). | 976 | 0.75 |
In triangle \(ABC\), we have \(AB = BC = 5\) and \(CA = 8\). What is the area of the region consisting of all points inside the triangle which are closer to \(AB\) than to \(AC\)? | \frac{60}{13} | 0.625 |
Find the maximum value of the expression \( ab + bc + ac + abc \) given that \( a + b + c = 12 \) (where \( a, b, \) and \( c \) are non-negative numbers). | 112 | 0.875 |
Given two linear functions $f(x)$ and $g(x)$ such that the graphs of $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6. | \frac{11}{2} | 0.875 |
Let the line \( L: lx + my + n = 0 \) intersect the parabola \( y^2 = 4ax \) at points \( P \) and \( Q \). \( F \) is the focus of the parabola. The line segments \( PF \) and \( QF \) intersect the parabola at points \( R \) and \( S \). Determine the equation of the line \( RS \). | nx - may + la^2 = 0 | 0.875 |
Altitudes \( BE \) and \( CF \) of acute triangle \( ABC \) intersect at \( H \). Suppose that the altitudes of triangle \( EHF \) concur on line \( BC \). If \( AB=3 \) and \( AC=4 \), then \( BC^2=\frac{a}{b} \), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a+b\). | 33725 | 0.125 |
Determine all positive integers \( n \) such that:
\[ 5^{n-1} + 3^{n-1} \mid 5^n + 3^n \] | 1 | 0.5 |
Find the probability that a randomly chosen number on the interval $[0 ; 5]$ is a solution to the equation $\sin (x+|x-\pi|)+2 \sin ^{2}(x-|x|)=0$. | \frac{\pi}{5} | 0.75 |
Let the set \( A = \{1, 2, 3, 4, 5, 6\} \) and the mapping \( f: A \rightarrow A \). If the triple composition \( f \cdot f \cdot f \) is an identity mapping, how many such functions \( f \) are there? | 81 | 0.625 |
A positive integer \( n \) is said to be good if \( 3n \) is a re-ordering of the digits of \( n \) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11. | 2475 | 0.875 |
Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\). | \frac{d}{2} | 0.5 |
Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \). | \frac{\sqrt{3}}{3} | 0.375 |
Given that the sequence $\left\{a_{n}\right\}$ is an arithmetic sequence with the first term and common difference both being positive, and that $a_{2}$, $a_{5}$, and $a_{9}$ form a geometric sequence in order, find the smallest positive integer value of $k$ such that $a_{1}+a_{2}+\cdots+a_{k}>100 a_{1}$. | 34 | 0.875 |
Several non-intersecting line segments are placed on a plane. Is it always possible to connect their endpoints with additional line segments so that a closed, non-self-intersecting polygonal chain is formed? | \text{No} | 0.25 |
A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task can be completed one hour earlier. Similarly, if the positions of workers $C$ and $D$ are swapped, the task can also be completed one hour earlier. How many minutes earlier can the task be completed if the positions of $A$ and $B$ as well as $C$ and $D$ are swapped at the same time, assuming other workers' efficiency remains unchanged? | 108 \text{ minutes} | 0.875 |
Given \( x \in [0, 3] \), find the maximum value of \( \frac{\sqrt{2 x^3 + 7 x^2 + 6 x}}{x^2 + 4 x + 3} \). | \frac{1}{2} | 0.625 |
In triangle \(ABC\), point \(K\) is taken on side \(AB\) such that \(AK: BK = 1: 2\), and point \(L\) is taken on side \(BC\) such that \(CL: BL = 2: 1\). Let \(Q\) be the point of intersection of lines \(AL\) and \(CK\). Find the area of triangle \(ABC\) if the area of triangle \(BQC\) is 1. | \frac{7}{4} | 0.125 |
Compute the definite integral:
$$
\int_{0}^{5} x^{2} \cdot \sqrt{25-x^{2}} \, dx
$$ | \frac{625 \pi}{16} | 0.625 |
In a sequence of 37 numbers, the sum of every six consecutive numbers is 29. The first number is 5. What could the last number be? | 5 | 0.75 |
A cylinder with a base diameter of 12 is cut by a plane that forms a $30^{\circ}$ angle with the base, forming an ellipse $S$. What is the eccentricity of the ellipse $S$? | \frac{1}{2} | 0.625 |
Find all positive integer solutions \((x, y, z, n)\) that satisfy the equation \(x^{2n+1} - y^{2n+1} = xyz + 2^{2n+1}\), where \(n \geq 2\) and \(z \leq 5 \cdot 2^{2n}\). | (3, 1, 70, 2) | 0.75 |
Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain? | 2118 | 0.375 |
In a sequence, all natural numbers from 1 to 2017 inclusive were written down. How many times was the digit 7 written? | 602 | 0.375 |
a) What is the maximum number of squares on an $8 \times 8$ board that can be colored black so that in every "corner" of three squares, at least one square remains uncolored?
b) What is the minimum number of squares on an $8 \times 8$ board that must be colored black so that in every "corner" of three squares, at least one square is black? | 32 | 0.5 |
Rumcajs teaches Cipísek to write numbers. They started from one and wrote consecutive natural numbers. Cipísek pleaded to stop, and Rumcajs promised that they would stop writing when Cipísek had written a total of 35 zeros. What is the last number Cipísek writes? | 204 | 0.125 |
There are 16 distinct points on a circle. Determine the number of different ways to draw 8 nonintersecting line segments connecting pairs of points so that each of the 16 points is connected to exactly one other point. | 1430 | 0.75 |
The set of positive odd numbers $\{1, 3, 5, \cdots\}$ is arranged in ascending order and grouped by the $n$th group having $(2n-1)$ odd numbers as follows:
$$
\begin{array}{l}
\{1\}, \quad \{3,5,7\}, \quad \{9,11,13,15,17\}, \cdots \\
\text{ (First group) (Second group) (Third group) } \\
\end{array}
$$
In which group does the number 1991 appear? | 32 | 0.75 |
All lateral edges of the pyramid are equal to $b$, and the height is $h$. Find the radius of the circle circumscribed around the base. | \sqrt{b^2 - h^2} | 0.875 |
Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers? | 18 | 0.375 |
If there is a positive integer \( m \) such that the factorial of \( m \) has exactly \( n \) trailing zeros, then the positive integer \( n \) is called a "factorial trailing number." How many positive integers less than 1992 are non-"factorial trailing numbers"? | 396 | 0.25 |
Write the equation of a plane that passes through point \( A \) and is perpendicular to vector \( \overrightarrow{BC} \).
\( A(-3, 6, 4) \)
\( B(8, -3, 5) \)
\( C(10, -3, 7) \) | x + z - 1 = 0 | 0.875 |
As shown in the figure, square \(ABCD\) and rectangle \(BEFG\) share a common vertex \(B\). The side length of square \(ABCD\) is \(6 \, \text{cm}\) and the length of rectangle \(BEFG\) is \(9 \, \text{cm}\). Find the width of the rectangle in centimeters. | 4 \text{ cm} | 0.625 |
Twenty-five coins are divided into piles in the following way. First, they are randomly split into two groups. Then any of the existing groups is split into two groups again, and this process continues until each group consists of just one coin. Each time a group is split into two groups, the product of the number of coins in the two resulting groups is recorded. What could the sum of all the recorded numbers be? | 300 | 0.625 |
Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.
In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$? | 1:3 | 0.625 |
A three-digit natural number was written on the board. We wrote down all other three-digit numbers that can be obtained by rearranging its digits. Therefore, there were three new numbers in addition to the original number on the board. The sum of the smallest two of all four numbers is 1088. What digits does the original number contain? | 0, 5, 8 | 0.125 |
Let \( P(X) \) be a monic polynomial of degree 2017 such that \( P(1) = 1 \), \( P(2) = 2 \), ..., \( P(2017) = 2017 \). What is the value of \( P(2018) \)? | 2017! + 2018 | 0.625 |
A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \). | \frac{4}{15} | 0.5 |
Can the square of any natural number begin with 1983 nines? | Yes | 0.875 |
The center of circle $k$ is $O$. Points $A, B, C, D$ are consecutive points on the circumference of $k$ such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^\circ$. Point $E$ is the projection of $D$ onto the diameter $AO$. Point $F$ is the trisection point of segment $DE$ closer to $E$. Finally, $G$ is the intersection point of $AO$ and $BF$. What is the measure of angle $OGD$ as $\alpha \rightarrow 60^\circ$? | 60^\circ | 0.125 |
Find the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+2 n+3}{2 n^{2}+2 n+1}\right)^{3 n^{2}-7}
$$ | e^3 | 0.75 |
At a joint conference of the Parties of Liars and Truth-lovers, 32 people were elected to the presidium and seated in four rows of 8 people each. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie, and truth-lovers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one of them is seated to the left, right, in front, or behind the other). | 8 | 0.625 |
In the triangular prism \(P-ABC\), \(\triangle ABC\) is an equilateral triangle with side length \(2\sqrt{3}\), \(PB = PC = \sqrt{5}\), and the dihedral angle \(P-BC-A\) is \(45^\circ\). Find the surface area of the circumscribed sphere around the triangular prism \(P-ABC\). | 25\pi | 0.375 |
Several points were marked on a line. After that, a point was added between each pair of neighboring points. This operation was repeated three times, and as a result, there were 65 points on the line. How many points were there initially? | 9 | 0.875 |
Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( BC = 5 \) and \( AB = 4 \). Find the length of side \( AC \) if the length of the vector \( 3 \overrightarrow{OA} - 4 \overrightarrow{OB} + \overrightarrow{OC} \) is 10. | 8 | 0.875 |
A group of 101 Dalmatians participate in an election, where they each vote independently for either candidate \( A \) or \( B \) with equal probability. If \( X \) Dalmatians voted for the winning candidate, the expected value of \( X^{2} \) can be expressed as \( \frac{a}{b} \) for positive integers \( a, b \) with \( \operatorname{gcd}(a, b)=1 \). Find the unique positive integer \( k \leq 103 \) such that \( 103 \mid a - b k \). | 51 | 0.375 |
Consider a fixed circle $\mathscr{C}$ passing through a point $A$ and a fixed line $\mathscr{D}$ passing through a point $B$. A variable circle passing through $A$ and $B$ intersects the circle $\mathscr{C}$ at $C$ and the line $\mathscr{D}$ at $E$. The line $CE$ intersects $\mathscr{C}$ at $K$. Show that the point $K$ is fixed. | K \text{ is fixed} | 0.75 |
How many even six-digit numbers exist where the same digits are not next to each other? | 265721 | 0.125 |
Let \(\pi\) be a permutation of the numbers from 2 through 2012. Find the largest possible value of \(\log_{2} \pi(2) \cdot \log_{3} \pi(3) \cdots \log_{2012} \pi(2012)\). | 1 | 0.75 |
A five-digit number \(abcde\) satisfies:
\[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \]
For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Find the total number of five-digit numbers that follow the sine rule.
Note: Please disregard any references or examples provided within the original problem if they are not part of the actual problem statement. | 2892 | 0.5 |
For an integer \( n>3 \), we use \( n ? \) to represent the product of all prime numbers less than \( n \) (called " \( n \)-question mark"). Solve the equation \( n ? = 2n + 16 \). | n = 7 | 0.625 |
Given a prime number \( p \) and a positive integer \( n \) where \( p \geq n \geq 3 \). The set \( A \) consists of sequences of length \( n \) with elements taken from the set \(\{1, 2, \cdots, p\}\). If for any two sequences \((x_{1}, x_{2}, \cdots, x_{n})\) and \((y_{1}, y_{2}, \cdots, y_{n})\) in set \( A \), there exist three distinct positive integers \( k, l, m \) such that \( x_{k} \neq y_{k}, x_{l} \neq y_{l}, x_{m} \neq y_{m} \), find the maximum number of elements in set \( A \). | p^{n-2} | 0.375 |
We rotate the square \(ABCD\) with a side length of 1 around its vertex \(C\) by \(90^\circ\). What area does side \(AB\) sweep out? | \frac{\pi}{4} | 0.875 |
Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days? | Yes | 0.875 |
Show that if the element \(a_{ik}\) of an \(n\)-th order determinant is equal to the smaller of the numbers \(i\) and \(k\) (or the common value of \(i\) and \(k\) when \(i = k\)), then the value of the determinant is 1. | 1 | 0.75 |
Find all solutions in positive integers to the following system of equations:
\[ a + b = \gcd(a, b)^2 \]
\[ b + c = \gcd(b, c)^2 \]
\[ c + a = \gcd(c, a)^2 \] | (2, 2, 2) | 0.5 |
1. Two people, A and B, are playing a game: A writes two rows, each with 10 numbers, arranged such that they satisfy the following rule: If $b$ is below $a$ and $d$ is below $c$, then $a+d=b+c$. After knowing this rule, B wants to determine all the written numbers. B can ask A questions like "What is the number at the third position in the first row?" or "What is the number at the ninth position in the second row?" How many such questions does B need to ask to determine all the numbers?
2. In an $m \times n$ grid filled with numbers, any rectangle formed by selecting two rows and two columns has the property that the sum of the numbers at two opposite corners equals the sum of the numbers at the other two opposite corners. Show that if some numbers are erased and the remaining numbers allow reconstruction of the erased numbers, at least $m+n-1$ numbers must remain. | m + n - 1 | 0.375 |
Seven members of the family are each to pass through one of seven doors to complete a challenge. The first person can choose any door to activate. After completing the challenge, the adjacent left and right doors will be activated. The next person can choose any unchallenged door among the activated ones to complete their challenge. Upon completion, the adjacent left and right doors to the chosen one, if not yet activated, will also be activated. This process continues until all seven members have completed the challenge. The order in which the seven doors are challenged forms a seven-digit number. How many different possible seven-digit numbers are there? | 64 | 0.5 |
If \(\alpha\) is a real root of the equation \(x^{5}-x^{3}+x-2=0\), find the value of \(\left\lfloor\alpha^{6}\right\rfloor\), where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\). | 3 | 0.5 |
Two symmetrical coins are flipped. What is the probability that both coins show numbers on their upper sides? | 0.25 | 0.25 |
There is a five-digit positive odd number \( x \). In \( x \), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x + 1) \), what is the value of \( x \)?
(Chinese Junior High School Mathematics Competition, 1987) | 29995 | 0.625 |
In triangle \( \triangle ABC \), the median \( BM \) is drawn. It is given that \( AB = 2BM \) and \( \angle MBA = 40^\circ \). Find \( \angle CBA \). | 110^\circ | 0.625 |
The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. | \sqrt{3} | 0.5 |
Calculate the volumes of solids formed by the rotation of figures bounded by the graphs of the functions around the x-axis.
$$
y=5 \cos x, \quad y=\cos x, \quad x=0, \quad x \geq 0
$$ | 6 \pi^2 | 0.75 |
As shown in the figure, in quadrilateral $ABCD$, $AB=7$, $BC=24$, $CD=20$, $DA=15$, and $\angle B=90^{\circ}$. What is the area of quadrilateral $ABCD$? | 234 | 0.75 |
Find all natural numbers \( k \) such that \( 3^{k} + 5^{k} \) is a power of a natural number with an exponent \(\geq 2\). | 1 | 0.75 |
From point \( O \), three rays \( OA, OB, OC \) are drawn, with the angles between each pair of rays being \( 60^{\circ}, 90^{\circ}, \) and \( 120^{\circ} \) respectively. What is the minimum angle between the angle bisectors of these three angles? | 45^\circ | 0.75 |
Given that the polynomial \(x^2 - x + a\) can divide \(x^{13} + x + 90\), determine the positive integer value of \(a\). | 2 | 0.75 |
Find three numbers forming a geometric progression, knowing that their product is 64 and their arithmetic mean is \( \frac{14}{3} \). | 8, 4, 2 | 0.125 |
The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle. | 6 \sqrt[4]{3} | 0.75 |
A certain project takes 24 days for person A to complete alone, 36 days for person B to complete alone, and 60 days for person C to complete alone. Given that all three teams worked for an exact integer number of days and completed the task within 18 days (including 18 days), how many days did person A work at the minimum? | 6 | 0.375 |
In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle? | 1 : \sqrt{2} | 0.5 |
How many solutions does the equation $\left\lfloor\frac{x}{10}\right\rfloor = \left\lfloor\frac{x}{11}\right\rfloor + 1$ have in integers? | 110 | 0.25 |
Let \( p \) be a prime number and the sequence \(\{a_n\}_{n \geq 0}\) satisfy \( a_0 = 0 \), \( a_1 = 1 \), and for any non-negative integer \( n \), \( a_{n+2} = 2a_{n+1} - p a_n \). If \(-1\) is a term in the sequence \(\{a_n\}\), find all possible values of \( p \). | 5 | 0.875 |
Let \( X = \{0, a, b, c\} \) and \( M(X) = \{ f \mid f: X \rightarrow X \} \) be the set of all functions from \( X \) to itself. Define the addition operation \( \oplus \) on \( X \) as given in the following table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\oplus & 0 & a & b & c \\
\hline
0 & 0 & a & b & c \\
\hline
a & a & 0 & c & b \\
\hline
b & b & c & 0 & a \\
\hline
c & c & b & a & 0 \\
\hline
\end{array}
\]
1. Determine the number of elements in the set:
\[
S = \{ f \in M(X) \mid f((x \oplus y) \oplus x) = (f(x) \oplus f(y)) \oplus f(x), \forall x, y \in X \}.
\]
2. Determine the number of elements in the set:
\[
I = \{ f \in M(X) \mid f(x \oplus x) = f(x) \oplus f(x), \forall x \in X \}.
\] | 64 | 0.375 |
Dima has 25 identical bricks of size $5 \times 14 \times 17$. Dima wants to construct a single tower using all of his bricks, each time adding one brick on top (each new brick adds 5, 14, or 17 to the current height of the tower). A number $n$ is called constructible if Dima can build a tower with a height of exactly $n$. How many constructible numbers exist? | 98 | 0.375 |
A king has eight sons, and they are all fools. Each night, the king sends three of them to guard the golden apples from the Firebird. The princes cannot catch the Firebird and blame each other, so no two of them agree to go on guard together a second time. What is the maximum number of nights this can continue? | 8 | 0.25 |
Given an arithmetic sequence where the sum of the first 4 terms is 26, the sum of the last 4 terms is 110, and the sum of all terms is 187, how many terms are in this sequence? | 11 | 0.375 |
From the numbers $1,2,3, \cdots, 2014$, select 315 different numbers (order does not matter) to form an arithmetic sequence. Among these, the number of ways to form an arithmetic sequence that includes the number 1 is ___. The total number of ways to form an arithmetic sequence is ___. | 5490 | 0.375 |
The rectangular spiral shown in the diagram is constructed as follows. Starting at \((0,0)\), line segments of lengths \(1,1,2,2,3,3,4,4, \ldots\) are drawn in a clockwise manner. The integers from 1 to 1000 are placed, in increasing order, wherever the spiral passes through a point with integer coordinates (that is, 1 at \((0,0)\), 2 at \((1,0)\), 3 at \((1,-1)\), and so on). What is the sum of all of the positive integers from 1 to 1000 which are written at points on the line \(y=-x\)? | 10944 | 0.125 |
Let \( R \) be the rectangle in the Cartesian plane with vertices at \((0,0)\), \((2,0)\), \((2,1)\), and \((0,1)\). \( R \) can be divided into two unit squares. Pro selects a point \( P \) uniformly at random in the interior of \( R \). Find the probability that the line through \( P \) with slope \(\frac{1}{2}\) will pass through both unit squares. | \frac{3}{4} | 0.625 |
Determine the seventy-third digit from the end in the square of the number consisting of 112 ones. | 0 | 0.25 |
Two different natural numbers end with 6 zeros and have exactly 56 divisors. Find their sum. | 7000000 | 0.625 |
An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain moment, there are 21 heads and 73 legs on the battlefield. How many monsters are there at this moment? | 13 | 0.875 |
Find the rational number which is the value of the expression
$$
2 \cos ^{6}\left(\frac{5 \pi}{16}\right) + 2 \sin ^{6}\left(\frac{11 \pi}{16}\right) + \frac{3 \sqrt{2}}{8}
$$ | \frac{5}{4} | 0.875 |
Let \( I \) be the center of the inscribed circle in triangle \( ABC \). It is given that \( CA + AI = BC \). Determine the value of the ratio \( \frac{\widehat{BAC}}{\widehat{CBA}} \). | 2 | 0.625 |
There are 100 boxes numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The viewer can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and answers all of them honestly without announcing the questions out loud. What is the minimum number of notes that need to be sent to definitely find out where the prize is? | 99 | 0.375 |
Xiao Ming places several chess pieces into a $3 \times 3$ grid of square cells. Each cell can have no pieces, one piece, or more than one piece. After calculating the total number of pieces in each row and each column, we obtain 6 different numbers. What is the minimum number of chess pieces required to achieve this? | 8 | 0.375 |
In a cube \( A B C D - A_{1} B_{1} C_{1} D_{1} \), let the points \( P \) and \( Q \) be the reflections of vertex \( A \) with respect to the plane \( C_{1} B D \) and the line \( B_{1} D \), respectively. Find the sine of the angle between the line \( P Q \) and the plane \( A_{1} B D \). | \frac{\sqrt{15}}{5} | 0.75 |
Find the smallest natural number ending in the digit 4 that becomes 4 times larger when its last digit is moved to the beginning of the number. | 102564 | 0.625 |
Let \( A \) and \( B \) be two fixed positive real numbers. The function \( f \) is defined by
\[ f(x, y)=\min \left\{ x, \frac{A}{y}, y+\frac{B}{x} \right\}, \]
for all pairs \( (x, y) \) of positive real numbers.
Determine the largest possible value of \( f(x, y) \). | \sqrt{A + B} | 0.875 |
Suppose that \(\{a_{n}\}_{n \geq 1}\) is an increasing arithmetic sequence of integers such that \(a_{a_{20}}=17\) (where the subscript is \(a_{20}\)). Determine the value of \(a_{2017}\). | 4013 | 0.875 |
\(\sin (x + 2\pi) \cos \left(2x - \frac{7\pi}{2}\right) + \sin \left(\frac{3\pi}{2} - x\right) \sin \left(2x - \frac{5\pi}{2}\right)\). | \cos 3x | 0.625 |
In three out of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles such that the product of the triplets along each of the three sides of the triangular diagram are the same? | 6 | 0.375 |
a) Show that if \( a^{2} + b^{2} = a + b = 1 \), then \( ab = 0 \).
b) Show that if \( a^{3} + b^{3} + c^{3} = a^{2} + b^{2} + c^{2} = a + b + c = 1 \), then \( abc = 0 \). | abc = 0 | 0.75 |
Let \( D \) be a point inside the acute triangle \( \triangle ABC \) such that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \). Find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \). | \sqrt{2} | 0.375 |
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