Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \( P \) is also considered a "good edge".

Given that 2003 non-intersecting diagonals divide \( P \) into several triangles, determine the maximum number of isosceles triangles, under this division, that have two "good edges".

Answer: 1003


Find the smallest three-digit number with the property that if you append to it a number that is 1 greater than it, the resulting six-digit number will be a perfect square.

Answer: 183


Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1,$what is the least possible total for the number of bananas?

Answer: 408


Let \(a, b, c, d\) be real numbers such that \(a \geq b \geq c \geq d\) and which satisfy the system of equations
\[
\begin{aligned}
a+b+c+d & =13 \\
a^{2}+b^{2}+c^{2}+d^{2} & =43
\end{aligned}
\]
Show that \(ab \geq 3 + cd\).

Answer: \ge3+cd


Consider a sequence of hexagons where each new hexagon adds two layers of dots instead of one. The first hexagon has 1 dot. The second hexagon has 1 central dot, 6 dots in the first layer, and 12 dots in the second layer, making a total of 19 dots. Determine the total number of dots in the third hexagon.

Answer: 61


Find the sum of squares of all distinct complex numbers \(x\) satisfying the equation
$$
0 = 4x^{10} - 7x^{9} + 5x^{8} - 8x^{7} + 12x^{6} - 12x^{5} + 12x^{4} - 8x^{3} + 5x^{2} - 7x + 4.
$$

Answer: -\frac{7}{16}


Given \(y z \neq 0\), and the set \(\{2x, 3z, xy\}\) can also be represented as \(\{y, 2x^2, 3xz\}\). Determine \(x\).

Answer: 1


Determine all three-digit numbers  $N$  having the property that  $N$  is divisible by 11, and  $\dfrac{N}{11}$  is equal to the sum of the squares of the digits of  $N$ .

Answer: 803


(1) Point $P$ is any point on the curve $y=x^{2}-\ln x$. The minimum distance from point $P$ to the line $x-y-4=0$ is ______.

(2) If the tangent line to the curve $y=g(x)$ at the point $(1,g(1))$ is $y=2x+1$, then the equation of the tangent line to the curve $f(x)=g(x)+\ln x$ at the point $(1,f(1))$ is ______.

(3) Given that the distance from point $P(1,0)$ to one of the asymptotes of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ is $\frac{1}{2}$, the eccentricity of the hyperbola $C$ is ______.

(4) A line passing through point $M(1,1)$ with a slope of $-\frac{1}{2}$ intersects the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ at points $A$ and $B$. If $M$ is the midpoint of segment $AB$, then the eccentricity of the ellipse $C$ is ______.

Answer: \frac{\sqrt{2}}{2}


In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)?

Answer: 200


Among 7 students, 6 are to be arranged to participate in social practice activities in two communities on Saturday, with 3 people in each community. How many different arrangements are there? (Answer with a number)

Answer: 140


Can the rounded values of $\sqrt{10^{2 n}-10^{n}}$ and $\sqrt{10^{2 n}-10^{n}+1}$ be equal? ( $n$ is a positive integer.)

Answer: No


Answer: \frac{1}{2}


Find all integer triplets $(x, y, z)$ such that $x^{2}+y^{2}+z^{2}-2xyz=0$.

Answer: (0,0,0)


Find \(\lim _{x \rightarrow 2}(2-x) \tan \frac{\pi}{4} x\).

Answer: \frac{4}{\pi}


Given the function $f(x) = |2x+1| + |3x-2|$, and the solution set of the inequality $f(x) \leq 5$ is $\left\{x \mid -\frac{4a}{5} \leq x \leq \frac{3a}{5}\right\}$, where $a, b \in \mathbb{R}$.
1. Find the values of $a$ and $b$;
2. For any real number $x$, the inequality $|x-a| + |x+b| \geq m^2 - 3m$ holds, find the maximum value of the real number $m$.

Answer: \frac{3 + \sqrt{21}}{2}


Let \( a_{i} = 0,1 \) for \( i = 1, 2, \cdots, n \). Find the number of sequences \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) that satisfy \( a_{1} \leqslant a_{2}, a_{2} \geqslant a_{3}, a_{3} \leqslant a_{4}, a_{4} \geqslant a_{5}, \cdots \).

Answer: F_{n+2}


A circle lies inside the parabola defined by the equation \( y = 4x^2 \) and is tangent to the parabola at two points. Determine how much higher the center of the circle is compared to the points of tangency.

Answer: \frac{1}{8}


Find the largest three-digit number that is a multiple of the sum of its digits and in which the first digit matches the third digit, but does not match the second digit.

Answer: 828


When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of seven consecutive positive integers, all of which are nonprime?

Answer: 53


Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). From there, without returning to the previous vertex, the ant selects one of the 3 adjacent vertices (excluding the one it came from) and continues to the next level. What is the probability that the ant reaches the bottom central vertex?

Answer: \frac{1}{3}


A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows. After finishing the drills, the lieutenant decides to arrange the recruits again but cannot remember how many rows there were initially. So, he orders them to form as many rows as his age. It turns out that each row again has the same number of soldiers, but in each row, there are 4 more soldiers than there were in the original arrangement. How old is the lieutenant?

Answer: 24


For which \( k \) is the number \( N = 101 \cdots 0101 \) with \( k \) ones a prime?

Answer: 2


The measure of angle $ACB$ is 30 degrees. If ray $CA$ is rotated 510 degrees about point $C$ clockwise, what will be the positive measure of the acute angle $ACB$, in degrees?

Answer: 120


Find the smallest positive integer \( n > 1 \) such that the arithmetic mean of \( 1^2, 2^2, 3^2, \cdots, n^2 \) is a perfect square.

Answer: 337


Let $A B C$ be a triangle with $A B=5, B C=8, C A=11$. The incircle $\omega$ and $A$-excircle $^{1} \Gamma$ are centered at $I_{1}$ and $I_{2}$, respectively, and are tangent to $B C$ at $D_{1}$ and $D_{2}$, respectively. Find the ratio of the area of $\triangle A I_{1} D_{1}$ to the area of $\triangle A I_{2} D_{2}$.

Answer: \frac{1}{9}


In a 6 by 6 grid, each of the 36 small squares measures 3 cm by 3 cm and is initially shaded. Five unshaded squares of side 1.5 cm are placed on the grid in such a way that some may overlap each other and/or the edges of the grid. If such overlap occurs, the overlapping parts are not visible as shaded. The area of the visible shaded region can be written in the form $A-B\phi$. What is the value $A+B$?

Answer: 335.25


The extension of the altitude \( BH \) of triangle \( ABC \) intersects the circle circumscribed around it at point \( D \) (points \( B \) and \( D \) lie on opposite sides of line \( AC \) ). The degree measures of arcs \( AD \) and \( CD \), not containing point \( B \), are \( 60^\circ \) and \( 90^\circ \) respectively. Determine the ratio in which segment \( BD \) is divided by side \( AC \).

Answer: \sqrt{3}:1


How many integers can be expressed as a sum of three distinct numbers if chosen from the set $\{4, 7, 10, 13, \ldots, 46\}$?

Answer: 37


In base \( R_1 \), the fractional expansion of \( F_1 \) is \( 0.373737 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.737373 \cdots \). In base \( R_2 \), the fractional expansion of \( F_1 \) is \( 0.252525 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.525252 \cdots \). What is the sum of \( R_1 \) and \( R_2 \) (both expressed in decimal)?

Answer: 19


Given Professor Chen has ten different language books, including two Arabic, four German, and four Spanish books, arranged on the shelf so that the Arabic books are together, the Spanish books are together, and two of the German books are treated as indistinguishable, calculate the number of ways to arrange the ten books.

Answer: 576


In the acute-angled triangle  $ABC$  the angle $ \angle B = 30^o$ , point  $H$  is the intersection point of its altitudes. Denote by  $O_1, O_2$  the centers of circles inscribed in triangles   $ABH ,CBH$   respectively. Find the degree of the angle between the lines  $AO_2$  and  $CO_1$ .

Answer: 45


Given natural numbers \(a, b,\) and \(c\), none of which is a multiple of the other, it is known that the number \(abc + 1\) is divisible by \(ab - b + 1\). Prove that \(c \geq b\).

(M. Antipov)

Answer: \geb


Given that $\cos α=-\dfrac{4}{5}\left(\dfrac{π}{2}<α<π\right)$, find $\cos\left(\dfrac{π}{6}-α\right)$ and $\cos\left(\dfrac{π}{6}+α\right)$.

Answer: -\dfrac{3+4\sqrt{3}}{10}


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 $BC$?

Answer: 15


Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

Answer: \frac{47}{256}


Let  $A$  be a given finite set with some of its subsets called pretty. Let a subset be called small, if it's a subset of a pretty set. Let a subset be called big, if it has a pretty subset. (A set can be small and big simultaneously, and a set can be neither small nor big.) Let  $a$  denote the number of elements of  $A$ , and let  $p$ ,  $s$  and  $b$  denote the number of pretty, small and big sets, respectively. Prove that  $2^a\cdot p\le s\cdot b$ .

*Proposed by András Imolay, Budapest*

Answer: 2^\cdotp\le\cdotb


In the diagram, \(ABCD\) is a parallelogram. \(E\) is on side \(AB\), and \(F\) is on side \(DC\). \(G\) is the intersection point of \(AF\) and \(DE\), and \(H\) is the intersection point of \(CE\) and \(BF\). Given that the area of parallelogram \(ABCD\) is 1, \(\frac{\mathrm{AE}}{\mathrm{EB}}=\frac{1}{4}\), and the area of triangle \(BHC\) is \(\frac{1}{8}\), find the area of triangle \(ADG\).

Answer: \frac{7}{92}


In a kindergarten, 5 children eat porridge every day, 7 children eat porridge every other day, and the rest never eat porridge. Yesterday, 9 children ate porridge. How many children will eat porridge today?

Answer: 8


In the Cartesian coordinate system $xOy$, point $F$ is a focus of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $B_1(0, -\sqrt{3})$ is a vertex of $C$, $\angle OFB_1 = \frac{\pi}{3}$.

$(1)$ Find the standard equation of $C$;

$(2)$ If point $M(x_0, y_0)$ is on $C$, then point $N(\frac{x_0}{a}, \frac{y_0}{b})$ is called an "ellipse point" of point $M$. The line $l$: $y = kx + m$ intersects $C$ at points $A$ and $B$, and the "ellipse points" of $A$ and $B$ are $P$ and $Q$ respectively. If the circle with diameter $PQ$ passes through point $O$, find the area of $\triangle AOB$.

Answer: \sqrt{3}


In the polar coordinate system, the curve $C_1$: $\rho=2\cos\theta$, and the curve $$C_{2}：\rho\sin^{2}\theta=4\cos\theta$$．Establish a Cartesian coordinate system xOy with the pole as the origin and the polar axis as the positive half-axis of x, the parametric equation of curve C is $$\begin{cases} x=2+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t\end{cases}$$（t is the parameter）．
（I）Find the Cartesian equations of $C_1$ and $C_2$；
（II）C intersects with $C_1$ and $C_2$ at four different points, and the sequence of these four points on C is P, Q, R, S. Find the value of $||PQ|-|RS||$．

Answer: \frac {11}{3}


During a year when Valentine's Day, February 14, falls on a Tuesday, what day of the week is Cinco de Mayo (May 5) and how many days are between February 14 and May 5 inclusively?

Answer: 81


Find a factor of 1464101210001 that lies between 1210000 and 1220000.

Answer: 1211101


A solid is formed by rotating a triangle with sides of lengths 3, 4, and 5 around the line containing its shortest side. Find the surface area of this solid.

Answer: 36\pi


In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced by any digit from $0, 1, 2, 3, 4, 5, 6, 7, 8$ (digits can repeat) so that the resulting 10-digit number is divisible by 45. In how many ways can this be done?

Answer: 1458


A regular $n$-gon is inscribed in a circle with radius $R$. By connecting the midpoints of the sides of this polygon, another regular $n$-gon is formed. Repeating this process infinitely, show that the sum of the areas of these polygons is equal to the area of a regular $n$-gon whose side lengths are each $2R$.

Answer: nR^2\cot(\frac{\pi}{n})


The world-renowned Marxist theorist *Joric* is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer  $n$ , he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the *defect* of the number  $n$ . Determine the average value of the defect (over all positive integers), that is, if we denote by  $\delta(n)$  the defect of  $n$ , compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\]
*Iurie Boreico*

Answer: \frac{1}{2}


Given an odd function \( f(x) \) defined on \(\mathbf{R}\), which satisfies \( f(x) + f(x+2) = a \) and \( f(1) = 0 \), where \( a \) is a constant, determine the minimum number of roots of the equation \( f(x) = 0 \) within the interval \((-3,7)\).

Answer: 9


In the acute triangle \(KLM\), \(V\) is the intersection of its heights, and \(X\) is the foot of the height onto side \(KL\). The angle bisector of angle \(XVL\) is parallel to side \(LM\), and the angle \(MKL\) measures \(70^\circ\).

What are the measures of the angles \(KLM\) and \(KML\)?

Answer: 55


The ellipse \(\frac{x^{2}}{2002^{2}}+\frac{y^{2}}{1949^{2}}=1\) intersects with two lines passing through the origin at points \(A, B, C,\) and \(D\). The slope of line \(AB\) is \(k\), and the slope of line \(CD\) is \(-\frac{1949^{2}}{2002^{2} \cdot k}\). Given that there exists a triangle \(\triangle PQR\) such that \(PQ=OA\), \(PR=OC\), and \(\angle QPR=\left|\angle AOC-\frac{\pi}{2}\right|\), find \(QR-\qquad\).

Answer: 53


Let  $S$  be the set of 81 points  $(x, y)$  such that  $x$  and  $y$  are integers from  $-4$  through  $4$ .  Let  $A$ ,  $B$ , and  $C$  be random points chosen independently from  $S$ , with each of the 81 points being equally likely.  (The points  $A$ ,  $B$ , and  $C$  do not have to be different.)  Let  $K$  be the area of the (possibly degenerate) triangle  $ABC$ .  What is the expected value (average value) of  $K^2$  ?

Answer: \frac{200}{3}


Answer: 32


Masha left home for school. A few minutes later, Vanya ran out of the same house towards the school. He overtook Masha one-third of the way, and when he reached the school, Masha still had half of the way to go. How many times faster does Vanya run compared to Masha's walking speed?

Answer: 4


How many different cubes are there with three faces colored red and three faces colored blue?

Answer: 2


Let the rational number  $p/q$  be closest to but not equal to  $22/7$  among all rational numbers with denominator  $< 100$ . What is the value of  $p - 3q$ ?

Answer: 14


To enhance and beautify the city, all seven streetlights on a road are to be changed to colored lights. If there are three colors available for the colored lights - red, yellow, and blue - and the installation requires that no two adjacent streetlights are of the same color, with at least two lights of each color, there are ____ different installation methods.

Answer: 114


Given triangle $ABC$ with midpoint $D$ on side $BC$, and point $G$ satisfies $\overrightarrow{GA}+ \overrightarrow{BG}+ \overrightarrow{CG}= \overrightarrow{0}$, and $\overrightarrow{AG}=\lambda \overrightarrow{GD}$, determine the value of $\lambda$.

Answer: -2


In triangle \(ABC\), the median \(AM\) is drawn. Circle \(\omega\) passes through point \(A\), touches line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On arc \(AD\) that does not contain point \(E\), point \(F\) is chosen such that \(\angle BFE = 72^{\circ}\). It is found that \(\angle DEF = \angle ABC\). Find angle \(\angle CME\).

Answer: 36


Given the positive integer \( A = \overline{a_{n} a_{n-1} \cdots a_{1} a_{0}} \), where \( a_{n}, a_{n-1}, \ldots, a_{0} \) are all non-zero and not all equal (with \( n \) being a positive integer), consider the following cyclic permutations of \( A \):
$$
\begin{array}{l}
A_{1}=\overline{a_{n-1} \cdots a_{1} a_{0} a_{n}}, \\
A_{2}=\overline{a_{n-2} \cdots a_{1} a_{0} a_{n} a_{n-1}}, \\
\cdots \cdots \\
A_{k}=\overline{a_{n-k} a_{n-k-1} \cdots a_{0} a_{n} \cdots a_{n-k+1}}, \\
\cdots \cdots \\
A_{n}=\overline{a_{0} a_{n} \cdots a_{1}}
\end{array}
$$

Determine \( A \) such that \( A \) divides each \( A_{k} \) for \( k=1,2, \cdots, n \).

Answer: 142857


In quadrilateral $ABCD$ with diagonals $AC$ and $BD$, intersecting at $O$, $BO=4$, $OD = 6$, $AO=8$, $OC=3$, and $AB=6$. The length of $AD$ is:
$\textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 6\sqrt{3}\qquad \textbf{(D)}\ 8\sqrt{2}\qquad \textbf{(E)}\ \sqrt{166}$

Answer: \textbf{(E)}~\sqrt{166}


In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals

Answer: 50^\circ


There are two positive integers \( A \) and \( B \). The sum of the digits of \( A \) is 19, and the sum of the digits of \( B \) is 20. When the two numbers are added together, there are two carries. What is the sum of the digits of \( (A+B) \)?

Answer: 21


Answer: \frac{3}{4}


If \( 998a + 1 = 999^2 \), find \( a \).

If \( \log_{10} a = \log_{2} b \), find \( b \).

The area of the triangle formed by the \( x \)-axis, the \( y \)-axis, and the line \( 2x + y = b \) is \( c \) square units. Find \( c \).

If \( 64t^2 + ct + d \) is a perfect square, find \( d \).

Answer: 1


Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$.

Answer: 0


A circle is drawn through the vertex \( M \) of an angle, intersecting the sides of the angle at points \( N \) and \( K \), and the angle bisector at point \( L \). Find the sum of the lengths of segments \( M N \) and \( M K \) if the area of quadrilateral \( M N L K \) is 25 and angle \( L M N \) is \( 30^\circ \).

Answer: 10\sqrt[4]{3}


Points \( A_1 \), \( B_1 \), and \( C_1 \) are taken on the sides \( BC \), \( CA \), and \( AB \) of triangle \( ABC \) respectively, such that the radii of the circles inscribed in triangles \( A_1BC_1 \), \( AB_1C_1 \), and \( A_1B_1C \) are all equal and are denoted by \( r \). The radius of the circle inscribed in triangle \( A_1B_1C_1 \) is \( r_1 \). Find the radius of the circle inscribed in triangle \( ABC \).

Answer: r+r_1


A certain real estate property is holding a lottery for homebuyers, with the following rules: For homeowners who purchase the property, they can randomly draw 2 balls from box $A$, which contains 2 red balls and 2 white balls, and 2 balls from box $B$, which contains 3 red balls and 2 white balls. If all 4 balls drawn are red, the homeowner wins the first prize and receives $10,000 in renovation funds; if exactly 3 balls are red, the homeowner wins the second prize and receives $5,000 in renovation funds; if exactly 2 balls are red, the homeowner wins the third prize and receives $3,000 in renovation funds; any other outcome is considered a consolation prize, awarding $1,500 in renovation funds.
$(Ⅰ)$ Three homeowners participate in the lottery. Find the probability that exactly one homeowner wins the second prize.
$(Ⅱ)$ Let $X$ denote the amount of renovation funds won by a homeowner in the lottery. Find the probability distribution and the expected value of $X$.

Answer: 3,675


20 shareholders are seated around a round table. What is the minimum total number of their shares if it is known that:
a) any three of them together have more than 1000 shares,
b) any three consecutive shareholders together have more than 1000 shares?

Answer: 6674


What is the minimum area of a triangle that contains a unit square?

Answer: 2


The coefficient of the $x^3$ term in the expansion of $(2-\sqrt{x})^8$ is $1120x^3$.

Answer: 112


In the expansion of $(x^2+ \frac{4}{x^2}-4)^3(x+3)$, find the constant term.

Answer: -240


Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function that satisfies

\[ f(1) = 2, \]
\[ f(2) = 1, \]
\[ f(3n) = 3f(n), \]
\[ f(3n + 1) = 3f(n) + 2, \]
\[ f(3n + 2) = 3f(n) + 1. \]

Find how many integers \( n \leq 2014 \) satisfy \( f(n) = 2n \).

Answer: 127


Convex quadrilateral \(E F G H\) has its vertices \(E, F, G, H\) located on the sides \(A B, B C, C D, D A\) respectively of convex quadrilateral \(A B C D\), such that
\[
\frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A} = 1
\]

Points \(A, B, C, D\) are located on the sides \(H_{1} E_{1}, E_{1} F_{1}, F_{1} G_{1}, G_{1} H_{1}\) respectively of convex quadrilateral \(E_{1} F_{1} G_{1} H_{1}\), with the conditions \(E_{1} F_{1} \parallel E F, F_{1} G_{1} \parallel F G, G_{1} H_{1} \parallel G H, H_{1} E_{1} \parallel H E\). Given \(\frac{E_{1} A}{A H_{1}} = \lambda\), find the value of \(\frac{F_{1} C}{C G_{1}}\).

Answer: \lambda


Given the positive numbers \( a, b, c, x, y, z \) that satisfy the equations \( cy + bz = a \), \( az + cx = b \), and \( bx + ay = c \), find the minimum value of the function
\[
f(x, y, z) = \frac{x^{2}}{1+x} + \frac{y^{2}}{1+y} + \frac{z^{2}}{1+z}.
\]

Answer: 1/2


In triangle \( \triangle ABC \), \( AB = 86 \), and \( AC = 97 \). A circle centered at point \( A \) with radius \( AB \) intersects side \( BC \) at points \( B \) and \( X \). Moreover, \( BX \) and \( CX \) have integer lengths. What is the length of \( BC \)?

Answer: 61


Given the lengths of the sides $a, b, c$ of a triangle and its area $S$, prove that $a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} S$. Also, determine the condition under which equality holds.

Answer: ^2+b^2+^2\ge4\sqrt{3}S


Let  $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $  be a function such that for all  $n > 1$  there is a prime divisor  $p$  of  $n$  such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that  $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $ . Determine  $ f(2014^2)+f(2015^3)+f(2016^5) $ .

Answer: \frac{49}{3}


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 \( f(84) \).

Answer: 997


The digital root of a positive integer is obtained by repeatedly taking the product of the digits until we get a single-digit number. For example, 24378 → 1344 → 48 → 32 → 6. Show that if \( n \) has a digital root of 1, then all its digits are 1.

Answer: 1


Let points \( D, E, F \) be on the three sides \( BC, CA, AB \) of \( \triangle ABC \) respectively, such that the incircles of \( \triangle AEF, \triangle BFD, \triangle CDE \) have equal radius \( r \). Let \( r_0 \) and \( R \) denote the inradius of \( \triangle DEF \) and \( \triangle ABC \) respectively. Prove that
\[ r + r_0 = R \]

Answer: r+r_0=R


In \(\triangle ABC\), let \(D\), \(E\), and \(F\) be points on sides \(BC\), \(CA\), and \(AB\), respectively, such that the segments \(AD\), \(BE\), and \(CF\) are concurrent at point \(P\). If \(AF: FB = 4: 5\) and the ratio of the area of \(\triangle APB\) to that of \(\triangle APC\) is \(1: 2\), determine \(AE: AC\).

Answer: \frac{2}{7}


In a closed right triangular prism ABC-A1B1C1 there is a sphere with volume $V$. If $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, then the maximum value of $V$ is \_\_\_\_\_\_.

Answer: \frac{9\pi}{2}


Let \( a, b, c > 0 \) and \( a + b + c = 2 \). Prove that:
\[
\sum \frac{ab}{c^{2} + 1} \leqslant 1
\]

Answer: 1


Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$.

Answer: -\frac{5}{4}


In the $5 \times 5$ grid shown, 15 cells contain X's and 10 cells are empty. What is the smallest number of X's that must be moved so that each row and each column contains exactly three X's?

Answer: 2


On the sides of triangle \(ABC\), points were marked: 10 on side \(AB\), 11 on side \(BC\), and 12 on side \(AC\). None of the vertices of the triangle were marked. How many triangles with vertices at the marked points exist?

Answer: 4951


Inside the acute-angled triangle \(ABC\), a point \(M\) is marked. The lines \(AM, BM\), and \(CM\) intersect the sides of the triangle at points \(A_1, B_1\), and \(C_1\) respectively. It is known that \(MA_1 = MB_1 = MC_1 = 3\) and \(AM + BM + CM = 43\). Find \(AM \cdot BM \cdot CM\).

Answer: 441


Let \(P(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \ldots + a_{1} x + a_{0}\), where \(n\) is a natural number. It is known that the numbers \(a_{0}, a_{1}, \ldots, a_{n}\) are integers, with \(a_{n} \neq 0\), \(a_{n-k} = a_{k}\) for all \(k = 0, 1, \ldots, n\), and \(a_{n} + a_{n-1} + \ldots + a_{1} + a_{0} = 0\). Prove that the number \(P(2022)\) is divisible by the square of some natural number greater than 1.

Answer: 2021^2


4. (7 points) Linlin copied a repeating decimal incorrectly as 0.123456. If the digits are correct but the repeating part indicator was mistakenly left out, how many possible original repeating decimals could there be?

Answer: 6


Points $A$ and $C$ lie on a circle centered at $O$, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle, and $\triangle ABC$ is equilateral. The circle intersects $\overline{BO}$ at $D$. What is $\frac{BD}{BO}$?

Answer: \frac{1}{2}


What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)?

Answer: 241


Sara baked 60 pies. Of these, one-third contained berries, half contained cream, three-fifths contained nuts, and one-fifth contained coconut. What is the largest possible number of pies that had none of these ingredients?

Answer: 24


If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - 2B$, then what is the value of $Y$?

Answer: 335


Find the number of ways to arrange the numbers 1 through 7 in a circle such that the numbers are increasing along each arc from 1.

Answer: 32


Let \( S \) be a finite set of points in the plane such that no three of them are on a line. For each convex polygon \( P \) whose vertices are in \( S \), let \( a(P) \) be the number of vertices of \( P \), and let \( b(P) \) be the number of points of \( S \) which are outside \( P \). Prove that for every real number \( x \)
$$
\sum_{P} x^{a(P)}(1-x)^{b(P)}=1,
$$
where the sum is taken over all convex polygons with vertices in \( S \). Note that a line segment, a point, and the empty set are considered as convex polygons with 2, 1, and 0 vertices, respectively.

Answer: 1


There are  $2019$  points given in the plane. A child wants to draw  $k$  (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal  $k$ , such that for any initial configuration of points it is possible to draw  $k$  discs with the above property?

Answer: 1010


\(-1 < a < 1\). The sequence \(x_1, x_2, x_3, \ldots\) is defined by \(x_1 = a\) and \(x_{n+1} = \frac{\sqrt{3 - 3x_n^2} - x_n}{2}\). Find necessary and sufficient conditions on \(a\) for all members of the sequence to be positive. Show that the sequence is always periodic.

Answer: 0<\frac{\sqrt{3}}{2}


All positive divisors of a positive integer  $N$  are written on a blackboard. Two players  $A$  and  $B$  play the following game taking alternate moves. In the firt move, the player  $A$  erases  $N$ . If the last erased number is  $d$ , then the next player erases either a divisor of  $d$  or a multiple of  $d$ . The player who cannot make a move loses. Determine all numbers  $N$  for which  $A$  can win independently of the moves of  $B$ .

*(4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)*

Answer: N


\(ABCD\) is a convex quadrilateral and \(E, F\) are the mid-points of \(BC\) and \(CD\) respectively. The line segments \(AE, AF\) and \(EF\) divide \(ABCD\) into four triangles, whose areas are four consecutive integers. Find the greatest possible area of \(\triangle ABD\).

Answer: 6


A positive integer  $n>1$  is called beautiful if  $n$  can be written in one and only one way as  $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$  for some positive integers  $a_1, a_2, \ldots, a_k$ , where  $k>1$  and  $a_1 \geq a_2 \geq \cdots \geq a_k$ . (For example 6 is beautiful since  $6=3 \cdot 2 \cdot 1=3+2+1$ , and this is unique. But 8 is not beautiful since  $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$  as well as  $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$ , so uniqueness is lost.) Find the largest beautiful number less than 100.

Answer: 95


In response to the national policy of expanding domestic demand, a manufacturer plans to hold a promotional event at the beginning of 2015. After investigation and estimation, the annual sales volume (i.e., the annual production volume $x$ in ten thousand units) and the annual promotional expenses $t$ (where $t > 0$) in ten thousand yuan satisfy $x=4- \frac {k}{t}$ (where $k$ is a constant). If the annual promotional expenses $t$ are 1 ten thousand yuan, the annual sales volume of the product is 1 ten thousand units. It is known that the fixed investment for the product in 2015 is 60 thousand yuan, and an additional investment of 120 thousand yuan is required to produce 1 ten thousand units of the product. The manufacturer sets the selling price of each unit to 1.5 times the average cost of the product (the product cost includes both fixed and additional investments).
- (Ⅰ) Express the profit $y$ (in ten thousand yuan) of the manufacturer for this product in 2015 as a function of the annual promotional expenses $t$ (in ten thousand yuan);
- (Ⅱ) How much should the manufacturer invest in annual promotional expenses in 2015 to maximize profit?

Answer: 3 \sqrt {2}


Jefferson Middle School has the same number of boys and girls. $\frac{3}{4}$ of the girls and $\frac{2}{3}$
of the boys went on a field trip. What fraction of the students on the field trip were girls?
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{9}{17}\qquad\textbf{(C) }\frac{7}{13}\qquad\textbf{(D) }\frac{2}{3}\qquad \textbf{(E) }\frac{14}{15}$

Answer: \textbf{(B)}\frac{9}{17}


Let  $S$  be the set of all permutations of  $\{1, 2, 3, 4, 5\}$ . For  $s = (a_1, a_2,a_3,a_4,a_5) \in S$ , define  $\text{nimo}(s)$  to be the sum of all indices  $i \in \{1, 2, 3, 4\}$  for which  $a_i > a_{i+1}$ . For instance, if  $s=(2,3,1,5,4)$ , then  $\text{nimo}(s)=2+4=6$ . Compute \[\sum_{s\in S}2^{\text{nimo}(s)}.\]

*Proposed by Mehtaab Sawhney*

Answer: 9765


The right focal point of the hyperbola \( x^{2}-\frac{y^{2}}{2}=1 \) draws a line \( l \), which intersects the hyperbola at points \( A \) and \( B \). If there exists a real number \( \lambda \) such that \( |AB|=\lambda \), and there are exactly 3 such lines \( l \), then \( \lambda = \quad \).

Answer: 4


In a sports complex, there is a rectangular area \(ABCD\) with the longer side \(AB\). The diagonals \(AC\) and \(BD\) intersect at an angle of \(60^\circ\). Runners train on the large circuit \(ACBDA\) or on the small track \(AD\). Mojmír ran ten times around the large circuit, and Vojta ran fifteen times around the small track, in one direction and then fifteen times in the opposite direction. Together they ran \(4.5 \text{ km}\). How long is the diagonal \(AC\)?

Answer: 100


How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls?

Answer: 25


Solve for $x$ in the equation $\frac{4}{7} \cdot \frac{1}{9} \cdot x = 14$.

Answer: 220.5


Evaluate  $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$

You may not use  $ \tan ^{ \minus{} 1} x$  or Complex Integral here.

Answer: \frac{\pi}{2a}


In the triangle \( \triangle ABC \), draw tangents to the incircle \( O \) of the triangle that are parallel to each side of the triangle. These tangents form three smaller triangles \( \triangle AED, \triangle BGF, \triangle CIH \) (where \( E, F \) are on \( AB \), \( G, H \) are on \( BC \), and \( I, D \) are on \( AC \)). The inradii of these smaller triangles are \( r_1, r_2, r_3 \) respectively, while the inradius of \( \triangle ABC \) is \( r \). Prove that \( r = r_1 + r_2 + r_3 \).

Answer: r_1+r_2+r_3


We need to color the three-element subsets of a seven-element set in such a way that if the intersection of two subsets is empty, then their colors are different. What is the minimum number of colors needed for this?

Answer: 3


The formula $N=8 \times 10^{8} \times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars.
The lowest income, in dollars, of the wealthiest $800$ individuals is at least:
$\textbf{(A)}\ 10^4\qquad \textbf{(B)}\ 10^6\qquad \textbf{(C)}\ 10^8\qquad \textbf{(D)}\ 10^{12} \qquad \textbf{(E)}\ 10^{16}$

Answer: \textbf{(A)}\10^4


A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?

Answer: 84


In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have?

Answer: 16


Given the numbers \( x_{1}, \ldots, x_{n} \) within the interval \(\left[0, \frac{\pi}{2}\right]\), find the maximum value of the expression

$$
A=\left(\sqrt{\sin x_{1}}+\ldots+\sqrt{\sin x_{n}}\right) \cdot\left(\sqrt{\cos x_{1}}+\ldots+\sqrt{\cos x_{n}}\right).
$$

Answer: \frac{n^2}{\sqrt{2}}


If an integer is divisible by $6$ and the sum of its last two digits is $15$, then what is the product of its last two digits?

Answer: 54


Answer: \frac{1}{3}


Answer: 18


Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?

*Proposed by Michael Ren*

Answer: 11


Given that the variables $a$ and $b$ satisfy the equation $b=-\frac{1}{2}a^2 + 3\ln{a} (a > 0)$, and that point $Q(m, n)$ lies on the line $y = 2x + \frac{1}{2}$, find the minimum value of $(a - m)^2 + (b - n)^2$.

Answer: \frac{9}{5}


Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]

Answer: \frac{7}{16}


All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$

Answer: \textbf{(D)}8


Let $y = ax^{2} + x - b$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$.
$(1)$ If $b = 1$ and the set $\{x | y = 0\}$ has exactly one element, find the set of possible values for the real number $a$.
$(2)$ Solve the inequality with respect to $x$: $y < (a-1)x^{2} + (b+2)x - 2b$.
$(3)$ When $a > 0$ and $b > 1$, let $P$ be the solution set of the inequality $y > 0$, and $Q = \{x | -2-t < x < -2+t\}$. If for any positive number $t$, $P \cap Q \neq \varnothing$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$.

Answer: \frac{1}{2}


Harry has 3 sisters and 5 brothers. His sister Harriet has $\text{S}$ sisters and $\text{B}$ brothers. What is the product of $\text{S}$ and $\text{B}$?

Answer: 10


On a 4x4 grid (where each unit distance is 1), calculate how many rectangles can be formed where each of the rectangle's vertices is a point on this grid.

Answer: 36


Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a $60 \%$ chance of winning each point, what is the probability that he will win the game?

Answer: 9 / 13


Let $\alpha$ and $\beta$ be complex numbers such that $|\beta| = 1$ and $\overline{\alpha} \beta \neq 1.$  Find the maximum value of
\[\left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right|.\]

Answer: 1


Let  $M$  be the point of intersection of the diagonals  $AC$  and  $BD$  of trapezoid  $ABCD$  ( $BC||AD$ ),  $AD>BC$ . Circle  $w_1$  passes through the point  $M$  and tangents  $AD$  at the point  $A$ . Circle  $w_2$  passes through the point  $M$  and tangents  $AD$  at the point  $D$ . Point  $S$  is the point of intersection of lines  $AB$  and  $DC$ . Line  $AS$  intersects  $w_1$  at the point  $X$ . Line  $DS$  intersects  $w_2$  at the point  $Y$ .  $O$  is a center of a circumcircle of  $\triangle ASD$ . Prove that  $SO\perp XY$

Answer: SO\perpXY


A continuous function  $f : R  \to R$  satisfies the conditions  $f(1000) = 999$  and  $f(x)f(f(x)) = 1$  for all real  $x$ . Determine  $f(500)$ .

Answer: \frac{1}{500}


In the convex quadrilateral $ABCD$, points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ respectively. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into 4 triangles whose areas are consecutive natural numbers. What is the greatest possible area of triangle $ABD$?

Answer: 6


Find the smallest positive integer  $k$  such that there is exactly one prime number of the form  $kx + 60$  for the integers  $0 \le x \le 10$ .

Answer: 17


Let  $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$  and  $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and  }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in  $S$ ?

Answer: 21


Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, let vectors $\overrightarrow{m}=(\cos B,\sin C)$ and $\overrightarrow{n}=(\cos C,-\sin B)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=\frac{1}{2}$.

$(1)$ Find the magnitude of the interior angle $A$;

$(2)$ If the radius of the circumcircle of $\triangle ABC$ is $R=2$, find the maximum value of the perimeter $L$ of $\triangle ABC$.

Answer: 2\sqrt{3}+4


Given that \(ABCD-A_{1}B_{1}C_{1}D_{1}\) is a cube and \(P-A_{1}B_{1}C_{1}D_{1}\) is a regular tetrahedron, find the cosine of the angle between the skew lines \(A_{1}P\) and \(BC_{1}\), given that the distance from point \(P\) to plane \(ABC\) is \(\frac{3}{2}AB\).

Answer: \frac{\sqrt{6}}{3}


Answer: 0


A square 10cm on each side has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of $\pi$.

[asy]
unitsize (1.5 cm);

draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle);
filldraw(arc((1,1),1,270,180)--arc((-1,1),1,360,270)--arc((-1,-1),1,90,0)--arc((1,-1),1,180,90)--cycle,gray);
[/asy]

Answer: 100-25\pi


Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.

Answer: 0.296


For a positive integer  $n$ , there is a school with  $2n$  people. For a set  $X$  of students in this school, if any two students in  $X$  know each other, we call  $X$  *well-formed*. If the maximum number of students in a well-formed set is no more than  $n$ , find the maximum number of well-formed set.

Here, an empty set and a set with one student is regarded as well-formed as well.

Answer: 3^n


Given thirteen points on a plane such that among any five of them, there are four that lie on a circle. Prove that at least six of the given points lie on a circle.

Answer: 6


Can it be asserted that for any values of the variable:
1) \(4 b^{2}\left(b^{3}-1\right)-3\left(1-2 b^{2}\right)>4\left(b^{5}-1\right)\);
2) \(a-a\left|-a^{2}-1\right|<1-a^{2}(a-1)\) ?

Answer: True


Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, calculate the value of $f\left(-\frac{{5π}}{{12}}\right)$.

Answer: \frac{\sqrt{3}}{2}


In triangle \( \triangle ABC \), prove that:
$$
\sin C = \cos A + \cos B \Leftrightarrow A=\frac{\pi}{2} \quad \text{or} \quad B=\frac{\pi}{2}.
$$

Answer: \frac{\pi}{2}\quad


Anna and Boris move simultaneously towards each other, from points A and B respectively. Their speeds are constant, but not necessarily equal. Had Anna started 30 minutes earlier, they would have met 2 kilometers nearer to B. Had Boris started 30 minutes earlier instead, they would have met some distance nearer to A. Can this distance be uniquely determined?

*(3 points)*

Answer: 2


A city is a point on the plane. Suppose there are  $n\geq 2$  cities. Suppose that for each city  $X$ , there is another city  $N(X)$  that is strictly closer to  $X$  than all the other cities. The government builds a road connecting each city  $X$  and its  $N(X)$ ; no other roads have been built. Suppose we know that, starting from any city, we can reach any other city through a series of road.

We call a city  $Y$  *suburban* if it is  $N(X)$  for some city  $X$ . Show that there are at least  $(n-2)/4$  suburban cities.

*Proposed by usjl.*

Answer: \frac{n-2}{4}


Answer: \frac{1}{2}


On a rectangular board with  $m$  rows and  $n$  columns, where  $m\leq n$ , some squares are coloured black in a way that no two rows are alike. Find the biggest integer  $k$  such that for every possible colouring to start with one can always color  $k$  columns entirely red in such a way that no two rows are still alike.

Answer: n-+1


Two people are playing a game. One person thinks of a ten-digit number, and the other can ask questions about which digits are in specific sets of positions in the number's sequence. The first person answers the questions without indicating which digits are in which exact positions. What is the minimum number of questions needed to reliably guess the number?

Answer: 10


Answer: 625


Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 626


For any non-empty subset \( X \) of the set \( M = \{1, 2, \ldots, 1000\} \), let \( \alpha_X \) represent the sum of the maximum and minimum numbers in \( X \). Find the arithmetic mean of all such \( \alpha_X \).

Answer: 1001


Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 2 meters. A bug, starting from vertex $A$, follows the rule that at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. What is the probability that the bug is at vertex $A$ after crawling exactly 10 meters?

Answer: \frac{20}{81}


A line $l$ is tangent to the circle $x^{2}+y^{2}=1$ and the sum of its intercepts on the two coordinate axes is equal to $\sqrt{3}$. Find the area of the triangle formed by line $l$ and the two coordinate axes.

Answer: \frac{3}{2}


Count the number of permutations \( a_{1} a_{2} \ldots a_{7} \) of 1234567 with the longest decreasing subsequence of length at most two (i.e. there does not exist \( i < j < k \) such that \( a_{i} > a_{j} > a_{k} \)).

Answer: 429


Let  $a$  and  $b$  be positive integers with the property that  $\frac{a}{b} > \sqrt2$ . Prove that  $$ \frac{a}{b} - \frac{1}{2ab} > \sqrt2 $$

Answer: \frac{}{b}-\frac{1}{2ab}>\sqrt{2}


Person A and person B depart simultaneously from locations A and B, respectively, and travel back and forth between A and B. At the moment when persons A and B first meet at location C, person C, who departed from A at the same time as person A, is still 100 meters away from location C. When person C reaches location C, person A has walked another 108 meters. When person C reaches location B, persons A and B meet each other for the second time. What is the distance between locations A and B in meters?

Answer: 3750


Let \( M \) be a set consisting of a finite number of positive integers, and suppose \( M = A_{1} \cup A_{2} \cup \cdots \cup A_{20} = B_{1} \cup B_{2} \cup \cdots \cup B_{20} \), where \( A_{i} \neq \varnothing, B_{i} \neq \varnothing \) for \( i = 1, 2, \ldots, 20 \). For any \( 1 \leq i < j \leq 20 \), we have \( A_{i} \cap A_{j} = \varnothing \) and \( B_{i} \cap B_{j} = \varnothing \).

Given that for any \( 1 \leq i \leq 20 \) and \( 1 \leq j \leq 20 \), if \( A_{i} \cap B_{j} = \varnothing \), then \( \left|A_{i} \cup B_{j}\right| \geq 18 \). Find the minimum number of elements in the set \( M \) (where \( |X| \) denotes the number of elements in set \( X \)).

Answer: 180


Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where  $\alpha,\beta$  and  $\gamma$  are positive and  $\alpha + \beta + \gamma = 180^{\circ}$ .

Answer: \frac{3\sqrt{3}}{2}


Let \(a_1, a_2, a_3, a_4\) be four rational numbers such that:
\[
\left\{a_i a_j \mid 1 \leq i < j \leq 4\right\} = \left\{-24, -2, -\frac{3}{2}, -\frac{1}{8}, 1, 3\right\}
\]
Find the value of \(a_1 + a_2 + a_3 + a_4\).


Let \( n \) be a positive integer. When \( n \) is divided by 902, 802, and 702, the remainders are 602, 502, and 402 respectively. Find the minimum remainder when \( n \) is divided by 2005.

Answer: 101


Find the value of the sum
$$
S = [\lg 2] + [\lg 3] + \cdots + [\lg 2008] + \left[\lg \frac{1}{2}\right] + \left[\lg \frac{1}{3}\right] + \cdots + \left[\lg \frac{1}{2008}\right]
$$
where $[x]$ denotes the greatest integer less than or equal to $x$.

Answer: -2004


A table consisting of 1861 rows and 1861 columns is filled with natural numbers from 1 to 1861 such that each row contains all numbers from 1 to 1861. Find the sum of the numbers on the diagonal that connects the top left and bottom right corners of the table if the filling of the table is symmetric with respect to this diagonal.

Answer: 1732591


Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$  Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points.  If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the focus of $\mathcal{P}$ to all four of the intersection points.

Answer: 1247


Find the smallest natural number that has exactly 12 different natural divisors, whose largest prime divisor is the number 101, and whose last digit is zero.

Answer: 2020


A truncated cone is circumscribed around a sphere, with its bases being great circles of two other spheres. Determine the total surface area of the truncated cone if the sum of the surface areas of the three spheres is $S$.

Answer: \frac{S}{2}


In triangle \(ABC\), the angle bisector \(BE\) and the median \(AD\) are equal and perpendicular. Find the area of triangle \(ABC\) if \(AB = \sqrt{13}\).

Answer: 12


Answer: a_{1}=a_{2}=\cdots=a_{n}


Given a line segment \( AB \), a line \( l \), and a point \( O \) on it, construct a point \( X \) on the line \( l \) such that \( OX = AB \) using a right angle.

Answer: X


Let the set \( S = \{ x \mid x \) is a 9-digit number in decimal representation, with the digits being \( 1, 2, 3 \} \). Define the mapping \( f: S \longrightarrow \{1, 2, 3\} \), such that for any pair of numbers \( x \) and \( y \) in \( S \) with all digits at the same positions being different, \( f(x) \neq f(y) \). Find the mapping \( f \) and the number of such mappings.

Answer: 54


A function \( F \) is defined on the set of triples of integers and takes real values. It is known that for any four integers \( a, b, c \), and \( n \), the following equalities hold:
\[
F(n a, n b, n c) = n \cdot F(a, b, c), \\
F(a + n, b + n, c + n) = F(a, b, c) + n, \\
F(a, b, c) = F(c, b, a).
\]
Find \( F(58, 59, 60) \).

Answer: 59


Given that $\cos(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{10}$, with $x \in (\frac{\pi}{2}, \frac{3\pi}{4})$.
(1) Find the value of $\sin x$;
(2) Find the value of $\cos(2x - \frac{\pi}{3})$.

Answer: -\frac{7 + 24\sqrt{3}}{50}


You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face.

Answer: 16


A pentagon $ABCDE$ is inscribed in a circle with a unit radius. It is known that $AB = \sqrt{2}$, $\angle ABE = 45^{\circ}$, $\angle EBD = 30^{\circ}$, and $BC = CD$. Find the area of the pentagon.

Answer: 1+\frac{3\sqrt{3}}{4}


A stationery store is running a promotion: each notebook has a sticker, and for every 5 stickers, a customer can receive another notebook (also with a sticker). Fifth-grader Katya believes she needs to buy as many notebooks as possible before the new semester. Each notebook costs 4 rubles, and Katya has 150 rubles. How many notebooks will Katya get?

Answer: 46


In a drawer Sandy has $5$ pairs of socks, each pair a different color.  On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer.  On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random.  The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers,  Find $m+n$.

Answer: 341


Seven people dine every Saturday night around a round table. How many times can they dine if each person wants to have two new neighbors each time? What is the result for eight people?

Answer: 3


A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calculate the sum of all possible numbers of members in this club.

Answer: 2807


The diagonals of an inscribed quadrilateral $ABCD$ intersect at point $P$, and triangle $APD$ is acute. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. A perpendicular from point $E$ is drawn to line $AC$, and a perpendicular from point $F$ is drawn to line $BD$. These perpendiculars intersect at point $Q$. Find the angle between lines $PQ$ and $BC$.

Answer: 90


From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will necessarily be two numbers whose difference is greater than 672 and less than 1344?

Answer: 674


Given that \(a\) and \(b\) are real numbers, and the polynomial \(x^{4} + a x^{3} + b x^{2} + a x + 1 = 0\) has at least one real root, determine the minimum value of \(a^{2} + b^{2}\).

Answer: 4/5


Square  $ABCD$  has center  $O$ ,  $AB=900$ ,  $E$  and  $F$  are on  $AB$  with  $AE<BF$  and  $E$  between  $A$  and  $F$ ,  $m\angle EOF =45^\circ$ , and  $EF=400$ . Given that  $BF=p+q\sqrt{r}$ , wherer  $p,q,$  and  $r$  are positive integers and  $r$  is not divisible by the square of any prime, find  $p+q+r$ .

Answer: 307


Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle.

Answer: 40\pi


The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0, 2x+y+3=0$ may be found by solving:
$\textbf{(A)}\ y^{2}+14y-7=0\qquad\textbf{(B)}\ y^{2}+8y+1=0\qquad\textbf{(C)}\ y^{2}+10y-7=0\qquad\\ \textbf{(D)}\ y^{2}+y-12=0\qquad \textbf{(E)}\ \text{None of these equations}$

Answer: \textbf{(C)}\y^{2}+10y-7=0


There are  $n$  sheep and a wolf in sheep's clothing . Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. First, the wolf chooses some sheep to make friend's with. In each of the following days, the wolf eats one of its friends. Whenever the wolf eats a sheep  $A$ :
(a) If a friend of  $A$  is originally a friend of the wolf, it un-friends the wolf.
(b) If a friend of  $A$  is originally not a friend of the wolf, it becomes a friend of the wolf.
Repeat the procedure until the wolf has no friend left.
Find the largest integer  $m$  in terms of  $n$  satisfying the following: There exists an initial friendsheep structure such that the wolf has  $m$  different ways of choosing initial sheep to become friends, so that the wolf has a way to eat all of the sheep.

Answer: 2^{n-1}


In how many different ways can 1,000,000 be represented as the product of three natural numbers? Products that differ only in the order of the factors are considered the same.

Answer: 139


Investigator Podberezovikov is pursuing Maxim Detochkin (each driving their own car). Initially, both were driving on the highway at a speed of 60 km/h, with Podberezovikov trailing Detochkin by 2 km. Upon entering the city, each reduced their speed to 40 km/h, and upon exiting the city onto a good road, each increased their speed to 70 km/h. When the road ended and transitioned to a dirt road, each had to reduce their speed again to 30 km/h. What was the distance between them on the dirt road?

Answer: 1


Two knights placed on distinct square of an  $8\times8$  chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is  $\sqrt{5}.$  In how many ways can two identical knights be placed on distinct squares of an  $8\times8$  chessboard such that they do NOT attack each other?

Answer: 1848


Each of the 10 dwarfs either always tells the truth or always lies. It is known that each of them likes exactly one type of ice cream: vanilla, chocolate, or fruit. First, Snow White asked those who like vanilla ice cream to raise their hands, and everyone raised their hands. Then she asked those who like chocolate ice cream to raise their hands, and half of the dwarfs raised their hands. Finally, she asked those who like fruit ice cream to raise their hand, and only one dwarf raised their hand. How many truthful dwarfs are there among the 10 dwarfs?

Answer: 4


In a certain country, there are 200 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that it should be possible to travel from any city to any other city using the flights of each airline (possibly with layovers). What is the maximum number of airlines for which this is possible?

Answer: 100


In $\triangle ABC$ points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$, respectively.  If $\overline{AD}$ and $\overline{BE}$ intersect at $T$ so that $AT/DT=3$ and $BT/ET=4$, what is $CD/BD$?

[asy]
pair A,B,C,D,I,T;
A=(0,0);
B=(6,8);
C=(11,0);
D=(9.33,2.66);
I=(7.5,0);
T=(6.5,2);
label("$T$",T,NW);
label("$D$",D,NE);
label("$E$",I,S);
label("$A$",A,S);
label("$C$",C,S);
label("$B$",B,N);
draw(A--B--C--cycle,linewidth(0.7));
draw(A--D,linewidth(0.7));
draw(B--I,linewidth(0.7));
[/asy]

Answer: \frac{4}{11}


A regular decagon $A_{0} A_{1} A_{2} \cdots A_{9}$ is given in the plane. Compute $\angle A_{0} A_{3} A_{7}$ in degrees.

Answer: 54^{\circ}


Among the following propositions, the true one is numbered \_\_\_\_\_\_.
(1) The negation of the proposition "For all $x>0$, $x^2-x\leq0$" is "There exists an $x>0$ such that $x^2-x>0$."
(2) If $A>B$, then $\sin A > \sin B$.
(3) Given a sequence $\{a_n\}$, "The sequence $a_n, a_{n+1}, a_{n+2}$ forms a geometric sequence" is a necessary and sufficient condition for $a_{n+1}^2=a_{n}a_{n+2}$.
(4) Given the function $f(x)=\lg x+ \frac{1}{\lg x}$, then the minimum value of $f(x)$ is 2.

Answer: (1)


Cut the letter E. Is it possible to cut the letter $\mathbf{E}$ into five parts so that they can form a square?

In the drawing, all dimensions are given in centimeters to avoid any doubts about the true proportions of the letter. In our case, the pieces are not allowed to be flipped upside down.

After solving this problem, think about whether it would be possible with four pieces if flipping the pieces is allowed.

Answer: 4


A mushroom is called "bad" if it contains at least 10 worms. In a basket, there are 90 bad mushrooms and 10 good mushrooms. Can all the mushrooms become good after some worms move from the bad mushrooms to the good mushrooms?

Answer: Yes


Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter $10\text{ cm}.$ A side view of the first four rows of each of the two different methods of packing is shown below.

[asy]
draw(circle((1,1),1),black+linewidth(1));
draw(circle((3,1),1),black+linewidth(1));
draw(circle((5,1),1),black+linewidth(1));
draw(circle((7,1),1),black+linewidth(1));
draw(circle((9,1),1),black+linewidth(1));
draw(circle((11,1),1),black+linewidth(1));
draw(circle((13,1),1),black+linewidth(1));
draw(circle((15,1),1),black+linewidth(1));
draw(circle((17,1),1),black+linewidth(1));
draw(circle((19,1),1),black+linewidth(1));
draw(circle((1,3),1),black+linewidth(1));
draw(circle((3,3),1),black+linewidth(1));
draw(circle((5,3),1),black+linewidth(1));
draw(circle((7,3),1),black+linewidth(1));
draw(circle((9,3),1),black+linewidth(1));
draw(circle((11,3),1),black+linewidth(1));
draw(circle((13,3),1),black+linewidth(1));
draw(circle((15,3),1),black+linewidth(1));
draw(circle((17,3),1),black+linewidth(1));
draw(circle((19,3),1),black+linewidth(1));
draw(circle((1,5),1),black+linewidth(1));
draw(circle((3,5),1),black+linewidth(1));
draw(circle((5,5),1),black+linewidth(1));
draw(circle((7,5),1),black+linewidth(1));
draw(circle((9,5),1),black+linewidth(1));
draw(circle((11,5),1),black+linewidth(1));
draw(circle((13,5),1),black+linewidth(1));
draw(circle((15,5),1),black+linewidth(1));
draw(circle((17,5),1),black+linewidth(1));
draw(circle((19,5),1),black+linewidth(1));
draw(circle((1,7),1),black+linewidth(1));
draw(circle((3,7),1),black+linewidth(1));
draw(circle((5,7),1),black+linewidth(1));
draw(circle((7,7),1),black+linewidth(1));
draw(circle((9,7),1),black+linewidth(1));
draw(circle((11,7),1),black+linewidth(1));
draw(circle((13,7),1),black+linewidth(1));
draw(circle((15,7),1),black+linewidth(1));
draw(circle((17,7),1),black+linewidth(1));
draw(circle((19,7),1),black+linewidth(1));
draw((0,15)--(0,0)--(20,0)--(20,15),black+linewidth(1));
dot((10,9));
dot((10,11));
dot((10,13));
label("Crate A",(10,0),S);
[/asy]

[asy]
draw(circle((1,1),1),black+linewidth(1));
draw(circle((3,1),1),black+linewidth(1));
draw(circle((5,1),1),black+linewidth(1));
draw(circle((7,1),1),black+linewidth(1));
draw(circle((9,1),1),black+linewidth(1));
draw(circle((11,1),1),black+linewidth(1));
draw(circle((13,1),1),black+linewidth(1));
draw(circle((15,1),1),black+linewidth(1));
draw(circle((17,1),1),black+linewidth(1));
draw(circle((19,1),1),black+linewidth(1));
draw(circle((2,2.75),1),black+linewidth(1));
draw(circle((4,2.75),1),black+linewidth(1));
draw(circle((6,2.75),1),black+linewidth(1));
draw(circle((8,2.75),1),black+linewidth(1));
draw(circle((10,2.75),1),black+linewidth(1));
draw(circle((12,2.75),1),black+linewidth(1));
draw(circle((14,2.75),1),black+linewidth(1));
draw(circle((16,2.75),1),black+linewidth(1));
draw(circle((18,2.75),1),black+linewidth(1));
draw(circle((1,4.5),1),black+linewidth(1));
draw(circle((3,4.5),1),black+linewidth(1));
draw(circle((5,4.5),1),black+linewidth(1));
draw(circle((7,4.5),1),black+linewidth(1));
draw(circle((9,4.5),1),black+linewidth(1));
draw(circle((11,4.5),1),black+linewidth(1));
draw(circle((13,4.5),1),black+linewidth(1));
draw(circle((15,4.5),1),black+linewidth(1));
draw(circle((17,4.5),1),black+linewidth(1));
draw(circle((19,4.5),1),black+linewidth(1));
draw(circle((2,6.25),1),black+linewidth(1));
draw(circle((4,6.25),1),black+linewidth(1));
draw(circle((6,6.25),1),black+linewidth(1));
draw(circle((8,6.25),1),black+linewidth(1));
draw(circle((10,6.25),1),black+linewidth(1));
draw(circle((12,6.25),1),black+linewidth(1));
draw(circle((14,6.25),1),black+linewidth(1));
draw(circle((16,6.25),1),black+linewidth(1));
draw(circle((18,6.25),1),black+linewidth(1));
draw((0,15)--(0,0)--(20,0)--(20,15),black+linewidth(1));
dot((10,9));
dot((10,11));
dot((10,13));
label("Crate B",(10,0),S);
[/asy]

Three pipes from Crate $B$ are shown.  Determine the height, $h,$ of this pile of $3$ pipes.

[asy]
draw(circle((10,10),10),black+linewidth(1));
draw(circle((30,10),10),black+linewidth(1));
draw(circle((20,27.5),10),black+linewidth(1));
draw((50,0)--(50,37.5),black+linewidth(1));
draw((49,0)--(51,0),black+linewidth(1));
draw((49,37.5)--(51,37.5),black+linewidth(1));
label("$h$",(50,0)--(50,37.5),E);
[/asy]

Answer: 10 + 5\sqrt{3}


A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path?
$\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5$

Answer: \textbf{(C)}10\sqrt{2}


The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______.

Answer: \frac{5}{4}


Let  $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$ . If the maximum element of  $S$  is  $p_0/q_0$  in reduced form, find  $p_0+q_0$ .

Answer: 595


Suppose  $50x$  is divisible by 100 and  $kx$  is not divisible by 100 for all  $k=1,2,\cdots, 49$  Find number of solutions for  $x$  when  $x$  takes values  $1,2,\cdots 100$ .
[list=1]
[*] 20
[*] 25
[*] 15
[*] 50
[/list]

Answer: 20


Find the largest integer \( a \) such that the expression

\[
a^2 - 15a - (\tan x - 1)(\tan x + 2)(\tan x + 5)(\tan x + 8)
\]

is less than 35 for all values of \( x \in (-\pi/2, \pi/2) \).

Answer: 10


A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$?

Answer: 6


The positive integer  $n$  is given and for all positive integers  $k$ ,  $1\leq k\leq n$ , denote by  $a_{kn}$  the number of all ordered sequences  $(i_1,i_2,\ldots,i_k)$  of positive integers which verify the following two conditions:

a)  $1\leq i_1<i_2< \cdots i_k \leq n$ ;

b)  $i_{r+1}-i_r \equiv 1 \pmod 2$ , for all  $r \in\{1,2,\ldots,k-1\}$ .

Compute the number  $a(n) = \sum\limits_{k=1}^n a_{kn}$ .

*Ioan Tomescu*

Answer: F_{n+3}-2


Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

Answer: \textbf{(B)}\336


Determine the smallest positive integer that has 144 positive divisors, and among these divisors, there are ten consecutive ones.

Answer: 110880


Let \( x \) and \( y \) satisfy the conditions:
\[
\left\{
\begin{array}{l}
2x + y \geq 4 \\
x - y \geq 1 \\
x - 2y \leq 2
\end{array}
\right.
\]
If \( z = ax + y \) achieves its minimum value only at point \( A(2,0) \), determine the range of the real number \( a \).

Answer: (-\frac{1}{2},2)


On an island of knights and liars (liars always lie, knights always tell the truth), each person supports exactly one football team. All island inhabitants participated in a survey. To the question "Do you support 'Rostov'?", 40% of the inhabitants answered "Yes". To a similar question about 'Zenit', 30% answered affirmatively, about 'Lokomotiv' - 50%, and about CSKA - 0%. What percentage of the island's inhabitants truly support 'Rostov'?

Answer: 30


On weekdays, the Absent-minded Scientist commutes to work on the ring light of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. Upon entering the station, he catches the first arriving train. It is known that in both directions trains run at approximately equal intervals. Via the northern route (through "Belorusskaya"), the train travels from "Kievskaya" to "Taganskaya" or back in 17 minutes, and via the southern route (through "Paveletskaya") in 11 minutes.

The Scientist has calculated through many years of observations that:
- A train going counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after a train going clockwise arrives. The same is true for "Taganskaya".
- The trip from home to work on average takes 1 minute less than the trip from work to home.

Find the expected interval between trains running in one direction.

Answer: 3


Given nine different numbers, how many different values of third-order determinants can be formed using all of them?

Answer: 10080


Let \( n \) be a non-negative integer. Define
\[
(1+4 \sqrt[3]{2}-4 \sqrt[3]{4})^{n} = a_{n} + b_{n} \sqrt[3]{2} + c_{n} \sqrt[3]{4},
\]
where \( a_{n}, b_{n}, c_{n} \) are integers. Prove that if \( c_{n} = 0 \), then \( n = 0 \).

Answer: 0


How many non-similar triangles have angles whose degree measures are distinct positive integers in  arithmetic progression?

Answer: 59


Suppose that  $2^{2n+1}+ 2^{n}+1=x^{k}$ , where  $k\geq2$  and  $n$  are positive integers. Find all possible values of  $n$ .

Answer: 4


What is the smallest sum of digits that a number of the form \( 3n^2 + n + 1 \) can have for a natural number \( n \)?

Answer: 3


Given that point \( P \) moves on the circle \( x^{2}+(y-4)^{2}=1 \) and point \( Q \) moves on the ellipse \( \frac{x^{2}}{9} + y^{2} = 1 \), find the maximum value of \( |PQ| \).

Answer: 1+3\sqrt{3}


Define a sequence of integers by  $T_1 = 2$  and for  $n\ge2$ ,  $T_n = 2^{T_{n-1}}$ . Find the remainder when  $T_1 + T_2 + \cdots + T_{256}$  is divided by 255.

*Ray Li.*

Answer: 20


Let $U = \{2, 4, 3-a^2\}$ and $P = \{2, a^2+2-a\}$. Given that the complement of $P$ in $U$, denoted as $\complement_U P$, is $\{-1\}$, find the value of $a$.

Answer: a = 2


A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle and the radius of the circle?

Answer: \frac{10}{\pi}


Mrs. Crabapple now teaches two different classes of British Literature. Her first class has 12 students and meets three times a week, while her second class has 9 students and meets twice a week. How many different sequences of crabapple recipients are possible in a week for both classes combined?

Answer: 139,968


Find the number of integers \( x \) such that the following three conditions all hold:
- \( x \) is a multiple of 5
- \( 121 < x < 1331 \)
- When \( x \) is written as an integer in base 11 with no leading 0s (i.e. no 0s at the very left), its rightmost digit is strictly greater than its leftmost digit.

Answer: 99


Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$?

$\text{(A)}\ \sqrt{10} \qquad \text{(B)}\ \sqrt{12} \qquad \text{(C)}\ \sqrt{13} \qquad \text{(D)}\ \sqrt{14} \qquad \text{(E)}\ \sqrt{15}$

Answer: (C)\\sqrt{13}


Using the minimum number of lines (circles and straight lines with a compass and straightedge), construct a line passing through a given point that is parallel to a given line.

Answer: 3


The average age of 8 people in a room is 25 years. A 20-year-old person leaves the room. Calculate the average age of the seven remaining people.

Answer: \frac{180}{7}


The maximum value of the function \( y = \cos x + \sin x + \cos x \sin x \) is ______.

Answer: \frac{1}{2}+\sqrt{2}


When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young.

Answer: 18


In the magical forest of Santa Claus, cedars grow one and a half times taller than firs and grow for 9 hours. Firs grow for 2 hours. Santa Claus planted cedar seeds at 12 o'clock and fir seeds at 2 o'clock in the afternoon. What time was it when the trees were of the same height? (The trees grow uniformly for the specified number of hours and then stop growing).

Answer: 15 or 18 hours.

Answer: 15


For any positive integer $n$, let $x_n = \mathrm{C}_{2n}^{n}$. Find all positive integers $h$ (where $h > 1$) such that the sequence $\left\{ x_n \right\}$, when taken modulo $h$, becomes a periodic sequence starting from a certain term.

Answer: 2


In an arithmetic sequence \(\left\{a_{n}\right\}\) with a common difference not equal to 0, if \(a_{4}=10\) and \(a_{3}, a_{6}, a_{10}\) form a geometric sequence, what is the general term formula of the sequence \(\left\{a_{n}\right\}\)?

Answer: a_n=n+6


A circle is divided into  $n$  sectors ( $n \geq 3$ ). Each sector can be filled in with either  $1$  or  $0$ . Choose any sector  $\mathcal{C}$  occupied by  $0$ , change it into a  $1$  and simultaneously change the symbols  $x, y$  in the two sectors adjacent to  $\mathcal{C}$  to their complements  $1-x$ ,  $1-y$ . We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a  $0$  in one sector and  $1$ s elsewhere. For which values of  $n$  can we end this process?

Answer: n\not\equiv0\pmod{3}


Let  $ S(n) $  be the sum of the squares of the positive integers less than and coprime to  $ n $ . For example,  $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $ , but  $ S(4) = 1^2 + 3^2 $ . Let  $ p = 2^7 - 1 = 127 $  and  $ q = 2^5 - 1 = 31 $  be primes. The quantity  $ S(pq) $  can be written in the form   $$  \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right)  $$   where  $ a $ ,  $ b $ , and  $ c $  are positive integers, with  $ b $  and  $ c $  coprime and  $ b < c $ . Find  $ a $ .

Answer: 7561


A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The probability that it will hit the coordinate axes at $(0,0)$ is $\frac{m}{3^n}$, where $m$ and $n$ are positive integers such that $m$ is not divisible by $3$. Find $m + n$.

Answer: 252


A $70 \times 70$ table is filled with numbers from 1 to 4900: in the first row, the numbers from 1 to 70 are written from left to right in ascending order; in the second row, the numbers from 71 to 140 are written in the same manner, and so on; in the last row, the numbers from 4831 to 4900 are written from left to right. Can a cell be found in this table such that the sum of the five numbers – the number in the cell and the four numbers in the cells adjacent to it on the sides – equals 2018? (Author of the problem: A. Solynin)

Answer: No


Let a convex n-gon \( A_1 A_2 \cdots A_n \) be inscribed in a unit circle. Find the maximum value of the sum of the squares of all its sides and diagonals, and determine when this maximum value is achieved.

Answer: n^2


Dots are placed two units apart both horizontally and vertically on a coordinate grid. Calculate the number of square units enclosed by the polygon formed by connecting these dots:
[asy]
size(90);
pair a=(0,0), b=(20,0), c=(20,20), d=(40,20), e=(40,40), f=(20,40), g=(0,40), h=(0,20);
dot(a);
dot(b);
dot(c);
dot(d);
dot(e);
dot(f);
dot(g);
dot(h);
draw(a--b--c--d--e--f--g--h--cycle);
[/asy]

Answer: 12


Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i + 1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $P_{3}$ is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 101


Let \( f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0} \) and \( g(x) = c_{n+1} x^{n+1} + c_{n} x^{n} + \cdots + c_{0} \) be non-zero polynomials with real coefficients. Suppose there exists a real number \(\gamma\) such that \( g(x) = (x+\gamma) f(x) \). Let \(a\) be the largest among \( |a_{n}|, |a_{n-1}|, \cdots, |a_{0}| \), and let \(c\) be the largest among \( |c_{n+1}|, |c_{n}|, \cdots, |c_{0}| \). Prove that \( \frac{a}{c} \leq n+1 \).

Answer: \frac{}{}\len+1


The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?

Answer: 432


Two circles are said to be *orthogonal* if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles  $\omega_1$  and  $\omega_2$  with radii  $10$  and  $13$ , respectively, are externally tangent at point  $P$ . Another circle  $\omega_3$  with radius  $2\sqrt2$  passes through  $P$  and is orthogonal to both  $\omega_1$  and  $\omega_2$ . A fourth circle  $\omega_4$ , orthogonal to  $\omega_3$ , is externally tangent to  $\omega_1$  and  $\omega_2$ . Compute the radius of  $\omega_4$ .

Answer: \frac{92}{61}


Aunt Martha bought some nuts. She gave Tommy one nut and a quarter of the remaining nuts, Bessie received one nut and a quarter of the remaining nuts, Bob also got one nut and a quarter of the remaining nuts, and finally, Jessie received one nut and a quarter of the remaining nuts. It turned out that the boys received 100 more nuts than the girls.

How many nuts did Aunt Martha keep for herself?

Answer: 321


From a 12 × 12 grid, a 4 × 4 square has been cut out, located at the intersection of horizontals from the fourth to the seventh and the same verticals. What is the maximum number of rooks that can be placed on this board such that no two rooks attack each other, given that rooks do not attack across the cut-out cells?

Answer: 15


The vertices of a regular $2n$-gon $A_{1} \ldots A_{2n}$ are divided into $n$ pairs. Prove that if $n=4m+2$ or $n=4m+3$, then two pairs of vertices are endpoints of equal segments.

Answer: 2


Find the smallest \( n \) such that if \( \{1, 2, \ldots, n\} \) is divided into two disjoint subsets, then we can always find three distinct numbers \( a, b, c \) in the same subset with \( ab = c \).

Answer: 96


If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$?

Answer: 18


Find the number of the subsets  $B$  of the set  $\{1,2,\cdots, 2005 \}$  such that the sum of the elements of  $B$  is congruent to  $2006$  modulo  $2048$

Answer: 2^{1994}


Find the volume $V$ of a truncated cone with height $h$ and base radii $R$ and $r$.

Answer: \frac{1}{3}\pi(R^2+Rr+r^2)


If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is:

Answer: \frac {\sqrt {8d^2 - p^2}}{2}


Let $A, B, C, D$ be points chosen on a circle, in that order. Line $B D$ is reflected over lines $A B$ and $D A$ to obtain lines $\ell_{1}$ and $\ell_{2}$ respectively. If lines $\ell_{1}, \ell_{2}$, and $A C$ meet at a common point and if $A B=4, B C=3, C D=2$, compute the length $D A$.

Answer: \sqrt{21}


Find the special solutions of the differential equation

$$
3 y = 2 x y' - \frac{2}{x} \left( y' \right)^2
$$

Answer: \frac{x^3}{6}


Find all integer  $n$  such that the following property holds: for any positive real numbers  $a,b,c,x,y,z$ , with  $max(a,b,c,x,y,z)=a$  ,  $a+b+c=x+y+z$  and  $abc=xyz$ , the inequality   $$ a^n+b^n+c^n \ge x^n+y^n+z^n $$   holds.

Answer: n\ge0


Fill in the numbers 1, 2, 3, ..., 15, 16 into the 16 squares in the table below, and satisfy the following conditions.

\[
\begin{array}{|c|c|c|c|}
\hline
A & B & C & D \\
\hline
E & F & G & H \\
\hline
J & K & L & M \\
\hline
N & P & Q & R \\
\hline
\end{array}
\]

1. \( A + C + F = 10 \)
2. \( B + H = R \)
3. \( D - C = 13 \)
4. \( E \times M = 126 \)
5. \( F + G = 21 \)
6. \( G \div J = 2 \)
7. \( H \times M = 36 \)
8. \( J \times P = 80 \)
9. \( K - N = Q \)

What is \( L \)?

Answer: 6


Let \( Q \) be the set of all rational numbers. Find all functions \( f: Q \to Q \) that satisfy the following conditions:
1. \( f(1) = 2 \);
2. For any \( x, y \in Q \), \( f(xy) = f(x) f(y) - f(x + y) + 1 \).

Answer: f(x)=x+1


Let a regular $n$-sided polygon have a side length $a$ and the longest and shortest diagonals be $b$ and $c$ respectively. If $a = b - c$, determine $n$.

Answer: 9


Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.

Answer: 486


How many four-digit positive integers \( x \) are there with the property that \( x \) and \( 3x \) have only even digits?

Answer: 82


An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find the ordered quadruple $(a, b, h, k)$.

(Enter your answer as an ordered list, for example, "1, 3, -9, 2".)

Answer: (8\sqrt3, 14, 2, 4)


For some complex number $z$ with $|z| = 3,$ there is some real $\lambda > 1$ such that $z,$ $z^2,$ and $\lambda z$ form an equilateral triangle in the complex plane. Find $\lambda.$

Answer: \frac{1 + \sqrt{33}}{2}


(a) Determine all real numbers \( a > 0 \) for which \( \sqrt{a^{2}+a}=\frac{2}{3} \).

(b) For each positive integer \( m \), determine the difference between \(\left(m+\frac{1}{2}\right)^{2}+\left(m+\frac{1}{2}\right) \) and the nearest perfect square.

(c) For every positive integer \( n \), prove that the number of positive integers \( c \) with \( n < \sqrt{c + \sqrt{c}} < n+1 \) is even.

Answer: \frac{1}{4}


The jury of an olympiad has 30 members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session, a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most 15 sessions there will be no more exclusions. (Note that nobody votes about his own competence.)

Answer: 15


There are  $ 25$  towns in a country. Find the smallest  $ k$  for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly  $ k$  direct routes to  $ k$  other towns;
2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

Answer: 6


A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ} = \frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$.

Answer: \sqrt{5 - \sqrt{5}}


Given \( x, y, z \in \mathbf{R}, m, n \in \mathbf{N}, m, n \geqslant 2 \), prove that:
\[ \sin^m x \cos^n y + \sin^m y \cos^n z + \sin^m z \cos^n x \leqslant 1 \]
and that 1 is the least upper bound.

Answer: 1


Calculate the sum of the first six terms of the geometric series $3 + \left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + \left(\frac{1}{3}\right)^4 + \left(\frac{1}{3}\right)^5$. Express your answer as a simplified fraction.

Answer: \frac{364}{81}


While there do not exist pairwise distinct real numbers  $a,b,c$  satisfying  $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let  $a,b,c$  be complex numbers such that  $a^2+b^2+c^2 = ab+bc+ca$  and  $|a+b+c| = 21$ . Given that  $|a-b| = 2\sqrt{3}$ ,  $|a| = 3\sqrt{3}$ , compute  $|b|^2+|c|^2$ .
<details><summary>Clarifications</summary>

- The problem should read  $|a+b+c| = 21$ .  An earlier version of the test read  $|a+b+c| = 7$ ; that value is incorrect.
- $|b|^2+|c|^2$  should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers  $m$  and  $n$ .  Find  $m+n$ .''
</details>

*Ray Li*

Answer: 132


Answer: 3011


Solve the equation

\[ f\left(\frac{x+1}{x-2}\right) + 2 f\left(\frac{x-2}{x+1}\right) = x \]

Answer: f(x)=\frac{4x+5}{3(1-x)}


Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\frac{a - b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

Answer: 146


In $\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively. Angle $\angle BAC = 60^\circ$ and $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?

Answer: 44


Let \( a_{1}, a_{2}, \cdots, a_{n} \) be an arithmetic sequence, and it is given that
$$
\sum_{i=1}^{n}\left|a_{i}+j\right|=2028 \text{ for } j=0,1,2,3.
$$
Find the maximum value of the number of terms \( n \).

Answer: 52


Answer: 2


Answer: \mathrm{(B)\\frac{3}{25}}


A set $\mathcal{T}$ of distinct positive integers has the following property: for every integer $y$ in $\mathcal{T},$ the arithmetic mean of the set of values obtained by deleting $y$ from $\mathcal{T}$ is an integer. Given that 1 belongs to $\mathcal{T}$ and that 1764 is the largest element of $\mathcal{T},$ what is the greatest number of elements that $\mathcal{T}$ can have?

Answer: 42


Let  $ABC$  be an isosceles triangle, and point  $D$  in its interior such that   $$ D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ $$

(a) Prove that  $\hat B=\hat C=80^\circ$ .
(b) Find the measure of the angle  $D \hat{A} C$ .

Answer: 5


Four positive integers $p$, $q$, $r$, $s$ satisfy $p \cdot q \cdot r \cdot s = 9!$ and $p < q < r < s$. What is the smallest possible value of $s-p$?

Answer: 12


Given positive integers \(a, b, c, x, y, z\) that satisfy the conditions
$$
a \geq b \geq c \geq 1, \quad x \geq y \geq z \geq 1,
$$
and
$$
\begin{cases}
2a + b + 4c = 4xyz, \\
2x + y + 4z = 4abc,
\end{cases}
$$
determine the number of six-tuples \((a, b, c, x, y, z)\) that satisfy the given conditions.

Answer: 0


Let \( A \) be a subset of \(\{1, 2, 3, \ldots, 2019\}\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \( A \)?

Answer: 505


Determine the minimum possible value of the sum \[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{6a},\] where \( a, b, \) and \( c \) are positive real numbers.

Answer: \frac{3}{\sqrt[3]{90}}


Given a triangle \(ABC\) with \(I\) as the incenter, suppose: \(CA + AI = BC\). Determine the ratio of the angles: \(\frac{\angle BAC}{\angle CAA}\).

Answer: 2


Given that each carbon pen costs 1 yuan 8 jiao, each notebook costs 3 yuan 5 jiao, and each pencil case costs 4 yuan 2 jiao, and Jing Jing spent exactly 20 yuan to buy these three kinds of stationery, find out how many notebooks she bought.

Answer: 4


Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square!

Answer: 8/5


Suppose \( f \) and \( g \) are differentiable functions such that
\[
x g(f(x)) f^{\prime}(g(x)) g^{\prime}(x)=f(g(x)) g^{\prime}(f(x)) f^{\prime}(x)
\]
for all real \( x \). Moreover, \( f \) is nonnegative and \( g \) is positive. Furthermore,
\[
\int_{0}^{a} f(g(x)) \, dx = 1 - \frac{e^{-2a}}{2}
\]
for all reals \( a \). Given that \( g(f(0)) = 1 \), compute the value of \( g(f(4)) \).

Answer: e^{-16}


Find all ordered pairs  $(a,b)$  of positive integers that satisfy  $a>b$   and the equation  $(a-b)^{ab}=a^bb^a$ .

Answer: (4,2)


\(a\) and \(b\) are positive reals and \(a > b\). Let \(C\) be the plane curve given by \(r = a - b \cos \theta\). For what values of \(\frac{b}{a}\) is \(C\) convex?

Answer: \frac{b}{}\le\frac{1}{2}


We say an integer  $n$  is naoish if  $n \geq 90$  and the second-to-last digit of  $n$  (in decimal notation) is equal to  $9$ . For example,  $10798$ ,  $1999$  and  $90$  are naoish, whereas  $9900$ ,  $2009$  and  $9$  are not. Nino expresses 2020 as a sum:
    \[
    2020=n_{1}+n_{2}+\ldots+n_{k}
    \]
    where each of the  $n_{j}$  is naoish.

    What is the smallest positive number  $k$  for which Nino can do this?

Answer: 8


For the real numbers \(a, b, c, d\) the following equations apply:

\[ a = \sqrt{45 - \sqrt{21 - a}}, \]
\[ b = \sqrt{45 + \sqrt{21 - b}}, \]
\[ c = \sqrt{45 - \sqrt{21 + c}}, \]
\[ d = \sqrt{45 + \sqrt{21 + d}}. \]

Show that \(abcd = 2004\).

Answer: 2004


On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$.

Answer: 200


In a mathematics competition conducted at a school, the scores $X$ of all participating students approximately follow the normal distribution $N(70, 100)$. It is known that there are 16 students with scores of 90 and above (inclusive of 90).
(1) What is the approximate total number of students who participated in the competition?
(2) If the school plans to reward students who scored 80 and above (inclusive of 80), how many students are expected to receive a reward in this competition?
Note: $P(|X-\mu| < \sigma)=0.683$, $P(|X-\mu| < 2\sigma)=0.954$, $P(|X-\mu| < 3\sigma)=0.997$.

Answer: 110


A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\angle ABC = \angle ACD$.

Answer: 25/16


A two-digit number is given in which the first digit is greater than the second. Its digits were rearranged to obtain a new two-digit number. The difference between the first and the second number turned out to be a perfect square. How many possible two-digit numbers satisfy this condition?

Answer: 13


There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.

What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)?

Answer: 13


Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:
$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 48\qquad \textbf{(E)}\ 96$

Answer: \textbf{(D)}\48


For which values of the constant \( c \) does the equation
\[
x^{2} - 2\left|x + \frac{1}{4}\right| + c = 0
\]
have exactly three distinct real roots?

Answer: \frac{1}{2}


Medians $\overline{DP}$ and $\overline{EQ}$ of isosceles $\triangle DEF$, where $DE=EF$, are perpendicular. If $DP= 21$ and $EQ = 28$, then what is ${DE}$?

Answer: \frac{70}{3}


In triangle  $ABC$ , find the smallest possible value of   $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$

Answer: \frac{8\sqrt{3}}{9}


Place the numbers $1, 2, 3, \cdots, 2001$ in a clockwise direction on a circle. First, eliminate the number 2. Then proceed to eliminate every second number in a clockwise direction until only one number remains. What is the last remaining number?
A. 1955
B. 2001
C. 81
D. 163

Answer: 1955


In a round-robin hockey tournament, 2016 teams participated. According to the tournament rules, 3 points are awarded for a win.

Answer: 6093360


The sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1} \in[0,2], \quad a_{n+1}=a_{n}+\frac{a_{n}^{2}}{n^{3}} \quad \left(n \in \mathbf{Z}_{+}\right).
$$

Prove: There exists a positive real number $M$ such that for any positive integer $n$, the inequality $a_{n} < M$ holds.

Answer: M


The three roots of the cubic  $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$  are distinct real numbers between  $ 0$  and  $ 1$ .  For every nonnegative integer  $ n$ , let  $ s_n$  be the sum of the  $ n$ th powers of these three roots.  What is the value of the infinite series
\[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?\]

Answer: 12


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.

Answer: \frac{4}{3}


Find the number of real roots of the equation:
\[ 4 \cos (2007a) = 2007a. \]

Answer: 3


Let  $ n \geq 3$  and consider a set  $ E$  of  $ 2n \minus{} 1$  distinct points on a circle. Suppose that exactly  $ k$  of these points are to be colored black.  Such a coloring is **good** if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly  $ n$  points from  $ E$ .  Find the smallest value of  $ k$  so that every such coloring of  $ k$  points of  $ E$  is good.

Answer: n


Consider a parabola with vertex V and a focus F. There exists a point B on the parabola such that BF = 25 and BV = 24. Determine the sum of all possible values of the length FV.

Answer: \frac{50}{3}


Two classes of elementary school students are going to the Children's Palace for an activity, but there is only one car for transportation. When the first class of students departs from the school by car, the second class starts walking. The car drops the first class off at some point along the way to continue walking and returns immediately to pick up the second class of students, then drives directly to the Children's Palace. The students walk at a speed of 4 kilometers per hour. The car drives at 40 kilometers per hour when carrying students and 50 kilometers per hour when empty. To ensure that both classes of students arrive at the Children's Palace at the same time, what fraction of the total distance does the first class walk? (Ignore the time taken for students to board and disembark the car.)

Answer: \frac{1}{7}


Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct.

Answer: 21840


Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements.

Answer: 48


A five-digit number $\overline{abcde}$ is given. When any two digits from this number are selected to form a two-digit number keeping the order as in the original five-digit number, there are 10 two-digit numbers: $33, 37, 37, 37, 38, 73, 77, 78, 83, 87$. What is the five-digit number $\overline{abcde}$?

Answer: 37837


What weights can be measured using a balance scale with weights of $1, 3, 9, 27$ grams? Generalize the problem!

Answer: 40


Given that $a, b \in R^{+}$ and $a + b = 1$, find the supremum of $- \frac{1}{2a} - \frac{2}{b}$.

Answer: -\frac{9}{2}


Define $\varphi^{k}(n)$ as the number of positive integers that are less than or equal to $n / k$ and relatively prime to $n$. Find $\phi^{2001}\left(2002^{2}-1\right)$. (Hint: $\phi(2003)=2002$.)

Answer: 1233


Calculate the value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} = \_\_\_\_\_\_$.

Answer: -\frac{1}{8}


Fix an integer  $ b \geq 2$ . Let  $ f(1) \equal{} 1$ ,  $ f(2) \equal{} 2$ , and for each  $ n \geq 3$ , define  $ f(n) \equal{} n f(d)$ , where  $ d$  is the number of base- $ b$  digits of  $ n$ . For which values of  $ b$  does
\[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)}
\]
converge?

Answer: 2


A sequence  $ (S_n), n \geq 1$  of sets of natural numbers with  $ S_1 = \{1\}, S_2 = \{2\}$  and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine  $ S_{1024}.$

Answer: 1024


The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&32&43&58&&\\
1&05&10&12&15&20\\
2&25&30&55&&\\
3&00&15&30&35&45\\
4&10&12&&&\\
\end{tabular}

Answer: 147.5


Find all integers $m$ such that $m^{2}+6 m+28$ is a perfect square.

Answer: 6,-12


Given a line $l$ intersects the hyperbola $x^2 - \frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$.

Answer: 4\sqrt{2}


1. Given $\sin\alpha + \cos\alpha = \frac{7}{13}$, with $\alpha \in (0, \pi)$, find the value of $\tan\alpha$.
2. Find the minimum value for $y=\sin 2x + 2\sqrt{2}\cos\left(\frac{\pi}{4}+x\right)+3$.

Answer: 2 - 2\sqrt{2}


For positive integers  $m$  and  $n$ , the decimal representation for the fraction  $\tfrac{m}{n}$  begins  $0.711$  followed by other digits. Find the least possible value for  $n$ .

Answer: 45


What is the minimum number of convex pentagons needed to form a convex 2011-gon?

Answer: 670


Let $\overline{AB}$ have a length of 8 units and $\overline{A'B'}$ have a length of 6 units. $D$ is located 3 units away from $A$ on $\overline{AB}$ and $D'$ is located 1 unit away from $A'$ on $\overline{A'B'}$. If $P$ is a point on $\overline{AB}$ such that $x$ (the distance from $P$ to $D$) equals $2$ units, find the sum $x + y$, given that the ratio of $x$ to $y$ (the distance from the associated point $P'$ on $\overline{A'B'}$ to $D'$) is 3:2.
A) $\frac{8}{3}$ units
B) $\frac{9}{3}$ units
C) $\frac{10}{3}$ units
D) $\frac{11}{3}$ units
E) $\frac{12}{3}$ units

Answer: \frac{10}{3}


Let \( x, y, z \) be positive numbers that satisfy the system of equations:

\[
\begin{cases}
x^{2}+xy+y^{2}=27 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=43
\end{cases}
\]

Find the value of the expression \( xy+yz+xz \).

Answer: 24


In the rectangular coordinate system, the symmetric point of point $A(-2,1,3)$ with respect to the $x$-axis is point $B$. It is also known that $C(x,0,-2)$, and $|BC|=3 \sqrt{2}$. Find the value of $x$.

Answer: -6


In an isosceles triangle, the center of the inscribed circle divides the altitude in the ratio $17: 15$. The base is 60. Find the radius of this circle.

Answer: 7.5


Given the set \( S = \{1, 2, \cdots, 100\} \), determine the smallest possible value of \( m \) such that in any subset of \( S \) with \( m \) elements, there exists at least one number that is a divisor of the product of the remaining \( m-1 \) numbers.

Answer: 26


A tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle.

Answer: 13


Find the smallest integer $k$ for which the conditions
 (1) $a_1,a_2,a_3\cdots$ is a nondecreasing sequence of positive integers
 (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
 (3) $a_9=k$
 are satisfied by more than one sequence.

Answer: 748


Determine the value of $$2002+\frac{1}{2}\left(2001+\frac{1}{2}\left(2000+\cdots+\frac{1}{2}\left(3+\frac{1}{2} \cdot 2\right)\right) \cdots\right)$$

Answer: 4002


A man has part of $$4500$ invested at $4$% and the rest at $6$%. If his annual return on each investment is the same, the average rate of interest which he realizes of the $4500 is:
$\textbf{(A)}\ 5\% \qquad \textbf{(B)}\ 4.8\% \qquad \textbf{(C)}\ 5.2\% \qquad \textbf{(D)}\ 4.6\% \qquad \textbf{(E)}\ \text{none of these}$

Answer: \textbf{(B)}\4.8


Positive integer  $n$  cannot be divided by  $2$  and  $3$ , there are no nonnegative integers  $a$  and  $b$  such that  $|2^a-3^b|=n$ . Find the minimum value of  $n$ .

Answer: 35


(a) Two circles with radii $\mathrm{R}$ and $\mathrm{r}$ are externally tangent. Demonstrate that the segment determined by the common external tangent $\ell$ measures $d = 2 \sqrt{\mathrm{Rr}}$.

(b) Consider, as illustrated, the three circles with radii $\mathrm{R}$, $\mathrm{r}$, and $x$, which are pairwise tangent and tangent to the line $\ell$. Show that

$$
\frac{1}{\sqrt{x}}=\frac{1}{\sqrt{R}}+\frac{1}{\sqrt{r}}
$$

Answer: \frac{1}{\sqrt{x}}=\frac{1}{\sqrt{R}}+\frac{1}{\sqrt{r}}