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For which integers $n > 2$ is the following statement true? "Any convex $n$-gon has a side such that neither of the two angles at its endpoints is an acute angle." | n \geq 7 | 0.375 |
Show that the last digits of the numbers $\left(n^{n}\right)^{n}$ for $n = 1, 2, 3, \ldots$ repeat periodically. Determine the period. | 10 | 0.75 |
Given real numbers \(x\) and \(y\) satisfying \(x^{2}+(y-2)^{2} \leq 1\), determine the range of values for \(\frac{x+\sqrt{3} y}{\sqrt{x^{2}+y^{2}}}\). | [1, 2] | 0.75 |
Find the functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(2n) = 2f(n) \) and \( f(2n+1) = 2f(n) + 1 \) for all \( n \in \mathbb{N} \). | f(n) = n | 0.25 |
Triangle \(ABC\) has \(AB = 10\), \(BC = 17\), and \(CA = 21\). Point \(P\) lies on the circle with diameter \(AB\). What is the greatest possible area of \( \triangle APC \)? | 94.5 | 0.625 |
Fifteen students numbered $1, 2, 3, \ldots, 15$ stand in a circle facing inward. On the first turn, the student with number 1 turns around to face outward. On the second turn, the students numbered 2 and 3 turn around. On the third turn, the students numbered $4, 5, 6$ turn around, and so on. On the 15th turn, all students turn around. After the 12th turn, how many students are still facing outward? | 12 | 0.125 |
The digits \(1, 2, 3, 4, 5, 6\) are randomly chosen (without replacement) to form the three-digit numbers \(M = \overline{ABC}\) and \(N = \overline{DEF}\). For example, we could have \(M = 413\) and \(N = 256\). Find the expected value of \(M \cdot N\). | 143745 | 0.375 |
In a certain meeting, there are 30 participants. Each person knows at most 5 other people among the remaining participants; for any group of five people, at least two people are not acquainted with each other. Find the largest positive integer $k$ such that among these 30 people, there exists a group of $k$ people where no two people are acquainted with each other. | 6 | 0.875 |
Each of the four volleyball teams has six players, including a captain and a setter, who are different people. In how many ways can a team of six players be formed from these four teams, where there is at least one player from each team and at least one captain-setter pair from one team? | 9720 | 0.375 |
For given numbers \( n \geq 2 \) and \( a > 0 \), find the maximum value of the sum \( \sum_{i=1}^{n-1} x_i x_{i+1} \) subject to the conditions \( x_i \geq 0 \) for \( i = 1, \ldots, n \) and \( x_1 + \ldots + x_n = a \). | \frac{a^2}{4} | 0.5 |
Determine all prime numbers \( p \) for which there exists a unique \( a \) in \( \{1, \ldots, p\} \) such that \( a^{3} - 3a + 1 \) is divisible by \( p \). | 3 | 0.875 |
What is the smallest possible area of a figure on the xy-plane, located between the lines \( x = -5 \) and \( x = 1 \), bounded below by the x-axis and above by the tangent line to the graph of the function \( y = 7 - 6x - x^2 \) at a point of tangency \( x_0 \), where \( -5 \leq x_0 \leq 1 \)? | 90 | 0.75 |
A 99-sided polygon has its sides painted in sequence with the colors red, blue, red, blue, $\cdots$, red, blue, yellow. Each side is painted in one color. Then, the following operation is allowed: ensuring that any two adjacent sides have different colors, one can change the color of one side each time. Can the coloring of the 99 sides be changed to red, blue, red, blue, $\cdots$, red, yellow, blue after a number of operations? | \text{No} | 0.375 |
A quadrilateral is inscribed in a circle with radius 1. Two opposite sides are parallel. The difference between their lengths is \( d > 0 \). The distance from the intersection of the diagonals to the center of the circle is \( h \). Find \(\sup \frac{d}{h}\) and describe the cases in which it is attained. | 2 | 0.5 |
All natural numbers from 1 to 20 are divided into pairs, and the numbers in each pair are summed. What is the maximum number of the resulting ten sums that can be divisible by 11? Justify your answer. | 9 | 0.125 |
Find the smallest possible sum of digits for a number of the form \(3n^2 + n + 1\) (where \(n\) is a positive integer). Does there exist a number of this form with a sum of digits 1999? | 3 | 0.75 |
A stack contains 300 cards: 100 white, 100 black, and 100 red. For each white card, the number of black cards below it is counted; for each black card, the number of red cards below it is counted; and for each red card, the number of white cards below it is counted. Find the maximum possible value of the sum of these 300 counts. | 20000 | 0.25 |
Four statements were made about triangle \(ABC\), and we know that out of these statements, two are true and two are false. What could be the perimeter of triangle \(ABC\) if \(BC = 1\)?
I. Triangle \(ABC\) is a right triangle.
II. The angle at \(A\) is \(30^\circ\).
III. \(AB = 2 \times BC\).
IV. \(AC = 2 \times BC\). | 3+\sqrt{5} | 0.625 |
In the Magic Land, one of the magical laws of nature states: "A flying carpet will only fly if it has a rectangular shape." Prince Ivan had a flying carpet of dimensions $9 \times 12$. One day, an evil dragon cut off a small piece of this carpet with dimensions $1 \times 8$. Prince Ivan was very upset and wanted to cut off another piece $1 \times 4$ to make a rectangle of $8 \times 12$, but the wise Vasilisa suggested a different approach. She cut the carpet into three parts and used magic threads to sew them into a square flying carpet of dimensions $10 \times 10$. Can you figure out how Vasilisa the Wise transformed the damaged carpet? | 10 \times 10 | 0.25 |
What is the greatest number of natural numbers not exceeding 2016 that can be selected such that the product of any two selected numbers is a perfect square? | 44 | 0.375 |
At the class reunion, 45 people attended. It turned out that any two of them, who have the same number of acquaintances among the attendees, do not know each other. What is the largest number of pairs of acquaintances that could have been among the attendees? | 870 | 0.125 |
In a triangle \(ABC\), angle \(A\) is twice the measure of angle \(B\), angle \(C\) is obtuse, and the side lengths are integers. What is the smallest possible perimeter of this triangle? | 77 | 0.5 |
For any real numbers \( a \) and \( b \), the inequality \( \max \{|a+b|,|a-b|,|2006-b|\} \geq C \) always holds. Find the maximum value of the constant \( C \). (Note: \( \max \{x, y, z\} \) denotes the largest among \( x, y, \) and \( z \).) | 1003 | 0.625 |
In triangle $ABC$, $AB = 6$ and $BC = 10$ units. At what distance from $B$ does the line connecting the foot of the angle bisector from $B$ and the midpoint of $AB$ intersect line $BC$? | 15 | 0.875 |
The set of five-digit numbers \( \{ N_1, \ldots, N_k \} \) is such that any five-digit number with all digits in increasing order coincides in at least one digit with at least one of the numbers \( N_1, \ldots, N_k \). Find the smallest possible value of \( k \). | 1 | 0.375 |
Find the maximum real number \(\lambda\) such that for the real-coefficient polynomial
$$
f(x) = x^3 + ax^2 + bx + c
$$
with all roots being non-negative real numbers, the inequality
$$
f(x) \geqslant \lambda(x - a)^3 \quad \text{for all} \; x \geqslant 0
$$
holds. Also, determine when equality holds in this inequality. | -\frac{1}{27} | 0.625 |
Vanya wrote the number 1 on the board and then added several more numbers. Each time Vanya writes a new number, Mitya calculates the median of the existing set of numbers and writes it down in his notebook. At a certain moment, the following numbers are recorded in Mitya's notebook: $1 ; 2 ; 3 ; 2.5 ; 3 ; 2.5 ; 2 ; 2 ; 2 ; 5$.
a) What number was written on the board fourth?
b) What number was written on the board eighth? | 2 | 0.625 |
In $\triangle ABC$, find the value of $a^{3} \sin (B-C) + b^{3} \sin (C-A) + c^{3} \sin (A-B)$. | 0 | 0.625 |
Let the following system of equations be satisfied for positive numbers \(x, y, z\):
\[
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 27 \\
y^{2} + y z + z^{2} = 25 \\
z^{2} + x z + x^{2} = 52
\end{array}
\right.
\]
Find the value of the expression \(x y + y z + x z\). | 30 | 0.75 |
A regular triangular prism \( ABC A_1 B_1 C_1 \) with a base \( ABC \) and side edges \( AA_1, BB_1, CC_1 \) is inscribed in a sphere of radius 3. The segment \( CD \) is the diameter of this sphere. Find the volume of the prism, given that \( AD = 2 \sqrt{6} \). | 6\sqrt{15} | 0.25 |
Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of 2, form a dihedral angle of 60 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. | 1 | 0.125 |
The smallest positive odd number that cannot be represented as \(7^{x}-3 \times 2^{y} (x, y \in \mathbb{Z}_{+})\) is $\qquad$ . | 3 | 0.5 |
Find the integers \( n \geq 1 \) for which \( n \) divides \( 2^{n} - 1 \). | n = 1 | 0.625 |
Given that in triangle \( \triangle ABC \), \( a = 2b \), \( \cos B = \frac{2 \sqrt{2}}{3} \), find the value of \( \sin \frac{A-B}{2} + \sin \frac{C}{2} \). | \frac{\sqrt{10}}{3} | 0.125 |
Given a regular tetrahedron $\mathrm{ABCD}$ with a point $\mathrm{P}$ inside it such that $P A=P B=\sqrt{11}$ and $P C=P D=\sqrt{17}$. What is the side length of the tetrahedron $\mathrm{ABCD}$? | 6 | 0.875 |
The side length \( BC \) of triangle \( ABC \) is 12 cm. A circle with radius 10 cm is circumscribed around the triangle. Find the lengths of sides \( AB \) and \( AC \) of the triangle, given that the radius \( OA \) of the circle bisects the side \( BC \) into two equal segments. | 2\sqrt{10} | 0.375 |
How many pairs of integers $(x, y)$ satisfy the equation \(\sqrt{x-\sqrt{x+23}}=2\sqrt{2}-y\)? | 1 | 0.625 |
Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), and point \(B\) lies on the parabola \(y=x^{2}\). What is the minimum length of segment \(AB\)? | \frac{189}{65} | 0.875 |
A mason has bricks with dimensions \( 2 \times 5 \times 8 \) and other bricks with dimensions \( 2 \times 3 \times 7 \). She also has a box with dimensions \( 10 \times 11 \times 14 \). The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box, filling its entire volume with no bricks sticking out. Find all possible values of the total number of bricks that she can pack. | 24 | 0.125 |
A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:
- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adventurer has rubies, they either have emeralds or diamonds (but not both simultaneously).
What is the minimum number of adventurers that could be in this group? | 17 | 0.375 |
How many ordered pairs of integers \((a, b)\) satisfy all of the following inequalities?
\[
\begin{array}{l}
a^2 + b^2 < 16 \\
a^2 + b^2 < 8a \\
a^2 + b^2 < 8b
\end{array}
\] | 6 | 0.75 |
In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may repeat), so that the resulting 12-digit number is divisible by 75. How many ways can this be done? | 2592 | 0.25 |
Tolya arranged in a row 101 coins of denominations 1, 2, and 3 kopecks. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins there are at least three coins. How many 3-kopeck coins could Tolya have? | 26 | 0.375 |
Given a plane intersects all 12 edges of a cube at an angle $\alpha$, find $\sin \alpha$. | \frac{\sqrt{3}}{3} | 0.625 |
Let \( S \) be the set of points \((x, y)\) in the plane such that the sequence \( a_n \) defined by \( a_0 = x \), \( a_{n+1} = \frac{a_n^2 + y^2}{2} \) converges. What is the area of \( S \)? | 4 + \pi | 0.125 |
When \( N \) takes all the values from 1, 2, 3, \ldots, 2015, how many numbers of the form \( 3^{n} + n^{3} \) are divisible by 7? | 288 | 0.5 |
What is the smallest number of digits that need to be appended to the right of the number 2014 so that the resulting number is divisible by all natural numbers less than 10? | 4 | 0.25 |
In the figure, a cake in the shape of a rectangle with sides $3 \times 3$ is shown. A straight line (cut) is illustrated, crossing the maximum number of squares, which is 5. The maximum number of pieces of the cake after the first cut is equal to $2 \cdot 5 + 4 = 14$. Thus, it is enough to use two straight lines to cut all 9 squares. | 2 \text{ cuts} | 0.375 |
Drop perpendiculars IP and IQ from point I onto sides BC and AB respectively. Points P and Q are points of tangency of these sides with the inscribed circle, and the quadrilateral PIQB is a square. The angles KIL and PIQ are right angles, hence angles PIK and QIL are equal. Consequently, right triangles PIK and QIL are congruent. According to Thales' theorem, the length KP=QL is half of the length BP=BQ, and the length AQ is twice the length BQ=BP.
Therefore, the length of side AB is \( AL + LB = \frac{6}{5}AL = \frac{6}{5}AO = \frac{3}{5}AC \). From the Pythagorean theorem, \( BC = \frac{4}{5}AC \). Hence, \( AB:BC:CA = 3:4:5 \). | 3:4:5 | 0.625 |
On a $5 \times 5$ board, two players alternately mark numbers on empty cells. The first player always marks 1's, the second 0's. One number is marked per turn, until the board is filled. For each of the nine $3 \times 3$ squares, the sum of the nine numbers on its cells is computed. Denote by \( A \) the maximum of these sums. How large can the first player make \( A \), regardless of the responses of the second player? | 6 | 0.125 |
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} | 0.25 |
Let $M$ be the midpoint of the base $AC$ of an isosceles triangle $ABC$. Points $E$ and $F$ are marked on sides $AB$ and $BC$ respectively, such that $AE \neq CF$ and $\angle FMC = \angle MEF = \alpha$. Find $\angle AEM$. | \alpha | 0.375 |
The trapezoid \(ABCD\) with base \(AD = 6\) is inscribed in a circle. The tangent to the circle at point \(A\) intersects lines \(BD\) and \(CD\) at points \(M\) and \(N\), respectively. Find \(AN\) if \(AB \perp MD\) and \(AM = 3\). | 12 | 0.25 |
We shuffle a deck of 52 French playing cards, and then draw one by one from the pile until we find an ace of black color. In what position is it most likely for the first black ace to appear? | 1 | 0.625 |
Let \( p \) be an odd prime. An integer \( x \) is called a quadratic non-residue if \( p \) does not divide \( x - t^{2} \) for any integer \( t \).
Denote by \( A \) the set of all integers \( a \) such that \( 1 \leq a < p \), and both \( a \) and \( 4 - a \) are quadratic non-residues. Calculate the remainder when the product of the elements of \( A \) is divided by \( p \). | 2 | 0.5 |
Points \( A, B, C, D \) lie on a circle in that order such that \(\frac{AB}{BC} = \frac{DA}{CD}\). If \(AC = 3\) and \(BD = BC = 4\), find \(AD\). | \frac{3}{2} | 0.5 |
In $\triangle ABC$, given that $x \sin A + y \sin B + z \sin C = 0$, find the value of $(y + z \cos A)(z + x \cos B)(x + y \cos C) + (y \cos A + z)(z \cos B + x)(x \cos C + y)$. | 0 | 0.75 |
Let \( x, y, z \) be the lengths of the midline segments connecting the midpoints of opposite edges of a tetrahedron \( P-ABC \). If the sum of the squares of the six edge lengths of the tetrahedron is 300, find \( x^{2} + y^{2} + z^{2} \). | 75 | 0.625 |
At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them away!" Find the largest possible value of $N$, given that no one lied. | 14 | 0.375 |
For any 4 distinct points \(P_{1}, P_{2}, P_{3}, P_{4}\) on a plane, find the minimum value of the ratio
\[
\frac{\sum_{1 \leq i < j \leq 4} P_{i}P_{j}}{\min_{1 \leq i < j \leq 4} P_{i}P_{j}}
\] | 5 + \sqrt{3} | 0.375 |
What is the smallest positive integer \( n \) such that \( n^{2} \) and \( (n+1)^{2} \) both contain the digit 7 but \( (n+2)^{2} \) does not? | 27 | 0.625 |
Given that
$$
S=\left|\sqrt{x^{2}+4 x+5}-\sqrt{x^{2}+2 x+5}\right|,
$$
for real values of \(x\), find the maximum value of \(S^{4}\). | 4 | 0.75 |
Using the digits $0$ to $9$ exactly once, form several composite numbers. What is the minimum sum of these composite numbers? | 99 | 0.125 |
A cryptarithm is given: ЛЯЛЯЛЯ + ФУФУФУ = ГГЫГЫЫР. Identical letters represent the same digits, different letters represent different digits. Find the sum of ЛЯ and ФУ. | 109 | 0.375 |
Five contestants $A, B, C, D, E$ participate in a "Voice" competition, and they stand in a row for a group appearance. Each contestant has a number badge on their chest, the sum of the five numbers is 35. It is known that the sum of the numbers of the contestants standing to the right of $\mathrm{E}$ is 13; the sum of the numbers of the contestants standing to the right of $D$ is 31; the sum of the numbers of the contestants standing to the right of $A$ is 21; and the sum of the numbers of the contestants standing to the right of $C$ is 7. What is the sum of the numbers of the contestants at the far left and the far right? | 11 | 0.375 |
Find the triangle. The sides and the height of a certain triangle are expressed by four consecutive integers. What is the area of this triangle? | 84 | 0.125 |
Given the sequence $\left\{a_{n}\right\}$ satisfying $a_{1}=1$ and $a_{n+1}=a_{n}+\frac{1}{2 a_{n}}$, find $\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)$. | 0 | 0.25 |
Given that 7 divides 111111. If \( b \) is the remainder when \(\underbrace{111111 \ldots 111111}_{a \text{-times }}\) is divided by 7, find the value of \( b \). If \( c \) is the remainder of \( \left\lfloor(b-2)^{4 b^{2}}+(b-1)^{2 b^{2}}+b^{b^{2}}\right\rfloor \) divided by 3, find the value of \( c \). If \( |x+1|+|y-1|+|z|=c \), find the value of \( d=x^{2}+y^{2}+z^{2} \). | 2 | 0.875 |
Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two of the same letter in a row. How many unpronounceable 7-letter words are there? | 86 | 0.25 |
Find the number of ordered pairs of integers \((a, b) \in \{1, 2, \ldots, 35\}^{2}\) (not necessarily distinct) such that \(ax + b\) is a quadratic residue modulo \(x^2 + 1\) and 35. | 225 | 0.25 |
From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board? | 23 | 0.875 |
Given that \( x, y, z \) are positive numbers such that \( x^2 + y^2 + z^2 = 1 \), find the minimum value of the expression:
\[
S = \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y}.
\] | \sqrt{3} | 0.75 |
Square \(CASH\) and regular pentagon \(MONEY\) are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have? | 8 | 0.25 |
Consider a $4 \times 4$ grid of squares, each of which is originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minutes it will take Piet to change the entire grid to blue? | 4 | 0.125 |
In a quadrilateral $ABCD$ lying in the plane, $AB=\sqrt{3}$, $AD=DC=CB=1$. The areas of triangles $ABD$ and $BCD$ are $S$ and $T$ respectively. What is the maximum value of $S^{2} + T^{2}$? | \frac{7}{8} | 0.375 |
Let \(a\) and \(b\) be two strictly positive integers with \((a, b) \neq (1,1)\) such that \(ab-1\) divides \(a^2 + b^2\). Show that \(a^2 + b^2 = 5ab - 5\). | a^2 + b^2 = 5ab - 5 | 0.875 |
$a$ and $b$ are natural numbers. Show that if $4ab - 1$ divides $(4a^2 - 1)^2$, then $a = b$. | a = b | 0.75 |
A needle lies on a plane. It is allowed to rotate the needle by $45^{\circ}$ around either of its ends. Is it possible, after making several such rotations, to return the needle to its original position with its ends swapped? | \text{No} | 0.75 |
Let \(R\) be the rectangle in the Cartesian plane with vertices at \((0,0), (2,0), (2,1)\), and \((0,1)\). The resulting figure has 7 segments of unit length, connecting neighboring lattice points (those lying on or inside \(R\)). Compute the number of paths from \((0,1)\) (the upper left corner) to \((2,0)\) (the lower right corner) along these 7 segments, where each segment can be used at most once. | 4 | 0.25 |
Let $a_{1}, a_{2}, a_{3}, \cdots$ be a non-decreasing sequence of positive integers. For $m \geqslant 1$, define
$$
b_{m}=\min \left\{n \mid a_{n} \geqslant m\right\},
$$
which means $b_{m}$ is the smallest $n$ such that $a_{n} \geqslant m$. Given that $a_{19}=85$, find the maximum value of $a_{1} + a_{2} + \cdots + a_{19} + b_{1} + b_{2} + \cdots + b_{85}$. | 1700 | 0.125 |
Given that \( f(x) \) is an odd function defined on \((-1, 1)\), is monotonically decreasing on the interval \([0, 1)\), and satisfies \( f(1-a) + f\left(1-a^2\right) < 0 \), determine the range of the real number \( a \). | (0, 1) | 0.625 |
On a plane, there are a finite number of polygons. If any two of these polygons have a line passing through the origin that intersects both of them, then these polygons are said to be properly placed. Find the smallest natural number \( m \) such that for any set of properly placed polygons, we can draw \( m \) lines passing through the origin such that each of these polygons intersects with at least one of these \( m \) lines. | 2 | 0.5 |
Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation? | 162 | 0.125 |
Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster than Fluffball, and seeds three times faster. How many seeds did Shaggy eat if Shaggy ate twice as many berries as Fluffball? | 54 | 0.375 |
Construct the polynomial $R(x)$ from problem $\underline{61019}$ if:
a) $P(x)=x^{6}-6 x^{4}-4 x^{3}+9 x^{2}+12 x+4$;
b) $P(x)=x^{5}+x^{4}-2 x^{3}-2 x^{2}+x+1$. | x^2 - 1 | 0.125 |
Find the probability that a randomly selected five-digit natural number with non-repeating digits, composed of the digits \(1,2,3,4,5,6,7,8\), is divisible by 8 without a remainder. | \frac{1}{8} | 0.875 |
What is the largest possible area of a triangle with sides \(a\), \(b\), and \(c\) that are within the following limits:
$$
0 < a \leq 1 \leq b \leq 2 \leq c \leq 3
$$ | 1 | 0.625 |
In $\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\angle A$. | 60^\circ | 0.875 |
Let \( N \) be the set of natural numbers \(\{1, 2, 3, \ldots \}\). Let \( Z \) be the integers. Define \( d : N \to Z \) by \( d(1) = 0 \), \( d(p) = 1 \) for \( p \) prime, and \( d(mn) = m d(n) + n d(m) \) for any integers \( m, n \). Determine \( d(n) \) in terms of the prime factors of \( n \). Find all \( n \) such that \( d(n) = n \). Define \( d_1(m) = d(m) \) and \( d_{n+1}(m) = d(d_n(m)) \). Find \( \lim_{n \to \infty} d_n(63) \). | \infty | 0.625 |
The opposite sides of a quadrilateral inscribed in a circle intersect at points \( P \) and \( Q \). Find the length of the segment \( |PQ| \), given that the tangents to the circle drawn from \( P \) and \( Q \) are \( a \) and \( b \) respectively. | \sqrt{a^2 + b^2} | 0.375 |
Find all natural values of \( n \) for which
$$
\cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9}, \text { and } \log _{2}^{2} n+45<\log _{2} 8 n^{13}
$$
Record the sum of the obtained values of \( n \) as the answer. | 644 | 0.625 |
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \). | 169 | 0.125 |
Using the equality \( \lg 11 = 1.0413 \), find the smallest number \( n > 1 \) for which among the \( n \)-digit numbers, there is not a single one equal to some natural power of the number 11. | 26 | 0.375 |
Positive numbers \( a, b, c \) satisfy \( a^2 b + b^2 c + c^2 a = 3 \). Find the minimum value of the expression:
\[ A = \frac{\sqrt{a^6 + b^4 c^6}}{b} + \frac{\sqrt{b^6 + c^4 a^6}}{c} + \frac{\sqrt{c^6 + a^4 b^6}}{a}. \] | 3 \sqrt{2} | 0.875 |
The following figure shows a cube.
Calculate the number of equilateral triangles that can be formed such that their three vertices are vertices of the cube. | 8 | 0.875 |
What is the minimum number of points that must be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point? | n-2 | 0.25 |
A function \( g \) is ever more than a function \( h \) if, for all real numbers \( x \), we have \( g(x) \geq h(x) \). Consider all quadratic functions \( f(x) \) such that \( f(1)=16 \) and \( f(x) \) is ever more than both \( (x+3)^{2} \) and \( x^{2}+9 \). Across all such quadratic functions \( f \), compute the minimum value of \( f(0) \). | \frac{21}{2} | 0.375 |
29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once in each pair). After the ball, each child told their parents how many times they danced. What is the maximum number of distinct numbers the children could have reported? | 29 | 0.25 |
When dividing the polynomial \( x^{1051} - 1 \) by \( x^4 + x^3 + 2x^2 + x + 1 \), what is the coefficient of \( x^{14} \) in the quotient? | -1 | 0.25 |
Fold a rectangular piece of paper which is 16 cm long and 12 cm wide as shown in the figure. Find the length of the crease \( GF \) in cm. | 15 | 0.5 |
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