Suppose (4-4): Coordinate Systems and Parametric Equations

In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the line $l$ is $\rho \sin (\theta - \frac{\mathrm{\pi }}{4})=\sqrt{2}$.

(I) Find the ordinary equation of $C$ and the slope angle of the line $l$;

(II) Let point $P(0,2)$, and $l$ intersects $C$ at points $A$ and $B$. Find $|PA|+|PB|$.

Answer: \frac{18\sqrt{2}}{5}


Find the largest natural number $k$ such that $3^k$ divides $2^{3^m} + 1$, where $m$ is any natural number.

Answer: 2


In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.)

Answer: 6


A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear?

Answer: 12


Is it possible to color 15 segments, as shown in the picture, in three colors so that no two segments of the same color share a common endpoint?

Answer: No


At vertex \( A \) of a triangle \( ABC \) inscribed in a circle, a tangent is drawn. A line parallel to side \( BC \) intersects the other two sides of the triangle at points \( M \) and \( N \), and the circle at points \( P \) and \( Q \), and the tangent at point \( S \). Prove that \( SM \cdot SN = SP \cdot SQ \).

Answer: SM\cdotSN=SP\cdotSQ


Given a triangular pyramid where two of the three lateral faces are isosceles right triangles and the third face is an equilateral triangle with a side length of 1, calculate the volume of this triangular pyramid.

Answer: \frac{\sqrt{3}}{12}


Find all values of \( n \in \mathbf{N} \) that have the following property: the fifth power of the sum of the digits of the decimal representation of the number \( n \) is equal to \( n^2 \).

Answer: 243


In a certain Italian city, there are 20 mafia clans, and it is known that each clan is in conflict with at least 14 other clans. Will there always be 4 clans that are mutually in conflict with each other?

Answer: Yes


With a monthly income of 30,000 rubles, the monthly essential expenses are as follows: 15,000 rubles for food, 5,000 rubles for utilities, and 2,500 rubles for phone, internet, and transportation. Assuming current savings of 10,000 rubles, how many months will it take to purchase a new television costing 25,000 rubles without using borrowed funds?

Answer: 2


For which positive integers \( n \) is \( n^3 - 18n^2 + 115n - 391 \) a cube?

Answer: 7,11,12,25


Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF=12$. Find the radius of the incircle of $\triangle DEF$.

Answer: 6(\sqrt{3}-1)


6.3 "If $r$ and $R$ are the radii of the incircle and circumcircle of a triangle, respectively, then $R \geq 2r$." A stronger form of this inequality is: If $r$ and $R$ are the radii of the incircle and circumcircle of a triangle, respectively, and $d$ is the distance between the incenter and the circumcenter, then $d^{2} = R^{2} - 2Rr$. Please explain the relationship between these two statements and provide a proof.

Answer: R\ge2r


On a shelf, there are 4 different comic books, 5 different fairy tale books, and 3 different story books, all lined up in a row. If the fairy tale books cannot be separated from each other, and the comic books also cannot be separated from each other, how many different arrangements are there?

Answer: 345600


Let \( P \) be the apex of a cone, and \( A, B, C \) be three points on the circular base, such that \( \angle ABC = 90^\circ \). \( M \) is the midpoint of \( AP \). Given \( AB = 1 \), \( AC = 2 \), \( AP = \sqrt{2} \), find the size of the dihedral angle \( M-BC-A \).

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


Given that the [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of $3x - 4$ to $y + 15$ is constant, and $y = 3$ when $x = 2$, then, when $y = 12$, $x$ equals:
$\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 8$

Answer: (B)\\frac{7}{3}


Answer: 49


Answer: 3


A, B, and C are guessing a two-digit number.

A says: The number has an even number of factors, and it is greater than 50.
B says: The number is odd, and it is greater than 60.
C says: The number is even, and it is greater than 70.

If each of them is only half correct, what is the number?

Answer: 64


What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, -1, 3, 5, 20\}$, where one of the numbers must be negative?

Answer: -0.05


Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\frac{2\pi}{3}$, and the circle's equation is $\rho=2\cos (\theta+\frac{\pi}{3})$:
(1) Find the parametric equation of line $l$;
(2) Let line $l$ intersect the circle at points $M$ and $N$, find the value of $|PM|\cdot|PN|$.

Answer: 6+2\sqrt{3}


Let $[x]$ denote the greatest integer less than or equal to the real number $x$, and let $\{x\}=x-[x]$. Find the sum of the squares of all natural numbers \( m \) that satisfy
$$
[(2 m+1)\{\sqrt{2 m+1}\}]=m
$$

Answer: 0


An engineer invested $\$15,\!000$ in a nine-month savings certificate that paid a simple annual interest rate of $9\%$. After nine months, she invested the total value of her investment in another nine-month certificate. After another nine months, the investment was worth $\$17,\!218.50$. If the annual interest rate of the second certificate is $s\%,$ what is $s$?

Answer: 10


A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sphere. Determine the probability that $Q$ lies inside one of the nine small spheres.

Answer: \frac{1}{3}


On the meadow, 25 dwarves gathered. It is known that: 1) each dwarf who wore a hat also wore shoes; 2) 12 dwarves came without hats; 3) 5 dwarves came barefoot. Which group of dwarves is larger and by how much: those who came wearing shoes but without a hat, or those who wore a hat?

Answer: 6


Do there exist real numbers $\alpha_{0}, \alpha_{1}, \ldots, \alpha_{2018}$, such that:
\[
\begin{cases}
\sum_{k=0}^{2018} \frac{\sin \alpha_{k}}{2^{k}} = 0 \\
\sum_{k=0}^{2018} \frac{\sin \left(\alpha_{k}+1\right)}{2^{k}} = 0
\end{cases}
\] ?

Answer: No


Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$.

Answer: 462


Consider a decreasing arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. If $a\_3a\_5=63$ and $a\_2+{a}\_{6} =16$,

(1) Find the general term formula of the sequence.
(2) For what value of $n$ does $S\_n$ reach its maximum value? Also, find the maximum value.
(3) Calculate $|a\_1|+|a\_2|+|a\_3|+…+|a\_n|$.

Answer: 66


In the quadratic equation $3x^{2}-6x-7=0$, the coefficient of the quadratic term is ____ and the constant term is ____.

Answer: -7


Find all real solutions to
\[\frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6}.\]Enter all solutions, separated by commas.

Answer: 7,-2


Let \( a_{1}, a_{2}, \ldots, a_{10} \) be natural numbers such that \( a_{1}<a_{2}<\ldots<a_{10} \). Let \( b_{k} \) be the greatest divisor of \( a_{k} \) that is less than \( a_{k} \). It turns out that \( b_{1}>b_{2}>\ldots>b_{10} \).

Prove that \( a_{10}>500 \).

Answer: a_{10}>500


Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) \((a>b>0)\) with eccentricity \(\frac{1}{2}\), where \(F_{1}\) and \(F_{2}\) are the left and right foci respectively, a line passing through \(F_{2}\) intersects the ellipse at points \(A\) and \(B\). If the maximum area of \(\triangle F_{1}AB\) is 6, find the equation of the ellipse.

Answer: \frac{x^2}{8}+\frac{y^2}{6}=1


Lines  $a_1, b_1, c_1$  pass through the vertices  $A, B, C$  respectively of a triange  $ABC$ ;  $a_2, b_2, c_2$  are the reflections of  $a_1, b_1, c_1$  about the corresponding bisectors of  $ABC$ ;  $A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1$ , and  $A_2, B_2, C_2$  are defined similarly. Prove that the triangles  $A_1B_1C_1$  and  $A_2B_2C_2$  have the same ratios of the area and circumradius (i.e.  $\frac{S_1}{R_1} = \frac{S_2}{R_2}$ , where  $S_i = S(\triangle A_iB_iC_i)$ ,  $R_i = R(\triangle A_iB_iC_i)$ )

Answer: \frac{S_1}{R_1}=\frac{S_2}{R_2}


On the leg \( AC \) of the right triangle \( ABC \) with hypotenuse \( AB \), a point \( P \) is marked. Point \( D \) is the foot of the perpendicular dropped from vertex \( A \) to line \( BP \), and point \( E \) is the foot of the perpendicular dropped from point \( P \) to side \( AB \). On the plane, a point \( T \) is chosen such that lines \( TA \) and \( TP \) are tangents to the circumcircle of triangle \( PAB \). The point \( O \) is the center of this circumcircle. The perpendicular dropped from point \( T \) to line \( DE \) intersects line \( BC \) at point \( Q \), and the line passing through point \( C \) parallel to \( OQ \) intersects segment \( BO \) at point \( K \). Find the angle \( \angle OKE \).

Answer: 90


How many ways are there to arrange the letters  $A,A,A,H,H$  in a row so that the sequence  $HA$  appears at least once?

*Author: Ray Li*

Answer: 9


Find all functions \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) such that
\[
f(-f(x) - f(y)) = 1 - x - y \quad \text{for all } x, y \in \mathbf{Z}.
\]

Answer: f(x)=x-1


Given the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) such that
\[
\begin{array}{l}
a_{1} = -1, \quad b_{1} = 2, \\
a_{n+1} = -b_{n}, \quad b_{n+1} = 2a_{n} - 3b_{n} \quad (n \in \mathbb{Z}_{+}).
\end{array}
\]
Find the value of \( b_{2015} + b_{2016} \).

Answer: -3\times2^{2015}


For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{2940}$ terminate?

Answer: 13


Given an isosceles triangle \( ABC \) with \( |AB| = |BC| \). \( AD \) is the angle bisector. A perpendicular erected to \( AD \) at point \( D \) intersects the extension of \( AC \) at point \( E \). The feet of the perpendiculars dropped from \( B \) and \( D \) to \( AC \) are \( M \) and \( N \) respectively. Find \( |MN| \) if \( |AE| = a \).

Answer: \frac{}{4}


The height of a right triangle, dropped onto its hypotenuse, divides the angle bisector of the acute angle in a ratio of 5:2, measured from the vertex. Find the measure of this angle.

Answer: \arccos\frac{5}{7}


When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?
$\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30$

Answer: \textbf{(B)}\10


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


The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers.

Answer: 10


The triangle  $ABC$  is right in  $A$  and  $R$  is the midpoint of the hypotenuse  $BC$  . On the major leg  $AB$  the point  $P$  is marked such that  $CP = BP$  and on the segment  $BP$  the point  $Q$  is marked such that the triangle  $PQR$  is equilateral.  If the area of triangle  $ABC$  is  $27$ , calculate the area of triangle  $PQR$  .

Answer: \frac{9}{2}


Let \( l \) and \( m \) be two intersecting lines, with points \( A \), \( B \), and \( C \) on line \( l \) such that \( AB = BC \). From points \( A \), \( B \), and \( C \), perpendiculars \( AD \), \( BE \), and \( CF \) are drawn to line \( m \), with feet at \( D \), \( E \), and \( F \) respectively. Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \).

Answer: \sqrt{6}


Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.)

Answer: 11


Let  $a_1, a_2, \ldots$  and  $b_1, b_2, \ldots$  be sequences such that  $a_ib_i - a_i - b_i = 0$  and  $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$  for all  $i \ge 1$ .  If  $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$ , then what is  $b_{6}$ ?

*Proposed by Andrew Wu*

Answer: 257


If \(\lceil{\sqrt{x}}\rceil=20\), how many possible integer values of \(x\) are there?

Answer: 39


The diagonals of quadrilateral \(ABCD\) intersect at point \(M\), and the angle between them is \(\alpha\). Let \(O_{1}, O_{2}, O_{3}, O_{4}\) be the centers of the circumcircles of triangles \(ABM, BCM, CDM\), and \(DAM\), respectively. Determine the ratio of the areas of quadrilateral \(ABCD\) and quadrilateral \(O_{1} O_{2} O_{3} O_{4}\).

Answer: 2\sin^2\alpha


Three not necessarily integer numbers are given. If each of these numbers is increased by 1, the product of the numbers also increases by 1. If each of these numbers is increased by 2, their product also increases by 2. Find these numbers.

Answer: -1


12 people sat around a round table; some of them are knights (who always tell the truth), and others are liars (who always lie). Then each of them said, "There is a liar among my neighbors." What is the maximum number of people sitting at the table who can say, "There is a knight among my neighbors?"

Answer: 8


Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads  $\textstyle\frac{1}{2}$  of the time and tails  $\textstyle\frac{1}{2}$  of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails.

Bob flips the VNC an infinite number of times. On the  $n$ th flip, Bob bets  $2^{-n}$  dollars that the VNC will show heads (so if the second flip shows heads, Bob wins  $\$ 0.25 $, and if the third flip shows tails, Bob loses $ \ $0.125$ ).

Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

*Proposed by Lewis Chen*

Answer: 34783


For the function \( f(x) \), if \( f(x) = x \), then \( x \) is called the "fixed point" of \( f(x) \). If \( f(f(x)) = x \), then \( x \) is called the "stable point" of \( f(x) \). The sets of "fixed points" and "stable points" of the function \( f(x) \) are denoted as \( A \) and \( B \), respectively, i.e., \( A = \{x \mid f(x) = x\} \) and \( B = \{x \mid f(f(x)) = x\} \).

(1) Prove that \( A \subseteq B \).

(2) If \( f(x) = a x^2 - 1 \) with \( a \in \mathbb{R} \) and \( x \in \mathbb{R} \), and \( A = B \neq \varnothing \), determine the range of the real number \( a \).

Answer: [-\frac{1}{4},\frac{3}{4}]


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

Answer: \frac{16\sqrt{3}}{9}


Let $a, b, c$ be the sides of a triangle. Prove the inequality $a^{3} + b^{3} + 3abc > c^{3}$.

Answer: ^3+b^3+3abc>^3


The sequence of positive integers $\{f(n)\}$ satisfies: $f(2)=15$ and
$$
\begin{aligned}
2 +\frac{4}{f(n)+1}<\frac{f(n)}{f(n)-4 n+2}+\frac{f(n)}{f(n+1)-4 n-2}<2+\frac{4}{f(n)-1},
\end{aligned}
$$
Calculate $S(n)= \frac{1}{f(1)}+\frac{1}{f(2)}+\cdots+\frac{1}{f(n)}.$

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


Given the sets \( A = \left\{x \left\lvert\, \frac{1}{4} \leqslant 2^{x} \leqslant \frac{1}{2}\right., x \in \mathbf{R}\right\} \) and \( B = \left\{x \mid x^{2} - 2 t x + 1 \leqslant 0\right\} \), if \( A \cap B = A \), determine the range of the real number \( t \).

Answer: (-\infty,-\frac{5}{4}]


How many six-digit numbers of the form ababab are there, which are the product of six different prime numbers?

Answer: 12


Given the two-variable function
$$
f(a, b)=\max _{x \in[-1,1]}\left\{\left|x^{2}-a x-b\right|\right\},
$$
find the minimum value of \( f(a, b) \).

Answer: \frac{1}{2}


How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 5, 6, 6, 7, 7\}$, with no digit used more times than it appears in the set?

Answer: 42


I am preparing for a French exam, which includes recalling vocabulary. There are 800 words in total on the exam, and the exam grade is based on the percentage of these words that I recall correctly. If I randomly guess the words I do not learn, my guessing has a 5% chance of being correct. What is the least number of words I should learn to ensure I score at least $90\%$ on the exam?

Answer: 716


Let $O$ be the origin, and let $(a,b,c)$ be a fixed point.  A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$  Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$  Find
\[\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.\]

Answer: 2


How many integer solutions does the equation \(\sqrt{x} + \sqrt{y} = \sqrt{1960}\) have?

Answer: 15


Given that \( f(x) \) is a polynomial of degree \( n \), and \( f(k) = \frac{k}{k+1} \) for \( k = 0,1, \ldots, n \), find \( f(n+1) \).

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


Given $a \in \mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.

1. Find the value of $a$.
2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$.

Answer: 50


The fraction  $\frac1{10}$  can be expressed as the sum of two unit fraction in many ways, for example,  $\frac1{30}+\frac1{15}$  and  $\frac1{60}+\frac1{12}$ .

Find the number of ways that  $\frac1{2007}$  can be expressed as the sum of two distinct positive unit fractions.

Answer: 7


**p1.** Find all real solutions of the system  $$ x^5 + y^5 = 1 $$    $$ x^6 + y^6 = 1 $$  **p2.** The centers of three circles of the radius  $R$  are located in the vertexes of equilateral triangle. The length of the sides of the triangle is  $a$  and  $\frac{a}{2}< R < a$ . Find the distances between the intersection points of the
circles, which are outside of the triangle.**p3.** Prove that no positive integer power of  $2$  ends with four equal digits.**p4.** A circle is divided in  $10$  sectors.  $90$  coins are located in these sectors,  $9$  coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly $ 2004$  moves?**p5.** Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer.**p6.** A troll tried to spoil a white and red  $8\times 8$  chessboard. The area of every square of the chessboard is one square foot. He randomly painted  $1.5\%$  of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer.

PS. You should use hide for answers.

Answer: (1,0)or(0,1)


Triangle  $ABC$  has  $AB=25$ ,  $AC=29$ , and  $BC=36$ .  Additionally,  $\Omega$  and  $\omega$  are the circumcircle and incircle of  $\triangle ABC$ .  Point  $D$  is situated on  $\Omega$  such that  $AD$  is a diameter of  $\Omega$ , and line  $AD$  intersects  $\omega$  in two distinct points  $X$  and  $Y$ .  Compute  $XY^2$ .

*Proposed by David Altizio*

Answer: 252


Mia buys 10 pencils and 5 erasers for a total of $2.00. Both a pencil and an eraser cost at least 3 cents each, and a pencil costs more than an eraser. Determine the total cost, in cents, of one pencil and one eraser.

Answer: 22


A pentagon \(ABCD E\) is inscribed in a circle of radius 10 such that \(BC\) is parallel to \(AD\) and \(AD\) intersects \(CE\) at \(M\). The tangents to this circle at \(B\) and \(E\) meet the extension of \(DA\) at a common point \(P\). Suppose \(PB = PE = 24\) and \(\angle BPD = 30^{\circ}\). Find \(BM\).

Answer: 13


Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \frac{a^{2}+1}{ab} - 2) \cdot c + \frac{\sqrt{2}}{c - 1}$.

Answer: 4 + 2\sqrt{2}


Given that $|\cos\theta|= \frac {1}{5}$ and $\frac {5\pi}{2}<\theta<3\pi$, find the value of $\sin \frac {\theta}{2}$.

Answer: -\frac{\sqrt{15}}{5}


Let \( \triangle ABC \) be a triangle with \( AB = 5 \), \( BC = 8 \), and \( CA = 7 \). Let \( \Gamma \) be a circle internally tangent to the circumcircle of \( \triangle ABC \) at \( A \) and also tangent to segment \( BC \). \( \Gamma \) intersects \( AB \) and \( AC \) at points \( D \) and \( E \), respectively. Determine the length of segment \( DE \).

Answer: \frac{40}{9}


Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: "Multiply my number by two and subtract two from the result," "Multiply my number by three and then add four," and lastly, "Add seven to my number!" The iPhone knows that initially, Vova's number was 1. How many four-digit numbers could the iPhone XXX theoretically achieve by obediently following Vova's commands?

Answer: 9000


A $50$-gon \(Q_1\) is drawn in the Cartesian plane where the sum of the \(x\)-coordinates of the \(50\) vertices equals \(150\). A constant scaling factor \(k = 1.5\) applies only to the \(x\)-coordinates of \(Q_1\). The midpoints of the sides of \(Q_1\) form a second $50$-gon, \(Q_2\), and the midpoints of the sides of \(Q_2\) form a third $50$-gon, \(Q_3\). Find the sum of the \(x\)-coordinates of the vertices of \(Q_3\).

Answer: 225


The parabola \( y^{2} = 2px \) (with \( p > 0 \)) has a focus \( F \) and a directrix \( l \). Points \( A \) and \( B \) are on the parabola and satisfy \( \angle AFB = \frac{\pi}{3} \). Let the projection of the midpoint \( M \) of segment \( AB \) onto \( l \) be \( N \). Find the maximum value of \( \frac{|MN|}{|AB|} \).

Answer: 1


Let $s(n)$ denote the sum of all odd digits of the number $n$. For example, $s(4)=0$, $s(173)=11$, and $s(1623)=4$. Calculate the value of the sum $s(1)+s(2)+s(3)+\ldots+s(321)$.

Answer: 1727


The lengths of the edges of a rectangular parallelepiped extending from one vertex are 8, 8, and 27. Divide the parallelepiped into four parts that can be assembled into a cube.

Answer: 12


The incircle of triangle \(ABC\) touches the sides \([AC]\) and \([AB]\) at \(E\) and \(F\). The lines \(BE\) and \(CF\) intersect at \(G\). Points \(R\) and \(S\) are defined such that \(BCER\) and \(BCSF\) are parallelograms.

Show that \(GR = GS\).

Answer: GR=GS


Older brother took Misha’s identical unpainted cubes and assembled a large cube from them. Then he painted some (but not all) faces of the large cube completely in red. When the paint dried, Misha disassembled the large cube and found that exactly 343 small cubes had no red faces. How many faces of the large cube did Misha’s brother paint? Justify your answer.

Answer: 3


Answer: 51


Let \(ABCD\) be a rectangle, and let \(E\) and \(F\) be points on segment \(AB\) such that \(AE = EF = FB\). If \(CE\) intersects the line \(AD\) at \(P\), and \(PF\) intersects \(BC\) at \(Q\), determine the ratio of \(BQ\) to \(CQ\).

Answer: 1/3


Inside the triangle \(ABC\), there is a point \(K\). It is known that \(AK = 1\), \(KC = \sqrt{3}\), and the angles \(\angle AKC\), \(\angle ABK\), and \(\angle KBC\) are \(120^\circ\), \(15^\circ\), and \(15^\circ\) respectively. Find \(BK\).

Answer: \frac{3\sqrt{2}+\sqrt{6}}{2}


Given real numbers $x$ and $y$ satisfying $x^{2}+4y^{2}\leqslant 4$, find the maximum value of $|x+2y-4|+|3-x-y|$.

Answer: 12


Xiao Zhang and Xiao Zhao can only take on the first two roles, while the other three can take on any of the four roles, calculate the total number of different selection schemes.

Answer: 48


A circle passes through the midpoints of the hypotenuse $AB$ and the leg $BC$ of the right triangle $ABC$ and touches the leg $AC$. In what ratio does the point of tangency divide the leg $AC$?

Answer: 1 : 3


Estimate the range of the submissions for this problem. Your answer must be between  $[0, 1000]$ . An estimate  $E$  earns  $\frac{2}{1+0.05|A-E|}$  points, where  $A$  is the actual answer.

*2022 CCA Math Bonanza Lightning Round 5.2*

Answer: 500


Given real numbers \\(x\\) and \\(y\\) satisfy the equation \\((x-3)^{2}+y^{2}=9\\), find the minimum value of \\(-2y-3x\\) \_\_\_\_\_\_.

Answer: -3\sqrt{13}-9


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

Answer: 139


A tetrahedron $PQRS$ has edges of lengths $8, 14, 19, 28, 37,$ and $42$ units. If the length of edge $PQ$ is $42$, determine the length of edge $RS$.

Answer: 14


Let  $n$  be an given positive integer, the set  $S=\{1,2,\cdots,n\}$ .For any nonempty set  $A$  and  $B$ , find the minimum of  $|A\Delta S|+|B\Delta S|+|C\Delta S|,$  where  $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$

Answer: n+1


Given that there are three points on the coordinate plane: \( O(0,0) \), \( A(12,2) \), and \( B(0,8) \). A reflection of \( \triangle OAB \) along the straight line \( y = 6 \) creates \( \triangle PQR \). If the overlapped area of \( \triangle OAB \) and \( \triangle PQR \) is \( m \) square units, find the value of \( m \).

Answer: 8


Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than  $180^{\circ}$ . Let  $C$  be a convex polygon with  $s$  sides. The interior region of  $C$  is the union of  $q$  quadrilaterals, none of whose interiors overlap each other.  $b$  of these quadrilaterals are boomerangs. Show that  $q\ge b+\frac{s-2}{2}$ .

Answer: q\ge\frac{-2}{2}


Answer: 1398100


In a village of 2011 dwarfs, each dwarf is either wearing a red hat or a blue hat. Dwarfs with red hats always tell the truth, while dwarfs with blue hats can lie. Additionally, they can change the color of their hats at any time. One day, each dwarf encountered every other dwarf exactly once, and each one claimed that the other was wearing a blue hat. What is the minimum number of times the dwarfs changed the color of their hats that day?

Answer: 2009


Let a constant $a$ make the equation $\sin x + \sqrt{3}\cos x = a$ have exactly three different solutions $x_{1}$, $x_{2}$, $x_{3}$ in the closed interval $\left[0,2\pi \right]$. The set of real numbers for $a$ is ____.

Answer: \{\sqrt{3}\}


In triangle $DEF$, $DE=130$, $DF=110$, and $EF=140$. The angle bisector of angle $D$ intersects $\overline{EF}$ at point $T$, and the angle bisector of angle $E$ intersects $\overline{DF}$ at point $S$. Let $R$ and $U$ be the feet of the perpendiculars from $F$ to $\overline{ES}$ and $\overline{DT}$, respectively. Find $RU$.

Answer: 60


If the sum of the digits of a natural number \( a \) is equal to 7, then \( a \) is called a "lucky number." Arrange all "lucky numbers" in ascending order \( a_{1}, a_{2}, a_{3}, \cdots \). If \( a_{n} = 2005 \), then \( a_{5n} = \) ______

Answer: 52000


A circle with radius \(R\) passes through the vertex \(A\) of an isosceles triangle \(ABC\), is tangent to the base \(BC\) at point \(B\), and intersects the lateral side \(AC\) at point \(D\). Find the lateral side \(AB\), given that \(\frac{AD}{DX} = k\).

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


In a course conducted by Professor Jones, each student is on average absent for one day out of a 40-day course. What is the probability that out of any two randomly selected students, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth.

Answer: 4.9\%


Let  $S$  be a subset of  $\{1, 2, 3, \cdots, 1989 \}$  in which no two members differ by exactly  $4$  or by exactly  $7$ . What is the largest number of elements  $S$  can have?

Answer: 905


Calculate the volume in cubic centimeters of a truncated cone formed by cutting a smaller cone from a larger cone. The larger cone has a diameter of 8 cm at the base and a height of 10 cm. The smaller cone, which is cut from the top, has a diameter of 4 cm and a height of 4 cm. Express your answer in terms of \(\pi\).

Answer: 48\pi


Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \).

Answer: 999


The ancient Greeks discovered that some natural numbers \( n \) have the remarkable property that the sum of the divisors of \( n \) is equal to the number \( n \) itself (the number itself is not considered a divisor). For example, \( n = 28 \) gives:

\[ 1 + 2 + 4 + 7 + 14 = 28 \]

Such numbers were called "perfect". Using the arithmetic function \( \sigma(n) \), which denotes the sum of all positive divisors of the number \( n \) (including the number \( n \) itself), one can write that a number \( n \) is perfect if \( \sigma(n) = 2n \). Perfect numbers are very rare. The first five of them are \( 6, 28, 496, 8128, 33550336 \). In 1976, only 24 perfect numbers were known, the largest being \( 2^{19936} \cdot (2^{19937} - 1) \), containing about 6000 digits.

In the eighteenth century, Euler proved that every even perfect number \( m \) can be expressed as \( m = 2^{n-1}(2^n - 1) \), where \( 2^n - 1 \) is a prime number.

Using these premises, in this problem, it is proposed to find all perfect numbers \( n \) for which \( \sigma[\sigma(n)] \) is also a perfect number.

Answer: 6


A pair of numbers is called magical if their sum is divisible by 7. What is the maximum number of magical pairs of adjacent numbers that can be obtained by writing all the numbers from 1 to 30 in a row in some order?

Answer: 26


The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$.

Answer: \frac{1}{2}


A tourist was returning from vacation by bicycle. On the first section of the route, which was 246 km, he traveled an average of 15 km less per day than on the last section of the route, which was 276 km. He arrived home exactly on the last day of his vacation. It is also known that it took him one more day than half the number of days left from that point until the end of the vacation to complete the first section. How many days before the end of his vacation did the tourist start his journey home?

Answer: 4


Given the parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$ and the intersection point $E$ of its directrix with the $x$-axis, let line $l$ pass through $E$.
$(1)$ If line $l$ is tangent to the parabola at point $M$, then $\angle EMF=$____.
$(2)$ Let $p=6$. If line $l$ intersects the parabola at points $A$ and $B$, and $AB\bot BF$, then $|AF|-|BF|=$____.

Answer: 12


In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\rho\cos(\theta+ \frac{\pi}{4})= \frac{\sqrt{2}}{2}$, and the parametric equation of curve $C$ is $\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$ (where $\theta$ is the parameter).

(1) Find the Cartesian equation of line $l$ and the ordinary equation of curve $C$;
(2) Curve $C$ intersects the x-axis at points $A$ and $B$, with the x-coordinate of point $A$ being less than that of point $B$. Let $P$ be a moving point on line $l$. Find the minimum value of the perimeter of $\triangle PAB$.

Answer: \sqrt{34}+2


In a larger geoboard grid of $7 \times 7$, points are evenly spaced vertically and horizontally. Points $A$ and $B$ are at $(3,3)$ and $(5,3)$ respectively. How many of the remaining points in the grid will allow for triangle $ABC$ to be isosceles?

Answer: 10


Color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly 2 black squares. How many different ways are there to color the chessboard?

Answer: 90


Let \( P \) be the apex of a cone, and \( A, B, C \) are three points on the circumference of its base, such that \(\angle ABC = 90^\circ\). Let \( M \) be the midpoint of \( AP \). Given \( AB = 1 \), \( AC = 2 \), and \( AP = \sqrt{2} \), determine the dihedral angle between planes \( M-BC-A \).

Answer: \arctan(\frac{2}{3})


Let a positive integer \( M \) in base \( n \) be represented such that all its digits are distinct, and each digit (except the leftmost one) differs by \(\pm 1\) from some digit to its left. Find the number of all such positive integers \( M \) in terms of \( n \).

Answer: 2^{n+1}-2n-2


Given a sequence $\{a_n\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\geqslant 2, n\in \mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then $S_{40}=$_______.

Answer: 440


[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(rightanglemark(D, C, B)); MP("A",(0,0),W); MP("B",(7,4.2),N); MP("C",(10,0),E); MP("D",(3,-5),S); MP("E",(3,0),N); MP("F",(7,0),S); [/asy]
In the figure $ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $\overline{AC}$, and $\overline{DE}$ and $\overline{BF}$ are perpendicual to $\overline{AC}$. If $AE=3, DE=5,$ and $CE=7$, then $BF=$
$\text{(A) } 3.6\quad \text{(B) } 4\quad \text{(C) } 4.2\quad \text{(D) } 4.5\quad \text{(E) } 5$

Answer: 4.2


Let \( a, b, c, d \) be complex numbers each with a modulus of 1. Prove that there exists a complex number \( z \) with a modulus of 1, such that
$$
\left|a z^{3}+b z^{2}+c z+d\right| \geqslant \sqrt{6} .
$$

Answer: \sqrt{6}


Given the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) where \(a, b \in \mathbb{R}^{+}\), with the semi-focal distance denoted by \(c\), and \(b^{2} = a c\). Let \(P\) and \(Q\) be any two points on the hyperbola, and let \(M\) be the midpoint of \(PQ\). When the slopes \(k_{PQ}\) of \(PQ\) and \(k_{OM}\) of \(OM\) exist, find the value of \(k_{PQ} \cdot k_{OM}\).

Answer: \frac{1+\sqrt{5}}{2}


At the CleverCat Academy, cats can learn to do three tricks: jump, play dead, and fetch. Of the cats at the academy:
- 60 cats can jump
- 35 cats can play dead
- 40 cats can fetch
- 20 cats can jump and play dead
- 15 cats can play dead and fetch
- 22 cats can jump and fetch
- 10 cats can do all three tricks
- 12 cats can do none of the tricks

How many cats are in the academy?

Answer: 100


Let \( k \) be a positive integer such that for any positive numbers \( a, b, c \) satisfying the inequality \( k(ab + bc + ca) > 5(a^2 + b^2 + c^2) \), there must exist a triangle with side lengths \( a, b, \) and \( c \). Find the minimum value of \( k \).

Answer: 6


Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube?

Answer: 294


In $\triangle ABC$, if $\angle B=30^{\circ}$, $AB=2 \sqrt {3}$, $AC=2$, find the area of $\triangle ABC$.

Answer: 2 \sqrt {3}


The circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $A$. The segment $O_{2}A$ intersects the circle $\omega_{1}$ again at point $K$, and the segment $O_{1}A$ intersects the circle $\omega_{2}$ again at point $L$. A line passing through point $A$ parallel to $KL$ intersects the circles $\omega_{1}$ and $\omega_{2}$ again at points $C$ and $D$ respectively. The segments $CK$ and $DL$ intersect at point $N$. Find the angle between the lines $O_{1}A$ and $O_{2}N$.

Answer: 90


Let  $x_1,x_2, \cdots,x_n$  and  $y_1,y_2, \cdots ,y_n$  be arbitrary real numbers satisfying  $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$ . Prove that
\[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\]
and find all cases of equality.

Answer: (x_1y_2-x_2y_1)^2\le2|1-\sum_{k=1}^nx_ky_k|


Determine the number of ordered pairs of positive integers \((a, b)\) satisfying the equation
\[ 100(a + b) = ab - 100. \]

Answer: 18


Does there exist a number \(\alpha\) (\(0 < \alpha < 1\)) such that there is an infinite sequence of positive numbers \(\{a_{n}\}\) satisfying
\[ 1 + a_{n+1} \leqslant a_{n} + \frac{\alpha}{n} \cdot a_{n}, \quad n = 1, 2, \cdots \]

Answer: No


Points \(D\) and \(E\) lie on sides \(AB\) and \(AC\) of \(\triangle ABC\) with area 1, respectively. Line segments \(BE\) and \(CD\) intersect at point \(P\). Points \(D\) and \(E\) move on \(AB\) and \(AC\) respectively, but satisfy the condition that the area of quadrilateral \(BCED\) is twice the area of \(\triangle PBC\). Find the maximum area of \(\triangle PDE\).

Answer: 5\sqrt{2}-7


For a positive integer \( n > 0 \), \( 3^n - 2^n \) is a power of a prime number. Show that \( n \) is prime.

Answer: n


A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set.

Answer: 2\sqrt{2}


Consider points arranged in a square of 20 columns and 20 rows, each colored either red or blue. Every time two points of the same color are adjacent in any row or column, they are connected by a segment of that same color. Adjacent points of different colors are connected by black segments. Among the points, there are 219 red ones, 39 of which are on the boundary of the square but none in the corners. There are also 237 black segments. How many blue segments are there?

Answer: 223


The base of the pyramid is a rhombus with diagonals \( d_{1} \) and \( d_{2} \). The height of the pyramid passes through the vertex of the acute angle of the rhombus. The area of the diagonal cross-section made through the smaller diagonal is \( Q \). Calculate the volume of the pyramid given that \( d_{1} > d_{2} \).

Answer: \frac{d_1}{12}\sqrt{16Q^2-d_1^2d_2^2}


Given a positive integer \( k \), find the largest positive integer \( m \) such that if no more than \( m \) of any set of \( 3k \) points in the plane are collinear, then the points can be divided into \( k \) sets of three points each, where the points in each set form the vertices of a triangle.

Answer: 2k


Determine the largest positive integer  $n$  which cannot be written as the sum of three numbers bigger than  $1$  which are pairwise coprime.

Answer: 17


A workshop produces products of types $A$ and $B$. Producing one unit of product $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while producing one unit of product $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling one unit of product $A$ is 80 thousand rubles, and for product $B$, it's 100 thousand rubles. The daily steel reserve is 700 kg, and the reserve of non-ferrous metals is 642 kg. How many units of products $A$ and $B$ should be produced per shift to maximize profit without exceeding the available resources? Record the maximum profit (in thousands of rubles) that can be obtained under these conditions as an integer without indicating the unit.

Answer: 2180


What is the last digit of the sum $4.1 \times 1 + 2 \times 2 + 3 \times 3 + \ldots + 2011 \times 2011 + 2012 \times 2012$?

Answer: 0


Determine the value of $x$ for which $(2010 + x)^2 = 4x^2$.

Answer: -670


A pedestrian left point $A$ for point $B$, and with some delay, a second pedestrian left. When the first pedestrian had covered half the distance, the second pedestrian had walked 15 km, and when the second pedestrian had covered half the distance, the first pedestrian had walked 24 km. The pedestrians arrived at point $B$ simultaneously. What is the distance between points $A$ and $B$?

Answer: 40


**Q.** Consider in the plane  $n>3$  different points. These have the properties, that all  $3$  points can be included in a triangle with maximum area  $1$ . Prove that all the  $n>3$  points can be included in a triangle with maximum area  $4$ .

*Proposed by TuZo*

Answer: 4


Three lines are drawn parallel to each of the three sides of  $\triangle ABC$  so that the three lines intersect in the interior of  $ABC$ . The resulting three smaller triangles have areas  $1$ ,  $4$ , and  $9$ . Find the area of  $\triangle ABC$ .

[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }
pair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);
draw(A--B--C--cycle);
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);
filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));
filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));
filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]

Answer: 36


Given an ellipse \(C_1\) centered at the origin \(O\) with foci \(F_1\) and \(F_2\) to the left and right respectively, and a parabola \(C_2\) with vertex at \(O\) and focus at \(F_2\). Let \(A\) be the intersection point of \(C_1\) and \(C_2\), where \(\angle A F_2 F_1\) is obtuse, \(\left|A F_1\right| = \frac{7}{2}\), and \(\left|A F_2\right| = \frac{5}{2}\). A line \(l\) passing through \(F_2\) but not perpendicular to the x-axis intersects \(C_1\) and \(C_2\) at points \(B, C, D,\) and \(E\) in sequence, where \(B\) and \(E\) are on \(C_1\), and \(C\) and \(D\) are on \(C_2\). Let \(G\) be the midpoint of segment \(CD\) and \(H\) be the midpoint of segment \(BE\). Find the value of \(\frac{|BE|\left|G F_2\right|}{|CD|\left|H F_2\right|}\).

Answer: 3


Given α ∈ (0,π), β ∈ (-π/2,π/2) satisfies sin(α + π/3) = 1/3, cos(β - π/6) = √6/6, determine sin(α + 2β).

Answer: \frac{2\sqrt{10}-2}{9}


Line segment $AC$ and line segment $BD$ intersect at point $P$, and randomly chosen circular arc passing through points $A$, $B$, $C$, and $D$ intersects line $l$ at points $R$ and $S$. Prove that:
$$
\frac{1}{OP} + \frac{1}{OQ} = \frac{1}{OR} + \frac{1}{OS}.
$$

Answer: \frac{1}{OP}+\frac{1}{OQ}=\frac{1}{OR}+\frac{1}{OS}


There are 6 types of plants: shooter, dual shooter, triple shooter, ice shooter, bidirectional shooter, and pod shooter. The sun cost for planting one of each type is 100, 200, 300, 150, 125, and 125 respectively. Feifei planted 10 plants and spent a total of 2500 suns. How many different ways $\qquad$ are there for her to plant these 10 plants? (For example, 7 triple shooters + 1 ice shooter + 1 pod shooter is a valid way.)

Answer: 8


Place the sequence $\{2n+1\}$ in parentheses sequentially, with the first parenthesis containing one number, the second two numbers, the third three numbers, the fourth four numbers, the fifth one number again, and then continuing in this cycle. Determine the sum of the numbers in the 104th parenthesis.

Answer: 2104


Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) on lines \( AB \) and \( BC \) respectively are such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) if \( \angle BAC = 17^{\circ} \).

Answer: 17


Define \( a_n = \frac{2(1 + \frac{1}{n})^{2n+1}}{(1 + \frac{1}{n})^n + (1 + \frac{1}{n})^{n+1}} \). Prove that \( a_n \) is strictly monotonic increasing.

Answer: a_n


Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the problem above. If $AB = 34$ units and $CD = 66$ units, what is the area of square $BCFE$?

Answer: 2244


Let \( p \) be a prime number.

a) Show that if \( a \equiv b (\bmod p) \), then \( a^{p} \equiv b^{p} (\bmod p^{2}) \).

b) How many integers \( n \in \{1, 2, \ldots, p^{2}\} \) are there for which the equation \( x^{p} \equiv n (\bmod p^{2}) \) has at least one solution?

Answer: p


Let \( f(x) = x^2 + 12x + 30 \). Find the largest real root of the equation \( f(f(f(x))) = 0 \).

Answer: -6 + \sqrt[8]{6}


Three pirates divided the diamonds they collected during the day: Bill and Sam got twelve each, and the rest went to John, who did not know how to count. At night, Bill stole one diamond from Sam, Sam stole one diamond from John, and John stole one diamond from Bill. As a result, the average mass of Bill's diamonds decreased by one carat, the average mass of Sam's diamonds decreased by two carats, but the average mass of John's diamonds increased by four carats. How many diamonds did John receive?

Answer: 9


A boy named Vasya wrote down the non-zero coefficients of a 9th degree polynomial \( P(x) \) in a notebook. Then, he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, continuing this process until he obtained a constant, which he also wrote down.

What is the minimum number of different numbers that he could have written down?

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

Answer: 9


Let $P(x)=x^{3}+a x^{2}+b x+2015$ be a polynomial all of whose roots are integers. Given that $P(x) \geq 0$ for all $x \geq 0$, find the sum of all possible values of $P(-1)$.

Answer: 9496


Given that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - tx - 2 = 0\), where \(\alpha < \beta\), and the function \(f(x) = \frac{4x - t}{x^2 + 1}\):

1. Find the value of \(\frac{f(\alpha) - f(\beta)}{\alpha - \beta}\).
2. Prove that for any positive numbers \(\lambda_1\) and \(\lambda_2\), \(\left| f\left( \frac{\lambda_1 \alpha + \lambda_2 \beta}{\lambda_1 + \lambda_2} \right) - f\left( \frac{\lambda_1 \beta + \lambda_2 \alpha}{\lambda_1 + \lambda_2} \right) \right| < 2|\alpha - \beta|\).

Answer: 2


A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located inside the square?

Answer: 1007


At the University of Bugelstein, you can study Languages, History, and Philosophy.
35% of students that study a language study English.
13% of all the university students study a language other than English.
No student studies more than one language.
What percentage of the university students study Languages?
A) 13%
B) 20%
C) 22%
D) 48%
E) 65%

Answer: 20


Let $\{a_n\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, given that $S_5 < S_6$, $S_6=S_7 > S_8$, then the correct conclusion(s) is/are \_\_\_\_\_\_
$(1) d < 0$
$(2) a_7=0$
$(3) S_9 > S_5$
$(4) S_6$ and $S_7$ are both the maximum value of $S_n$.

Answer: (1)(2)(4)


Let  $f: [0, 1] \to \mathbb{R}$  be a continuous strictly increasing function such that
\[ \lim_{x \to 0^+} \frac{f(x)}{x}=1. \]
(a) Prove that the sequence  $(x_n)_{n \ge 1}$  defined by
\[ x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx \]
is convergent.
(b) Find the limit of the sequence  $(y_n)_{n \ge 1}$  defined by
\[ y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right). \]

Answer: \ln2021


We are given a convex quadrilateral  $ABCD$  in the plane.
(*i*) If there exists a point  $P$  in the plane such that the areas of  $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$  are equal, what condition must be satisfied by the quadrilateral  $ABCD$ ?
(*ii*) Find (with proof) the maximum possible number of such point  $P$  which satisfies the condition in (*i*).

Answer: 1


Let \(ABCD\) be a square with side length 5, and let \(E\) be the midpoint of side \(AB\). Let \(P\) and \(Q\) be the feet of perpendiculars from \(B\) and \(D\) to \(CE\), respectively, and let \(R\) be the foot of the perpendicular from \(A\) to \(DQ\). The segments \(CE, BP, DQ\), and \(AR\) partition \(ABCD\) into five regions. What is the median of the areas of these five regions?

Answer: 5


Given that the coordinates of points $M$ and $N$ satisfy the system of inequalities $\left\{\begin{array}{l}x \geqslant 0, \\ y \geqslant 0, \\ x+2 y \leqslant 6, \\ 3 x + y \leqslant 12,\end{array}\right.$ and $\vec{a}=(1,-1)$, find the range of values for $\overrightarrow{M N} \cdot \vec{a}$.

Answer: [-7,7]


In the spring college entrance examination of Shanghai in 2011, there were 8 universities enrolling students. If exactly 3 students were admitted by 2 of these universities, the number of ways this could happen is ____.

Answer: 168


If the angles $A$ and $C$ in $\triangle ABC$ satisfy the equation $5(\cos A+\cos C)+4(\cos A \cos C+1)=0$, then $\tan \frac{A}{2} \cdot \tan \frac{C}{2} = \ \qquad$ .

Answer: 3


A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse.

Answer: 12


Given a circle $x^2 + (y-1)^2 = 1$ with its tangent line $l$, which intersects the positive x-axis at point A and the positive y-axis at point B. Determine the y-intercept of the tangent line $l$ when the distance AB is minimized.

Answer: \frac{3+\sqrt{5}}{2}


A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Answer: 506


A magician guesses the area of a convex 2008-gon \( A_{1} A_{2} \ldots A_{2008} \) located behind a screen. He announces two points on the perimeter of the polygon; the audience marks these points, draws a line through them, and tells the magician the smaller area of the two parts into which the 2008-gon is divided by this line. The magician can name a point either as a vertex or a point dividing a specified side in a specified numerical ratio. Prove that with 2006 questions, the magician can guess the area of the polygon.

Answer: 2006


Given \( a_{1} = 1 \) and \( a_{n+1} = \lambda a_{n}^{2} + 2 \left(n \in \mathbf{N}^{*}\right) \), if the sequence \(\{a_{n}\}\) is bounded, i.e., there exists a constant \( M \) such that \( a_{n} \leq M \) for all \( n \in \mathbf{N}^{*} \), find the maximum value of the real number \( \lambda \).

Answer: \frac{1}{8}


At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.

At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them away!" Find the largest possible value of $N$, given that no one lied.

Answer: 14


In the coordinate plane, a set of  $2000$  points  $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$  is called *good* if  $0\leq x_i \leq 83$ ,  $0\leq y_i \leq 83$  for  $i = 1, 2, \dots, 2000$  and  $x_i \not= x_j$  when  $i\not=j$ . Find the largest positive integer  $n$  such that, for any good set, the interior and boundary of some unit square contains exactly  $n$  of the points in the set on its interior or its boundary.

Answer: 25


Given that circles $C_{1}$ and $C_{2}$ intersect at two points, one of which has coordinates $(9,6)$, and the product of their radii is 68. If the $x$-axis and the line $y = mx$ (where \(m > 0\)) are both tangent to the two circles, find the value of $m$.

Answer: \frac{12\sqrt{221}}{49}


9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost?

Answer: 1.11


Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $a_{n+1}=\frac{(n+1) a_{n}}{2 n+a_{n}} \left(n \in \mathbb{N}_{+}\right)$, calculate the value of $\sum_{k=1}^{2017} \frac{k}{a_{k}}$.

Answer: 2^{2018}-2019


A \( 5 \mathrm{~cm} \) by \( 5 \mathrm{~cm} \) pegboard and a \( 10 \mathrm{~cm} \) by \( 10 \mathrm{~cm} \) pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in \( 1 \mathrm{~cm} \) intervals from each edge. Pegs are placed into the holes on the two main diagonals of both pegboards. The \( 5 \mathrm{~cm} \) by \( 5 \mathrm{~cm} \) pegboard is shown; it has 16 holes. The 8 shaded holes have pegs, and the 8 unshaded holes do not. How many empty holes does the \( 10 \mathrm{~cm} \) by \( 10 \mathrm{~cm} \) pegboard have?

Answer: 100


Among the five-digit numbers formed using the digits 0, 1, 2, 3, 4, how many have the first and last digits the same, and the three middle digits all different?

Answer: 240


All numbers \( a_{1}, a_{2}, \ldots, a_{100} \) are either 1 or -1, and their sum is 0. Prove that there exists an integer \( 0 \leq n \leq 75 \) such that
\[
a_{n+1}+a_{n+2}+\cdots+a_{n+25}=1
\]

Answer: 1


Prove that $\sin \theta \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right)=\frac{1}{4} \sin 3 \theta$.

Answer: \sin\theta\sin(60-\theta)\sin(60+\theta)=\frac{1}{4}\sin3\theta


On a circumference of 1956, what is the minimum number of points that must be selected such that each point has exactly one point at a distance of 1 and exactly one point at a distance of 2 (the distance between two points is measured by arc length)?

Answer: 1304


Let  $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$  for real number  $ p > 0$ . Answer the following questions.

(1) Find the value of  $ x \equal{} s_p$  for which  $ f(x)$  is minimal and draw the graph of  $ y \equal{} f(x)$ .

(2) Let  $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$ . Find the value of  $ t \equal{} t_p$  for which  $ g(t)$  is minimal.

(3) Use the fact  $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$  to find the limit  $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$ .

Answer: -\frac{1}{2}


On an island, there are chameleons of five colors. When one chameleon bites another, the color of the bitten chameleon changes according to a certain rule, and the new color depends only on the color of the biting chameleon and the color of the bitten chameleon. It is known that 2023 red chameleons can agree on a sequence of bites, after which all of them will become blue. What is the smallest $k$ such that $k$ red chameleons can guarantee to agree on a sequence of bites so that they all become blue?

For example, the rules could be such: if a red chameleon bites a green one, the bitten chameleon changes color to blue; if a green chameleon bites a red one, the bitten chameleon remains red, that is, "changes color to red"; if a red chameleon bites a red one, the bitten chameleon changes color to yellow, and so on. (Specific rules for color changes may be different.)

Answer: 5


The integers  $1, 2, \dots, 64$  are written in the squares of a  $8 \times 8$  chess board, such that for each  $1 \le i < 64$ , the numbers  $i$  and  $i+1$  are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?

Answer: 456


A cuckoo clock chimes "cuckoo" on the hour, with the number of "cuckoo" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger until he moved the clock forward by 7 hours. How many times did the clock chime "cuckoo" during this period?

Answer: 43


The height of the large cylinder is twice that of the small cylinder. The lateral surface area of the large cylinder is 12 times the lateral surface area of the small cylinder. The volume of the large cylinder is $\qquad$ times the volume of the small cylinder.

Answer: 72


In triangle \(ABC\), \(D\) and \(E\) are the midpoints of \(BC\) and \(CA\), respectively. \(AD\) and \(BE\) intersect at \(G\). Given that \(GECD\) is cyclic, \(AB=41\), and \(AC=31\), compute \(BC\).

Answer: 49


An omino is a 1-by-1 square or a 1-by-2 horizontal rectangle. An omino tiling of a region of the plane is a way of covering it (and only it) by ominoes. How many omino tilings are there of a 2-by-10 horizontal rectangle?

Answer: 7921


How many of the natural numbers from 1 to 700, inclusive, contain the digit 5 at least once?

Answer: 214


Show that for any sequence \( a_0, a_1, \ldots, a_n, \ldots \) consisting of ones and minus ones, there exist such \( n \) and \( k \) such that

\[ |a_0 a_1 \cdots a_k + a_1 a_2 \cdots a_{k+1} + \cdots + a_n a_{n+1} \cdots a_{n+k}| = 2017. \]

Answer: 2017


The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.)

Answer: 2048


Triangle $ABC^{}_{}$ has $AB=9^{}_{}$ and $BC: AC=40: 41^{}_{}$. What's the largest area that this triangle can have?

Answer: 820


Let $\alpha$ and $\beta$ be real numbers such that for any real numbers $x, y, z$, the inequality $\alpha(x y+y z+z x) \leqslant M \leqslant \beta\left(x^{2}+y^{2}+z^{2}\right)$ always holds, where $M=\sqrt{x^{2}+x y+y^{2}} \cdot \sqrt{y^{2}+y z+z^{2}}+\sqrt{y^{2}+y z+z^{2}} \cdot \sqrt{z^{2}+z x+x^{2}}+\sqrt{z^{2}+z x+x^{2}} \cdot \sqrt{x^{2}+x y+y^{2}}$. Find the maximum value of $\alpha$ and the minimum value of $\beta$.

Answer: 3


Suppose  $\overline{a_1a_2...a_{2009}}$  is a  $2009$ -digit integer such that for each  $i = 1,2,...,2007$ , the  $2$ -digit integer  $\overline{a_ia_{i+1}}$  contains  $3$  distinct prime factors. Find  $a_{2008}$
(Note:  $\overline{xyz...}$  denotes an integer whose digits are  $x, y,z,...$ .)

Answer: 6


Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers.

Answer: 11


Points  $ K$ ,  $ L$ ,  $ M$ , and  $ N$  lie in the plane of the square  $ ABCD$  so that  $ AKB$ ,  $ BLC$ ,  $ CMD$ , and  $ DNA$  are equilateral triangles. If  $ ABCD$  has an area of  $ 16$ , find the area of  $ KLMN$.

Answer: 32 + 16\sqrt{3}


Does there exist a natural number $k$ such that from any 180 vertices of a regular 360-gon centered at point $O$, one can choose two vertices $A$ and $B$ such that the angle $AOB$ is equal to $k$ degrees?

Answer: 120


Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 99$

Answer: \textbf{(B)}\10


On the table lies a rectangular sheet of paper. Sasha cuts it along a straight line into two parts and places the parts back on the table. Then, he takes one of the parts, cuts it again along a straight line into two parts, and places the parts back on the table. He repeats the process of taking a part from the table and cutting it along a straight line. What is the minimum number of cuts Sasha needs to make so that there are at least 252 eleven-sided polygons on the table?

Answer: 2015


Real numbers \( x \) and \( y \) are such that for any distinct odd prime numbers \( p \) and \( q \), the number \( x^{p} + y^{q} \) is rational.

Prove that \( x \) and \( y \) are rational numbers.

Answer: x


There are 3 female and 2 male volunteers, a total of 5 volunteers, who need to be distributed among 3 communities to participate in volunteer services. Each community can have 1 to 2 people. Female volunteers A and B must be in the same community, and male volunteers must be in different communities. The number of different distribution methods is \_\_\_\_\_\_.

Answer: 12


The quadrilateral  $ABCD$  is an isosceles trapezoid with  $AB = CD = 1$ ,  $BC = 2$ , and  $DA = 1+ \sqrt{3}$ .  What is the measure of  $\angle ACD$  in degrees?

Answer: 90


Below the standard representation of a positive integer  $n$  is the representation understood by  $n$  in the decimal system, where the first digit is different from  $0$ . Everyone positive integer n is now assigned a number  $f(n)$  by using the standard representation of  $n$  last digit is placed before the first.
Examples:  $f(1992) = 2199$ ,   $f(2000) = 200$ .
Determine the smallest positive integer  $n$  for which  $f(n) = 2n$  holds.

Answer: 105263157894736842


Given Karl's rectangular garden measures \(30\) feet by \(50\) feet with a \(2\)-feet wide uniformly distributed pathway and Makenna's garden measures \(35\) feet by \(55\) feet with a \(3\)-feet wide pathway, compare the areas of their gardens, assuming the pathways take up gardening space.

Answer: 225


An integer-valued function $f$ is called tenuous if $f(x) + f(y) > y^2$ for all positive integers $x$ and $y.$  Let $g$ be a tenuous function such that $g(1) + g(2) + \dots + g(20)$ is as small as possible.  Compute the minimum possible value for $g(14).$

Answer: 136


The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:
1. The sum of the four numbers in each "田" (four cells in a square) is equal.
2. 华 $\times$ 华 $=$ 英 $\times$ 英 + 赛 $\times$ 赛.
3. 数 > 学

According to the above conditions, find the product of the numbers represented by 华, 杯, and 赛.

Answer: 120


Let positive integers $a$, $b$, $c$ satisfy $ab + bc = 518$ and $ab - ac = 360$. The maximum value of $abc$ is ____.

Answer: 1008


A circle passing through the vertex \(M\) of a certain angle intersects the sides of the angle at points \(N\) and \(K\), and the angle bisector at point \(L\). Find the sum of the lengths of segments \(MN\) and \(MK\) if the area of \(MNKL\) is 49, and the angle \(LMN\) is \(30^{\circ}\).

Answer: 14\sqrt[4]{3}


The opposite of $-23$ is ______; the reciprocal is ______; the absolute value is ______.

Answer: 23


Define the sequence $a_{1}, a_{2} \ldots$ as follows: $a_{1}=1$ and for every $n \geq 2$, $a_{n}= \begin{cases}n-2 & \text { if } a_{n-1}=0 \\ a_{n-1}-1 & \text { if } a_{n-1} \neq 0\end{cases}$. A non-negative integer $d$ is said to be jet-lagged if there are non-negative integers $r, s$ and a positive integer $n$ such that $d=r+s$ and that $a_{n+r}=a_{n}+s$. How many integers in $\{1,2, \ldots, 2016\}$ are jet-lagged?

Answer: 51


What is the sum of the digits of the integer equal to \(3 \times 10^{500} - 2022 \times 10^{497} - 2022\)?

Answer: 4491


Let $A_{12}$ denote the answer to problem 12. There exists a unique triple of digits $(B, C, D)$ such that $10>A_{12}>B>C>D>0$ and $$\overline{A_{12} B C D}-\overline{D C B A_{12}}=\overline{B D A_{12} C}$$ where $\overline{A_{12} B C D}$ denotes the four digit base 10 integer. Compute $B+C+D$.

Answer: 11


What is the minimum number of cells that need to be painted in a 35-cell side square (35 x 35—a total of 1225 cells) so that from any unpainted cell, it is not possible to reach another unpainted cell in one knight's move?

Answer: 612


A triangle was cut into two triangles. Find the maximum value of $N$ such that among the 6 angles of these two triangles, exactly $N$ are the same.

Answer: 4


In parallelogram $ABCD$, where $\angle BAD$ is $60^{\circ}$, $AB = 2$, $AD = 5$, the bisector of $\angle BAD$ intersects the bisector of $\angle ABC$ at point $K$ and the bisector of $\angle CDA$ at point $L$. The bisector of $\angle BCD$ intersects the bisector of $\angle CDA$ at point $M$ and the bisector of $\angle ABC$ at point $N$. Find the ratio of the area of quadrilateral $KLMN$ to the area of parallelogram $ABCD$.

Answer: \frac{9}{20}


Find the functions  $ f:\mathbb{N}\longrightarrow\mathbb{N} $  that satisfy the following relation:  $$  \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} .  $$

Answer: f(x)=x


The sequence \(\{a_n\}\) is a geometric sequence with a common ratio of \(q\), where \(|q| > 1\). Let \(b_n = a_n + 1 (n \in \mathbb{N}^*)\), if \(\{b_n\}\) has four consecutive terms in the set \(\{-53, -23, 19, 37, 82\}\), find the value of \(q\).

Answer: -\dfrac{3}{2}


The bar graph shows the grades in a mathematics class for the last grading period.  If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?
$\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{9}{10}$

Answer: C


Positive integers less than 900 that can be written as a product of two or more consecutive prime numbers. Find their count.

Answer: 14


Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \le 0$

Answer: \textbf{(E)}


What is the maximum number of finite roots that the equation

$$
\left|x - a_{1}\right| + \ldots + |x - a_{50}| = \left|x - b_{1}\right| + \ldots + |x - b_{50}|
$$

can have, where $a_{1}, a_{2}, \ldots, a_{50}, b_{1}, b_{2}, \ldots, b_{50}$ are distinct numbers?

Answer: 49


Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 30^\circ$ and $m\angle BAC = 45^\circ$, the ratio of the area of $ABCD$ to the area of the circle can be expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, where $a,$ $b,$ and $c$ are positive integers. What is the value of $a + b + c$?

Answer: 7


There are more than 20 and fewer than 30 children in Miss Tree's class. They are all standing in a circle. Anna notices that there are six times as many children between her and Zara going round the circle clockwise, as there are going round anti-clockwise. How many children are there in the class?

Answer: 23


Given a ball with a diameter of 6 inches rolling along the path consisting of four semicircular arcs, with radii $R_1 = 100$ inches, $R_2 = 60$ inches, $R_3 = 80$ inches, and $R_4 = 40$ inches, calculate the distance traveled by the center of the ball from the start to the end of the track.

Answer: 280\pi


A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a shape is the sum of its side lengths.

Answer: 52


Answer: 200203_{-4}


Answer: \cos\frac{\pi}{n+1}


Eight knights are randomly placed on a chessboard (not necessarily on distinct squares). A knight on a given square attacks all the squares that can be reached by moving either (1) two squares up or down followed by one square left or right, or (2) two squares left or right followed by one square up or down. Find the probability that every square, occupied or not, is attacked by some knight.

Answer: 0


In a right-angled triangle $ABC$ with the right angle at $C$, the angle bisector $BD$ and the altitude $CH$ are drawn. A perpendicular $CK$ is dropped from vertex $C$ to the angle bisector $BD$. Find the angle $HCK$ if $BK: KD = 3:1$.

Answer: 30


It is known that \( a^2 + 2 = b^4, \, b^2 + 2 = c^4, \, c^2 + 2 = a^4 \). What is the value of the product \( (a^2 - 1)(b^2 - 1)(c^2 - 1) \)? Find all possible values and prove that there are no other solutions.

Answer: 1


Let  $a,b,c$  be positive real numbers such that   $$ \frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1. $$   Prove that  $abc \le \frac{1}{8}$ .

Answer: abc\le\frac{1}{8}


Given \(0 < a < b\), draw two lines \(l\) and \(m\) through two fixed points \(A(a, 0)\) and \(B(b, 0)\), respectively, such that they intersect the parabola \(y^2 = x\) at four distinct points. When these four points lie on a common circle, find the locus of the intersection point \(P\) of the two lines \(l\) and \(m\).

Answer: \frac{b}{2}


The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$?
$\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17$

Answer: \textbf{(C)}12


On the coordinate plane, the graphs of three reduced quadratic polynomials intersect the y-axis at the points $-15,-6,-27$ respectively. For each of the polynomials, the coefficient at $x$ is a natural number, and the larger root is a prime number. Find the sum of all roots of these polynomials.

Answer: -9


How many nonnegative integers can be expressed as \[b_6\cdot4^6+b_5\cdot4^5+b_4\cdot4^4+b_3\cdot4^3+b_2\cdot4^2+b_1\cdot4^1+b_0\cdot4^0,\] where $b_i \in \{-1,0,1\}$ for $0 \leq i \leq 6$?

Answer: 5462


Inside the triangle \(ABC\), on the bisector of its angle \(B\), a point \(M\) is chosen such that \(AM = AC\) and \(\angle BCM = 30^{\circ}\).

Prove that \(\angle AMB = 150^{\circ}\).

Answer: 150


We call a set of professors and committees on which they serve a university if:
1. Given two distinct professors, there is one and only one committee on which they both serve.
2. Given any committee \( C \) and any professor \( P \) not on that committee, there is exactly one committee on which \( P \) serves, and no professors on committee \( C \) serve.
3. There are at least two professors on each committee and there are at least two committees.

What is the smallest number of committees a university can have?

Answer: 6


For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$?

Answer: 26


Let \( P\left(x_{0}, y_{0}\right) \) be any point on the ellipse \(\frac{x^{2}}{8}+\frac{y^{2}}{2}=1\). A line \( y = kx + m \) (where \( km \neq 0 \)) intersects the ellipse \( C_{1}: \frac{x^{2}}{4} + y^{2} = 1 \) at points \( M \) and \( N \). The chord \( MN \) is bisected by the line \( OP \) and satisfies \( m y_{0} > -1 \). If the area of triangle \( \triangle PMN \) is 1, determine the shape of the quadrilateral \( OMNP \) and the number of common points between line \( PM \) and the ellipse \( C_{1} \). Provide proof for your conclusions.

Answer: 1


Let $S = \{1, 2, \dots, n\}$ for some integer $n > 1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if \begin{enumerate} \item[(i)] $\pi(k) > \pi(k+1)$ for $k=1$; \item[(ii)] $\pi(k-1) < \pi(k)$ and $\pi(k) > \pi(k+1)$ for $1 < k < n$; \item[(iii)] $\pi(k-1) < \pi(k)$ for $k=n$. \end{enumerate} (For example, if $n=5$ and $\pi$ takes values at $1, 2, 3, 4, 5$ of $2, 1, 4, 5, 3$, then $\pi$ has a local maximum of 2 at $k=1$, and a local maximum of 5 at $k=4$.) What is the average number of local maxima of a permutation of $S$, averaging over all permutations of $S$?

Answer: \frac{n+1}{3}


Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?

Answer: 28


Given an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) with left and right foci \( F_{1}, F_{2} \) respectively, let \( P \) be a point on the ellipse \( C \) in the first quadrant. The extended lines \( P F_{1}, P F_{2} \) intersect the ellipse \( C \) at points \( Q_{1}, Q_{2} \) respectively. Find the maximum value of the difference of areas of triangles \( \triangle P F_{1} Q_{2} \) and \( \triangle P F_{2} Q_{1} \).

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


A 60-degree angle contains five circles, where each subsequent circle (starting from the second) touches the previous one. By how many times is the sum of the areas of all five circles greater than the area of the smallest circle?

Answer: 7381


$ABCD$ is a square with side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. In square units this maximum area is:
$\textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ \frac{9}{16}\qquad\textbf{(C)}\ \frac{19}{32}\qquad\textbf{(D)}\ \frac{5}{8}\qquad\textbf{(E)}\ \frac{2}3$

Answer: \textbf{(D)}


Sandra has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do?

Answer: \frac{5}{9}


Given are sheets and the numbers  $00, 01, \ldots, 99$  are written on them. We must put them in boxes  $000, 001, \ldots, 999$  so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?

Answer: 34


The population of a small town is $480$.  The graph indicates the number of females and males in the town, but the vertical scale-values are omitted.  How many males live in the town?

Answer: 160


Does there exist a sequence of \(2011^{2011}\) consecutive positive integers that contains exactly 2011 prime numbers? Prove your answer.

Answer: Yes


Point $P$ moves on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, and points $A$ and $B$ move respectively on the circles $x^2 + (y-4)^2 = 16$ and $x^2 + (y+4)^2 = 4$. The maximum value of $PA + PB$ is \_\_\_\_\_\_.

Answer: 16


In the convex quadrilateral \(ABCD\), the length of the segment connecting the midpoints of sides \(AB\) and \(CD\) is 1. The lines \(BC\) and \(AD\) are perpendicular. Find the length of the segment connecting the midpoints of the diagonals \(AC\) and \(BD\).

Answer: 1


For the function $f(x)= \sqrt {2}(\sin x+\cos x)$, the following four propositions are given:
$(1)$ There exists $\alpha\in\left(- \frac {\pi}{2},0\right)$, such that $f(\alpha)= \sqrt {2}$;
$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=- \frac {3\pi}{4}$;
$(3)$ There exists $\phi\in\mathbb{R}$, such that the graph of the function $f(x+\phi)$ is centrally symmetric about the origin;
$(4)$ The graph of the function $f(x)$ can be obtained by shifting the graph of $y=-2\cos x$ to the left by $ \frac {\pi}{4}$.
Among these, the correct propositions are \_\_\_\_\_\_.

Answer: (2)(3)


Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?

Answer: \frac{8}{17}


In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin.

Answer: 3(\sqrt{3} - 1)


Given the function $f(x) = x^2 + ax + b$ ($a, b \in \mathbb{R}$).

(Ⅰ) Given $x \in [0, 1]$,

(i) If $a = b = 1$, find the range of the function $f(x)$;

(ii) If the range of the function $f(x)$ is $[0, 1]$, find the values of $a$ and $b$;

(Ⅱ) When $|x| \geq 2$, it always holds that $f(x) \geq 0$, and the maximum value of $f(x)$ in the interval $(2, 3]$ is 1, find the maximum and minimum values of $a^2 + b^2$.

Answer: 74


Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis?

Answer: \frac{1}{2}


When $0.42\overline{153}$ is expressed as a fraction in the form $\frac{x}{99900}$, what is the value of $x$?

Answer: 42111


Prove: A necessary and sufficient condition for \( x \mid f^{k}(x) \) is \( x \mid f(x) \), where \( k \in \mathbf{N} \).

Answer: x\midf(x)


There are three piles of peaches with the following counts: $93, 70, 63$. A monkey moves peaches among the 3 piles. It is known that the monkey can move at most 5 peaches at a time, and will eat 1 peach while carrying it from one pile to another. When the number of peaches in the 3 piles is equal, how many peaches will the monkey have eaten at least?

Answer: 4


Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$.

Answer: 11


Given an ellipse \(\frac{x^{2}}{4} + y^{2} = 1\) and three points \(A, B, C\) on it, the line \(BC\) with a negative slope intersects the \(y\)-axis at point \(M\). If the origin \(O\) is the centroid of triangle \(\triangle ABC\), and the area ratio of triangles \(\triangle BMA\) to \(\triangle CMO\) is \(3:2\), determine the slope of line \(BC\).

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


Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are eliminated. The remaining team wins. How many different possible competition sequences can occur?

Answer: 3432


If 3400 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?

Answer: 38


How many different positive three-digit integers can be formed using only the digits in the set $\{3, 3, 4, 4, 4, 7, 8\}$ if no digit may be used more times than it appears in the given set of available digits?

Answer: 43


Insert a square into an isosceles triangle with a lateral side of 10 and a base of 12.

Answer: 4.8


A parallelogram is circumscribed around a circle with radius \( R \). The area of the quadrilateral formed by the points of tangency of the circle with the parallelogram is \( S \). Find the sides of the parallelogram.

Answer: \frac{4R^3}{S}


Let \( \triangle ABC \) be a triangle with \( AB=13, BC=14 \), and \( CA=15 \). Let \( \Gamma \) be the circumcircle of \( \triangle ABC \), let \( O \) be its circumcenter, and let \( M \) be the midpoint of minor arc \( \widehat{BC} \). Circle \( \omega_{1} \) is internally tangent to \( \Gamma \) at \( A \), and circle \( \omega_{2} \), centered at \( M \), is externally tangent to \( \omega_{1} \) at a point \( T \). Ray \( AT \) meets segment \( BC \) at point \( S \), such that \( BS - CS = \frac{4}{15} \). Find the radius of \( \omega_{2} \).

Answer: \frac{1235}{108}


The diagonal \( AC \) of the inscribed quadrilateral \( ABCD \) is the diameter of the circumscribed circle \( \omega \). From point \( D \), a line is drawn perpendicular to the segment \( BC \), and it intersects the circle \( \omega \) again at point \( E \). Find the ratio of the areas of triangle \( BCD \) and quadrilateral \( ABEC \).

Answer: 1


The parametric equation of line $l$ is
$$
\begin{cases}
x= \frac { \sqrt {2}}{2}t \\
y= \frac { \sqrt {2}}{2}t+4 \sqrt {2}
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of circle $c$ is $\rho=2\cos(\theta+ \frac{\pi}{4})$. Tangent lines are drawn from the points on the line to the circle; find the minimum length of these tangent lines.

Answer: 2\sqrt{6}


A ball is dropped from a height of 150 feet and rebounds to three-fourths of the distance it fell on each bounce. How many feet will the ball have traveled when it hits the ground the fifth time?

Answer: 765.234375


Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 097


Given that $y$ is a multiple of $42522$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+17)$ and $y$?

Answer: 102


At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the contest?
$\textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30$

Answer: \textbf{(C)}\28


Each cell of a \(2017 \times 2017\) board contains a chip. In one operation, you can remove a chip that has a nonzero even number of neighbors (neighbors are chips located in cells adjacent by side or corner). What is the minimum number of chips that can be left on the board using such operations?

Answer: 2


Let $d(n)$ denote the number of positive divisors of the positive integer $n$. What is the smallest positive real value of $c$ such that $d(n) \leq c \cdot \sqrt{n}$ holds for all positive integers $n$?

Answer: \sqrt{3}


We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$?

Answer: 2/5


Calculate the value of $\dfrac{\sqrt[3]{81}}{\sqrt[4]{81}}$ in terms of 81 raised to what power?

Answer: \frac{1}{12}


We defined an operation denoted by $*$ on the integers, which satisfies the following conditions:

1) $x * 0 = x$ for every integer $x$;
2) $0 * y = -y$ for every integer $y$;
3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.

Determine the result of the operation $19 * 90$.

Answer: 1639


Given that \(a, b, c, d\) are positive real numbers and \(a + b + c + d = 4\), prove that \(a^{2}bc + b^{2}da + c^{2}da + d^{2}bc \leq 4\).

Answer: 4


If
\[\frac{\sin x}{\cos y} + \frac{\sin y}{\cos x} = 1 \quad \text{and} \quad \frac{\cos x}{\sin y} + \frac{\cos y}{\sin x} = 6,\]then find $\frac{\tan x}{\tan y} + \frac{\tan y}{\tan x}.$

Answer: \frac{124}{13}


In the Chinese length measurement units, 1 meter = 3 chi, 1 zhang = 10 chi, and 1 kilometer = 2 li. How many zhang are in 1 li?

Answer: 150


In a rectangular coordinate system, the ellipse \(\Omega: \frac{x^{2}}{4}+y^{2}=1\) has a moving point \(P\) on \(\Omega\), and \(A\) and \(B\) are two fixed points with \(B\) at coordinates \((0,3)\). If the minimum area of triangle \(PAB\) is 1 and the maximum area is 5, what is the length of segment \(AB\)?

Answer: \sqrt{7}


In a rectangular  $57\times 57$  grid of cells,  $k$  of the cells are colored black. What is the smallest positive integer  $k$  such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?

[i]Proposed by James Lin

Answer: 457


Compute the prime factorization of  $1007021035035021007001$ .  (You should write your answer in the form  $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$  where  $p_1,\ldots,p_k$  are distinct prime numbers and  $e_1,\ldots,e_k$  are positive integers.)

Answer: 7^7 \times 11^7 \times 13^7


Answer: 13


The vector $\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is rotated $90^\circ$ about the origin.  During the rotation, it passes through the $x$-axis.  Find the resulting vector.

Answer: \begin{pmatrix} 2 \sqrt{2} \\ -\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix}


Given three unit vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ in a plane that are not collinear and satisfy $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = \mathbf{0}$. If $0 \leq t \leq 1$, what is the range of values of $|-2 \boldsymbol{a} + t \boldsymbol{b} + (1-t) \boldsymbol{c}|$?

Answer: [\frac{5}{2},\sqrt{7}]


If $N$ is a positive integer between 1000000 and 10000000, inclusive, what is the maximum possible value for the sum of the digits of $25 \times N$?

Answer: 67


Given a triangle \( \triangle ABC \) with side lengths 13, 14, and 15, there are 4 circles with the same radius \( r \) placed inside \( \triangle ABC \). Circle \( O_{1} \) is tangent to sides \( AB \) and \( AC \), circle \( O_{2} \) is tangent to sides \( BA \) and \( BC \), and circle \( O_{3} \) is tangent to sides \( CB \) and \( CA \). Circle \( O \) is tangent to \( O_{1}, O_{2}, \) and \( O_{3} \). Find the radius \( r \).

Answer: \frac{260}{129}


Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\sqrt{3}y$.

(Ⅰ) Find the standard equation of the ellipse $C$;

(Ⅱ) If the line $x=-2$ intersects the ellipse at points $P$ and $Q$, and $A$, $B$ are points on the ellipse located on either side of the line $x=-2$.

(i) If the slope of line $AB$ is $\frac{1}{2}$, find the maximum area of the quadrilateral $APBQ$;

(ii) When the points $A$, $B$ satisfy $\angle APQ = \angle BPQ$, does the slope of line $AB$ have a fixed value? Please explain your reasoning.

Answer: \frac{1}{2}


Given the tetrahedron \(OPQR\), let the points \(P, Q, R\) be reflected over \(O\) to respectively form points \(P', Q', R'\). The edges of tetrahedron \(ABCD\) are bisected by the points \(P, Q, R, P', Q', R'\). How many times larger is the volume of tetrahedron \(ABCD\) compared to the volume of tetrahedron \(OPQR\)?

Answer: 16


The Gauss family has three boys aged 7, a girl aged 14, and a boy aged 15. What is the mean (average) of the ages of the children?
(A) 9
(B) 7
(C) 11
(D) 14
(E) 10

Answer: 10


Given that \(a, b, c\) are integers such that \(abc = 60\), and that the complex number \(\omega \neq 1\) satisfies \(\omega^3 = 1\), find the minimum possible value of \(\left| a + b\omega + c\omega^2 \right|\).

Answer: \sqrt{3}


16 progamers are playing in a single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Each time a progamer wins, he will have a ceremony to celebrate. A player's first ceremony is ten seconds long, and each subsequent ceremony is ten seconds longer than the previous one. What is the total length in seconds of all the ceremonies over the entire tournament?

Answer: 260


One hundred rectangles are arranged edge-to-edge in a continuation of the pattern shown. Each rectangle measures $3 \mathrm{~cm}$ by $1 \mathrm{~cm}$. What is the perimeter, in $\mathrm{cm}$, of the completed shape?

Answer: 602


In the expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\dfrac{1}{2}}$ is:
$\textbf{(A)}\ -7 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ -21 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 35$

Answer: \textbf{(C)}\-21


In a variant of SHORT BINGO, a $5\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:
- First column contains 5 distinct numbers from the set $1-15$.
- Second column contains 5 distinct numbers from the set $16-30$.
- Third column (excluding the FREE middle square) contains 4 distinct numbers from $31-45$.
- Fourth column contains 5 distinct numbers from the set $46-60$.
- Fifth column contains 5 distinct numbers from the set $61-75$.

How many distinct possibilities are there for the values in the first column of this modified SHORT BINGO card?

Answer: 360360


Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(m)^2 + f(n) \) is a divisor of \( (m^2 + n)^2 \) for all \( m, n \in \mathbb{N} \).

Answer: f(n)=n


Let \omega=\cos \frac{2 \pi}{727}+i \sin \frac{2 \pi}{727}$. The imaginary part of the complex number $$\prod_{k=8}^{13}\left(1+\omega^{3^{k-1}}+\omega^{2 \cdot 3^{k-1}}\right)$$ is equal to $\sin \alpha$ for some angle $\alpha$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, inclusive. Find $\alpha$.

Answer: \frac{12 \pi}{727}


What is the smallest α such that two squares with a total area of 1 can always be placed inside a rectangle of area α, with sides parallel to those of the rectangle and with no overlap of their interiors?

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


The cells of a $5 \times 7$ table are filled with numbers so that in each $2 \times 3$ rectangle (either vertical or horizontal) the sum of the numbers is zero. By paying 100 rubles, you can choose any cell and find out what number is written in it. What is the smallest amount of rubles needed to determine the sum of all the numbers in the table with certainty?

Answer: 100


Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, determine the total area in square centimeters of the football-shaped area of regions II and III combined. Express your answer as a decimal to the nearest tenth.

Answer: 9.1


The polynomial $x^8 - 1$ is factored as
\[x^8 - 1 = p_1(x) p_2(x) \dotsm p_k(x),\]where each factor $p_i(x)$ is a non-constant polynomial with real coefficients.  Find the largest possible value of $k.$

Answer: 5


How many possible bijections \( f \) on \(\{1, 2, \dots, n\}\) are there such that for each \( i = 2, 3, \dots, n \), we can find \( j < i \) with \( f(i) - f(j) = \pm 1\)?

Answer: 2^{n-1}


Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \frac{321}{400}$?

Answer: 12


In  $\triangle ABC$  points  $M,O$  are midpoint of  $AB$  and circumcenter. It is true, that  $OM=R-r$ . Bisector of external  $\angle A$  intersect  $BC$  at  $D$  and bisector of external  $\angle C$  intersect  $AB$  at  $E$ .
Find possible values of  $\angle CED$ *D. Shiryaev*

Answer: 90


Let  $ABC$  be a triangle with  $BC = a$ ,  $CA = b$ , and  $AB = c$ . The  $A$ -excircle is tangent to  $\overline{BC}$  at  $A_1$ ; points  $B_1$  and  $C_1$  are similarly defined.

Determine the number of ways to select positive integers  $a$ ,  $b$ ,  $c$  such that

- the numbers  $-a+b+c$ ,  $a-b+c$ , and  $a+b-c$  are even integers at most 100, and
- the circle through the midpoints of  $\overline{AA_1}$ ,  $\overline{BB_1}$ , and  $\overline{CC_1}$  is tangent to the incircle of  $\triangle ABC$ .

Answer: 125000


Find the greatest  $n$  such that  $(z+1)^n = z^n + 1$  has all its non-zero roots in the unitary circumference, e.g.  $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$  implies  $|\alpha| = 1.$

Answer: 7


The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

Answer: 108


The product of the ages of a father and his son is 2015. What is the difference between their ages?
A) 29
B) 31
C) 34
D) 36
E) None of these

Answer: 34


There are 64 small wooden cubes with an edge length of 1. Each cube has two faces painted red, and the other faces are white. Among these cubes, 20 have two adjacent faces painted red, and 44 have two opposite faces painted red. If these cubes are assembled to form one large cube, what is the maximum possible area of the red surface on the large cube?

Answer: 76


On an island, there are 100 knights and 100 liars. Every resident has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident said either the phrase "All my friends are knights" or the phrase "All my friends are liars," with exactly 100 people saying each phrase. Find the minimum possible number of pairs of friends where one is a knight and the other is a liar.

Answer: 50


The side length of the cube \(A B C D - A_{1} B_{1} C_{1} D_{1}\) is 1. Determine the distance \(d\) between the lines \(A_{1} C_{1}\) and \(B D_{1}\).

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


Given \(a_{1}, a_{2}, \cdots, a_{2014}, b_{1}, b_{2}, \cdots, b_{2014}\) as distinct real numbers, they are filled into a \(2014 \times 2014\) grid according to the following method: at the intersection of the \(i\)-th row and the \(j\)-th column, the number \(a_{i} + b_{j}\) is placed. It is known that the product of the numbers in any row of the grid is 2014. Prove that the product of the numbers in any column of the grid is also 2014.

Answer: 2014


In a fixed $3 \times 3$ grid, 4 identical chess pieces are to be placed into the cells. No more than one piece can be placed in any single cell. If each row and each column must contain at least one piece, how many different ways can this be done?

Answer: 45


Answer: 16\sqrt[4]{2}


A severe earthquake in Madrid caused a total of €50 million in damages. At the time of the earthquake, the exchange rate was such that 2 Euros were worth 3 American dollars. How much damage did the earthquake cause in American dollars?

Answer: 75,000,000


Let \( A B C \) be a triangle with circumcenter \( O \). The points \( P \) and \( Q \) are interior points of the sides \( C A \) and \( A B \), respectively. The circle \( k \) passes through the midpoints of the segments \( B P \), \( C Q \), and \( P Q \). Prove that if the line \( P Q \) is tangent to circle \( k \) then \( O P = O Q \).

Answer: OP=OQ


Consider a function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that for every integer \( n \geq 0 \), there are at most \( 0.001 n^{2} \) pairs of integers \( (x, y) \) for which \( f(x + y) \neq f(x) + f(y) \) and \( \max \{|x|, |y|\} \leq n \). Is it possible that for some integer \( n \geq 0 \), there are more than \( n \) integers \( a \) such that \( f(a) \neq a \cdot f(1) \) and \( |a| \leq n \)?

Answer: No


A binary operation  $*$  on real numbers has the property that  $(a * b) * c = a+b+c$  for all  $a$ ,  $b$ ,  $c$ . Prove that  $a * b = a+b$ .

Answer: b


Through the center of the circumscribed circle around triangle \(ABC\), lines perpendicular to sides \(AC\) and \(BC\) are drawn. These lines intersect the altitude \(CH\) of the triangle or its extension at points \(P\) and \(Q\). It is known that \(CP = p\) and \(CQ = q\). Find the radius of the circumscribed circle around triangle \(ABC\).

Answer: \sqrt{pq}


The ratio $AC:CB$ is $3:4$, in $\triangle ABC$. The external angle bisector of $\angle C$ intersects the extension of $BA$ at $P$, where $A$ is between $P$ and $B$. Find the ratio $PA:AB$.

Answer: 3:1


Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF.

Answer: 2\sqrt{65}


Given the random variable $\eta\sim B(n,p)$, and $E(2\eta)=8$, $D(4\eta)=32$, find the respective values of $n$ and $p$.

Answer: 0.5


The roots of the equation \(x^{3}-10x+11=0\) are \(u\), \(v\), and \(w\). Determine the value of

\[
\operatorname{arctan} u + \operatorname{arctan} v + \operatorname{arctan} w.
\]

Answer: \frac{\pi}{4}


Consider the graph in 3-space of
$$
0 = xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x).
$$
This graph divides 3-space into \( N \) connected regions. What is \( N \)?

Answer: 48


In triangle  $\triangle ABC$ , the points  $A', B', C'$  are on sides  $BC, AC, AB$  respectively. Also,  $AA', BB', CC'$  intersect at the point  $O$ (they are concurrent at  $O$ ). Also,  $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$ . Find the value of  $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$ .

Answer: 94


Given \( a_n = 1 + 2 + \cdots + n \) where \( n \in \mathbf{Z}_{+} \), and
\[ S_m = a_1 + a_2 + \cdots + a_m \text{ for } m = 1, 2, \cdots, \]
find the number of terms among \( S_1, S_2, \cdots, S_{2017} \) that are divisible by 2 but not by 4.

Answer: 252


Given the vertices of a regular 100-gon \( A_1, A_2, A_3, \ldots, A_{100} \). How many ways can three vertices be chosen to form an obtuse-angled triangle? (10 points)

Answer: 117600


Let $G$ be a simple graph with 20 vertices and 100 edges, and there are 4050 ways to find a pair of non-intersecting edges in it. Prove that $G$ is regular.
(Iranian Mathematics Olympiad 2004)

Answer: G


The physical education teacher lined up the class so that everyone was facing him. There are 12 people to the right of Kolya, 20 people to the left of Sasha, and 8 people to the right of Sasha. How many people are to the left of Kolya?

Answer: 16


In the diagram, $ABCD$ is a parallelogram with an area of 27. $CD$ is thrice the length of $AB$. What is the area of $\triangle ABC$?

[asy]
draw((0,0)--(2,3)--(10,3)--(8,0)--cycle);
draw((2,3)--(0,0));
label("$A$",(0,0),W);
label("$B$",(2,3),NW);
label("$C$",(10,3),NE);
label("$D$",(8,0),E);
[/asy]

Answer: 13.5


The opposite number of $2- \sqrt{3}$ is ______, and its absolute value is ______.

Answer: 2- \sqrt{3}


In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\triangle CDE$ is $22,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $CE$ is $9.$ What is the length of $AB?$

Answer: 11


Given the ellipse $\frac{x^2}{3} + y^2 = 1$ and the line $l: y = kx + m$ intersecting the ellipse at two distinct points $A$ and $B$.

(1) If $m = 1$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ ($O$ is the origin), find the value of $k$.

(2) If the distance from the origin $O$ to the line $l$ is $\frac{\sqrt{3}}{2}$, find the maximum area of $\triangle AOB$.

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


How many roots does the equation

$$
\overbrace{f(f(\ldots f}^{10 \text{ times }}(x) \ldots))+\frac{1}{2}=0
$$

where $f(x)=|x|-1$ have?

Answer: 20


In triangle \( \triangle ABC \), the median \( BM \) is drawn. It is given that \( AB = 2BM \) and \( \angle MBA = 40^\circ \). Find \( \angle CBA \).

Answer: 110


Quadrilateral \(ABCD\) is circumscribed about a circle \(\omega\). Prove that the diameter of the circle \(\omega\) does not exceed the length of the segment connecting the midpoints of sides \(BC\) and \(AD\).

Answer: MN\ge2r


Given \(2n\) real numbers \(a_{1}, a_{2}, \cdots, a_{2n}\) satisfying the condition
$$
\sum_{i=1}^{2 n-1}\left(a_{i+1}-a_{i}\right)^{2}=1,
$$
find the maximum value of \(\left(a_{n+1}+a_{n+2}+\cdots+a_{2n}\right) - \left(a_{1}+a_{2}+\cdots+a_{n}\right)\).

Answer: \sqrt{\frac{n(2n^2+1)}{3}}