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13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, with $a=2$, and $(2+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum area of $\triangle ABC$. | \sqrt{3} | 6/8 |
Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 399, what is the sum of $a$ and $b$? | 21 | 6/8 |
Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \, \text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both boys spent 10 minutes at the respective relative's house. On their way back, they met again $400 \, \text{m}$ away from Károly's house. How far apart do the two families live? What assumptions can we make to answer this question? | 1760 | 4/8 |
Find the number of pairs of integers $(x, y)$ that satisfy the equation $x^{2} + xy = 30000000$. | 256 | 5/8 |
Let $\{x\}$ denote the smallest integer not less than the real number \(x\). Find the value of the expression $\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}$. | 19854 | 5/8 |
Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$. | 100902018 | 1/8 |
A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | 2/3 | 3/8 |
A convex $n$ -gon $P$ , where $n > 3$ , is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed? | 4 | 4/8 |
$ABC$ is a triangle with points $D$ , $E$ on $BC$ with $D$ nearer $B$ ; $F$ , $G$ on $AC$ , with $F$ nearer $C$ ; $H$ , $K$ on $AB$ , with $H$ nearer $A$ . Suppose that $AH=AG=1$ , $BK=BD=2$ , $CE=CF=4$ , $\angle B=60^\circ$ and that $D$ , $E$ , $F$ , $G$ , $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$ . | \sqrt{3} | 3/8 |
Compute the number of sets \( S \) such that every element of \( S \) is a nonnegative integer less than 16, and if \( x \in S \) then \( (2 x \bmod 16) \in S \). | 678 | 1/8 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}$$ | -\sqrt{\frac{2}{7}} | 2/8 |
Find the smallest composite number that has no prime factors less than 20. | 529 | 7/8 |
Find the number of units in the length of diagonal $DA$ of the regular hexagon shown. Express your answer in simplest radical form. [asy]
size(120);
draw((1,0)--(3,0)--(4,1.732)--(3,3.464)--(1,3.464)--(0,1.732)--cycle);
draw((1,0)--(1,3.464));
label("10",(3.5,2.598),NE);
label("$A$",(1,0),SW);
label("$D$",(1,3.464),NW);
[/asy] | 10\sqrt{3} | 3/8 |
Given that \( n \) is an integer, let \( p(n) \) denote the product of its digits (in decimal representation).
1. Prove that \( p(n) \leq n \).
2. Find all \( n \) such that \( 10 p(n) = n^2 + 4n - 2005 \). | 45 | 2/8 |
Find the integer $n$ such that $-150 < n < 150$ and $\tan n^\circ = \tan 286^\circ$. | -74 | 3/8 |
Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$ . | 4 | 1/8 |
In our daily life, we often use passwords, such as when making payments through Alipay. There is a type of password generated using the "factorization" method, which is easy to remember. The principle is to factorize a polynomial. For example, the polynomial $x^{3}+2x^{2}-x-2$ can be factorized as $\left(x-1\right)\left(x+1\right)\left(x+2\right)$. When $x=29$, $x-1=28$, $x+1=30$, $x+2=31$, and the numerical password obtained is $283031$.<br/>$(1)$ According to the above method, when $x=15$ and $y=5$, for the polynomial $x^{3}-xy^{2}$, after factorization, what numerical passwords can be formed?<br/>$(2)$ Given a right-angled triangle with a perimeter of $24$, a hypotenuse of $11$, and the two legs being $x$ and $y$, find a numerical password obtained by factorizing the polynomial $x^{3}y+xy^{3}$ (only one is needed). | 24121 | 4/8 |
Let \([AB]\) and \([CD]\) be two segments such that \(AB = CD\). Points \(P\) and \(Q\) are taken on these segments such that \(AP = CQ\). The lines \( (AC) \) and \((BD) \) intersect at \(X\), and the line \((PQ)\) intersects the lines \((AC)\) and \((BD)\) at \(E\) and \(F\) respectively. Show that as point \(P\) moves, the circumcircle of triangle \(XEF\) passes through a fixed point other than \(X\). | Y | 1/8 |
The left and right foci of the ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1$ are $F_{1}$ and $F_{2}$, respectively. There is a point $P$ on the ellipse such that $\angle F_{1}PF_{2} = 30^{\circ}$. Find the area of triangle $F_{1}PF_{2}$. | 18 - 9\sqrt{3} | 4/8 |
Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 47 | 7/8 |
When the repeating decimal $0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is:
$\textbf{(A)}\ 15 \qquad \textbf{(B) }\ 45 \qquad \textbf{(C) }\ 114 \qquad \textbf{(D) }\ 135 \qquad \textbf{(E) }\ 150$ | \textbf{(A)}\15 | 1/8 |
In a bag containing 7 apples and 1 orange, the probability of randomly picking an apple is ______, and the probability of picking an orange is ______. | \frac{1}{8} | 4/8 |
Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\circ}$. Find the ratio of the radii of the circles. | 1/3 | 3/8 |
Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$ , where $i$ is the imaginary unit. | p\equiv1\pmod{4} | 7/8 |
How many graphs are there on 10 vertices labeled \(1,2, \ldots, 10\) such that there are exactly 23 edges and no triangles? | 42840 | 2/8 |
Suppose $a$ and $b$ are positive integers with a curious property: $(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n$ is an integer for at least $3$ , but at most finitely many different choices of positive integers $n$ . What is the least possible value of $a+b$ ? | 6 | 1/8 |
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup? | D | 2/8 |
Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$ . | 111 | 6/8 |
Compute the line integral of the vector field given in spherical coordinates:
\[ 2 = e^{r} \sin \theta \mathbf{e}_{r} + 3 \theta^{2} \sin \varphi \mathbf{e}_{\theta} + \tau \varphi \theta \mathbf{e}_{\varphi} \]
along the line \( L: \left\{ r=1, \varphi=\frac{\pi}{2}, 0 \leqslant 0 \leqslant \frac{\pi}{2} \right\} \) in the direction from point \( M_{0}\left(1,0, \frac{\pi}{2}\right) \) to point \( M_{1}\left(1, \frac{\pi}{2}, \frac{\pi}{2}\right) \). | \frac{\pi^3}{8} | 7/8 |
Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : c \). | 2:\sqrt{5}:\sqrt{5} | 2/8 |
Given \(2n\) real numbers \(a_{1}, a_{2}, \cdots, a_{2n}\) such that \(\sum_{i=1}^{2n-1}\left(a_{i+1}-a_{i}\right)^{2} = 1\), find the maximum value of \(\left(a_{n-1}+a_{n-2}+\cdots+a_{2n}\right)-\left(a_{1}+a_{2}+\cdots+a_{n}\right)\). | \sqrt{\frac{n(2n^2+1)}{3}} | 1/8 |
How many quadratic polynomials (i.e., polynomials of degree two) with integer coefficients exist such that they take values only from the interval \([0,1]\) on the segment \([0,1]\)? | 12 | 1/8 |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatest number of elements that $\mathcal{S}$ can have? | 30 | 2/8 |
Given $\cos\alpha= \frac {4}{5}$, $\cos\beta= \frac {3}{5}$, $\beta\in\left(\frac {3\pi}{2}, 2\pi\right)$, and $0<\alpha<\beta$, calculate the value of $\sin(\alpha+\beta)$. | -\frac{7}{25} | 7/8 |
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$.
(1) Find the smallest positive period and the interval where the function is decreasing;
(2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, where $a=7$. If acute angle $A$ satisfies $f(\frac{A}{2}-\frac{\pi}{6})=\sqrt{3}$, and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of triangle $ABC$. | 10\sqrt{3} | 7/8 |
A number \( a \) is randomly chosen from \( 1, 2, 3, \cdots, 10 \), and a number \( b \) is randomly chosen from \( -1, -2, -3, \cdots, -10 \). What is the probability that \( a^{2} + b \) is divisible by 3? | 37/100 | 7/8 |
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught?
*Author: Anderson Wang* | 25 | 1/8 |
On the side $BC$ of rectangle $ABCD$, a point $K$ is marked. There is a point $H$ on the segment $AK$ such that $\angle AHD=90^{\circ}$. It turns out that $AK=BC$. How many degrees is $\angle ADH$ if $\angle CKD=71^\circ$? | 52 | 6/8 |
Compose the equation of the tangent line to the given curve at the point with abscissa \( x_{0} \).
\[ y = \frac{-2 \left( x^{8} + 2 \right)}{3 \left( x^{4} + 1 \right)}, \quad x_{0} = 1 \] | -\frac{2}{3}x-\frac{1}{3} | 7/8 |
Let $PQRS$ be a quadrilateral that has an incircle and $PQ\neq QR$ . Its incircle touches sides $PQ,QR,RS,$ and $SP$ at $A,B,C,$ and $D$ , respectively. Line $RP$ intersects lines $BA$ and $BC$ at $T$ and $M$ , respectively. Place point $N$ on line $TB$ such that $NM$ bisects $\angle TMB$ . Lines $CN$ and $TM$ intersect at $K$ , and lines $BK$ and $CD$ intersect at $H$ . Prove that $\angle NMH=90^{\circ}$ . | \angleNMH=90 | 1/8 |
Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$
Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$. | 3 - 2\sqrt{2} | 1/8 |
In $\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$?
$\textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \ 61 \qquad \textbf{(E)} \ 72$ | \textbf{(D)}61 | 1/8 |
In a convex quadrilateral \(ABCD\), \(AB = AC = AD = BD\) and \(\angle BAC = \angle CBD\). Find \(\angle ACD\). | 70 | 1/8 |
Define \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) by \( f(x, y) = (2x - y, x + 2y) \). Let \( f^{0}(x, y) = (x, y) \) and, for each \( n \in \mathbb{N} \), \( f^{n}(x, y) = f(f^{n-1}(x, y)) \). Determine the distance between \( f^{2016} \left( \frac{4}{5}, \frac{3}{5} \right) \) and the origin. | 5^{1008} | 6/8 |
The left and right foci of the hyperbola $C$ are respectively $F_1$ and $F_2$, and $F_2$ coincides with the focus of the parabola $y^2=4x$. Let the point $A$ be an intersection of the hyperbola $C$ with the parabola, and suppose that $\triangle AF_{1}F_{2}$ is an isosceles triangle with $AF_{1}$ as its base. Then, the eccentricity of the hyperbola $C$ is _______. | \sqrt{2} + 1 | 1/8 |
Given two moving points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) on the parabola \(y^2 = 6x\), where \(x_1 \neq x_2\) and \(x_1 + x_2 = 4\). If the perpendicular bisector of segment \(AB\) intersects the \(x\)-axis at point \(C\), find the maximum area of triangle \(ABC\). | \frac{14\sqrt{7}}{3} | 6/8 |
Let $ABC$ be a triangle and $M$ be the midpoint of $[AB]$. A line is drawn through vertex $C$, and two perpendiculars, $[AP]$ and $[BP]$, are dropped to this line. Show that $MP = MQ$. | MP=MQ | 5/8 |
In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$. | 148 | 2/8 |
A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$ ? | 6 | 3/8 |
At an interview, ten people were given a test consisting of several questions. It is known that any group of five people together answered all the questions (i.e., for each question, at least one of the five gave the correct answer), but any group of four did not. What is the minimum number of questions this could have been? | 210 | 4/8 |
Find an integer \( x \) such that \(\left(1+\frac{1}{x}\right)^{x+1}=\left(1+\frac{1}{2003}\right)^{2003}\). | -2004 | 1/8 |
Is there a square number that has the same number of positive divisors of the form \( 3k+1 \) as of the form \( 3k+2 \)? | No | 3/8 |
Calculate:<br/>$(1)-4^2÷(-32)×(\frac{2}{3})^2$;<br/>$(2)(-1)^{10}÷2+(-\frac{1}{2})^3×16$;<br/>$(3)\frac{12}{7}×(\frac{1}{2}-\frac{2}{3})÷\frac{5}{14}×1\frac{1}{4}$;<br/>$(4)1\frac{1}{3}×[1-(-4)^2]-(-2)^3÷\frac{4}{5}$. | -10 | 6/8 |
What is the value of $\frac{20 \times 21}{2+0+2+1}$?
Options:
A) 42
B) 64
C) 80
D) 84
E) 105 | 84 | 5/8 |
Some students are sitting around a circular table passing a small bag containing 100 pieces of candy. Each person takes one piece upon receiving the bag and then passes it to the person to their right. If one student receives the first and the last piece, how many students could there be?
(A) 10
(B) 11
(C) 19
(D) 20
(E) 25 | 11 | 1/8 |
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$. | \frac{4\pi}{3} | 2/8 |
Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a<b$ and there are no other five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b-a$? | 100, 110, 11 | 1/8 |
Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
*Author: Anderson Wang* | 1025 | 6/8 |
Find the smallest \( n \) such that the sequence of positive integers \( a_1, a_2, \ldots, a_n \) has each term \( \leq 15 \) and \( a_1! + a_2! + \ldots + a_n! \) has last four digits 2001. | 3 | 2/8 |
Does there exist a set \( H \) consisting of 2006 points in 3-dimensional space such that the following properties are satisfied:
(a) The points of \( H \) do not lie in a single plane,
(b) No three points of \( H \) lie on a single straight line, and
(c) For any line connecting two points of \( H \), there exists another line, parallel to it and different from it, connecting two different points of \( H \)? | Yes | 6/8 |
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 | 6/8 |
Let \( A \) and \( B \) be two sets, and \((A, B)\) be called a "pair". If \( A \neq B \), then \((A, B)\) and \((B, A)\) are considered different "pairs". Find the number of different pairs \((A, B)\) that satisfy the condition \( A \cup B = \{1,2,3,4\} \). | 81 | 7/8 |
A school table tennis championship was held using the Olympic system. The winner won 6 matches. How many participants in the championship won more matches than they lost? (In the first round of the championship, conducted using the Olympic system, participants are divided into pairs. Those who lost the first match are eliminated from the championship, and those who won in the first round are again divided into pairs for the second round. The losers are again eliminated, and winners are divided into pairs for the third round, and so on, until one champion remains. It is known that in each round of our championship, every participant had a pair.) | 16 | 5/8 |
The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \). | \frac{1}{2} | 1/8 |
If the cotangents of the three interior angles \(A, B, C\) of triangle \(\triangle ABC\), denoted as \(\cot A, \cot B, \cot C\), form an arithmetic sequence, then the maximum value of angle \(B\) is \(\frac{\pi}{3}\). | \frac{\pi}{3} | 7/8 |
Given that A and B are any two points on the line l, and O is a point outside of l. If there is a point C on l that satisfies the equation $\overrightarrow {OC}= \overrightarrow {OA}cosθ+ \overrightarrow {OB}cos^{2}θ$, find the value of $sin^{2}θ+sin^{4}θ+sin^{6}θ$. | \sqrt {5}-1 | 7/8 |
The polynomial $P(x)$ is a monic, quartic polynomial with real coefficients, and two of its roots are $\cos \theta + i \sin \theta$ and $\sin \theta + i \cos \theta,$ where $0 < \theta < \frac{\pi}{4}.$ When the four roots of $P(x)$ are plotted in the complex plane, they form a quadrilateral whose area is equal to half of $P(0).$ Find the sum of the four roots. | 1 + \sqrt{3} | 6/8 |
Let $ABC$ be a triangle with $\angle A = 60^\circ$ , $AB = 12$ , $AC = 14$ . Point $D$ is on $BC$ such that $\angle BAD = \angle CAD$ . Extend $AD$ to meet the circumcircle at $M$ . The circumcircle of $BDM$ intersects $AB$ at $K \neq B$ , and line $KM$ intersects the circumcircle of $CDM$ at $L \neq M$ . Find $\frac{KM}{LM}$ . | \frac{13}{8} | 1/8 |
Two circles touch internally. A line passing through the center of the smaller circle intersects the larger circle at points \( A \) and \( D \), and the smaller circle at points \( B \) and \( C \). Find the ratio of the radii of the circles, given that \( AB: BC: CD = 2: 4: 3 \). | 3 | 1/8 |
In the arithmetic sequence $\{a_n\}$, it is known that $a_1=10$, and the sum of the first $n$ terms is $S_n$. If $S_9=S_{12}$, find the maximum value of $S_n$ and the corresponding value of $n$. | 55 | 5/8 |
Solve the following inequalities for $x$:
(1) $x^{2}-2a|x|-3a^{2}<0$
(2) $ax^{2}-(2a+1)x+2>0$
(where $a \in \mathbf{R}$). | (\frac{1}{},2) | 4/8 |
An apple, pear, orange, and banana were placed in four boxes (one fruit in each box). Labels were made on the boxes:
1. An orange is here.
2. A pear is here.
3. If a banana is in the first box, then either an apple or a pear is here.
4. An apple is here.
It is known that none of the labels correspond to reality.
Determine which fruit is in which box. In the answer, record the box numbers sequentially, without spaces, where the apple, pear, orange, and banana are located, respectively (the answer should be a 4-digit number). | 2431 | 7/8 |
Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$ , the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$ . | 4 | 2/8 |
Let $\left\{a_{n}\right\}$ (for $n \geqslant 1$) be an increasing sequence of positive integers such that:
1. For all $n \geqslant 1$, $a_{2n} = a_{n} + n$;
2. If $a_{n}$ is a prime number, then $n$ is a prime number.
Prove that for all $n \geqslant 1$, $a_{n} = n$. | a_n=n | 3/8 |
Consider the sequences \(\{a_n\}\) and \(\{b_n\}\) defined as follows:
\[
\begin{array}{l}
a_{0}=\frac{\sqrt{2}}{2}, \quad a_{n+1}=\frac{\sqrt{2}}{2} \sqrt{1-\sqrt{1-a_{n}^{2}}}, \quad n=0, 1, 2, \ldots \\
b_{0}=1, \quad b_{n+1}=\frac{\sqrt{1+b_{n}^{2}}-1}{b_{n}}, \quad n=0, 1, 2, \ldots
\end{array}
\]
Prove that for any non-negative integer \( n \), the following inequality holds:
\[ 2^{n+2} a_{n} < \pi < 2^{n+2} b_{n}. \] | 2^{n+2}a_n<\pi<2^{n+2}b_n | 6/8 |
Find all functions \( f: \mathbb{R} \longrightarrow \mathbb{R} \) such that, for any real number \( x \),
\[ f(f(x)) = x^{2} - 1996 \] | Nosuchfunctionexists | 1/8 |
In a village with at least one inhabitant, there are several associations. Each inhabitant of the village is a member of at least $k$ clubs and each two different clubs have at most one common member. Show that at least $k$ of these clubs have the same number of members. | k | 2/8 |
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 | 1/8 |
In the polar coordinate system, given the curve $C: \rho = 2\cos \theta$, the line $l: \left\{ \begin{array}{l} x = \sqrt{3}t \\ y = -1 + t \end{array} \right.$ (where $t$ is a parameter), and the line $l$ intersects the curve $C$ at points $A$ and $B$.
$(1)$ Find the rectangular coordinate equation of curve $C$ and the general equation of line $l$.
$(2)$ Given the polar coordinates of point $P$ as $({1, \frac{3\pi}{2}})$, find the value of $\left(|PA|+1\right)\left(|PB|+1\right)$. | 3 + \sqrt{3} | 7/8 |
An equivalent of the expression
$\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right)+\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right)$, $xy \not= 0$,
is: | 2xy+\frac{2}{xy} | 5/8 |
Given a trapezoid \(ABCD\) with bases \(AD = a\) and \(BC = b\). Points \(M\) and \(N\) lie on sides \(AB\) and \(CD\) respectively, with the segment \(MN\) parallel to the bases of the trapezoid. Diagonal \(AC\) intersects this segment at point \(O\). Find \(MN\), given that the areas of triangles \(AMO\) and \(CNO\) are equal. | \sqrt{} | 4/8 |
Let $L O V E R$ be a convex pentagon such that $L O V E$ is a rectangle. Given that $O V=20$ and $L O=V E=R E=R L=23$, compute the radius of the circle passing through $R, O$, and $V$. | 23 | 7/8 |
In space, there are 9 points where any 4 points are not coplanar. Connect several line segments among these 9 points such that no tetrahedron exists in the graph. What is the maximum number of triangles in the graph? $\qquad$ . | 27 | 4/8 |
Let \(\alpha\) and \(\beta\) satisfy the equations \(\alpha^3 - 3\alpha^2 + 5\alpha - 4 = 0\) and \(\beta^3 - 3\beta^2 + 5\beta - 2 = 0\) respectively. Find the value of \(\alpha + \beta\). | 2 | 6/8 |
There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such moves, it is possible to achieve a configuration where each chip is shifted one position clockwise relative to its initial position? | 50 | 1/8 |
An object is moving towards a converging lens with a focal length of \( f = 10 \ \mathrm{cm} \) along the line defined by the two focal points at a speed of \( 2 \ \mathrm{m/s} \). What is the relative speed between the object and its image when the object distance is \( t = 30 \ \mathrm{cm} \)? | 1.5 | 1/8 |
Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy] | 150 | 7/8 |
On the sides \(AB, BC, CD, DA\) of parallelogram \(ABCD\), points \(M, N, P, Q\) are given such that \(\frac{AM}{MB}=k_{1}, \frac{BN}{NC}=k_{2}, \frac{CP}{PD}=k_{1}, \frac{DQ}{QA}=k_{2}\). Prove that the lines \(AP, BQ, CM, DN\), upon intersecting, form another parallelogram and that the ratio of the area of this parallelogram to the area of the given parallelogram is
$$
\frac{k_{1} \cdot k_{2}}{\left(1+k_{1}\right)\left(1+k_{2}\right)+1}
$$ | \frac{k_{1}\cdotk_{2}}{(1+k_{1})(1+k_{2})+1} | 1/8 |
On the line segment AB, point O is marked, and from it, rays OC, OD, OE, and OF are drawn in the given order in one half-plane of line AB (ray OC lies between rays OA and OD). Find the sum of all angles with vertex O, whose sides are the rays OA, OC, OD, OE, OF, and OB, if \(\angle COF = 97^\circ\) and \(\angle DOE = 35^\circ\). | 1226 | 1/8 |
Given the function $f(x)=\sin \left( \frac {5\pi}{6}-2x\right)-2\sin \left(x- \frac {\pi}{4}\right)\cos \left(x+ \frac {3\pi}{4}\right).$
$(1)$ Find the minimum positive period and the intervals of monotonic increase for the function $f(x)$;
$(2)$ If $x_{0}\in\left[ \frac {\pi}{3}, \frac {7\pi}{12}\right]$ and $f(x_{0})= \frac {1}{3}$, find the value of $\cos 2x_{0}.$ | - \frac {2 \sqrt {6}+1}{6} | 5/8 |
Inside an isosceles triangle \( ABC \) (\( AB = AC \)), a point \( K \) is marked. Point \( L \) is the midpoint of segment \( BK \). It turns out that \(\angle AKB = \angle ALC = 90^\circ \), and \( AK = CL \). Find the angles of triangle \( ABC \). | 60 | 1/8 |
In the circumscribed quadrilateral \(ABCD\), the measures of the angles \(CDA\), \(DAB\), and \(ABC\) are \(90^{\circ}, 120^{\circ}\), and \(120^{\circ}\) respectively, and the length of side \(BC\) is 1 cm. Find the length of side \(AB\). | 2-\sqrt{3} | 3/8 |
The lengths of two skew edges of a tetrahedron \(ABCD\) are \(a\) and \(b\). The length of the segment connecting their midpoints is \(k\). What is the maximum possible surface area and volume of the tetrahedron? | \frac{abk}{6} | 1/8 |
From 125 sugar cubes, a $5 \times 5 \times 5$ cube was made. Ponchik picked all the cubes that have an odd number of neighbors and ate them (neighbors are those cubes that share a face). How many cubes did Ponchik eat in total? | 62 | 7/8 |
Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$ | 375 | 5/8 |
In a convex $n$-gon $P$, each side and each diagonal are colored with one of $n$ colors. For which values of $n$ is it possible to color such that for any three distinct colors among the $n$ colors, there exists a triangle with vertices at the vertices of $P$ and its three sides colored with these three colors? | n | 3/8 |
$A$ and $B$ play the following game: random \( k \) numbers are chosen from the first 100 positive integers. If the sum of these is even, $A$ wins; otherwise, $B$ wins. For which values of \( k \) are the winning chances of \( A \) and \( B \) equal? | k | 4/8 |
Let $ABCD$ be a regular tetrahedron, and let $O$ be the centroid of triangle $BCD$. Consider the point $P$ on $AO$ such that $P$ minimizes $PA+2(PB+PC+PD)$. Find $\sin \angle PBO$. | \frac{1}{6} | 7/8 |
Let \( f(n) \) be the integer closest to \( \sqrt[4]{n} \). Then, \( \sum_{k=1}^{2018} \frac{1}{f(k)} = \) ______. | \frac{2823}{7} | 6/8 |
The center \(O\) of the circle circumscribed around the quadrilateral \(ABCD\) is inside it. Find the area of the quadrilateral if \(\angle B A O = \angle D A C\), \(AC = m\), and \(BD = n\). | \frac{mn}{2} | 2/8 |
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