A square is drawn on a plane with its sides horizontal and vertical. Several line segments parallel to its sides are drawn inside the square, such that no two segments lie on the same line and do not intersect at a point that is internal for both segments. The segments divide the square into rectangles, with each vertical line crossing the square and not containing any of the segments intersecting exactly $k$ rectangles, and each horizontal line crossing the square and not containing any of the segments intersecting exactly $l$ rectangles. What could be the number of rectangles in the partition?

Answer: kl


In triangle \( \triangle ABC \),
\[ \tan A, \ (1+\sqrt{2}) \tan B, \ \tan C \]
form an arithmetic sequence. What is the minimum value of angle \( B \)?

Answer: \frac{\pi}{4}


When \((1+x)^{38}\) is expanded in ascending powers of \(x\), \(N_{1}\) of the coefficients leave a remainder of 1 when divided by 3, while \(N_{2}\) of the coefficients leave a remainder of 2 when divided by 3. Find \(N_{1} - N_{2}\).

Answer: 4


A polynomial is called compatible if all its coefficients are equal to $0$, $1$, $2$, or $3$. Given a natural number $n$, find the number of compatible polynomials that satisfy the condition $p(2) = n$.

Answer: \lfloor\frac{n}{2}\rfloor+1


Let \( p(x) \) be a polynomial of degree at most \( 2n \), and for every integer \( k \) in the interval \([-n, n]\), it holds that \( |p(k)| \leq 1 \). Prove that for any \( x \in [-n, n] \), \( |p(x)| \leq 2^{2n} \).

Answer: 2^{2n}


There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$.

Answer: 9999


The dimensions of a rectangular parallelepiped are 2, 3, and 6 cm. Find the edge length of a cube such that the ratios of their volumes are equal to the ratios of their surface areas.

Answer: 3\,


Find the sum of all integer values of $h$ for which the equation $||r+h|-r|-4r=9|r-3|$ with respect to $r$ has at most one root.

Answer: -93


Lines $L_1,L_2,\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other.
All lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\{L_1,L_2,\dots,L_{100}\}$ is
$\textbf{(A) }4350\qquad \textbf{(B) }4351\qquad \textbf{(C) }4900\qquad \textbf{(D) }4901\qquad  \textbf{(E) }9851$

Answer: \textbf{(B)}4351


How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have?

Answer: 6


Two circles intersect at points \( A \) and \( B \). Chords \( A C \) and \( A D \) are drawn in each of these circles such that each chord in one circle is tangent to the other circle. Find \( A B \) if \( C B = a \) and \( D B = b \).

Answer: \sqrt{}


Given points $P(\cos \alpha, \sin \alpha)$, $Q(\cos \beta, \sin \beta)$, and $R(\cos \alpha, -\sin \alpha)$ in a two-dimensional space, where $O$ is the origin, if the cosine distance between $P$ and $Q$ is $\frac{1}{3}$, and $\tan \alpha \cdot \tan \beta = \frac{1}{7}$, determine the cosine distance between $Q$ and $R$.

Answer: \frac{1}{2}


A rectangular grid \(7 \times 14\) (cells) is given. What is the maximum number of three-cell corners that can be cut out from this rectangle?

Answer: 32


Let the sequence  $\{x_n\}$  be defined by  $x_1 \in \{5, 7\}$  and, for  $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$ . For example, the possible values of  $x_3$  are  $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and  $7^{7^7}$ . Determine the sum of all possible values for the last two digits of  $x_{2012}$ .

Answer: 75


Find the total number of solutions to the following system of equations:

 $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37}
b(a + d)\equiv b \pmod{37}
c(a + d)\equiv c \pmod{37}
bc + d^2\equiv d \pmod{37}
ad - bc\equiv 1 \pmod{37} \end{array}$

Answer: 1


If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point
$\textbf{(A)}\ (0,1997)\qquad\textbf{(B)}\ (0,-1997)\qquad\textbf{(C)}\ (19,97)\qquad\textbf{(D)}\ (19,-97)\qquad\textbf{(E)}\ (1997,0)$

Answer: \textbf{(E)}


A secret facility is in the shape of a rectangle measuring $200 \times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths along the external perimeter (while the intruder remained in place). Three guards ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?

Answer: 150


Find the functions \( f: \mathbb{Q}_{+}^{*} \rightarrow \mathbb{Q}_{+}^{*} \) such that for all \( x \in \mathbb{Q}_{+}^{*} \), we have \( f(x+1) = f(x) + 1 \) and \( f(1/x) = 1 / f(x) \).

Answer: f(x)=x


A sequence of numbers is defined by the conditions: \(a_{1}=1\), \(a_{n+1}=a_{n}+\left\lfloor\sqrt{a_{n}}\right\rfloor\).

How many perfect squares appear among the first terms of this sequence that do not exceed 1,000,000?

Answer: 10


Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), prove that \(\cos \alpha + \cos \beta + \sqrt{2} \sin \alpha \sin \beta \leq \frac{3 \sqrt{2}}{2}\).

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


In the diagram, \(C\) lies on \(AE\) and \(AB=BC=CD\). If \(\angle CDE=t^{\circ}, \angle DEC=(2t)^{\circ}\), and \(\angle BCA=\angle BCD=x^{\circ}\), determine the measure of \(\angle ABC\).

Answer: 60


Answer: 6


Given a line \( l \) intersects an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) at points \( A \) and \( B \), and \( O \) is the origin of the coordinates.
1. Find the maximum area of triangle \(\triangle AOB\) and the equation of the line \( l \) when this maximum area is achieved.
2. Let \( L=\{l \mid \) the line \( l \) gives the maximum area of \(\triangle AOB \) \}. If \( l_{1}, l_{2}, l_{3}, l_{4} \in L \) such that \( l_{1} \parallel l_{2}, l_{3} \parallel l_{4} \), and \( k_{1}, k_{2}, k_{3}, k_{4} \) are the slopes of the corresponding lines such that \( k_{1} + k_{2} + k_{3} + k_{4} = 0 \), find the minimum area of the quadrilateral formed by these four lines.

Answer: 2\sqrt{2}


Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\square$ is unclearly printed.
$(1)$ She guessed $\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find the value of the expression when $x=-2$;
$(2)$ Her father said she guessed wrong, the standard answer's simplification does not contain quadratic terms. Please calculate and determine the value of $\square$ in the original question.

Answer: -2


Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \).

Answer: 1475


A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$.

Answer: 15


A line divides the length of an arc of a circle in the ratio 1:3. In what ratio does it divide the area of the circle?

Answer: \frac{\pi - 2}{3\pi + 2}


Let the two foci of the conic section $C$ be $F_1$ and $F_2$, respectively. If there exists a point $P$ on curve $C$ such that the ratio $|PF_1| : |F_1F_2| : |PF_2| = 4 : 3 : 2$, determine the eccentricity of curve $C$.

Answer: \frac{3}{2}


Liu and Li, each with one child, go to the park together to play. After buying tickets, they line up to enter the park. For safety reasons, the first and last positions must be occupied by fathers, and the two children must stand together. The number of ways for these 6 people to line up is \_\_\_\_\_\_.

Answer: 24


Let  $r$  be a fixed positive real number. It is known that for some positive integer  $n$  the following statement is true: for any positive real numbers  $a_1,...,a_n$  satisfying the equation  $a_1+...+a_n=r(\frac{1}{a_1}+...+\frac{1}{a_n})$  they also satisfy the equation  $\frac{1}{\sqrt{r}-a_1}+...+\frac{1}{\sqrt{r}-a_n}=\frac{1}{\sqrt{r}}$  ( $a_i\neq\sqrt{r}$ ). Find  $n$ .

Answer: 2


A clock has an hour, minute, and second hand, all of length  $1$ . Let  $T$  be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of  $T$ ?

*Proposed by Dylan Toh*

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


Two regular triangular pyramids with the same base are inscribed in the same sphere. Knowing that the side length of the base of the triangular pyramids is \(a\) and the radius of the sphere is \(R\), and that the angles between the lateral faces and the base of the two pyramids are \(\alpha\) and \(\beta\) respectively, find the value of \(\tan (\alpha+\beta)\).

Answer: -\frac{4\sqrt{3}R}{3a}


An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
$\textbf{(A)}\ 190  \qquad\textbf{(B)}\  191 \qquad\textbf{(C)}\  192 \qquad\textbf{(D)}\  195 \qquad\textbf{(E)}\ 196$

Answer: \textbf{(B)}\191


Let  $m_n$  be the smallest value of the function  ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$
Show that  $m_n  \to \frac{1}{2}$ , as  $n \to \infty.$

Answer: \frac{1}{2}


Through the point \( P \), which lies on the common chord \( AB \) of two intersecting circles, chord \( KM \) of the first circle and chord \( LN \) of the second circle are drawn. Prove that the quadrilateral \( KLMN \) is cyclic.

Answer: KLMN


The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this   service center?

$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 110 \qquad \text{(D)}\ 120 \qquad \text{(E)}\ 130$

Answer: (E)\130


In the right triangle \( \triangle ABC \),

\[
\angle A = 90^\circ, \, AB = AC
\]

\( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively. \( D \) is an arbitrary point on the segment \( MN \) (excluding points \( M \) and \( N \)). The extensions of \( BD \) and \( CD \) intersect \( AC \) and \( AB \) at points \( F \) and \( E \) respectively. If

\[
\frac{1}{BE} + \frac{1}{CF} = \frac{3}{4}
\]

then find the length of \( BC \).

Answer: 4\sqrt{2}


Let  $P$  be a point outside a circle  $\Gamma$  centered at point  $O$ , and let  $PA$  and  $PB$  be tangent lines to circle  $\Gamma$ . Let segment  $PO$  intersect circle  $\Gamma$  at  $C$ . A tangent to circle  $\Gamma$  through  $C$  intersects  $PA$  and  $PB$  at points  $E$  and  $F$ , respectively. Given that  $EF=8$  and  $\angle{APB}=60^\circ$ , compute the area of  $\triangle{AOC}$ .

*2020 CCA Math Bonanza Individual Round #6*

Answer: 12 \sqrt{3}


An \( n \)-digit number \( x \) has the following property: if the last digit of \( x \) is moved to the front, the result is \( 2x \). Find the smallest possible value of \( n \).

Answer: 18


Answer: 1552


Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?

Answer: 1


How many three-digit numbers remain if we exclude all three-digit numbers in which all digits are the same or the middle digit is different from the two identical end digits?

Answer: 810


A circle is tangent to both branches of the hyperbola \( x^{2} - 20y^{2} = 24 \) as well as the \( x \)-axis. Compute the area of this circle.

Answer: 504\pi


The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is:

Answer: 1 : 9


Consider a regular \( n \)-gon with \( n > 3 \), and call a line acceptable if it passes through the interior of this \( n \)-gon. Draw \( m \) different acceptable lines, so that the \( n \)-gon is divided into several smaller polygons.
(a) Prove that there exists an \( m \), depending only on \( n \), such that any collection of \( m \) acceptable lines results in one of the smaller polygons having 3 or 4 sides.
(b) Find the smallest possible \( m \) which guarantees that at least one of the smaller polygons will have 3 or 4 sides.

Answer: n-4


The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number?

Answer: 78


Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$.

Answer: 134


The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$

Answer: 2


On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.

The first person said: "Exactly 1 percent of the people present in this room are liars."

The second person said: "Exactly 2 percent of the people present in this room are liars."

and so on.

The person with number $n$ said: "Exactly $n$ percent of the people present in this room are liars."

How many people could be in the room, given that it is known that at least one of them is a knight?

Answer: 100


The sum of ten numbers is zero. The sum of all their pairwise products is also zero. Find the sum of their fourth powers.

Answer: 0


Find the number of ordered pairs  $(x,y)$  of positive integers that satisfy  $x\le 2y\le 60$  and  $y\le 2x\le 60.$

Answer: 480


A running competition on an unpredictable distance is conducted as follows. On a circular track with a length of 1 kilometer, two points \( A \) and \( B \) are randomly selected (using a spinning arrow). The athletes then run from \( A \) to \( B \) along the shorter arc. Find the median value of the length of this arc, that is, a value \( m \) such that the length of the arc exceeds \( m \) with a probability of exactly 50%.

Answer: 0.25


The Wolf and the three little pigs wrote a detective story "The Three Little Pigs-2", and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remaining 2100 gold coins to the Wolf. The fee for each book is divided equally among its authors. How much money should the Wolf take for himself?

Answer: 700


$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine $n$.

Answer: 4


Calculate the limit of the function:

$$\lim _{x \rightarrow \pi}\left(\operatorname{ctg}\left(\frac{x}{4}\right)\right)^{1 / \cos \left(\frac{x}{2}\right)}$$

Answer: e


In triangle \( KLM \), with all sides distinct, the bisector of angle \( KLM \) intersects side \( KM \) at point \( N \). A line through point \( N \) intersects side \( LM \) at point \( A \), such that \( MN = AM \). Given that \( LN = a \) and \( KL + KN = b \), find \( AL \).

Answer: \frac{^2}{b}


Tanya wrote a certain two-digit number on a piece of paper; to Sveta, who was sitting opposite her, the written number appeared different and was 75 less. What number did Tanya write?

Answer: 91


In a  $k$ -player tournament for  $k > 1$ , every player plays every other player exactly once. Find with proof the smallest value of  $k$  such that it is possible that for any two players, there was a third player who beat both of them.

Answer: 7


Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?

Answer: 3


Our Slovak grandmother shopped at a store where they had only apples, bananas, and pears. Apples were 50 cents each, pears were 60 cents each, and bananas were cheaper than pears. Grandmother bought five pieces of fruit, with exactly one banana among them, and paid 2 euros and 75 cents.

How many cents could one banana cost? Determine all possibilities.

Answer: 35,45,55


The hourly teaching fees of Teacher Huanhuan and Teacher Xixi are in the ratio of 5:4. The company decided to accelerate the training for these two assistant teachers, and increased their hourly teaching fees by 20 yuan each. After the increase, the ratio of their hourly teaching fees became 6:5. After the increase, the sum of their hourly teaching fees is $\qquad$ yuan.

Answer: 220\,


Let \( n \) be a natural number. Decompose \( n \) into sums of powers of \( p \) (where \( p \) is a positive integer greater than 1), in such a way that each power \( p^k \) appears at most \( p^2 - 1 \) times. Denote by \( C(n, p) \) the total number of such decompositions. For example, for \( n = 8 \) and \( p = 2 \):

\[ 8 = 4 + 4 = 4 + 2 + 2 = 4 + 2 + 1 + 1 = 2 + 2 + 2 + 1 + 1 = 8 \]

Thus \( C(8, 2) = 5 \). Note that \( 8 = 4 + 1 + 1 + 1 + 1 \) is not counted because \( 1 = 2^0 \) appears 4 times, which exceeds \( 2^2 - 1 = 3 \). Then determine \( C(2002, 17) \).

Answer: 118


Let \( P \) be a point on the ellipse \(\frac{y^2}{4} + \frac{x^2}{3} = 1\). Given points \( A(1,1) \) and \( B(0,-1) \), find the maximum value of \( |PA| + |PB| \).

Answer: 5


Given that $\overrightarrow {a}|=4$, $\overrightarrow {e}$ is a unit vector, and the angle between $\overrightarrow {a}$ and $\overrightarrow {e}$ is $\frac {2π}{3}$, find the projection of $\overrightarrow {a}+ \overrightarrow {e}$ on $\overrightarrow {a}- \overrightarrow {e}$.

Answer: \frac {5 \sqrt {21}}{7}


Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\).

Answer: \frac{}{2}


A circle of radius 1 is tangent to a circle of radius 2. The sides of $\triangle ABC$ are tangent to the circles as shown, and the sides $\overline{AB}$ and $\overline{AC}$ are congruent. What is the area of $\triangle ABC$?

[asy]
unitsize(0.7cm);
pair A,B,C;
A=(0,8);
B=(-2.8,0);
C=(2.8,0);
draw(A--B--C--cycle,linewidth(0.7));
draw(Circle((0,2),2),linewidth(0.7));
draw(Circle((0,5),1),linewidth(0.7));
draw((0,2)--(2,2));
draw((0,5)--(1,5));
label("2",(1,2),N);
label("1",(0.5,5),N);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
[/asy]

Answer: 16\sqrt{2}


Cagney can frost a cupcake every 15 seconds and Lacey can frost a cupcake every 45 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes.

Answer: 53


Given sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) satisfy:
\[
b_{n}=\begin{cases} a_{\frac{n+1}{2}}, & n \text{ is odd;} \\
\sqrt{a_{n+1}}, & n \text{ is even.}
\end{cases}
\]
If \(\{b_{n}\}\) is a geometric sequence and \(a_{2}+b_{2}=108\), find the general term formula for the sequence \(\{a_{n}\}\).

Answer: a_n=9^n


Suppose \( S = \{1, 2, \cdots, 2005\} \). If any subset of \( S \) containing \( n \) pairwise coprime numbers always includes at least one prime number, find the minimum value of \( n \).

Answer: 16


In how many ways can we place 8 digits equal to 1 and 8 digits equal to 0 on a 4x4 board such that the sums of the numbers written in each row and column are the same?

Answer: 90


Let $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $C$ in the line $y = 2x$.  Let the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) of $C^*$ be written in the form
\[12x^2 + bxy + cy^2 + d = 0.\]
Find the product $bc$.

Answer: 84


In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ²(3+sin²θ)=12.

1. Find the general equation of line l and the rectangular coordinate equation of curve C.
2. If line l intersects curve C at points A and B, and point P is defined as (2,1), find the value of $$\frac {|PB|}{|PA|}+ \frac {|PA|}{|PB|}$$.

Answer: \frac{86}{7}


Raashan, Sylvia, and Ted play the following game. Each starts with $$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $$1$? (For example, Raashan and Ted may each decide to give $$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $$0$, Sylvia will have $$2$, and Ted will have $$1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $$1$ to, and the holdings will be the same at the end of the second round.)
$\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$

Answer: \textbf{(B)}\frac{1}{4}


There are a certain number of red balls, green balls, and blue balls in a bag. Of the balls in the bag, $\frac{1}{3}$ are red and $\frac{2}{7}$ are blue. The number of green balls in the bag is 8 less than twice the number of blue balls. The number of green balls in the bag is
(A) 12
(B) 16
(C) 20
(D) 24
(E) 28

Answer: 16


A company occupies three floors in a tower: the 13th, 14th, and 25th floors. A cleaning crew is hired to clean these spaces. They work for four hours on the 13th and 14th floors. Then, the crew splits in half: one half cleans the 25th floor while the other half continues cleaning the 13th and 14th floors. After four more hours, the cleaning of the lower two floors is complete, but the 25th floor requires an additional cleaner who will finish it in 8 more hours the next day. Assuming that the cleaning time is directly proportional to the area being cleaned, how many cleaners were in the crew initially?

Answer: 8


Given that the equation \( x^{4} - px^{3} + q = 0 \) has an integer root, find the prime numbers \( p \) and \( q \).

Answer: 3,2


Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0.

Answer: 2014


Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\mathcal{T}_{12}$.

Answer: 1705


Let \(\mathbb{Z}_{\geq 0}\) be the set of non-negative integers, and let \(f: \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0} \rightarrow \mathbb{Z}_{\geq 0}\) be a bijection such that whenever \(f(x_1, y_1) > f(x_2, y_2)\), we have \(f(x_1+1, y_1) > f(x_2+1, y_2)\) and \(f(x_1, y_1+1) > f(x_2, y_2+1)\).

Let \(N\) be the number of pairs of integers \((x, y)\), with \(0 \leq x, y < 100\), such that \(f(x, y)\) is odd. Find the smallest and largest possible values of \(N\).

Answer: 5000


In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}-4x=0$. The parameter equation of curve $C_{2}$ is $\left\{\begin{array}{l}x=\cos\beta\\ y=1+\sin\beta\end{array}\right.$ ($\beta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive $x$-axis as the polar axis.
$(1)$ Find the polar coordinate equations of curves $C_{1}$ and $C_{2}$;
$(2)$ If the ray $\theta =\alpha (\rho \geqslant 0$, $0＜\alpha＜\frac{π}{2})$ intersects curve $C_{1}$ at point $P$, the line $\theta=\alpha+\frac{π}{2}(\rho∈R)$ intersects curves $C_{1}$ and $C_{2}$ at points $M$ and $N$ respectively, and points $P$, $M$, $N$ are all different from point $O$, find the maximum value of the area of $\triangle MPN$.

Answer: 2\sqrt{5} + 2


A store distributes 9999 cards among its customers. Each card has a 4-digit number, ranging from 0001 to 9999. If the sum of the first 2 digits is equal to the sum of the last 2 digits, the card is a winning card. For example, card 0743 is a winning card. Prove that the sum of the numbers of all the winning cards is divisible by 101.

Answer: 101


Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
[/asy]

Answer: 8


(1) The definite integral $\int_{-1}^{1}(x^{2}+\sin x)dx=$ ______.

(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.

(3) Given the function $f(x)=\begin{cases}1-\log_{a}(x+2), & x\geqslant 0 \\ g(x), & x < 0\end{cases}$ is an odd function, then the root of the equation $g(x)=2$ is ______.

(4) Given the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(0 < b < a < \sqrt{2}b)$, its foci are $F_{1}$ and $F_{2}$ respectively. Circle $N$ has $F_{2}$ as its center, and its minor axis length as the diameter. A tangent line to circle $N$ passing through point $F_{1}$ touches it at points $A$ and $B$. If the area of quadrilateral $F_{1}AF_{2}B$ is $S= \frac{2}{3}a^{2}$, then the eccentricity of ellipse $M$ is ______.

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


What is the minimum number of points in which 5 different non-parallel lines, not passing through a single point, can intersect?

Answer: 10


Let the isosceles right triangle  $ABC$  with   $\angle A= 90^o$  . The points   $E$  and  $F$  are taken on the ray  $AC$  so that  $\angle ABE   = 15^o$  and  $CE = CF$ . Determine the measure of the angle  $CBF$ .

Answer: 15


Prove that if the polynomial \( x^4 + ax^3 + bx + c \) has all real roots, then \( ab \leq 0 \).

Answer: \le0


A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car not rented out, the maintenance cost is 50 yuan per month. To maximize the monthly revenue of the rental company, the monthly rent for each car should be set at ______.

Answer: 4050


Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the number of possible values of $s$?

Answer: 22


Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time.

Answer: 1.25


A $2$ by $2$ square is divided into four $1$ by $1$ squares.  Each of the small squares is to be painted either green or red.  In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square?  There may be as few as zero or as many as four small green squares.

Answer: 6


A fishing vessel illegally fishes in a foreign country's territorial waters, resulting in an identical loss of value for the foreign country with each cast of the net. The probability that the vessel will be detained by the foreign coast guard during each cast is $1 / k$, where $k$ is a natural number. Assume that the event of the vessel being detained or not during each cast is independent of the previous fishing activities. If the vessel is detained by the foreign coast guard, all previously caught fish are confiscated, and it can no longer fish in these waters. The captain plans to leave the foreign territorial waters after casting the net for the $n$-th time. Since the possibility that the vessel might be detained by the foreign coast guard cannot be disregarded, the profit from fishing is a random variable. Find the number $n$ that maximizes the expected value of the fishing profit.

Answer: k-1


Determine all composite positive integers \( n \) with the following property: If \( 1 = d_1 < d_2 < \ldots < d_k = n \) are all the positive divisors of \( n \), then

\[
\left(d_2 - d_1\right) : \left(d_3 - d_2\right) : \cdots : \left(d_k - d_{k-1}\right) = 1 : 2 : \cdots : (k-1).
\]

Answer: 4


An infinite tree graph has edges that connect lattice points such that the distance between two connected points is at most 1998. Every lattice point in the plane is a vertex of the graph. Prove that there exists a pair of points in the plane at a unit distance from each other that are connected in the tree graph by a path of length at least \( 10^{1998} \).

Answer: 10^{1998}


I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.

Answer: \frac{1}{63}


Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $\dots$ by $a_1 = 1$ and $a_{n + 1}^3 = 99a_n^3$ for all $n \geq 1$. Then $a_{100}$ equals
$\textbf{(A)}\ 33^{33} \qquad  \textbf{(B)}\ 33^{99} \qquad  \textbf{(C)}\ 99^{33} \qquad  \textbf{(D)}\ 99^{99} \qquad  \textbf{(E)}\ \text{none of these}$

Answer: \textbf{(C)}\99^{33}


The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible.

Answer: 127


Given \\(|a|=1\\), \\(|b|= \sqrt{2}\\), and \\(a \perp (a-b)\\), the angle between vector \\(a\\) and vector \\(b\\) is ______.

Answer: \frac{\pi}{4}


How many different real numbers $x$ satisfy the equation \[(x^{2}-5)^{2}=16?\]
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$

Answer: \textbf{(D)}4


A plane is at a distance $a$ from the center of a unit sphere. Find the edge length of a cube, one face of which lies in this plane, with the vertices of the opposite face on the sphere.

Answer: \frac{\sqrt{6-2a^2}-2a}{3}


Answer: .426


What is the smallest number, \( n \), which is the product of 3 distinct primes where the mean of all its factors is not an integer?

Answer: 130


Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head.

Answer: 48


Let $p$ be an odd prime number, and define $d_{p}(n)$ as the remainder of the Euclidean division of $n$ by $p$. A sequence $\left(a_{n}\right)_{n \geqslant 0}$ is called a $p$-sequence if for all $n \geq 0$, $a_{n+1} = a_{n} + d_{p}(a_{n})$.

- Does there exist an infinite number of prime numbers $p$ for which there exist two $p$-sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ such that $a_{n} > b_{n}$ for infinitely many $n$, and $a_{n} < b_{n}$ for infinitely many $n$?
- Does there exist an infinite number of prime numbers $p$ for which there exist two $p$-sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ such that $a_{0} < b_{0}$ but $a_{n} > b_{n}$ for all $n \geq 1$?

Answer: Yes


Given a parallelogram \(A B C D\) with \(\angle B = 111^\circ\) and \(B C = B D\). A point \(H\) is marked on the segment \(B C\) such that \(\angle B H D = 90^\circ\). Point \(M\) is the midpoint of side \(A B\). Find the angle \(A M H\). Provide the answer in degrees.

Answer: 132


For what values of the real parameter \( t \) does the system of inequalities
\[ x \geq y^2 + t y \geq x^2 + t \]
have exactly one solution in the set of real number pairs?

Answer: \frac{1}{4}


Given a geometric sequence $\{a_n\}$ with a common ratio $q=-5$, and $S_n$ denotes the sum of the first $n$ terms of the sequence, the value of $\frac{S_{n+1}}{S_n}=\boxed{?}$.

Answer: -4


Call a natural number  $n{}$  *interesting* if any natural number not exceeding  $n{}$  can be represented as the sum of several (possibly one) pairwise distinct positive divisors of  $n{}$ .
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other than the powers of two.
[/list]
*Proposed by N. Agakhanov*

Answer: 992


Let point \(P\) be any point on the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) (where \(a > 0\) and \(b > 0\)). A line passing through point \(P\) intersects the asymptotes \(l_{1}: y = \frac{b}{a} x\) and \(l_{2}: y = -\frac{b}{a} x\) at points \(P_{1}\) and \(P_{2}\), respectively. Let \(\lambda = \frac{P_{1} P}{P P_{2}}\). Prove that the area of the triangle \(\triangle O P_{1} P_{2}\) is \( \frac{(1 + \lambda)^{2}}{4 |\lambda|} a b \).

Answer: \frac{(1+\lambda)^2}{4|\lambda|}


In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.)
 The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.

Answer: 792


Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.
$(1)$ Find $p$;
$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$.

Answer: 20\sqrt{5}


The table below displays the grade distribution of the $30$ students in a mathematics class on the last two tests.  For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2.  What percent of the students received the same grade on both tests?

Answer: 40\%


Let the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfy the condition
\[
f\left(x^{3}+y^{3}\right)=(x+y)\left((f(x))^{2}-f(x) \cdot f(y)+(f(y))^{2}\right), \quad x, y \in \mathbf{R}.
\]

Prove that for all \( x \in \mathbf{R} \),
\[
f(1996 x) = 1996 f(x).
\]

Answer: f(1996x)=1996f(x)


We colored the faces of a regular dodecahedron with red, blue, yellow, and green so that any two adjacent faces have different colors. How many edges of the dodecahedron have one face colored blue and the other green?

Answer: 5


Two sons inherit a herd of cows after their father dies. They sell the herd, receiving as many dollars per head as there were heads in the herd. With the money from the sale, the sons buy sheep at $10 each and one lamb that costs less than $10. They then divide the sheep and the lamb such that each son gets an equal number of animals. How much should the son who received only sheep pay his brother to ensure each has an equal share?

Answer: 2


The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere?

Answer: 100\pi


Let  $n$  denote the product of the first  $2013$  primes.  Find the sum of all primes  $p$  with  $20 \le p \le 150$  such that
(i)  $\frac{p+1}{2}$  is even but is not a power of  $2$ , and
(ii) there exist pairwise distinct positive integers  $a,b,c$  for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by  $p$  but not  $p^2$ .

*Proposed by Evan Chen*

Answer: 431


Let $x_{1} > 0$ be a root of the equation $a x^{2} + b x + c = 0$.

Prove: there exists a root $x_{2}$ of the equation $c x^{2} + b x + a = 0$ such that $x_{1} + x_{2} \geq 2$.

Answer: x_1+x_2\ge2


If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers  $p,q$ , find  $p+q$ .

*Proposed by Michael Kural*

Answer: 9901


Given a convex quadrilateral \( EFGH \) with vertices \( E \), \( F \), \( G \), and \( H \) respectively on the edges \( AB \), \( BC \), \( CD \), and \( DA \) of another convex quadrilateral \( ABCD \), such that \(\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1\). Additionally, the points \( A \), \( B \), \( C \), and \( D \) lie on the edges \( H_1E_1 \), \( E_1F_1 \), \( F_1G_1 \), and \( G_1H_1 \) of yet another convex quadrilateral \( E_1F_1G_1H_1 \), such that \( E_1F_1 \parallel EF \), \( F_1G_1 \parallel FG \), \( G_1H_1 \parallel GH \), and \( H_1E_1 \parallel HE \). Given \(\frac{E_1A}{AH_1} = \lambda\), find the value of \(\frac{F_1C}{CG_1}\).

Answer: \lambda


The number in the parentheses that satisfies the following equation is equal to $\qquad$
$$
\frac{\frac{(}{128}+1 \frac{2}{7}}{5-4 \frac{2}{21} \times 0.75} \div \frac{\frac{1}{3}+\frac{5}{7} \times 1.4}{\left(4-2 \frac{2}{3}\right) \times 3}=4.5
$$

Answer: \frac{1440}{7}


Xiao Xiao did an addition problem, but he mistook the second addend 420 for 240, and the result he got was 390. The correct result is ______.

Answer: 570


Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$?
$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

Answer: \textbf{(A)}45


Divide a circle with a circumference of 24 into 24 equal segments. Select 8 points from the 24 segment points such that the arc length between any two chosen points is not equal to 3 or 8. How many different ways are there to choose such a set of 8 points? Provide reasoning.

Answer: 258


Calculate the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, with no repeated digits.

Answer: 259980


The solution of the equation $\dfrac{1+2^{x}}{1+x^{-x}}= \dfrac{1}{4}$ is $x=$                  .

Answer: -2


In the dream market, a Sphinx offered a traveler seven illusions, two naps, and one nightmare for four dreams. To another traveler, the same Sphinx offered four illusions, four naps, and two nightmares for seven dreams. The Sphinx measures equally for all travelers.

How many illusions did one dream cost?

Answer: 10


Let  $\{u_n \}$  be the sequence defined by its first two terms  $u_0, u_1$   and the recursion formula
\[u_{n+2 }= u_n - u_{n+1}.\]**(a)** Show that  $u_n$  can be written in the form  $u_n = \alpha a^n + \beta b^n$ , where  $a, b, \alpha, \beta$  are constants independent of  $n$  that have to be determined.**(b)** If  $S_n = u_0 + u_1 + \cdots + u_n$ , prove that  $S_n + u_{n-1}$  is a constant independent of  $n.$  Determine this constant.

Answer: 2u_0+u_1


Find the smallest natural number $n$ such that if $p$ is a prime number and $n$ is divisible by $p-1$, then $n$ is also divisible by $p$.

Answer: 1806


Find the largest \( \mathrm{C} \) such that for all \( \mathrm{y} \geq 4 \mathrm{x}>0 \), the inequality \( x^{2}+y^{2} \geq \mathrm{C} x y \) holds.

Answer: \frac{17}{4}


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


Monika is thinking of a four-digit number that has the following properties:

- The product of the two outer digits is 40,
- The product of the two inner digits is 18,
- The difference between the two outer digits is the same as the difference between the two inner digits,
- The difference between the thought number and the number written in reverse order (i.e., the number with the same digits but in reverse order) is the largest possible.

Determine the number Monika is thinking of.

Answer: 8635


Find all real values of $x$ that satisfy $\frac{x^2+x^3-2x^4}{x+x^2-2x^3} \ge -1.$ (Give your answer in interval notation.)

Answer: [-1,-\frac{1}{2})\cup(-\frac{1}{2},0)\cup(0,1)\cup(1,\infty)


$A, B, G$ are three points on a circle with $\angle A G B=48^\circ$. The chord $AB$ is trisected by points $C$ and $D$ with $C$ closer to $A$. The minor arc $AB$ is trisected by points $E$ and $F$ with $E$ closer to $A$. The lines $EC$ and $FD$ meet at $H$. If $\angle A H B = x^\circ$, find $x$.

Answer: 32


For a positive integer \( n \) and a real number \( x \) (where \( 0 \leq x < n \)), define
$$
f(n, x) = (1 - \{x\}) \binom{n}{[x]} + \{x\} \binom{n}{[x] + 1},
$$
where \( [x] \) is the floor function which gives the greatest integer less than or equal to \( x \), and \( \{x\} = x - [x] \).

If integers \( m \) and \( n \) (where \( m, n \geq 2 \)) satisfy
$$
f\left(m, \frac{1}{n}\right) + f\left(m, \frac{2}{n}\right) + \cdots + f\left(m, \frac{mn-1}{n}\right) = 123,
$$
find the value of \( f\left(n, \frac{1}{m}\right) + f\left(n, \frac{2}{m}\right) + \cdots + f\left(n, \frac{mn-1}{m}\right) \).

Answer: 74


Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0,
$$
and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$.

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


Given the numbers \( x, y, z \in [0, \pi] \), find the minimum value of the expression

$$
A = \cos (x - y) + \cos (y - z) + \cos (z - x)
$$

Answer: -1


Consider the set  $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$  where  $a, b, c, d, e$  are integers. If  $D$  is the average value of the fourth element of such a tuple in the set, taken over all the elements of  $\mathcal{S}$ , find the largest integer less than or equal to  $D$ .

Answer: 66


In how many distinct ways can I arrange my five keys on a keychain, if I want to put my house key next to my car key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that is, by reflection or rotation).

Answer: 6


Given the parameterized equation of a line is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{cases}$$ (where $t$ is the parameter), determine the angle of inclination of the line.

Answer: \frac{\pi}{3}


Answer: 125000


Given the function $f(x) = \ln x - ax$, where $a \in \mathbb{R}$.
(1) If the line $y = 3x - 1$ is a tangent line to the graph of the function $f(x)$, find the value of the real number $a$.
(2) If the maximum value of the function $f(x)$ on the interval $[1, e^2]$ is $1 - ae$ (where $e$ is the base of the natural logarithm), find the value of the real number $a$.

Answer: \frac{1}{e}


An integer $N$ is selected at random in the range $1\leq N \leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)}\ \frac{1}{5}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{4}{5}\qquad\textbf{(E)}\ 1$

Answer: \textbf{(D)}\\frac{4}{5}


Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$

Answer: 360


Determine the range of the function

\[ f(x)=\frac{4-3 \sin ^{6} x-3 \cos ^{6} x}{\sin x \cos x} \]

in the interval \(\left(0, \frac{\pi}{2}\right)\).

Answer: [6,\infty)


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

\[
A=\frac{\cos ^{2} x_{1}+\ldots+\cos ^{2} x_{n}}{\sqrt{n}+\sqrt{\operatorname{ctg}^{4} x_{1}+\ldots+\operatorname{ctg}^{4} x_{n}}}
\]

Answer: \frac{\sqrt{n}}{4}


For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n?$

Answer: 16


Answer: 4


The ratio of the number of games won to the number of games lost by the High School Hurricanes is $7/3$ with 5 games ended in a tie. Determine the percentage of games lost by the Hurricanes, rounded to the nearest whole percent.

Answer: 24\%


Evaluate the integral \(\int_{0}^{1} \ln x \ln (1-x) \, dx\).

Answer: 2 - \frac{\pi^2}{6}


Given 10 points in the plane where no three points are collinear, we draw 4 line segments, each connecting two points on the plane. The choice of these line segments is arbitrary, and each line segment has an equal chance of being chosen. Find the probability that any three of these line segments will form a triangle with vertices among the given 10 points. Express this probability in the form \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Determine \(m+n\).

Answer: 489


The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are

Answer: 63,65


In $\Delta XYZ$, $\overline{MN} \parallel \overline{XY}$, $XM = 5$ cm, $MY = 8$ cm, and $NZ = 9$ cm. What is the length of $\overline{YZ}$?

Answer: 23.4


The bar graph shows the results of a survey on color preferences.  What percent preferred blue?

Answer: 24\%


There are 5 different positive integers, and the product of any two of them is a multiple of 12. What is the minimum value of the sum of these 5 numbers?

Answer: 62


Znayka told Neznaika that to convert kilolunas to kilograms, one needs to divide the mass in kilolunas by 4 and then decrease the resulting number by $4\%$. Neznaika decided that to convert from kilograms to kilolunas, one needs to multiply the mass in kilograms by 4 and then increase the resulting number by $4\%$. By what percentage of the correct value in kilolunas will he be mistaken if he uses this method?

Answer: 0.16


A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library?

Answer: 95/159


Determine the smallest positive integer $n \geq 3$ for which $$A \equiv 2^{10 n}\left(\bmod 2^{170}\right)$$ where $A$ denotes the result when the numbers $2^{10}, 2^{20}, \ldots, 2^{10 n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A=10241048576$).

Answer: 14


There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\]What is $k?$

Answer: 137


In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$?

Answer: 2


Let $T$ be the set of numbers of the form $2^{a} 3^{b}$ where $a$ and $b$ are integers satisfying $0 \leq a, b \leq 5$. How many subsets $S$ of $T$ have the property that if $n$ is in $S$ then all positive integer divisors of $n$ are in $S$ ?

Answer: 924


Given an ordered  $n$ -tuple  $A=(a_1,a_2,\cdots ,a_n)$  of real numbers, where  $n\ge 2$ , we define  $b_k=\max{a_1,\ldots a_k}$  for each k. We define  $B=(b_1,b_2,\cdots ,b_n)$  to be the “*innovated tuple*” of  $A$ . The number of distinct elements in  $B$  is called the “*innovated degree*” of  $A$ .
Consider all permutations of  $1,2,\ldots ,n$  as an ordered  $n$ -tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to  $2$

Answer: n-\frac{n-1}{H_{n-1}}


Find all functions  $f:\mathbb{N} \rightarrow \mathbb{N}$  such that for every prime number  $p$  and natural number  $x$ ,  $$ \{ x,f(x),\cdots f^{p-1}(x) \}  $$  is a complete residue system modulo  $p$ . With  $f^{k+1}(x)=f(f^k(x))$  for every natural number  $k$  and  $f^1(x)=f(x)$ .

*Proposed by IndoMathXdZ*

Answer: f(x)=x+1


Calculate the lengths of the arcs of the curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$

Answer: \frac{8\pi^3}{3}


Answer: 4\sqrt{6}


The height of a regular quadrilateral prism \( A B C D A_{1} B_{1} C_{1} D_{1} \) is twice as small as the side of its base. Find the maximum value of the angle \( \angle A_{1} M C_{1} \), where \( M \) is a point on edge \( AB \).

Answer: 90


The roots of the equation $x^{2}-2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
$\textbf{(A)}\ y = x^{2}, y = 2x\qquad\textbf{(B)}\ y = x^{2}-2x, y = 0\qquad\textbf{(C)}\ y = x, y = x-2\qquad\textbf{(D)}\ y = x^{2}-2x+1, y = 1$
$\textbf{(E)}\ y = x^{2}-1, y = 2x-1$
[Note: Abscissas means x-coordinate.]

Answer: \textbf{(C)}


Can a circle be circumscribed around the quadrilateral \( A B C D \) if \( \angle A D C=30^{\circ} \), \( A B=3 \), \( B C=4 \), and \( A C=6 \)?

Answer: No


Answer: 1


Given \\(a < 0\\), \\((3x^{2}+a)(2x+b) \geqslant 0\\) holds true over the interval \\((a,b)\\), then the maximum value of \\(b-a\\) is \_\_\_\_\_\_.

Answer: \dfrac{1}{3}


On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$, points $B$ and $C$ are chosen on the circle such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is 12 and $\cos \alpha = \frac{5}{6}$.

Answer: 20


Two ships are moving in straight lines at a constant speed towards the same port. Initially, the positions of the ships and the port form an equilateral triangle. After the second ship has traveled 80 km, a right triangle is formed. When the first ship arrives at the port, the second ship still has 120 km to go. Find the initial distance between the ships.

Answer: 240\,


A set \( S \) has a relation \( \rightarrow \) defined on it for pairs of elements from the set \( S \), and it possesses the following properties:
1) For any two distinct elements \( a, b \in S \), exactly one of the relations \( a \rightarrow b \) or \( b \rightarrow a \) holds.
2) For any three distinct elements \( a, b, c \in S \), if the relations \( a \rightarrow b \) and \( b \rightarrow c \) hold, then the relation \( c \rightarrow a \) also holds.

What is the maximum number of elements that the set \( S \) can contain?

Answer: 3


Given the user satisfaction ratings: 1 person with 10 points, 1 person with 9 points, 2 people with 8 points, 4 people with 7 points, 1 person with 5 points, and 1 person with 4 points, calculate the average, mode, median, and 85% percentile of the user satisfaction rating for this product.

Answer: 8.5


Vasya was driving from Sosnovka to Petrovka. Along the way, he saw a sign indicating "70 km to Petrovka". After traveling another 20 km, Vasya saw a sign indicating "130 km to Sosnovka". What is the distance (in kilometers) from Sosnovka to Petrovka?

Option 1: 140 km
Option 2: 170 km
Option 3: 160 km
Option 4: 180 km

Answer: 180


In triangle \(ABC\), a point \(D\) is marked on side \(BC\) such that \(BD: DC = 1:3\). Additionally, points \(E\) and \(K\) are marked on side \(AC\) such that \(E\) lies between \(A\) and \(K\). Segment \(AD\) intersects segments \(BE\) and \(BK\) at points \(M\) and \(N\) respectively, and it is given that \(BM: ME = 7:5\) and \(BN: NK = 2:3\). Find the ratio \(MN: AD\).

Answer: \frac{11}{45}


On a spherical surface with an area of $60\pi$, there are four points $S$, $A$, $B$, and $C$, and $\triangle ABC$ is an equilateral triangle. The distance from the center $O$ of the sphere to the plane $ABC$ is $\sqrt{3}$. If the plane $SAB$ is perpendicular to the plane $ABC$, then the maximum volume of the pyramid $S-ABC$ is \_\_\_\_\_\_.

Answer: 27


The route from point $B$ to $C$ begins with an uphill climb of $3 \text{ km}$, followed by a $5 \text{ km}$ horizontal section and ends with a downhill slope of $6 \text{ km}$ that has the same gradient as the initial uphill. A traveler started from $B$, turned back halfway, and returned to $B$ after $3$ hours and $36$ minutes. After resting, they took $3$ hours and $27$ minutes to reach $C$ and finally made the return trip in $3$ hours and $51$ minutes. What were the traveler's speeds on the horizontal path, uphill, and downhill?

Answer: 5


In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that $\overline{UV}\parallel\overline{BC}$, $\overline{WX}\parallel\overline{AB}$, and $\overline{YZ}\parallel\overline{CA}$. Right angle folds are then made along $\overline{UV}$, $\overline{WX}$, and $\overline{YZ}$. The resulting figure is placed on a level floor to make a table with triangular legs. Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor. Then $h$ can be written in the form $\frac{k\sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime. Find $k+m+n$.
[asy] unitsize(1 cm); pair translate; pair[] A, B, C, U, V, W, X, Y, Z; A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]); translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate; draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]); dot("$A$",A[0],N); dot("$B$",B[0],SE); dot("$C$",C[0],SW); dot("$U$",U[0],NE); dot("$V$",V[0],NW); dot("$W$",W[0],NW); dot("$X$",X[0],S); dot("$Y$",Y[0],S); dot("$Z$",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("$U$",U[1],NE); dot("$V$",V[1],NW); dot("$W$",W[1],NW); dot("$X$",X[1],dir(-70)); dot("$Y$",Y[1],dir(250)); dot("$Z$",Z[1],NE);[/asy]

Answer: 318


Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$.

Answer: 30


In February of a non-leap year, Kirill and Vova decided to eat ice cream according to the following rules:
1. If the day of the month was even and the day of the week was Wednesday or Thursday, they would each eat seven servings of ice cream.
2. If the day of the week was Monday or Tuesday and the day of the month was odd, they would each eat three servings of ice cream.
3. If the day of the week was Friday, the number of servings each of them ate would be equal to the day of the month.

On all other days and under other conditions, eating ice cream was prohibited. What is the maximum number of servings of ice cream that Vova could eat in February under these conditions?

Answer: 110


Let  $A_1B_1C_1$ ,  $A_2B_2C_2$ , and  $A_3B_3C_3$  be three triangles in the plane. For  $1 \le i \le3$ , let  $D_i $ ,  $E_i$ , and  $F_i$  be the midpoints of  $B_iC_i$ ,  $A_iC_i$ , and  $A_iB_i$ , respectively. Furthermore, for  $1 \le i \le 3$  let  $G_i$  be the centroid of  $A_iB_iC_i$ .
Suppose that the areas of the triangles  $A_1A_2A_3$ ,  $B_1B_2B_3$ ,  $C_1C_2C_3$ ,  $D_1D_2D_3$ ,  $E_1E_2E_3$ , and  $F_1F_2F_3$  are  $2$ ,  $3$ ,  $4$ ,  $20$ ,  $21$ , and  $2020$ , respectively. Compute the largest possible area of  $G_1G_2G_3$ .

Answer: 917


Suppose that there are initially eight townspeople and one goon. One of the eight townspeople is named Jester. If Jester is sent to jail during some morning, then the game ends immediately in his sole victory. (However, the Jester does not win if he is sent to jail during some night.) Find the probability that only the Jester wins.

Answer: \frac{1}{3}


 How many (possibly empty) sets of lattice points  $\{P_1, P_2, ... , P_M\}$ , where each point  $P_i =(x_i, y_i)$  for  $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line  $P_iP_j$  is positive for each  $1 \le i < j \le M$  ? An infinite slope, e.g.  $P_i$  is vertically above  $P_j$  , does not count as positive.

Answer: 3432


Let \( f(x) = x^{2} + a \). Define \( f^{1}(x) = f(x) \) and \( f^{n}(x) = f\left(f^{n-1}(x)\right) \) for \( n = 2, 3, \ldots \). Given that \( M = \{ a \in \mathbb{R} \mid |f^{n}(0)| \leq 2 \text{ for all positive integers } n \} \).

Prove that \( M = \left[-2, \frac{1}{4}\right] \).

Answer: [-2,\frac{1}{4}]


Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters?

Answer: 28


\( \binom{n}{0} \binom{n}{\left\lfloor \frac{n}{2} \right\rfloor} + 2 \binom{n}{1} \binom{n-1}{\left\lfloor \frac{n-1}{2} \right\rfloor} + 2^2 \binom{n}{2} \binom{n-2}{\left\lfloor \frac{n-2}{2} \right\rfloor} + \ldots + 2^{n-1} \binom{n}{n-1} \binom{1}{\left\lfloor \frac{1}{2} \right\rfloor} + 2^n = \binom{2n+1}{n} \)

Show that the above equation holds true.

Answer: \binom{2n+1}{n}


On an island, there are knights, who always tell the truth, and liars, who always lie.

One day, 80 islanders gathered, each wearing a T-shirt numbered from 1 to 80 (with different numbers for each resident). Each of them said one of two phrases:

- "Among the gathered, at least 5 liars have a T-shirt number greater than mine."
- "Among the gathered, at least 5 liars have a T-shirt number less than mine."

What is the minimum number of knights that could be among these 80 residents?

Answer: 70


Given an arithmetic sequence $\{a\_n\}$, where $a\_n \in \mathbb{N}^*$, and $S\_n = \frac{1}{8}(a\_n + 2)^2$. If $b\_n = \frac{1}{2}a\_n - 30$, find the minimum value of the sum of the first $\_\_\_\_\_\_$ terms of the sequence $\{b\_n\}$.

Answer: 15


Given $$∫_{ 0 }^{ 2 }(\cos \frac {π}{4}x+ \sqrt {4-x^{2}})dx$$, evaluate the definite integral.

Answer: \pi+\frac{4}{\pi}


Admiral Ackbar needs to send a 5-character message through hyperspace to the Rebels. Each character is a lowercase letter, and the same letter may appear more than once in a message. When the message is beamed through hyperspace, the characters come out in a random order. Ackbar chooses his message so that the Rebels have at least a \(\frac{1}{2}\) chance of getting the same message he sent. How many distinct messages could he send?

Answer: 26


In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find the length of $GH$ and $GI$.

Answer: 18.7


Given that $\alpha$ and $\beta$ are acute angles, $\tan\alpha= \frac {1}{7}$, $\sin\beta= \frac { \sqrt {10}}{10}$, find $\alpha+2\beta$.

Answer: \frac {\pi}{4}


There are two types of containers: 27 kg and 65 kg. How many containers of the first and second types were there in total, if the load in the containers of the first type exceeds the load of the container of the second type by 34 kg, and the number of 65 kg containers does not exceed 44 units?

Answer: 66


Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book?

Answer: 298


In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other?

Answer: \frac{7}{9}


From the numbers \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), two numbers are selected to form an ordered pair of real numbers \\((x, y)\\). The probability that \\(\dfrac{x}{y+1}\\) is an integer is equal to \_\_\_\_\_\_

Answer: \dfrac{4}{15}


Answer: 88572


An investor placed $\$15,000$ in a three-month term deposit that offered a simple annual interest rate of $8\%$. After three months, the total value of the deposit was reinvested in another three-month term deposit. After these three months, the investment amounted to $\$15,735$. Determine the annual interest rate, $s\%$, of the second term deposit.

Answer: 11.36


A boy named Vasya wrote down the nonzero coefficients of a tenth-degree polynomial \( P(x) \) in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, and continued this process until he arrived at a constant, which he also wrote down.

What is the minimum number of different numbers he could have ended up with?

Coefficients are written down with their signs, and constant terms are also recorded. If there is a term of the form \(\pm x^n\), \(\pm 1\) is written down.

Answer: 10


Answer: 4620


Given the ellipse Q: $$\frac{x^{2}}{a^{2}} + y^{2} = 1 \quad (a > 1),$$ where $F_{1}$ and $F_{2}$ are its left and right foci, respectively. A circle with the line segment $F_{1}F_{2}$ as its diameter intersects the ellipse Q at exactly two points.
(1) Find the equation of ellipse Q;
(2) Suppose a line $l$ passing through point $F_{1}$ and not perpendicular to the coordinate axes intersects the ellipse at points A and B. The perpendicular bisector of segment AB intersects the x-axis at point P. The range of the x-coordinate of point P is $[-\frac{1}{4}, 0)$. Find the minimum value of $|AB|$.

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


A basket of apples is divided into two parts, A and B. The ratio of the number of apples in A to the number of apples in B is $27: 25$. Part A has more apples than Part B. If at least 4 apples are taken from A and added to B, then Part B will have more apples than Part A. How many apples are in the basket?

Answer: 156


Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that
\begin{align*}
a^2 + b^2 + c^2 + d^2 &= 4, \\
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16.
\end{align*}

Answer: 15


Calculate the areas of the regions bounded by the curves given in polar coordinates.

$$
r=\cos 2 \phi
$$

Answer: \frac{\pi}{2}


Triangle $PQR$ has side-lengths $PQ = 20, QR = 40,$ and $PR = 30.$ The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ What is the perimeter of $\triangle PXY?$

Answer: 50


Given $\boldsymbol{a} = (\cos \alpha, \sin \alpha)$ and $\boldsymbol{b} = (\cos \beta, \sin \beta)$, the relationship between $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $|k \boldsymbol{a} + \boldsymbol{b}| - \sqrt{3}|\boldsymbol{a} - k \boldsymbol{b}|$, where $k > 0$. Find the minimum value of $\boldsymbol{a} \cdot \boldsymbol{b}$.

Answer: \frac{1}{2}


Simplify first, then find the value of the algebraic expression $\frac{a}{{{a^2}-2a+1}}÷({1+\frac{1}{{a-1}}})$, where $a=\sqrt{2}$.

Answer: \sqrt{2}+1


How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$?

Answer: 2


Determine the number of decreasing sequences of positive integers \(b_1 \geq b_2 \geq b_3 \geq \cdots \geq b_7 \leq 1500\) such that \(b_i - i\) is divisible by 3 for \(1 \leq i \le 7\). Express the number of such sequences as \({m \choose n}\) for some integers \(m\) and \(n\), and compute the remainder when \(m\) is divided by 1000.

Answer: 506


Mr. Wang, a math teacher, is preparing to visit a friend. Before leaving, Mr. Wang calls the friend's house, and the phone number is 27433619. After the call, Mr. Wang realizes that this phone number is exactly the product of 4 consecutive prime numbers. What is the sum of these 4 prime numbers?

Answer: 290


In a circle there are 101 numbers written. It is known that among any five consecutive numbers, there are at least two positive numbers. What is the minimum number of positive numbers that can be among these 101 written numbers?

Answer: 41


Let \( n \) be a fixed positive integer. Let \( S \) be any finite collection of at least \( n \) positive reals (not necessarily all distinct). Let \( f(S) = \left( \sum_{a \in S} a \right)^n \), and let \( g(S) \) be the sum of all \( n \)-fold products of the elements of \( S \) (in other words, the \( n \)-th symmetric function). Find \( \sup_S \frac{g(S)}{f(S)} \).

Answer: \frac{1}{n!}


Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent.

Answer: 67.5


Let the sides of two rectangles be \(\{a, b\}\) and \(\{c, d\}\) with \(a < c \leq d < b\) and \(ab < cd\). Prove that the first rectangle can be placed within the second one if and only if

$$
\left(b^{2} - a^{2}\right)^{2} \leq (bd - ac)^{2} + (bc - ad)^{2}.
$$

Answer: (b^{2}-^{2})^{2}\le(bd-ac)^{2}+(-ad)^{2}


Given \( A_{1}, A_{2}, \cdots, A_{n} \) are \( n \) non-empty subsets of the set \( A=\{1,2,3, \cdots, 10\} \), if for any \( i, j \in \{1,2,3, \cdots, n\} \), we have \( A_{i} \cup A_{j} \neq A \), then the maximum value of \( n \) is \(\qquad\).

Answer: 511


Two rectangles, one $8 \times 10$ and the other $12 \times 9$, are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction.

Answer: 65


The probability of an event happening is $\frac{1}{2}$, find the relation between this probability and the outcome of two repeated experiments.

Answer: 50\%


Given that \( p \) is a prime number and \( r \) is the remainder when \( p \) is divided by 210, if \( r \) is a composite number that can be expressed as the sum of two perfect squares, find \( r \).

Answer: 169


Given a right triangle \( ABC \) with legs \( AC = 3 \) and \( BC = 4 \). Construct triangle \( A_1 B_1 C_1 \) by successively translating point \( A \) a certain distance parallel to segment \( BC \) to get point \( A_1 \), then translating point \( B \) parallel to segment \( A_1 C \) to get point \( B_1 \), and finally translating point \( C \) parallel to segment \( A_1 B_1 \) to get point \( C_1 \). If it turns out that angle \( A_1 B_1 C_1 \) is a right angle and \( A_1 B_1 = 1 \), what is the length of segment \( B_1 C_1 \)?

Answer: 12


Multiply $2$ by $54$. For each proper divisor of $1,000,000$, take its logarithm base $10$. Sum these logarithms to get $S$, and find the integer closest to $S$.

Answer: 141


Given the sequence \(\{a_n\}\) defined by \(a_{1}=\frac{3}{2}\), and \(a_{n}=\frac{3 n a_{n-1}}{2 a_{n-1}+n-1}\) for \(n \geq 2\), \(n \in \mathbb{N}_{+}\).

(1) Find the general term of the sequence \(\{a_n\}\).

(2) For all positive integers \(n\), the inequality \(a_{1} a_{2} a_{3} \cdots a_{n}<\lambda \cdot n!\) always holds. Determine the smallest positive integer \(\lambda\).

Answer: 2


Given an ellipse $\frac{x^{2}}{8} + \frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine the slope $k$ of line $BC$.

Answer: \frac{1}{2}


A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the corresponding segments of the legs.

Answer: \frac{3 \sqrt{15}}{2}


On a segment of length 1, several smaller segments are colored, and the distance between any two colored points is not equal to 0.1. Prove that the sum of the lengths of the colored segments does not exceed 0.5.

Answer: 0.5


In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Answer: 574


Circle \(\omega\) is inscribed in rhombus \(H M_{1} M_{2} T\) so that \(\omega\) is tangent to \(\overline{H M_{1}}\) at \(A\), \(\overline{M_{1} M_{2}}\) at \(I\), \(\overline{M_{2} T}\) at \(M\), and \(\overline{T H}\) at \(E\). Given that the area of \(H M_{1} M_{2} T\) is 1440 and the area of \(E M T\) is 405, find the area of \(A I M E\).

Answer: 540


Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.

Answer: 217


Given an arithmetic sequence $\{a_n\}$ with the common difference $d$ being an integer, and $a_k=k^2+2$, $a_{2k}=(k+2)^2$, where $k$ is a constant and $k\in \mathbb{N}^*$
$(1)$ Find $k$ and $a_n$
$(2)$ Let $a_1 > 1$, the sum of the first $n$ terms of $\{a_n\}$ is $S_n$, the first term of the geometric sequence $\{b_n\}$ is $l$, the common ratio is $q(q > 0)$, and the sum of the first $n$ terms is $T_n$. If there exists a positive integer $m$, such that $\frac{S_2}{S_m}=T_3$, find $q$．

Answer: \frac{\sqrt{13}-1}{2}


The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\boxed{32768}$.

Answer: 16384


Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.

1. Determine whether the function $f(x) = x^2$ belongs to set $M$ and explain your reasoning.
2. If $f(x) = \sin x \in M$, find all companion pairs $(a,k)$ for the function $f(x)$.
3. If $(1,1)$ and $(2,-1)$ are both companion pairs of the function $f(x)$, where $f(x) = \cos(\frac{\pi}{2}x)$ for $1 \leq x < 2$ and $f(x) = 0$ for $x=2$. Find all zeros of the function $y=f(x)$ when $2014 \leq x \leq 2016$.

Answer: 2016


Given the number $200 \ldots 002$ (100 zeros). We need to replace two of the zeros with non-zero digits such that the resulting number is divisible by 66. How many ways can this be done?

Answer: 27100


Consider  $n \geq 2$  distinct points in the plane  $A_1,A_2,...,A_n$  . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?

Answer: 2n-3


Find the least positive integer $n$ such that $$\frac 1{\sin 30^\circ\sin 31^\circ}+\frac 1{\sin 32^\circ\sin 33^\circ}+\cdots+\frac 1{\sin 88^\circ\sin 89^\circ}+\cos 89^\circ=\frac 1{\sin n^\circ}.$$

Answer: n = 1


A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=\frac m n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Answer: 512


You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled $1,2, \ldots$. Each ball has probability $1 / 2^{n}$ of being put into box $n$. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls?

Answer: 5 / 7


Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered 1, 2, 3, ..., 20. Starting with the player who has the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends. Given that Yaiza lost and 5 cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player's hand does not matter.)

Answer: 324


A sphere with radius $R$ is tangent to one base of a truncated cone and also tangent to its lateral surface along a circle that is the circumference of the other base of the cone. Find the volume of the body consisting of the cone and the sphere, given that the surface area of this body is $S$.

Answer: \frac{SR}{3}


Consider an arithmetic sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms denoted as $S_{n}$. It is given that $a_{3} = 12$, $S_{12} > 0$, and $S_{13} < 0$.
(1) Determine the range of values for the common difference $d$.
(2) Identify which of $S_{1}, S_{2}, \cdots, S_{12}$ is the largest value and provide a justification.

Answer: S_6


In the sequence $\left\{x_{n}\right\}$, where $x_{1}=a$, $x_{2}=b$, and $x_{n}=\frac{x_{n-1}+x_{n-2}}{2}$, prove that $\lim _{n \rightarrow \infty} x_{n}$ exists and determine its value.

Answer: \frac{2b}{3}


In the geometric sequence ${a_n}$ where $q=2$, if the sum of the series $a_2 + a_5 + \dots + a_{98} = 22$, calculate the sum of the first 99 terms of the sequence $S_{99}$.

Answer: 77


A line drawn through the circumcenter and orthocenter of triangle \(ABC\) cuts off equal segments \(C A_{1}\) and \(C B_{1}\) from its sides \(CA\) and \(CB\). Prove that angle \(C\) is equal to \(60^\circ\).

Answer: 60


Let \( A, B, C \) be distinct points on the parabola \( y = x^2 \). \( R \) is the radius of the circumcircle of triangle \( \triangle ABC \). Find the range of values for \( R \).

Answer: (\frac{1}{2},+\infty)


Let \( n \) be a positive integer greater than 2. A function \( f: \mathbf{R}^{2} \rightarrow \mathbf{R} \) satisfies the following condition: For any regular \( n \)-gon \( A_{1} A_{2} \cdots A_{n} \), we have \(\sum_{i=1}^{n} f(A_{i}) = 0 \). Prove that \( f \equiv 0 \).

Answer: f\equiv0


In the expansion of  $(2x +3y)^{20}$ , find the number of coefficients divisible by  $144$ .

*Proposed by Hannah Shen*

Answer: 15


A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally scheduled for 9 days, but one day is now a rest day, how many matches will be completed?

Answer: 40


Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\).

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


Calculate the limit of the numerical sequence:

$$\lim _{n \rightarrow \infty}\left(\frac{7 n^{2}+18 n-15}{7 n^{2}+11 n+15}\right)^{n+2}$$

Answer: e


Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved.

Answer: 16.67\%


Answer: 709


Suppose that \((a_1, b_1), (a_2, b_2), \ldots, (a_{100}, b_{100})\) are distinct ordered pairs of nonnegative integers. Let \(N\) denote the number of pairs of integers \((i, j)\) satisfying \(1 \leq i < j \leq 100\) and \(\left|a_i b_j - a_j b_i\right| = 1\). Determine the largest possible value of \(N\) over all possible choices of the 100 ordered pairs.

Answer: 197


A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color?

Answer: \frac{1}{126}


In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2 \tan^{-1} (1/3)$. Find $\alpha$.

Answer: \frac{\pi}{2}


Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are the midpoints of $AD$ and $BC$, respectively. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form and find $p+q$, where the ratio is $p:q$.

Answer: 12


Calculate the definite integral:

$$
\int_{1}^{8} \frac{5 \sqrt{x+24}}{(x+24)^{2} \cdot \sqrt{x}} \, dx
$$

Answer: \frac{1}{8}


On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The inhabitants answered one by one in such a way that the others could hear. Knights told the truth, liars lied. Each follower answered the same as the majority of those who had answered before them, and if the number of "Yes" and "No" answers was equal, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island inhabitants?

Answer: 1009


Given the integers \( 1, 2, 3, \ldots, 40 \), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3.

Answer: 58


Let an integer  $ n > 3$ . Denote the set  $ T\equal{}\{1,2, \ldots,n\}.$  A subset S of T is called *wanting set* if S has the property: There exists a positive integer  $ c$  which is not greater than  $ \frac {n}{2}$  such that  $ |s_1 \minus{} s_2|\ne c$  for every pairs of arbitrary elements  $ s_1,s_2\in S$ . How many does a *wanting set* have at most are there ?

Answer: \lfloor\frac{2n}{3}\rfloor


In the country of Wonderland, an election campaign is being held for the title of the best tea lover, with the candidates being the Mad Hatter, March Hare, and Dormouse. According to a poll, $20\%$ of the residents plan to vote for the Mad Hatter, $25\%$ for the March Hare, and $30\%$ for the Dormouse. The remaining residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter needs to secure to guarantee not losing to both the March Hare and the Dormouse (regardless of how the undecided votes are allocated), given that each undecided voter will vote for one of the candidates. The winner is determined by a simple majority. Justify your answer.

Answer: 70


Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer: 200


Find the maximum value of the function
$$
f(x) = \sqrt{3} \sin 2x + 2 \sin x + 4 \sqrt{3} \cos x.
$$

Answer: \frac{17}{2}


Let \(ABCD\) be an isosceles trapezoid such that \(AD = BC\), \(AB = 3\), and \(CD = 8\). Let \(E\) be a point in the plane such that \(BC = EC\) and \(AE \perp EC\). Compute \(AE\).

Answer: 2\sqrt{6}


Given $f(x) = 2\sqrt{3}\sin x \cos x + 2\cos^2x - 1$,

(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;

(2) In $\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, $B$, and $C$ respectively, with $a=1$, $b=\sqrt{3}$, and $f(A) = 2$, determine the angle $C$.

Answer: \frac{\pi}{2}


Given $f(x)= \begin{cases} 2a-(x+ \frac {4}{x}),x < a\\x- \frac {4}{x},x\geqslant a\\end{cases}$.
(1) When $a=1$, if $f(x)=3$, then $x=$ \_\_\_\_\_\_;
(2) When $a\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=$ \_\_\_\_\_\_.

Answer: - \frac {11}{6}


The area of trapezoid $ABCD$ with bases $AD$ and $BC$ ($AD > BC$) is 48, and the area of triangle $AOB$, where $O$ is the intersection point of the diagonals of the trapezoid, is 9. Find the ratio of the bases of the trapezoid $AD: BC$.

Answer: 3


Joey wrote a system of equations on a blackboard, where each of the equations was of the form  $a + b = c$  or  $a \cdot b = c$  for some variables or integers  $a, b, c$ . Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads: $x\,\,\,\, z = 15$  $x\,\,\,\, y = 12$  $x\,\,\,\, x = 36$ If  $x, y, z$  are integer solutions to the original system, find the sum of all possible values of  $100x+10y+z$ .

Answer: 2037


Let the polynomial \( P(x) = x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n \) have complex roots \( x_1, x_2, \cdots, x_n \). Define \( \alpha = \frac{1}{n} \sum_{k=1}^{n} x_k \) and \( \beta^2 = \frac{1}{n} \sum_{k=1}^{n} |x_k|^2 \), with \( \beta^2 < 1 + |\alpha|^2 \). If a complex number \( x_0 \) satisfies \( |\alpha - x_0|^2 < 1 - \beta^2 + |\alpha|^2 \), prove that \( |P(x_0)| < 1 \).

Answer: |P(x_0)|<1


Pictured below is a rectangular array of dots with the bottom left point labelled \( A \). In how many ways can two points in the array be chosen so that they, together with point \( A \), form a triangle with positive area?

[Image of a rectangular array of dots with point \( A \) at the bottom left.]

(Note: Include the actual image if providing this to an end-user, as visual context is critical for solving the problem).

Answer: 256


Let \( Q \) be the set of some permutations of the numbers \( 1, 2, \ldots, 100 \) such that for any \( 1 \leq a, b \leq 100 \) with \( a \neq b \), there is at most one \( \sigma \in Q \) where \( b \) immediately follows \( a \) in \( \sigma \). Find the maximum number of elements in the set \( Q \).

Answer: 100


Find all natural values of \( n \) for which
$$
\cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9}, \text { and } \log _{2}^{2} n+45<\log _{2} 8 n^{13}
$$

Record the sum of the obtained values of \( n \) as the answer.

Answer: 644


All edges of a tetrahedron are \(12 \, \text{cm}\). Can it be placed in a box shaped as a rectangular parallelepiped with sides \(9 \, \text{cm}\), \(13 \, \text{cm}\), and \(15 \, \text{cm}\)?

Answer: Yes


On a table, there are 100 piles of pebbles, with 1, 2, ..., up to 100 pebbles in each pile respectively. In one step, you can reduce any number of selected piles, provided that you remove the same number of pebbles from each selected pile.

What is the minimum number of steps needed to remove all the pebbles from the table?

Answer: 7


Given that F is the right focus of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, and A is one endpoint of the ellipse's minor axis. If F is the trisection point of the chord of the ellipse that passes through AF, calculate the eccentricity of the ellipse.

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


Joe has  $1729$  randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths  $9$  and  $10$  or of sidelengths  $1$  and  $12$  (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths  $1$  and  $12$  or  $9$  and  $10$  are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

Answer: 7


Given $(a+i)i=b+ai$, solve for $|a+bi|$.

Answer: \sqrt{2}


Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[\frac{(a - b)(c - d)}{(b - c)(d - a)} = \frac{3}{7}.\] Find the product of all possible values of
\[\frac{(a - c)(b - d)}{(a - b)(c - d)}.\]

Answer: -\frac{4}{3}


The real numbers  $x$ ,  $y$ ,  $z$ , and  $t$  satisfy the following equation:
\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for  $t$ .

Answer: 125


The function defined on the set of real numbers, \(f(x)\), satisfies \(f(x-1) = \frac{1 + f(x+1)}{1 - f(x+1)}\). Find the value of \(f(1) \cdot f(2) \cdot f(3) \cdots f(2008) + 2008\).

Answer: 2009


The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value?

Answer: 45


Answer: \frac{-1-\sqrt{21}\\sqrt{2\sqrt{21}+6}}{4}


In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters?

Answer: 89


A grid sheet of size \(5 \times 7\) was cut into \(2 \times 2\) squares, L-shaped pieces covering 3 cells, and \(1 \times 3\) strips. How many \(2 \times 2\) squares could be obtained?

Answer: 5


Answer: \frac{\pi}{4} - \frac{1}{2}


The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$.

Answer: 555


Let ($a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive integers such that for each $2\le$ $i$ $\le10$ either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?

$\textbf{(A)}\ \ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ \ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ \ 362,880$

Answer: \textbf{(B)}\512


Let \( n \) be the product of the first 10 primes, and let
\[ S = \sum_{xy \mid n} \varphi(x) \cdot y, \]
where \( \varphi(x) \) denotes the number of positive integers less than or equal to \(x\) that are relatively prime to \( x \), and the sum is taken over ordered pairs \((x, y)\) of positive integers for which \( xy \) divides \( n \). Compute \(\frac{S}{n}\).

Answer: 1024


Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e., there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of the players can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\).

Answer: 1011


Let  $ n $  be a natural number. How many numbers of the form  $ \pm 1\pm 2\pm 3\pm\cdots\pm n $  are there?

Answer: \frac{n(n+1)}{2}+1


Wang Wei walks from location A to location B, while Zhang Ming rides a bicycle from location B to location A. They meet on the road after half an hour. After reaching location A, Zhang Ming immediately returns to location B and catches up with Wang Wei 20 minutes after their first meeting. After arriving at location B, Zhang Ming turns back, and they meet for the third time $\qquad$ minutes after their second meeting.

Answer: 40


Given that for all \( x \in [-1, 1] \), \( x^{3} - a x + 1 \geqslant 0 \), find the range of the real number \( a \).

Answer: [0,\frac{3\sqrt[3]{2}}{2}]


Place each of the digits 4, 5, 6, and 7 in exactly one square to make the smallest possible product. The grid placement is the same as described before.

Answer: 2622


The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the other three can take on any of the four tasks, then the total number of different dispatch plans is \_\_\_\_\_\_ (The result should be expressed in numbers).

Answer: 36


A circle is drawn through two adjacent vertices of a square such that the length of the tangent to it from the third vertex of the square is three times the side length of the square. Find the area of the circle if the side length of the square is \(a\).

Answer: \frac{65\pi^2}{4}


A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \mapsto 15 + 2 \times 7 = 29$, $5 \mapsto 0 + 2 \times 5 = 10$. A natural number is called ‘good’ if after repeatedly applying this operation, the resulting number stops changing. Find the smallest such good number.

Answer: 19


A book has a total of 100 pages, numbered sequentially from 1, 2, 3, 4…100. The digit “2” appears in the page numbers a total of \_\_\_\_\_\_ times.

Answer: 20


Let \( ABC \) be a triangle such that \( AB = AC = 182 \) and \( BC = 140 \). Let \( X_1 \) lie on \( AC \) such that \( CX_1 = 130 \). Let the line through \( X_1 \) perpendicular to \( BX_1 \) at \( X_1 \) meet \( AB \) at \( X_2 \). Define \( X_2, X_3, \ldots \) as follows: for \( n \) odd and \( n \geq 1 \), let \( X_{n+1} \) be the intersection of \( AB \) with the perpendicular to \( X_{n-1} X_n \) through \( X_n \); for \( n \) even and \( n \geq 2 \), let \( X_{n+1} \) be the intersection of \( AC \) with the perpendicular to \( X_{n-1} X_n \) through \( X_n \). Find \( BX_1 + X_1X_2 + X_2X_3 + \ldots \)

Answer: \frac{1106}{5}


A stacking of circles in the plane consists of a base, or some number of unit circles centered on the $x$-axis in a row without overlap or gaps, and circles above the $x$-axis that must be tangent to two circles below them (so that if the ends of the base were secured and gravity were applied from below, then nothing would move). How many stackings of circles in the plane have 4 circles in the base?

Answer: 14


A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid?

Answer: 139.5


For a given increasing function \( f: \mathbb{R} \rightarrow \mathbb{R} \), define the function \( g(x, y) \) as follows:

\[ g(x, y) = \frac{f(x+y) - f(x)}{f(x) - f(x-y)}, \quad x \in \mathbb{R}, \; y > 0. \]

Suppose that for \( x=0 \) and all \( y > 0 \), and for \( x \neq 0 \) and all \( y \in (0, |x|] \), the following inequality holds:

\[ 2^{-1} < g(x, y) < 2. \]

Prove that for all \( x \in \mathbb{R} \) and \( y > 0 \), the following inequality holds:

\[ 14^{-1} < g(x, y) < 14. \]

Answer: \frac{1}{14}<(x,y)<14


Show that \(\sin^n(2x) + (\sin^n(x) - \cos^n(x))^2 \leq 1\).

Answer: \sin^n(2x)+(\sin^n(x)-\cos^n(x))^2\le1


What is the distance from Boguli to Bolifoyn?

Answer: 10


A parabola is inscribed in an equilateral triangle \( ABC \) of side length 1 such that \( AC \) and \( BC \) are tangent to the parabola at \( A \) and \( B \), respectively. Find the area between \( AB \) and the parabola.

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


What is the maximum number of checkers that can be placed on an $8 \times 8$ board so that each one is being attacked?

Answer: 32


What is the probability of having  $2$  adjacent white balls or  $2$  adjacent blue balls in a random arrangement of  $3$  red,  $2$  white and  $2$  blue balls?

Answer: $\dfrac{10}{21}$


Sector $OAB$ is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form. [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
draw(Arc(origin,3,90,180));
draw((-3,0)--(origin)--(0,3));
real x = 3/(1 + sqrt(2));
draw(Circle((-x,x),x)); label("$B$",(0,3),N); label("$A$",(-3,0),W);
label("$O$",(0,0),SE); label("3 cm",(0,0)--(-3,0),S);
[/asy]

Answer: 3\sqrt{2}-3


The sequence \( a_n \) is defined as follows: \( a_1 = 1 \), \( a_{n+1} = a_n + \frac{1}{a_n} \) for \( n \geq 1 \). Prove that \( a_{100} > 14 \).

Answer: a_{100}>14


In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$

Answer: \textbf{(B)}154


If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth.

Answer: 0.16


Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their original heights?

Answer: 5:4


Answer: \frac{\pi}{2}


Given the set \( A = \{0, 1, 2, \cdots, 9\} \), and a collection of non-empty subsets \( B_1, B_2, \cdots, B_k \) of \( A \) such that for \( i \neq j \), \( B_i \cap B_j \) has at most two elements, determine the maximum value of \( k \).

Answer: 175


Answer: \sqrt[3]{112}


In certain cells of a rectangular board of size 101 by 99, there is a turtle in each cell. Every minute, each turtle simultaneously crawls to an adjacent cell of the board, sharing a side with the one they are currently in. Each subsequent move is made perpendicular to the previous one: if the previous move was horizontal - left or right, then the next one will be vertical - up or down, and vice versa. What is the maximum number of turtles that can move indefinitely on the board such that at each moment no cell contains more than one turtle?

Answer: 9800


A laptop is originally priced at $\$1200$. It is on sale for $15\%$ off. John applies two additional coupons: one gives $10\%$ off the discounted price, and another gives $5\%$ off the subsequent price. What single percent discount would give the same final price as these three successive discounts?

Answer: 27.325\%


The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$.  The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s.  Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\,$, where $L \in \{A, B, C\}$, and $P_n\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\overline{P_n L}$.  Given that $P_7 = (14,92)\,$, what is $k + m\,$?

Answer: 344


Consider the integer\[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$.

Answer: 342