$x$  and  $y$  are chosen at random (with uniform density) from the interval  $(0, 1)$ . What is the probability that the closest integer to  $x/y$  is even?

Answer: \frac{5-\pi}{4}


A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist?

Answer: 10


For positive integer  $n$  let  $a_n$  be the integer consisting of  $n$  digits of  $9$  followed by the digits  $488$ . For example,  $a_3 = 999,488$  and  $a_7 = 9,999,999,488$ . Find the value of  $n$  so that an is divisible by the highest power of  $2$ .

Answer: 6


If one vertex of an ellipse and its two foci form the vertices of an equilateral triangle, then find the eccentricity $e$ of the ellipse.

Answer: \dfrac{1}{2}


Given that α > 0 is real and n is a positive integer, determine the maximum possible value of the expression α^a + α^b + α^c + ... , where a, b, c, ... is any sequence of positive integers whose sum is n.

Answer: \max(n\alpha,\alpha^n)


Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.

Answer: 63


A person A departs from location $A$ and travels at a constant speed to location $B$. At the same time, person B departs from location $B$ and travels at a constant speed to location $A$. They meet at location $C$. After meeting, person A travels 150 meters further and then turns back to chase person B. When person A catches up with person B, they are 540 meters away from location $C$. Person A then immediately turns back and heads to location $B$. When person A arrives at location $B$, person B also arrives at location $A$. Find the distance between locations $A$ and $B$ in meters.

Answer: 2484


A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\frac{3}{4}$ of its former rate and arrives $3\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. The length of the trip in miles was:
$\textbf{(A)}\ 400 \qquad \textbf{(B)}\ 465 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 640 \qquad \textbf{(E)}\ 550$

Answer: \textbf{(C)}\600


Masha has three identical dice, each face of which has one of six different prime numbers with a total sum of 87.

Masha rolled all three dice twice. The first time, the sum of the numbers rolled was 10, and the second time, the sum of the numbers rolled was 62.

Exactly one of the six numbers never appeared. What number is it?

Answer: 17


The velocity of the free fall of a ball is given by the law of conservation of energy $V=\sqrt{2 g r \cos \alpha}$. The free fall of the ball is described by the following equations:

$$
x(t)=R \sin \alpha+V \cos \alpha t, \quad y(t)=R(1-\cos \alpha)+V \sin \alpha t-\frac{g t^{2}}{2}
$$

The flight time $T$ is found from the quadratic equation $y(T)=0$:

$$
T=\sqrt{\frac{2 R}{g}}\left[\sin \alpha \sqrt{\cos \alpha}+\sqrt{1-\cos ^{3} \alpha}\right]
$$

From this, we find the desired coordinate of the point of impact:

$$
x(T)=R\left[\sin \alpha+\sin 2 \alpha+\sqrt{\cos \alpha\left(1-\cos ^{3} \alpha\right)}\right]
```

Answer: R[\sin\alpha+\sin2\alpha+\sqrt{\cos\alpha(1-\cos^3\alpha)}]


Find $AB$ in the triangle below.

[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0.5,sqrt(3)/2);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18$", (A+C)/2,W);
label("$30^\circ$", (0.3,0),N);
[/asy]

Answer: 18\sqrt{3}


Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing at his original speed. By the time the document is completed, A and B have typed an equal number of characters. What is the total number of characters in the document?

Answer: 18000


On some planet, there are  $2^N$  countries  $(N \geq 4).$  Each country has a flag  $N$  units wide and one unit high composed of  $N$  fields of size  $1 \times 1,$  each field being either yellow or blue. No two countries have the same flag. We say that a set of  $N$  flags is diverse if these flags can be arranged into an  $N \times N$  square so that all  $N$  fields on its main diagonal will have the same color. Determine the smallest positive integer  $M$  such that among any  $M$  distinct flags, there exist  $N$  flags forming a diverse set.

*Proposed by Tonći Kokan, Croatia*

Answer: 2^{N-2}+1


Find a three-digit number whose square is a six-digit number, such that each subsequent digit from left to right is greater than the previous one.

Answer: 367


Does there exist 7 distinct lines in a plane such that among their intersection points, there are at least 6 points where exactly 3 lines intersect, and at least 4 points where exactly 2 lines intersect?

Answer: No


Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

Answer: 5


A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct barcodes are possible?

Answer: 116


$n$ mushroom gatherers went to the forest and brought back a total of 338 mushrooms (it is possible that some of the gatherers did not bring back any mushrooms). A boy named Petya, upon learning this, stated, "Some two of them must have brought an identical number of mushrooms!" For what smallest $n$ will Petya certainly be correct? Provide a justification for your answer.

Answer: 27


Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 15x\]for all integers $x.$ Find the smallest possible value of $a.$

Answer: 15


In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, it is known that $a_1=4$ and $a_7^2=a_1a_{10}$. The sum of the first $n$ terms is $S_n$.
1. Find the general formula for the sequence $\{a_n\}$.
2. Find the maximum value of $S_n$ and the value of $n$ when the maximum is achieved.

Answer: 26


Given the function $f(x)={e}^{-x}+ \frac{nx}{mx+n}$.

(1) If $m=0$, $n=1$, find the minimum value of the function $f(x)$;

(2) If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac{m}{n}$.

Answer: \frac{1}{2}


From the interval \(\left(2^{2n}, 2^{3n}\right)\), \(2^{2n-1} + 1\) odd numbers were selected.

Prove that among the selected numbers, there are two numbers such that the square of each is not divisible by the other.

Answer: 2


Let the sides of a triangle be \(a, b, c\) and the semi-perimeter be \(p\). Prove that \(\sqrt{p-a} + \sqrt{p-b} + \sqrt{p-c} \leq \sqrt{3p}\).

Answer: \sqrt{p-}+\sqrt{p-b}+\sqrt{p-}\le\sqrt{3p}


A circle is inscribed in the trapezoid \(ABCD\) and touches the side \(AD\) at the point \(K\). Find the area of the trapezoid if \(AK = 16\), \(DK = 4\), and \(CD = 6\).

Answer: 432


A cube is suspended in space with its top and bottom faces horizontal. The cube has one top face, one bottom face, and four side faces. Determine the number of ways to move from the top face to the bottom face, visiting each face at most once, without moving directly from the top face to the bottom face, and not moving from side faces back to the top face.

Answer: 20


Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola  $ xy\equal{}1$  and both branches of the hyperbola  $ xy\equal{}\minus{}1.$  (A set  $ S$  in the plane is called *convex* if for any two points in  $ S$  the line segment connecting them is contained in  $ S.$ )

Answer: 4


Given a moving large circle $\odot O$ tangent externally to a fixed small circle $\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\odot O_{1}$ at point $C$ and intersects $\odot O$ at points $D$ and $E$. Find $C D \cdot C E = \quad$.

Answer: 36


Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

Answer: 5


A tetrahedron \( P-ABC \) has edge lengths \( PA = BC = \sqrt{6} \), \( PB = AC = \sqrt{8} \), and \( PC = AB = \sqrt{10} \). Find the radius of the circumsphere of this tetrahedron.

Answer: \sqrt{3}


A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ -9 \end{pmatrix} + u \begin{pmatrix} 4 \\ 2 \end{pmatrix}.\]Find the point where the lines intersect.

Answer: \begin{pmatrix} 7 \\ -8 \end{pmatrix}


How can you cut 50 cm from a string that is $2 / 3$ meters long without any measuring tools?

Answer: 50


A particular integer is the smallest multiple of 72, each of whose digits is either 0 or 1. How many digits does this integer have?

Answer: 12


Find a ten-digit number where the first digit indicates how many times the digit 0 appears in the number, the second digit indicates how many times the digit 1 appears, and so forth, with the tenth digit indicating how many times the digit 9 appears in the number.

Generalize and solve the problem for a number system with base $n$.

Answer: 6210001000


A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 391


In the triangle \( \triangle ABC \), if \(\sin^2 A + \sin^2 B + \sin^2 C = 2\), calculate the maximum value of \(\cos A + \cos B + 2 \cos C\).

Answer: \sqrt{5}


For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \]

Answer: -1


A wooden cube with edges of length $3$ meters has square holes, of side one meter, centered in each face, cut through to the opposite face. Find the entire surface area, including the inside, of this cube in square meters.

Answer: 72


Quadrilateral $A B C D$ satisfies $A B=8, B C=5, C D=17, D A=10$. Let $E$ be the intersection of $A C$ and $B D$. Suppose $B E: E D=1: 2$. Find the area of $A B C D$.

Answer: 60


The table shows the vertical drops of six roller coasters in Fibonacci Fun Park:
\begin{tabular}{|l|c|}
\hline
Speed Demon & 150 feet \\
\hline
Looper & 230 feet \\
\hline
Dare Devil & 160 feet \\
\hline
Giant Drop & 190 feet \\
\hline
Sky Scream & 210 feet \\
\hline
Hell Spiral & 180 feet \\
\hline
\end{tabular}
What is the positive difference between the mean and the median of these values?

Answer: 1.67


Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:

Answer: \text{(1) and (4)}


Using 70 yuan, four types of books A, B, C, and D were bought, with a total of 10 books. Knowing that the prices per book for A, B, C, and D are 3 yuan, 5 yuan, 7 yuan, and 11 yuan respectively, and that at least one of each book was bought, how many different purchasing methods are there in total?

Answer: 4


Given the line $y=kx+b\left(k \gt 0\right)$ is tangent to the circle $x^{2}+y^{2}=1$ and the circle $\left(x-4\right)^{2}+y^{2}=1$, find $k=$____ and $b=$____.

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


Given that \(a\) and \(d\) are non-negative numbers, \(b\) and \(c\) are positive numbers, and \(b+c \geqslant a+d\), find the minimum value of the following expression:
$$
\frac{b}{c+d} + \frac{c}{a+b}
$$

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


Points A and B are on a circle of radius 7 and AB = 8. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC?

Answer: \sqrt{98 - 14\sqrt{33}}


If both $a$ and $b$ in the fraction $\frac{a+b}{a^2+b^2}$ are enlarged by a factor of $2$, then calculate the new value of the fraction.

Answer: \frac{1}{2}


The angles at the larger base of a trapezoid are 30 degrees and 60 degrees, and the shorter leg is 5. Find the difference between the bases.

Answer: 10


On each face of a cube, the midlines are divided into points in a $1:3$ ratio. What is the ratio of the volume of the convex shape determined by these points to the volume of the cube?

Answer: \frac{1}{2}


The function $f(x)$ satisfies
\[ f(xy) = 3xf(y) \]
for all real numbers $x$ and $y$. If $f(1) = 10$, find $f(5)$.

Answer: 150


In a $4 \times 4$ grid, numbers are written in such a way that the sum of the numbers in the neighboring cells (cells that share a side) of each number is equal to 1.

Find the sum of all the numbers in the grid.

Answer: 6


How many different patterns can be made by shading exactly two of the nine squares?  Patterns that can be matched by flips and/or turns are not considered different.  For example, the patterns shown below are not considered different.

Answer: 8


In triangle \(ABC\), the angle \(ABC\) is 120 degrees, and points \(K\) and \(M\) are marked on the side \(AC\) such that \(AK = AB\) and \(CM = CB\). From point \(K\), a perpendicular \(KH\) is drawn to the line \(BM\). Find the ratio \(BK : KH\).

Answer: 2:1


The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased?

Answer: 1598


The stem-and-leaf plot shows the duration of songs (in minutes and seconds) played during a concert by a band. There are 15 songs listed in the plot. In the stem-and-leaf plot, $3 \ 15$ represents $3$ minutes, $15$ seconds, which is the same as $195$ seconds. Find the median duration of the songs. Express your answer in seconds.

\begin{tabular}{c|ccccc}
1&30&45&50&&\\
2&10&20&30&35&50\\
3&00&15&15&30&45\\
\end{tabular}

Answer: 170


Every member of the entertainment team can either sing, dance, or both. It is known that 2 people can sing and 5 people can dance. Now, 2 people are selected from the entertainment team. Let $\xi$ be the number of people selected who can both sing and dance, and $P(\xi>0) = \frac{7}{10}$. Calculate $E \xi$.

Answer: 0.8


Petya and Vasya competed in a 60-meter race. When Petya finished, Vasya was 9 meters behind him. In the second race, Petya stood exactly 9 meters behind Vasya at the starting point. Who finished first in the second race and by how many meters did he lead his opponent? (Assume that each boy ran both times at his own constant speed).

Answer: 1.35


On the parade ground, 2018 soldiers are lined up in a single row. The commander can order either all soldiers standing in even positions or all soldiers standing in odd positions to leave the formation. After this order, the soldiers remaining in formation close ranks into a single row. In how many ways can the commander issue a series of 8 orders so that there are exactly 7 soldiers left in formation?

Answer: 30


In the plane \( Oxy \), point \( A \) with coordinates \( (x, y) \) satisfies the equation \( 2a^2 + 2ax + x^2 - 2xy + 2y^2 = 0 \), and a circle centered at point \( B \) is given by the equation \( a^2 x^2 + a^2 y^2 - 2a^3 x - 6a x - 2a^2 y + a^4 + 9 = 0 \). Find all values of parameter \( a \) for which points \( A \) and \( B \) lie on opposite sides of the line \( x = 4 \) (points \( A \) and \( B \) do not lie on this line).

Answer: (-\infty,-2)\cup(0,1)\cup(3,\infty)


Answer: 132


Answer: 2\sqrt{2}+2


While taking the SAT, you become distracted by your own answer sheet.  Because you are not bound to the College Board's limiting rules, you realize that there are actually  $32$  ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could mark  $AB$ , or  $AC$ , or  $ABD$ , or even  $ABCDE$ , or nothing at all!

You begin to wonder how many ways you could mark off the 10 questions you haven't yet answered.  To increase the challenge, you wonder how many ways you could mark off the rest of your answer sheet without ever marking the same letter twice in a row.  (For example, if  $ABD$  is marked for one question,  $AC$  cannot be marked for the next one because  $A$  would be marked twice in a row.)  If the number of ways to do this can be expressed in the form  $2^m p^n$ , where  $m,n > 1$  are integers and  $p$  is a prime, compute  $100m+n+p$ .

*Proposed by Alexander Dai*

Answer: 2013


There are 48 students participating in three sports activities. Each student only participates in one activity, and the number of participants in each activity is different. Each number contains the digit "6". How many students are participating in each of the three sports activities?

Answer: 6,16,26


In an equilateral triangular prism \( S-ABC \), the lateral edges are equal in length to the edges of the base. If \( E \) and \( F \) are the midpoints of \( SC \) and \( AB \) respectively, what is the angle between the skew lines \( EF \) and \( SA \)?

Answer: 45


A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?

Answer: 1462


A quadratic trinomial  $P(x)$  with the  $x^2$  coefficient of one is such, that  $P(x)$  and  $P(P(P(x)))$  share a root. Prove that  $P(0)*P(1)=0$ .

Answer: P(0)\cdotP(1)=0


A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?

Answer: 401


In the diagram below, $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{3}{5}$ and $\sin C = \frac{1}{4}$, what is the length of $DC$?

Answer: 18\sqrt{15}


Let \( f(x) = x^3 + 3x - 1 \) have roots \( a, b, c \). Given that
\[
\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}
\]
can be written as \( \frac{m}{n} \), where \( m \) and \( n \) are positive integers and \( \operatorname{gcd}(m, n) = 1 \), find \( 100m + n \).

Answer: 3989


The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
$\textbf{(A)}\ \frac{10}{3}\qquad\textbf{(B)}\ \frac{20}{3}\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ \frac{15}{2}\qquad\textbf{(E)}\ 8$

Answer: \textbf{(D)}\\frac{15}{2}


In the isosceles triangle \(ABC\) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\). In what ratio does the line \(AD\) divide the altitude \(BE\) of triangle \(ABC\), counting from vertex \(B\)?

Answer: 1:2


Points \( K \) and \( M \) are marked on the sides \( BC \) and \( AD \) of the inscribed quadrilateral \( ABCD \), respectively, such that \( BK : KC = AM : MD \). A point \( L \) is chosen on the segment \( KM \) such that \( KL : LM = BC : AD \). Find the ratio of the areas of triangles \( ACL \) and \( BDL \), given that \( AC = p \) and \( BD = q \).

Answer: \frac{p}{q}


Let \( p \) and \( q \) be positive real numbers with \( p + q < 1 \). Teams \( A \) and \( B \) play a series of games. For each game, \( A \) wins with probability \( p \), \( B \) wins with probability \( q \), and they tie with probability \( 1-p-q \). The series ends when one team has won two more games than the other, that team being declared the winner of the series. What is the probability that \( A \) wins the series?

Answer: \frac{p^2}{p^2+q^2}


Given an equilateral triangle \( ABC \). Point \( K \) is the midpoint of side \( AB \), and point \( M \) lies on side \( BC \) such that \( BM : MC = 1 : 3 \). A point \( P \) is chosen on side \( AC \) such that the perimeter of triangle \( PKM \) is minimized. In what ratio does point \( P \) divide side \( AC \)?

Answer: 2/3


In a tetrahedron \(ABCD\), one edge has a length of 3, and the other five edges each have a length of 2. Find the radius of its circumscribed sphere.

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


The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10.

Answer: 400


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

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


Let \( x, y, z, w \) be different positive real numbers such that \( x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{w}=w+\frac{1}{x}=t \). Find \( t \).

Answer: \sqrt{2}


The sides of triangle \( T \) are parallel to the medians of triangle \( T_1 \). Prove that the medians of triangle \( T \) are parallel to the sides of triangle \( T_1 \).

Answer: ThemediansoftriangleTparalleltothesidesoftriangleT_{1}


Find all positive integers  $a,b,c$  satisfying  $(a,b)=(b,c)=(c,a)=1$  and \[ \begin{cases} a^2+b\mid b^2+c b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of  $a^2+b$  are congruent to  $1$  modulo  $7$

Answer: (1,1,1)


Person A departs from point $A$ to meet person B, traveling 80 kilometers to reach point $B$. At that moment, person B had already left point $B$ for point $C$ half an hour earlier. Person A had been traveling for 2 hours by the time they left point $A$, and then decides to proceed to point $C$ at twice their original speed. After 2 more hours, both A and B arrive at point $C$ simultaneously. What is the speed of person B in kilometers per hour?

Answer: 64


On a plane, a line  $\ell$  and two circles  $c_1$  and  $c_2$  of different radii are given such that   $\ell$   touches both circles at point  $P$ . Point  $M \ne P$  on  $\ell$  is chosen so that the angle  $Q_1MQ_2$  is as large as possible where  $Q_1$  and  $Q_2$  are the tangency points of the tangent lines drawn from  $M$  to  $c_i$  and  $c_2$ , respectively, differing from  $\ell$  . Find  $\angle PMQ_1 + \angle PMQ_2$ ·

Answer: 180


Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probability that Bob wins can be written as a fraction \( \frac{a}{b} \) in lowest terms. What is \( a + b \)?

Answer: 3895


Answer: 2+\sqrt{5}


In a math competition, there are three problems: A, B, and C. Among the 25 participating students, each student solves at least one problem. Among all the students who did not solve problem A, the number of students who solved problem B is twice the number of students who solved problem C. The number of students who only solved problem A is one more than the number of students who solved problem A among the remaining students. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B?

Answer: 6


Find the smallest positive integer $k$ such that, for all $a$ satisfying $0 \leqslant a \leqslant 1$ and for all positive integers $n$, the inequality $a^{k}(1-a)^{n}<\frac{1}{(n+1)^{3}}$ holds.

Answer: 4


If the numbers $x$ and $y$ are inversely proportional and when the sum of $x$ and $y$ is 54, $x$ is three times $y$, find the value of $y$ when $x = 5$.

Answer: 109.35


A fair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6 is rolled twice. Let $a$ and $b$ denote the outcomes of the first and second rolls, respectively.

(1) Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.

(2) Find the probability that the segments with lengths $a$, $b$, and $5$ form an isosceles triangle.

Answer: \frac{7}{18}


Find the point $P$ inside the triangle $ABC$ for which $PA + PB + PC$ is maximum, and also find the point $Q$ for which $QA^2 + QB^2 + QC^2$ is maximum. Prove that $P = Q$.

Answer: Q


Find all integers  $n$  satisfying  $n \geq 2$  and  $\dfrac{\sigma(n)}{p(n)-1} = n$ , in which  $\sigma(n)$  denotes the sum of all positive divisors of  $n$ , and  $p(n)$  denotes the largest prime divisor of  $n$ .

Answer: 6


Consider the identity \( 1+2+\cdots+n=\frac{1}{2} n(n+1) \). If we set \( P_{1}(x)=\frac{1}{2} x(x+1) \), then it is the unique polynomial such that for all positive integer \( n \), \( P_{1}(n)=1+2+\cdots+n \). In general, for each positive integer \( k \), there is a unique polynomial \( P_{k}(x) \) such that
\[ P_{k}(n)=1^{k}+2^{k}+3^{k}+\cdots+n^{k} \quad \text { for each } n=1,2, \ldots . \]

Find the value of \( P_{2010}\left(-\frac{1}{2}\right) \).

Answer: 0


Let \( ABC \) be a triangle and \( D \) the foot of the angle bisector from \( A \). Denote \( O \) (resp. \( O_1 \), resp. \( O_2 \)) as the circumcenter of \( \triangle ABC \) (resp. \( \triangle ABD \), resp. \( \triangle ADC \)). Show that \( OO_1 = OO_2 \).

Answer: OO_1=OO_2


On a ring road, there are three cities: $A$, $B$, and $C$. It is known that the path from $A$ to $C$ along the arc not containing $B$ is three times longer than the path through $B$. The path from $B$ to $C$ along the arc not containing $A$ is four times shorter than the path through $A$. By what factor is the path from $A$ to $B$ shorter along the arc not containing $C$ than the path through $C$?

Answer: 19


Let  $n$  be some positive integer and  $a_1 , a_2 , \dots , a_n$  be real numbers. Denote  $$ S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2}, $$  where  $a_{n+1} = a_1$  and  $a_{n+2} = a_2.$ 1. Show that  $S_0 - S_1 \geq 0$ .
2. Show that  $3$  is the minimum value of  $C$  such that for any  $n$  and  $a_1 , a_2 , \dots , a_n,$  there holds  $C(S_0 - S_1) \geq S_1 - S_2$ .

Answer: 3


Determine, with proof, all integers \( n \) for which it is possible to completely dissect a regular \( n \)-sided polygon into integer-ratio right triangles.

Answer: 4


Find the derivative.

\[ y = \frac{1}{\sqrt{2}} \cdot \operatorname{arctg} \frac{3 x-1}{\sqrt{2}}+\frac{1}{3} \cdot \frac{3 x-1}{3 x^{2}-2 x+1} \]

Answer: \frac{4}{3(3x^2-2x+1)^2}


Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence.

Answer: 1039


On the board, there are two-digit numbers. Each number is composite, but any two numbers are coprime. What is the maximum number of such numbers that can be written?

Answer: 4


Find a natural number \( N \) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \( N \).

Answer: 12005


A sequence of positive integers is defined by  $a_0=1$  and  $a_{n+1}=a_n^2+1$  for each  $n\ge0$ . Find  $\text{gcd}(a_{999},a_{2004})$ .

Answer: 677


In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form.

Answer: 2\sqrt{5}


Marisela is putting on a juggling show! She starts with  $1$  ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.

Answer: 390


The base of a regular tetrahedron is $ABC$, and its apex is $D$. Show that a plane can pass through the midpoints of edges $DC$, $DB$, $AC$, and $AB$. Cut the tetrahedron with a plane parallel to this plane and calculate the perimeter of the resulting cross-section.

Answer: 2a


A particle is located at some point on a line. During the first second, it splits in half and the halves move away from each other in opposite directions by a distance of 1 from the original position. During the next second, the resulting particles again split in half and the halves move away from each other in opposite directions by a distance of 1 from their previous positions. When any two particles collide, they annihilate each other. For example, after two seconds, only two particles remain. How many particles will remain after 129 seconds?

Answer: 4


Given $\tan 2\alpha= \frac {3}{4}$, $\alpha\in(- \frac {\pi}{2}, \frac {\pi}{2})$, $f(x)=\sin (x+\alpha)+\sin (\alpha-x)-2\sin \alpha$, and for any $x\in\mathbb{R}$, it always holds that $f(x)\geqslant 0$, find the value of $\sin (\alpha- \frac {\pi}{4})$.

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


A square has sides of length 3 units. A second square is formed having sides that are $120\%$ longer than the sides of the first square. This process is continued sequentially to create a total of five squares. What will be the percent increase in the perimeter from the first square to the fifth square? Express your answer to the nearest tenth.

Answer: 107.4\%


A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x$.

Answer: 166


Show that

$$
\varlimsup_{t \rightarrow \infty}|\varphi(t)|=1
$$

for the characteristic function $\varphi(t)$ of an arbitrary discrete random variable $\xi$.

To prove this, first consider the Fejér kernel

$$
F_{n}(t)=\frac{1}{n} \sum_{k=0}^{n-1} \sum_{|j| \leqslant k} e^{i j t}, \quad n \geqslant 1, \quad t \in \mathbb{R}
$$

and verify that $F_{n}(t) \geqslant 0$ for all $t$. Then show that if $\xi$ takes values $0, x_{1}, x_{2}, \ldots$ with probabilities $p_{0}, p_{1}, p_{2}, \ldots$, then

$$
\overline{\lim }_{t \rightarrow \infty}|\varphi(t)| \geqslant \lim_{T\to\infty} \frac{1}{T} \int_{0}^{T} \varphi(t) \overline{K_{N}(t, n)} \, dt \geqslant p_{0}+\frac{n-1}{n} \sum_{k=1}^{N} p_{k}
$$

where $K_{N}(t, n)=\prod_{k=1}^{N} F_{n}\left(x_{k} t\right)$, $n, N \geqslant 1$, and then take the limit as $n, N \rightarrow \infty$ to establish that the equality holds.

Answer: 1


12 teachers each made a guess about the answer to this question. Their guesses were: "not less than 1", "not more than 2", "not less than 3", "not more than 4", "not less than 11", "not more than 12" (with "not less than" followed by an odd number and "not more than" followed by an even number). How many teachers guessed the correct answer?

Answer: 7


For what values of the parameter \( a \) does the equation \( |f(x) - 4| = p(x) \), where \( f(x) = \left| \frac{x^2 + 3x}{x+3} - \frac{x^2 - 4x + 4}{2-x} \right| \) and \( p(x) = a \), have three solutions? If there is more than one value for the parameter, give the product of these values.

Answer: 2


A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?

Answer: 9


In the Cartesian coordinate plane $(xOy)$, if the line $ax + y - 2 = 0$ intersects the circle centered at $C$ with the equation $(x - 1)^2 + (y - a)^2 = 16$ at points $A$ and $B$, and $\triangle ABC$ is a right triangle, then the value of the real number $a$ is _____.

Answer: -1


There are some playing cards on the table. Person A took a prime number of cards, leaving the remaining number of cards as a multiple of 5. Person B then took a prime number of cards, leaving the remaining number of cards as a multiple of 3. Person C took a prime number of cards, leaving the remaining number of cards as a multiple of 2. Person D took a prime number of cards, and a prime number of cards were left for Person E. It is known that the numbers of cards taken by A, B, C, D, and E are in decreasing order. What is the minimum number of cards originally on the table?

Answer: 63


Answer: \frac{1}{^2}-\frac{1}{b^2}=\frac{1}{}


Calculate the base five product of the numbers $131_5$ and $12_5$.

Answer: 2122_5


Lord Moneybag said to his grandson, "Bill, listen carefully! Christmas is almost here. I have taken an amount between 300 and 500 pounds, which is a multiple of 6. You will receive 5 pounds in 1-pound coins. When I give you each pound, the remaining amount will first be divisible by 5, then by 4, then by 3, then by 2, and finally by 1 and itself only. If you can tell me how much money I have, you'll get an extra ten." How much money did the lord take?

Answer: 426


Find the total length of the intervals on the number line where the inequalities $x < 1$ and $\sin (\log_{2} x) < 0$ hold.

Answer: \frac{2^{\pi}}{1+2^{\pi}}


A section of a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ by a plane passing through its vertex $D$ intersects the lateral edges $A A_{1}$, $B B_{1}$, and $C C_{1}$ at points $K$, $M$, and $N$, respectively. Find the ratio of the volume of the pyramid with vertex at point $P$ and base $D K M N$ to the volume of the given parallelepiped, if it is known that point $P$ divides the edge $D D_{1}$ in the ratio $D P : P D_{1} = m : n$.

Answer: \frac{}{3(+n)}


Let \(A B C D E F\) be a regular hexagon. A frog starts at vertex \(A\). Each time, it can jump to one of the two adjacent vertices. If the frog reaches vertex \(D\) within 5 jumps, it stops jumping. If it does not reach vertex \(D\) within 5 jumps, it stops after completing the 5 jumps. How many different sequences of jumps are possible from the start until the frog stops?

Answer: 26


How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$)

Answer: 2


We are given a cone with height 6, whose base is a circle with radius  $\sqrt{2}$ . Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?

![Image](https://i.imgur.com/AHqHHP6.png)

Answer: \frac{3}{2}


There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers  $a$  and  $b$  appear on the two packed cards, the number  $a + b + ab$  is set on the new card.
If we start with ten cards with the numbers  $1, 2, 3, 4, 5, 6, 7, 8, 9$  and  $10$  respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer

Answer: 11!-1


Given an ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a perpendicular line to the x-axis through the left focus $F_{1}(-2,0)$ intersects the ellipse at points $P$ and $Q$. The line $PF_{2}$ intersects the y-axis at $E(0, \dfrac {3}{2})$. $A$ and $B$ are points on the ellipse located on either side of $PQ$.
- (I) Find the eccentricity $e$ and the standard equation of the ellipse;
- (II) When $\angle APQ=\angle BPQ$, is the slope $K_{AB}$ of line $AB$ a constant value? If so, find this constant value; if not, explain why.

Answer: - \dfrac {1}{2}


Given that Liliane has $30\%$ more cookies than Jasmine and Oliver has $10\%$ less cookies than Jasmine, and the total number of cookies in the group is $120$, calculate the percentage by which Liliane has more cookies than Oliver.

Answer: 44.44\%


A certain store sells a product, and because the purchase price decreased by 6.4% compared to the original purchase price, the profit margin increased by 8 percentage points. What was the original profit margin for selling this product?

Answer: 17\%


In the 3rd grade, the boys wear blue swim caps, and the girls wear red swim caps. The male sports commissioner says, "I see 1 more blue swim cap than 4 times the number of red swim caps." The female sports commissioner says, "I see 24 more blue swim caps than red swim caps." Based on the sports commissioners' statements, calculate the total number of students in the 3rd grade.

Answer: 37


A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. Determine the minimum number of turns required to remove exactly  $2012$  rocks from the table.

Answer: 18


A subset  $B$  of  $\{1, 2, \dots, 2017\}$  is said to have property  $T$  if any three elements of  $B$  are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property  $T$  may contain.

Answer: 1009


Answer: 2026


Find the greatest natural number $n$ such that $n\leq 2008$ and $(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]$ is a perfect square.

Answer: 1921


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c\sin A= \sqrt {3}a\cos C$ and $(a-c)(a+c)=b(b-c)$, consider the function $f(x)=2\sin x\cos ( \frac {π}{2}-x)- \sqrt {3}\sin (π+x)\cos x+\sin ( \frac {π}{2}+x)\cos x$.
(1) Find the period and the equation of the axis of symmetry of the function $y=f(x)$.
(2) Find the value of $f(B)$.

Answer: \frac {5}{2}


A company sells a brand of cars in two places, A and B, with profits (in units of 10,000 yuan) of $L_{1}=-x^{2}+21x$ and $L_{2}=2x$ respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 cars in both places, what is the maximum profit it can achieve?

Answer: 120


Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse?

Answer: 15


For how many integers \( n \) between 1 and 20 (inclusive) is \( \frac{n}{18} \) a repeating decimal?

Answer: 14


King Arthur's bodyguards had a knight tournament. At the end of the tournament, it was revealed that for any two bodyguards, there is a third one who defeated both of them. What is the minimum number of bodyguards who participated in the tournament?

Answer: 7


A point $P$ is chosen at random in the interior of a unit square $S$. Let $d(P)$ denote the distance from $P$ to the closest side of $S$. The probability that $\frac{1}{5}\le d(P)\le\frac{1}{3}$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 281


A set of six edges of a regular octahedron is called Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles?

Answer: 6


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

Answer: 341


A triangle has side lengths \(a\), \(b\), and \(c\). Prove that \(\sqrt{a + b - c} + \sqrt{b + c - a} + \sqrt{c + a - b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}\). When do you have equality?

Answer: \sqrt{}+\sqrt{}+\sqrt{b}\le\sqrt{}+\sqrt{b}+\sqrt{}


Given that point $P$ lies on the line $3x+4y+8=0$, and $PA$ and $PB$ are the two tangents drawn from $P$ to the circle $x^{2}+y^{2}-2x-2y+1=0$. Let $A$ and $B$ be the points of tangency, and $C$ be the center of the circle. Find the minimum possible area of the quadrilateral $PACB$.

Answer: 2\sqrt{2}


Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 20/3 & 4/3 \\ -8/3 & 8/3 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$

Answer: \begin{pmatrix} 20/3 & 4/3 \\ -8/3 & 8/3 \end{pmatrix}


Two spheres are inscribed in a dihedral angle and touch each other. The radius of one sphere is twice the radius of the other, and the line connecting their centers forms a \(45^{\circ}\) angle with the edge of the dihedral angle. Find the measure of the dihedral angle. In your answer, write the cosine of this angle, rounding if necessary to two decimal places.

Answer: 0.56


 $\triangle ABC$  has area  $240$ . Points  $X, Y, Z$  lie on sides  $AB$ ,  $BC$ , and  $CA$ , respectively. Given that  $\frac{AX}{BX} = 3$ ,  $\frac{BY}{CY} = 4$ , and  $\frac{CZ}{AZ} = 5$ , find the area of  $\triangle XYZ$ .

[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label(" $A$ ",A,N);
label(" $B$ ",B,S);
label(" $C$ ",C,E);
label(" $X$ ",X,W);
label(" $Y$ ",Y,S);
label(" $Z$ ",Z,NE);[/asy]

Answer: 122


In the acute-angled triangle  $ABC$ , the altitudes  $BP$  and  $CQ$  were drawn, and the point  $T$  is the intersection point  of the altitudes of  $\Delta PAQ$ . It turned out that  $\angle CTB = 90 {} ^ \circ$ . Find the measure of  $\angle BAC$ .

(Mikhail Plotnikov)

Answer: 45


Compute the number of subsets  $S$  of  $\{0,1,\dots,14\}$  with the property that for each  $n=0,1,\dots,
6$ , either  $n$  is in  $S$  or both of  $2n+1$  and  $2n+2$  are in  $S$ .

*Proposed by Evan Chen*

Answer: 2306


Let $$ A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}} $$ Find the largest integer less than or equal to $\frac{1}{A}$.

Answer: 1


What is the largest real number \(\theta\) less than \(\pi\) (i.e. \(\theta < \pi\)) such that
$$
\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0
$$
and
$$
\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1?
$$

Answer: \frac{2046\pi}{2047}


For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have?

Answer: 4022


The hares are sawing a log. They made 10 cuts. How many pieces of wood did they get?

Answer: 11


The faces of a dice are labeled with the numbers $6, 7, 8, 9, 10, 11$. The dice was rolled twice. The first time, the sum of the numbers on the four "vertical" faces (i.e., excluding the bottom and top faces) was 33, and the second time the sum of these numbers was 35. What number can be on the face opposite the face with the number 7? Find all possible options.

Answer: 911


A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?
$\textbf{(A)}\ \dfrac{\pi}{3}-1 \qquad \textbf{(B)}\ \dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3} \qquad \textbf{(C)}\ \dfrac{\pi}{18} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{2\pi}{9}$

Answer: \textbf{(B)}\\frac{2\pi}{9}-\frac{\sqrt{3}}{3}


A sequence \(\{a_{n}\}\) has nine terms, \(a_{1} = a_{9} = 1\), and for each \(i \in \{1, 2, \cdots, 8\}\), it holds that \(\frac{a_{i+1}}{a_{i}} \in \{2, 1, -\frac{1}{2}\}\). Determine the number of such sequences.

Answer: 491


Let $f$ be a function taking the positive integers to the positive integers, such that

(i) $f$ is increasing (i.e. $f(n + 1) > f(n)$ for all positive integers $n$)
(ii) $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $f(m) = n$ or $f(n) = m.$

Find the sum of all possible values of $f(30).$

Answer: 900


Let \(a, b\), and \(c\) be positive real numbers such that
\[
\begin{aligned}
a^{2}+a b+b^{2} & =9, \\
b^{2}+b c+c^{2} & =52, \\
c^{2}+c a+a^{2} & =49.
\end{aligned}
\]
Compute the value of \(\frac{49 b^{2}-33 b c+9 c^{2}}{a^{2}}\).

Answer: 52


Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a.$

Answer: 501


Let  $M$  be a finite subset of the plane such that for any two different points  $A,B\in M$  there is a point  $C\in M$  such that  $ABC$  is equilateral. What is the maximal number of points in  $M?$

Answer: 3


Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$?

Answer: 2592


Suppose the quadratic function $f(x)=ax^2+bx+c$ has a maximum value $M$ and a minimum value $m$ in the interval $[-2,2]$, and the set $A={x|f(x)=x}$.

(1) If $A={1,2}$ and $f(0)=2$, find the values of $M$ and $m$.
(2) If $A={2}$ and $a\geqslant 1$, let $g(a)=M+m$, find the minimum value of $g(a)$.

Answer: \frac{63}{4}


The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.

Answer: 120^\circ


Find the largest natural number \( n \) with the following property: for any odd prime number \( p \) less than \( n \), the difference \( n - p \) is also a prime number.

Answer: 10


At each vertex of a tetrahedron, there is an ant. At a given moment, each ant randomly chooses an edge and moves along it to the neighboring vertex. What is the probability that two ants will meet either midway on the edge or at the end of their journey?

Answer: \frac{25}{27}


Given triangle \( \triangle ABC \) with midpoint \( N \) of side \( AB \), where \( \angle A > \angle B \). Point \( D \) lies on ray \( AC \) such that \( CD = BC \). Point \( P \) lies on ray \( DN \) on the same side of \( BC \) as point \( A \), such that \( \angle PBC = \angle A \). Segment \( PC \) intersects \( AB \) at point \( E \) and segment \( DP \) intersects \( BC \) at point \( T \). Find the value of the expression \( \frac{BC}{TC} - \frac{EA}{EB} \).

Answer: 2


In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$?

Answer: 34


What is the smallest positive integer \( n \) which cannot be written in any of the following forms?
- \( n = 1 + 2 + \cdots + k \) for a positive integer \( k \).
- \( n = p^k \) for a prime number \( p \) and integer \( k \).
- \( n = p + 1 \) for a prime number \( p \).
- \( n = pq \) for some distinct prime numbers \( p \) and \( q \).

Answer: 40


In triangle \( ABC \), the angle bisectors \( BP \) and \( CT \) intersect at point \( O \). It is known that points \( A, P, O \), and \( T \) lie on the same circle. Find angle \( A \).

Answer: 60


We know that  $201$  and  $9$  give the same remainder when divided by  $24$ . What is the smallest positive integer  $k$  such that  $201+k$  and  $9+k$  give the same remainder when divided by  $24$ ?

*2020 CCA Math Bonanza Lightning Round #1.1*

Answer: 1


Let \(ABC\) be a triangle with \(AB=2021, AC=2022\), and \(BC=2023\). Compute the minimum value of \(AP+2BP+3CP\) over all points \(P\) in the plane.

Answer: 6068


Given \(\alpha, \beta, \gamma \in \left(0, \frac{\pi}{4}\right)\) and \(\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1\). Prove that \(\frac{\pi}{2} < \alpha + \beta + \gamma \leq 3 \arcsin \frac{\sqrt{3}}{3}\).

Answer: \frac{\pi}{2}<\alpha+\beta+\gamma\le3\arcsin\frac{\sqrt{3}}{3}


Answer: 4


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( f\left(x^{2}+f(y)\right)=y+f(x)^{2} \).

Answer: f(x)=x


Let $n$ be a positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ is an integer. Which of the following statements is not  true:
$\mathrm{(A)}\ 2\ \text{divides\ }n \qquad\mathrm{(B)}\ 3\ \text{divides\ }n \qquad\mathrm{(C)}$ $\ 6\ \text{divides\ }n  \qquad\mathrm{(D)}\ 7\ \text{divides\ }n \qquad\mathrm{(E)}\ n > 84$

Answer: \mathrm{(E)}\n>84


In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$.  If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$?

Answer: 259


Let \( A B C \) be an isosceles triangle with \( A B = A C \). Let \( D \) and \( E \) be the feet of the perpendiculars from \( B \) and \( C \) to \( \overline{A C} \) and \( \overline{A B} \), respectively. Suppose that \( \overline{C E} \) and \( \overline{B D} \) intersect at point \( H \). If \( E H = 1 \) and \( A D = 4 \), find \( D E \).

Answer: \frac{8\sqrt{17}}{17}


In right triangle $A B C$, a point $D$ is on hypotenuse $A C$ such that $B D \perp A C$. Let $\omega$ be a circle with center $O$, passing through $C$ and $D$ and tangent to line $A B$ at a point other than $B$. Point $X$ is chosen on $B C$ such that $A X \perp B O$. If $A B=2$ and $B C=5$, then $B X$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.

Answer: 8041


Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 50\qquad\textbf{(E)}\ 60$

Answer: 40\textbf{(C)}


Given the function $f(x)=|x-2|+|x-a^{2}|$.
$(1)$ If the inequality $f(x)\leqslant a$ has solutions for $x$, find the range of the real number $a$;
$(2)$ If the positive real numbers $m$, $n$ satisfy $m+2n=a$, when $a$ takes the maximum value from $(1)$, find the minimum value of $\left( \dfrac {1}{m}+ \dfrac {1}{n}\right)$.

Answer: \dfrac {3}{2}+ \sqrt {2}


Given \( m = \frac{\sin x}{\sin (y-z)}, n = \frac{\sin y}{\sin (z-x)}, p = \frac{\sin z}{\sin (x-y)} \), find the value of \( m n + n p + p m \).

Answer: -1


There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result?

Answer: 197


Find the largest  $a$  for which there exists a polynomial  $$ P(x) =a x^4 +bx^3 +cx^2 +dx +e $$  with real coefficients which satisfies  $0\leq P(x) \leq 1$  for  $-1 \leq x \leq 1.$

Answer: 4


For integers \(a, b, c, d\), let \(f(a, b, c, d)\) denote the number of ordered pairs of integers \((x, y) \in \{1,2,3,4,5\}^{2}\) such that \(ax + by\) and \(cx + dy\) are both divisible by 5. Find the sum of all possible values of \(f(a, b, c, d)\).

Answer: 31


Solve in positive integers the equation
\[ n \equal{} \varphi(n) \plus{} 402 ,
\]
where  $ \varphi(n)$  is the number of positive integers less than  $ n$  having no common prime factors with  $ n$ .

Answer: 802


Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?

Answer: 17


Two concentric circles have radii 2006 and 2007. $ABC$ is an equilateral triangle inscribed in the smaller circle and $P$ is a point on the circumference of the larger circle. Given that a triangle with side lengths equal to $PA, PB$, and $PC$ has area $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a+b+c$.

Answer: 4020


Through the point with the coordinates $(9,9)$, lines (including those parallel to the coordinate axes) are drawn, dividing the plane into angles of $9^{\circ}$. Find the sum of the x-coordinates of the points where these lines intersect with the line $y = 10 - x$.

Answer: 95


Given that  $5^{2018}$  has  $1411$  digits and starts with  $3$  (the leftmost non-zero digit is  $3$ ), for how many integers  $1\leq n\leq2017$  does  $5^n$  start with  $1$ ?

*2018 CCA Math Bonanza Tiebreaker Round #3*

Answer: 607


Given a cone with vertex $P$, base radius 2, and height 1. A point $Q$ is taken on the base of the cone such that the angle between the line $PQ$ and the base does not exceed $45^{\circ}$. What is the area of the region formed by all such points $Q$?

Answer: 3\pi


Suppose $x$, $y$, and $z$ are all positive real numbers, and $x^{2}+y^{2}+z^{2}=1$, then the minimum value of $\frac{(z+1)^{2}}{2xyz}$ is $\_\_\_\_\_\_$.

Answer: 3+2 \sqrt{2}


Altitudes $B E$ and $C F$ of acute triangle $A B C$ intersect at $H$. Suppose that the altitudes of triangle $E H F$ concur on line $B C$. If $A B=3$ and $A C=4$, then $B C^{2}=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Answer: 33725


Given that \( n \) is a natural number less than 50, find all \( n \) such that \( 3n + 5 \) and \( 5n + 4 \) have a common divisor greater than 1.

Answer: 7,20,33,46


A positive integer is called friendly if it is divisible by the sum of its digits. For example, 111 is friendly but 123 is not. Find the number of all two-digit friendly numbers.

Answer: 23


On the height $B H$ of triangle $A B C$, a certain point $D$ is marked. Line $A D$ intersects side $BC$ at point $E$, line $C D$ intersects side $A B$ at point $F$. It is known that $B H$ divides segment $F E$ in the ratio $1: 3$, starting from point $F$. Find the ratio $F H: H E$.

Answer: 1:3


There are nine cards, each with the numbers $2, 3, 4, 5, 6, 7, 8, 9, 10$. Four people, A, B, C, and D, each draw two of these cards.

Person A says: "The two numbers I drew are relatively prime because they are consecutive."
Person B says: "The two numbers I drew are not relatively prime and are not multiples of each other."
Person C says: "The two numbers I drew are both composite numbers and they are relatively prime."
Person D says: "The two numbers I drew are in a multiple relationship and they are not relatively prime."

Assuming all four people are telling the truth, what is the number on the remaining card? $\quad \quad$

Answer: 7


Let \( x \) and \( y \) be real numbers. Determine the maximum value of \( \frac{2x + \sqrt{2}y}{2x^4 + 4y^4 + 9} \).

Answer: \frac{1}{4}


Let  $n$  be a positive integer. Suppose there are exactly  $M$  squarefree integers  $k$  such that  $\left\lfloor\frac nk\right\rfloor$  is odd in the set  $\{ 1, 2,\ldots, n\}$ . Prove  $M$  is odd.

An integer is *squarefree* if it is not divisible by any square other than  $1$ .

Answer: M


Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2512, \\
pr&=&qs&=&1225.
\end{array}
\]
If \( T = p+q+r+s \), compute the value of \( \lfloor T \rfloor \).

Answer: 140


In the country of Taxonia, each person pays as many thousandths of their salary in taxes as the number of tugriks that constitutes their salary. What salary is most advantageous to have?

(Salary is measured in a positive number of tugriks, not necessarily an integer.)

Answer: 500


A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of different colours, they are then joined by a green line segment. In total, there are 52 red dots. There are 2 red dots at corners with an additional 16 red dots on the edges of the array. The remainder of the red dots are inside the array. There are 98 green line segments. The number of blue line segments is

Answer: 37


Find the number of strictly increasing sequences of nonnegative integers with the following properties: - The first term is 0 and the last term is 12. In particular, the sequence has at least two terms. - Among any two consecutive terms, exactly one of them is even.

Answer: 144


I planned to work 25 hours a week for 15 weeks to earn $3750$ for a vacation. However, due to a family emergency, I couldn't work for the first three weeks. How many hours per week must I work for the remaining weeks to still afford the vacation?

Answer: 31.25


Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$.

Answer: 32


A construction company is building a tunnel. After completing $\frac{1}{3}$ of the tunnel at the original speed, they use new equipment, resulting in a $20\%$ increase in construction speed and daily working hours reduced to $80\%$ of the original. The entire tunnel is completed in 185 days. If the new equipment had not been used, how many days would it take to complete the tunnel at the original speed?

Answer: 180


In an acute-angled triangle \(ABC\), heights \(AP\) and \(CQ\) are drawn from vertices \(A\) and \(C\) to sides \(BC\) and \(AB\), respectively. Given that the area of triangle \(ABC\) is 18, the area of triangle \(BPQ\) is 2, and \(PQ = 2\sqrt{2}\), find the radius of the circumcircle of triangle \(ABC\).

Answer: \frac{9}{2}


Given that events A and B are independent, and both are mutually exclusive with event C. It is known that $P(A) = 0.2$, $P(B) = 0.6$, and $P(C) = 0.14$. Find the probability that at least one of A, B, or C occurs, denoted as $P(A+B+C)$.

Answer: 0.82


How many integers $m$ are there from 1 to 1996, such that $\frac{m^{2}+7}{m+4}$ is not a reduced fraction?

Answer: 86


The points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( S A B, S A C, S B C, \) and \( A B C \) of the tetrahedron \( S A B C \). It is known that \( A B = S C = 5 \), \( A C = S B = 7 \), \( B C = S A = 8 \). Find the volume of the tetrahedron \( K L M N \). If necessary, round your answer to two decimal places.

Answer: 0.66


The traffic police brigade of our county is carrying out a comprehensive road traffic safety rectification campaign "Hundred-Day Battle" throughout the county, which strictly requires riders of electric bicycles and motorcycles to comply with the rule of "one helmet, one belt". A certain dealer purchased a type of helmet at a unit price of $30. When the selling price is $40, the monthly sales volume is 600 units. On this basis, for every $1 increase in the selling price, the monthly sales volume will decrease by 10 units. In order for the dealer to achieve a monthly profit of $10,000 from selling this helmet and to minimize inventory as much as possible, what should be the actual selling price of this brand of helmet? Explain your reasoning.

Answer: 50


Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{|6a - 4b| + |3(a + b\sqrt{3}) + 2(a\sqrt{3} - b)|}{\sqrt{a^2 + b^2}}
$$

Answer: \sqrt{39}


Given a set of positive real numbers \(A=\left\{a_{1}, a_{2}, a_{3}, \cdots, a_{100}\right\}\), define the set \(S=\{(a, b) \mid a \in A, b \in A, a - b \in A\}\). Determine the maximum number of elements in the set \(S\).

Answer: 4950


Given \( n, k \in \mathbf{N}^{+} \), let \( S \) be a set of \( n \) points in the plane (with no three points collinear), and for any point \( P \) in \( S \), there exist \( k \) points in \( S \) that are equidistant from \( P \). Prove that \( k \leq \frac{1}{2} + \sqrt{2n} \).

Answer: k\le\frac{1}{2}+\sqrt{2n}


In the encoded equality $A B + A B + A B + A B + A B + A B + A B + A B + A B = A A B$, digits are replaced with letters: the same digits with the same letter, and different digits with different letters. Find all possible decipherings. (I. Rubanov)

Answer: 25


Given positive integers \( a, b, \) and \( c \) such that \( a < b < c \). If the product of any two numbers minus 1 is divisible by the third number, what is \( a^{2} + b^{2} + c^{2} \)?

Answer: 38


Find the remainder when   $$ \left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor $$   is divided by  $10^4$ .

*Proposed by Vijay Srinivasan*

Answer: 7800


Given right triangle \( \triangle ABC \), where \(\angle ACB = 90^{\circ}, BC = a, CA = b\). Points \( D, E, F \) lie on sides \( BC, CA, AB \) respectively, such that \( \triangle DEF \) is an equilateral triangle. Find the minimum area of \( \triangle DEF \).

Answer: \frac{\sqrt{3}^2b^2}{4(^2+b^2+\sqrt{3})}


Determine all integers $n > 1$ such that $\frac{2^{n}+1}{n^{2}}$ is an integer.

Answer: 3


When $10^{95} - 95 - 2$ is expressed as a single whole number, calculate the sum of the digits.

Answer: 840


How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row?

Answer: 199776


Let  $A$  be the set  $\{k^{19}-k: 1<k<20, k\in N\}$ . Let  $G$  be the GCD of all elements of  $A$ .
Then the value of  $G$  is?

Answer: 798


A school volleyball team played several matches. After winning the next match, the share of victories increased by $1 / 6$. In order to increase the share of victories by another $1 / 6$, the team had to win two more consecutive matches. What is the minimum number of wins the team needs to achieve to increase the share of victories by another $1 / 6$?

Answer: 6


130 trees are planted in a circle: birches and lindens (both types are present). Each tree has a sign that reads: "Next to this tree, two different trees grow." It is known that all the lindens have false statements and exactly one birch has a false statement. How many birches could have been planted? List all possible options.

Answer: 87


Two circles have radii  $15$  and  $95$ . If the two external tangents to the circles intersect at  $60$  degrees, how far apart are the centers of the circles?

Answer: 160


If: (1) \(a, b, c, d\) are all elements of the set \(\{1,2,3,4\}\); (2) \(a \neq b\), \(b \neq c\), \(c \neq d\), \(d \neq a\); (3) \(a\) is the smallest among \(a, b, c, d\). Then, how many different four-digit numbers \(\overline{abcd}\) can be formed?

Answer: 24


In a parallelogram  $ABCD$ , points  $E$  and  $F$  are the midpoints of  $AB$  and  $BC$ , respectively, and  $P$  is the intersection of  $EC$  and  $FD$ . Prove that the segments  $AP,BP,CP$  and  $DP$  divide the parallelogram into four triangles whose areas are in the ratio  $1 : 2 : 3 : 4$ .

Answer: 1:2:3:4


Given the function $f(x)=-\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$.

Answer: -4


How many $3$-digit positive integers have digits whose product equals $24$?

$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

Answer: \textbf{(D)}\21


Three real numbers $x, y, z$ are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of $x-y$ and $x-z$ is greater than $-\frac{1}{2}$ and less than $\frac{1}{2}$?

Answer: \frac{7}{12}


For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$.

Answer: 26


Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that $|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.$ What is the smallest possible value of $n$?

Answer: 20


A rectangle with sides $a$ and $b$ rotates around an axis passing through its vertex, parallel to the diagonal that does not pass through this vertex. Find the volume of the resulting solid of revolution. Given: $ABCD$ is a rectangle, $|AB|=a$, $|AD|=b$, $l$ is the axis of rotation, $A \in l$, $l \parallel [BD]$. Find: $V_{ABCD}$, the volume of the solid of revolution.

Answer: \frac{2\pi^2b^2}{\sqrt{^2+b^2}}


Given the function $f(x)=x^3-3x-a$, find the value of $(M-N)$, where $M$ and $-N$ are the maximum and minimum values of $f(x)$ on the interval $[0, 3]$.

Answer: 20


From a container filled to the brim with $100\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water. On the third day, after mixing the contents again, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water. The next morning, her parents discovered that the volume of water in the container was 1.5 liters more than the volume of the remaining juice. How many liters of juice did Masha ultimately drink? If the answer is ambiguous, provide the sum of all possible values of the desired quantity.

Answer: 1.75


Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, with $g(x) \neq 0$, $f(x)g'(x) > f'(x)g(x)$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \neq 1$), $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$. For the finite sequence $\frac{f(n)}{g(n)} = (n = 1, 2, \ldots, 0)$, find the probability that the sum of the first $k$ terms is greater than $\frac{15}{16}$ for any positive integer $k$ ($1 \leq k \leq 10$).

Answer: \frac{3}{5}


For positive integers  $n$ , let  $S_n$  be the set of integers  $x$  such that  $n$  distinct lines, no three concurrent, can divide a plane into  $x$  regions (for example,  $S_2=\{3,4\}$ , because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum  $i$  such that  $S_i$  contains at least 4 elements?

Answer: 4


Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win of 30 rubles, and there are no other prizes. The Scientist decided to take a risk since he has nothing to lose. Find the probability that the Absent-Minded Scientist will be able to win enough money to buy a bus ticket.

Answer: 0.19


Diagonals of a quadrilateral  $ABCD$  are equal and meet at point  $O$ . The perpendicular bisectors to segments  $AB$  and  $CD$  meet at point  $P$ , and the perpendicular bisectors to  $BC$  and  $AD$  meet at point  $Q$ . Find angle  $\angle POQ$ .

by A.Zaslavsky

Answer: 90


How many five-digit numbers are there, not divisible by 1000, in which the first, third, and last digits are even?

Answer: 9960


The numbers \( p \) and \( q \) are chosen such that the parabola \( y = p x - x^2 \) intersects the hyperbola \( x y = q \) at three distinct points \( A, B, \) and \( C \). The sum of the squares of the sides of triangle \( ABC \) is 324, and the intersection point of its medians is at a distance of 2 from the origin. Find the product \( pq \).

Answer: 42


Vasya wrote down 11 natural numbers in one notebook. Petya wrote down the greatest common divisors of each pair of numbers recorded in Vasya's notebook in another notebook. It turned out that every number written in one of the two notebooks is also in the other notebook. What is the maximum number of distinct numbers that could have been written in Vasya's notebook?

Answer: 10


Bill buys a stock for $100. On the first day, the stock decreases by $25\%$, on the second day it increases by $35\%$ from its value at the end of the first day, and on the third day, it decreases again by $15\%$. What is the overall percentage change in the stock's value over the three days?

Answer: -13.9375\%


Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcasting methods.

Answer: 36


Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
\[P(1) = P(3) = P(5) = P(7) = a\]and
\[P(2) = P(4) = P(6) = P(8) = -a.\]What is the smallest possible value of $a$?

Answer: 315


Let  $V=[(x,y,z)|0\le x,y,z\le 2008]$  be a set of points in a 3-D space.
If the distance between two points is either  $1, \sqrt{2}, 2$ , we color the two points differently.
How many colors are needed to color all points in  $V$ ?

Answer: 7


Let $S$ be the set \{1,2, \ldots, 2012\}. A perfectutation is a bijective function $h$ from $S$ to itself such that there exists an $a \in S$ such that $h(a) \neq a$, and that for any pair of integers $a \in S$ and $b \in S$ such that $h(a) \neq a, h(b) \neq b$, there exists a positive integer $k$ such that $h^{k}(a)=b$. Let $n$ be the number of ordered pairs of perfectutations $(f, g)$ such that $f(g(i))=g(f(i))$ for all $i \in S$, but $f \neq g$. Find the remainder when $n$ is divided by 2011 .

Answer: 2


It is given polygon with  $2013$  sides  $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any  $9$  consecutive vertices is constant and its value is  $300$ . If we know that  $A_{13}$  is marked with  $13$  and  $A_{20}$  is marked with  $20$ , determine with which number is marked  $A_{2013}$

Answer: 67


Given that point O is the center of the regular octagon ABCDEFGH, and Y is the midpoint of the side CD, determine the fraction of the area of the octagon that is shaded if the shaded region includes triangles DEO, EFO, and half of triangle CEO.

Answer: \frac{5}{16}


Let \( A = (a_{ij}) \), where \( i, j = 1, 2, \ldots , n \), be a square matrix with all \( a_{ij} \) non-negative integers. For each \( i, j \) such that \( a_{ij} = 0 \), the sum of the elements in the \( i \)th row and the \( j \)th column is at least \( n \). Prove that the sum of all the elements in the matrix is at least \( \frac{n^2}{2} \).

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


Let $p, q, r$, and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2500,\\
pr&=&qs&=&1200.
\end{array}
\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$.

Answer: 140


Find the lateral surface area of a right triangular prism with height \( h \), given that a line passing through the center of the top base and the midpoint of a side of the bottom base is inclined at an angle of \( 60^{\circ} \) to the plane of the base.

Answer: 6h^2


In the quadrilateral \(ABCD\), it is given that \(\cos \angle BAD = \frac{3}{4}\), \(\angle BAC = \angle DAC\), \(AD < AB\), and \(AB = 5\), \(AC = BD = \sqrt{14}\). If \(\overrightarrow{AC} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AD}\) (\(\lambda, \mu \in \mathbf{R}\)), find \(\lambda + \mu\).

Answer: \frac{7}{5}


From a square with a side length of $6 \text{ cm}$, identical isosceles right triangles are cut off from each corner so that the area of the square is reduced by $32\%$. What is the length of the legs of these triangles?

Answer: 2.4


Let \(ABCD\) be a parallelogram with an area of 1, and let \(M\) be a point on the segment \([BD]\) such that \(MD = 3MB\). Let \(N\) be the point of intersection of the lines \( (AM) \) and \( (CB) \). Calculate the area of the triangle \(MND\).

Answer: \frac{1}{8}


Define the integer sequences $\{x_{n}\}$ and $\{y_{n}\}$ as follows:
$$
\begin{array}{l}
x_{0}=1, x_{1}=1, x_{n+1}=x_{n}+2x_{n-1}\; (n \geqslant 1); \\
y_{0}=1, y_{1}=7, y_{n+1}=2y_{n}+3y_{n-1}\; (n \geqslant 1).
\end{array}
$$

Prove that these two sequences have no common terms other than 1.

Answer: 1


Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{3-\sqrt{2}} \sin x + 1) \cdot (3 + 2\sqrt{7-\sqrt{2}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.

Answer: -9


Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF.

Answer: 10


The symbols $(a,b,\ldots,g)$ and $[a,b,\ldots, g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $a,b,\ldots, g$. For example, $(3,6,18)=3$ and $[6,15]=30$. Prove that

$\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$.

Answer: \frac{[,b,]^2}{[,b][b,][,]}=\frac{(,b,)^2}{(,b)(b,)(,)}


Let \( X \) be the set of rational numbers excluding 0 and \(\pm1\). Define \( f: X \to X \) by \( f(x) = x - \frac{1}{x} \). For each positive integer \( n \), let \( f_1(x) = f(x) \), \( f_2(x) = f(f(x)) \), \( f_3(x) = f(f_2(x)) \), and so forth. Does there exist a value \( x \) in \( X \) such that for any positive integer \( n \), there exists \( y \) in \( X \) with \( f_n(y) = x \)?

Answer: No


Let \( S \) be a set consisting of \( m \) pairs \((a, b)\) of positive integers with the property that \( 1 \leq a < b \leq n \). Show that there are at least
\[ 4m \frac{\left(m-\frac{n^2}{4}\right)}{3n} \]
triples \((a, b, c)\) such that \((a, b)\), \((a, c)\), and \((b, c)\) belong to \( S \).

Answer: 4m\frac{(-\frac{n^2}{4})}{3n}


Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \_\_\_\_\_ kg.

Answer: 70


If the set $\{1, a, \frac{b}{a}\}$ equals the set $\{0, a^2, a+b\}$, then find the value of $a^{2017} + b^{2017}$.

Answer: -1


How many different values can $\angle ABC$ take, where $A, B, C$ are distinct vertices of a cube?

Answer: 5


Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.

Answer: 20


Numbers 1 through 9 are written in the cells of a $3 \times 3$ table. All possible sums of the numbers in adjacent (by side) cells are then listed in a notebook. What is the minimum number of different sums that could be listed in the notebook?

Answer: 4


Prove that the polynomial with rational coefficients $f(x)=\frac{1}{5} x^{5}+\frac{1}{2} x^{4}+\frac{1}{3} x^{3}-\frac{1}{30} x$ is an integer-valued polynomial.

Answer: f(x)


In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho^{2}-2\rho\cos \theta-4\rho\sin \theta+4=0$, and the equation of line $l$ is $x-y-1=0$.
$(1)$ Write the parametric equation of curve $C$;
$(2)$ Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value.

Answer: 1+ \sqrt{2}


Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'm rounding. For example, there are 17 people. I say that a third turned. Should one person split in your opinion? No, with rounding, six people turned. From whole numbers, the closest to the fraction $17 / 3$ is 6. And if I say that half of the 17 people turned, it means 8 or 9 people." It is known that Baron Munchausen never lies. What is the largest number of people that could have been in the crowd?

Answer: 37


Let $\mathrm{P}$ be a polynomial with integer coefficients, such that there exist three distinct integers $a, b, c$ for which $|P(a)|=|P(b)|=|P(c)|=1$. Show that $P$ has no integer roots.

Answer: P


Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, given that $a^{2} + b^{2} - 3c^{2} = 0$, where $c$ is the semi-latus rectum, find the value of $\frac{a + c}{a - c}$.

Answer: 3 + 2\sqrt{2}


In the plane rectangular coordinate system \( xOy \), the orthocenter of a triangle \( \triangle ABC \) whose vertices lie on the parabola \( \Gamma: y^{2}=2px \) (where \( p > 0 \)) is the focus \( F \). Given that the centroid of \( \triangle ABC \) is \( G \), find the range of possible values for the slope \( k \) of the line \( FG \).

Answer: [-\frac{\sqrt{7}}{7},\frac{\sqrt{7}}{7}]


A man chooses two positive integers \( m \) and \( n \). He defines a positive integer \( k \) to be good if a triangle with side lengths \( \log m \), \( \log n \), and \( \log k \) exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of \( mn \).

Answer: 134


A nut has the shape of a regular hexagonal prism. Each lateral face of the nut is painted in one of three colors: white, red, or blue, with adjacent faces painted in different colors. How many different nut paintings are possible? (It is not necessary to use all three colors in the painting.)

Answer: 13


The rules of a card arrangement game are as follows: Arrange nine cards marked with $1,2, \cdots, 9$ randomly in a row. If the number on the first card (from the left) is $k$, then reverse the order of the first $k$ cards, which is called an operation. The game stops when no operation can be performed (i.e., the number on the first card is "1"). If an arrangement cannot be operated on and can be obtained by performing one operation on a unique other arrangement, it is called a "second stop arrangement". Find the probability of a second stop arrangement occurring among all possible arrangements.

Answer: \frac{103}{2520}


Five points are placed on a sphere of radius 1. What is the largest possible value for the shortest distance between two of the points? Find all configurations for which the maximum is attained.

Answer: \sqrt{2}


Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$, where $J$ is the center of the octagon. By connecting every second vertex starting from $A$, we obtain a larger equilateral triangle $\triangle ACE$. Compute the ratio of the area of $\triangle ABJ$ to the area of $\triangle ACE$.

Answer: \frac{1}{4}


Given that for all \(\theta \in \mathbb{R}\), the modulus of the complex number \((a+\cos \theta) + (2a - \sin \theta)i\) does not exceed 2, determine the range of values for the real number \(a\).

Answer: [-\frac{\sqrt{5}}{5},\frac{\sqrt{5}}{5}]


Given that \(\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)\) and \(\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1\), prove that \(\frac{\sin^3 \alpha}{\sin \beta}+\frac{\sin^3 \beta}{\sin \gamma}+\frac{\sin^3 \gamma}{\sin \alpha} \geq 1\).

Answer: 1


Let \( AEF \) be a triangle with \( EF = 20 \) and \( AE = AF = 21 \). Let \( B \) and \( D \) be points chosen on segments \( AE \) and \( AF \), respectively, such that \( BD \) is parallel to \( EF \). Point \( C \) is chosen in the interior of triangle \( AEF \) such that \( ABCD \) is cyclic. If \( BC = 3 \) and \( CD = 4 \), then the ratio of areas \(\frac{[ABCD]}{[AEF]}\) can be written as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).

Answer: 5300


The obtuse angle formed by the hands of a clock at $10:30$ is $(100+a)^{\circ}$. Find $a$.

The lines $a x + b y = 0$ and $x - 5 y + 1 = 0$ are perpendicular to each other. Find $b$.

If $(b+1)^{4}=2^{c+2}$, find $c$.

If $c-9=\log _{c}(6d-2)$, find $d$.

Answer: 2


Let \( a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6} \) be non-negative integers and satisfy
\[
\begin{cases}
a_{1} + 2a_{2} + 3a_{3} + 4a_{4} + 5a_{5} + 6a_{6} = 26 \\
a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} = 5
\end{cases}
\]
If \( c \) is the number of solutions to the system of equations, determine the value of \( c \).

Answer: 5


There are 1959 positive numbers $a_{1}, a_{2}, \ldots, a_{1959}$, the sum of which is equal to 1. All possible combinations of 1000 numbers are considered, with combinations being regarded as identical if they differ only in the order of the numbers. For each combination, the product of the included numbers is considered. Prove that the sum of all these products is less than 1.

Answer: <1


Let  $n$  be a positive integer. Let  $B_n$  be the set of all binary strings of length  $n$ . For a binary string  $s_1\hdots s_n$ , we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by  $b$ . Then, we replace  $s_b$  with  $1-s_b$ . A string  $a$  is said to be a *descendant* of  $b$  if  $a$  can be obtained from  $b$  through a finite number of twists. A subset of  $B_n$  is called *divided* if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of  $B_n$ .

*Remark.* Here is an example of a twist:  $101100 \rightarrow 101000$  because  $1\mid 0\mid 11\mid 00$  has  $4$  blocks of consecutive digits.

*Viktor Simjanoski*

Answer: 2^{n-2}


Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives?

Answer: \frac{11}{72}


Consider an equilateral triangular grid $G$ with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points. Let $S$ be a closed non-selfintersecting polygon which has 210 vertices, using each point in $G$ exactly once. Find the sum of all possible values of the area of $S$.

Answer: 52 \sqrt{3}


Let $A B C$ be a triangle with $A B=8, B C=15$, and $A C=17$. Point $X$ is chosen at random on line segment $A B$. Point $Y$ is chosen at random on line segment $B C$. Point $Z$ is chosen at random on line segment $C A$. What is the expected area of triangle $X Y Z$ ?

Answer: 15


Let \( f, g: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*} \) be strictly increasing functions such that
\[ f(\mathbf{N}^{*}) \cup g(\mathbf{N}^{*}) = \mathbf{N}^{*} \]
and
\[ f(\mathbf{N}^{*}) \cap g(\mathbf{N}^{*}) = \varnothing \]
with
\[ g(n) = f(f(n)) + 1. \]

Find \( f(240) \).

Answer: 388


Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as "division exponentiation," denoted as $a^{ⓝ}$, read as "$a$ circle $n$ times." For example, $2\div 2\div 2$ is denoted as $2^{③}$, read as "$2$ circle $3$ times"; $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$ is denoted as $\left(-3\right)^{④}$, read as "$-3$ circle $4$ times".
$(1)$ Write down the results directly: $2^{③}=$______, $(-\frac{1}{2})^{④}=$______;
$(2)$ Division exponentiation can also be converted into the form of powers, such as $2^{④}=2\div 2\div 2\div 2=2\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=(\frac{1}{2})^{2}$. Try to directly write the following operation results in the form of powers: $\left(-3\right)^{④}=$______; ($\frac{1}{2})^{⑩}=$______; $a^{ⓝ}=$______;
$(3)$ Calculate: $2^{2}\times (-\frac{1}{3})^{④}\div \left(-2\right)^{③}-\left(-3\right)^{②}$.

Answer: -73


What is the least positive integer  $n$  such that  $25^n +16^n$  leaves a remainder of  $1$  when divided by  $121$ ?

Answer: 32


A store has a safe with the password $3854 \square 942$. The fifth digit of the password is forgotten, but it is known that the password is the product of $5678 \times 6789$. What should be filled in the blank $\square$?

Answer: 7


A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59.

Answer: 24


Specify a six-digit number $N$ consisting of distinct digits such that the numbers $2N$, $3N$, $4N$, $5N$, and $6N$ are permutations of its digits.

Answer: 142857


In triangle $ABC$ where $AB = 3 \cdot BC$, point $M$ is the midpoint of side $AB$ and $BD$ is the angle bisector. Find the angle $MDB$.

Answer: 90


An isosceles triangle \(ABC\) has an inscribed rhombus \(DECF\) such that vertex \(E\) lies on side \(BC\), vertex \(F\) on side \(AC\), and vertex \(D\) on side \(AB\). Find the side length of the rhombus if \(AB = BC = 12\) and \(AC = 6\).

Answer: 4


Determine all positive integers  $a,b,c$  such that  $ab + ac + bc$  is a prime number and  $$ \frac{a+b}{a+c}=\frac{b+c}{b+a}. $$

Answer: (1,1,1)


Find the functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that

$$
\forall x, y \in \mathbb{Q}, f(1)=2 \text{ and } f(x y)=f(x) f(y)-f(x+y)+1
$$

Answer: f(x)=x+1


Points $A, B, C, D$ lie on a circle in that order such that $\frac{A B}{B C}=\frac{D A}{C D}$. If $A C=3$ and $B D=B C=4$, find $A D$.

Answer: \frac{3}{2}


Given a regular tetrahedron with edge length $a$, prove that the sum of the squares of the lengths of the projections (on any plane) of the segments joining its center to its vertices is equal to $a^{2}$.

Answer: ^2


The simple quadrilateral \(ABCD\) has sides \(AB\), \(BC\), and \(CD\) with lengths 4, 5, and 20, respectively. If the angles \(B\) and \(C\) are obtuse, and \(\sin C = -\cos B = \frac{3}{5}\), calculate the length of the side \(AD\).

Answer: 25


In the sequence $\left\{a_n\right\}$, each term $a_n$ represents the largest odd divisor of the positive integer $n$. For example, the divisors of 5 are $1$ and $5$, so $a_5 = 5$. The divisors of 6 are $1, 2, 3, 6$, so $a_6 = 3$.

Let $S_n$ be the sum of the first $n$ terms of the sequence $\left\{a_n\right\}$. What is $S_{2^{2000}-1}$?

Answer: \frac{4^{2000}-1}{3}


Let  $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$ .

a) Show that  $\sum\limits_{n=1}^{+\infty}a_nx^n$  converges for all  $x \in (-4, 4)$  and that the function  $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$  satisfies the differential equation  $x(x - 4)f'(x) + (x + 2)f(x) = -x$ .

b) Prove that  $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$ .

Answer: \frac{1}{3}+\frac{2\pi\sqrt{3}}{27}


In right triangle  $ ABC$  with right angle at  $ C$ ,  $ \angle BAC < 45$  degrees and  $ AB \equal{} 4$ .  Point  $ P$  on  $ AB$  is chosen such that  $ \angle APC \equal{} 2\angle ACP$  and  $ CP \equal{} 1$ .  The ratio  $ \frac{AP}{BP}$  can be represented in the form  $ p \plus{} q\sqrt{r}$ , where  $ p,q,r$  are positive integers and  $ r$  is not divisible by the square of any prime.  Find  $ p\plus{}q\plus{}r$ .

Answer: 7


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

Answer: 49


Consider a larger grid extending from point $A$ to point $B$, now divided into a 3x2 grid. You can still only move right or down along the drawn segments. How many different routes are there from point $A$ to point $B$?

[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,10)--(0,10)--cycle);
draw((5,0)--(5,10));
draw((10,0)--(10,10));
draw((0,5)--(15,5));
dot((0,10));
dot((15,0));
label("$A$",(0,10),NW);
label("$B$",(15,0),SE);
[/asy]

Answer: 10


Given point \( P \), circle \( k \), and the secant line \( AB \) passing through \( P \) such that \( PA = AB = 1 \). The tangents from \( P \) to circle \( k \) touch the circle at points \( C \) and \( D \). The intersection of \( AB \) and \( CD \) is point \( M \). What is the distance \( PM \)?

Answer: \frac{4}{3}


In the arithmetic sequence $\{a_n\}$, it is given that $a_{15}+a_{16}+a_{17}=-45$ and $a_{9}=-36$. Let $S_n$ be the sum of the first $n$ terms.
$(1)$ Find the minimum value of $S_n$ and the corresponding value of $n$;
$(2)$ Calculate $T_n=|a_1|+|a_2|+\cdots+|a_n|$.

Answer: -630


Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 929


Given positive numbers \( a \) and \( b \) that satisfy \( 2 + \log_{2} a = 3 + \log_{3} b = \log_{6}(a+b) \), find the value of \( \frac{1}{a} + \frac{1}{b} \).

Answer: 108


The school organized a picnic with several attendees. The school prepared many empty plates. Each attendee who arrives will count the empty plates and then take one plate for food (no one can take more than one plate). The first attendee will count all the empty plates, the second will count one fewer, and so on. The last attendee will find 4 empty plates remaining. It is known that the sum of the total number of plates prepared by the school and the total number of attendees is 2015. How many people attended the picnic in total?

Answer: 1006


In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base.

Answer: 6/5


Given the parabola $y=x^{2}+2x-3$ intersects the line $l_{1}$: $y=-x+m$ at points $A$ and $C$, and the line $l_{2}$ is symmetric with respect to the axis of symmetry of the parabola. Line $l_{2}$ intersects the parabola at points $B$ and $D$, where points $A$ and $D$ are above the $x$-axis, and points $B$ and $C$ are below the $x$-axis. If $AC \cdot BD = 26$, then the value of $m$ is ______.

Answer: -2


Find all values of the parameter \( a \) for which the equation

\[ 8|x-4 a|+\left|x-a^{2}\right|+7 x-2 a=0 \]

has no solution.

Answer: (-\infty,-22)\cup(0,\infty)


Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes was the bicycle not in motion?

Answer: 75


Given a parallelogram with area  $1$  and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the  $8$  lines formed we have  a octagon inside of the parallelogram. Determine the area of this octagon

Answer: \frac{1}{2}


In a convex quadrilateral \(ABCD\) with parallel sides \(AD\) and \(BC\), a line \(L\) is drawn parallel to \(AD\) and intersects sides \(AB\) and \(CD\) at points \(M\) and \(N\), respectively. It is known that quadrilaterals \(AMND\) and \(MBNS\) are similar, and the sum of the lengths of sides \(AD\) and \(BC\) is no more than 4. Find the maximum possible length of segment \(MN\).

Answer: 2


A triangle has a perimeter \( 2p \) and an inscribed circle. A tangent to this circle, parallel to one side of the triangle, is drawn. Find the maximum possible length of the segment of this tangent that lies within the triangle.

Answer: \frac{p}{4}


Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis.

Answer: 17


Given real numbers \( x, y, z, w \) such that \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \).

Answer: 3/2


A circle $\omega$ with center $O$ is circumscribed around triangle $ABC$. Circle $\omega_1$ is tangent to line $AB$ at point $A$ and passes through point $C$, and circle $\omega_2$ is tangent to line $AC$ at point $A$ and passes through point $B$. A line through point $A$ intersects circle $\omega_1$ at point $X$ and circle $\omega_2$ at point $Y$ such that $X$ and $Y$ are distinct from $A$. Point $M$ is the midpoint of segment $XY$. Find the angle $OMX$.

Answer: 90


Given triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) intersect at point \( K \), find the value of \( \frac{|KO|}{|KH|} \).

Answer: 3/2


Given that \( x, y, z \) are positive real numbers and \( x + y + z = 1 \). If \(\frac{a}{xyz} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} - 2\), then the range of the real number \( a \) is \(\qquad\).

Answer: (0,\frac{7}{27}]


Given the line \( l \) passing through \( A(2,1) \) intersects the hyperbola \( x^2 - \frac{y^2}{2} = 1 \) at two points \( P_1 \) and \( P_2 \). Find the equation of the locus of the midpoint of the line segment \( P_1 P_2 \).

Answer: 2x^2-y^2-4x+y=0


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

Answer: -\frac{7}{16}


After watching a movie, viewers rated the film one by one with an integer score from 0 to 10. At any given moment, the movie’s rating was calculated as the sum of all scores given, divided by their count. At a particular moment $T$, the rating turned out to be an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers who could have voted after moment $T$?

Answer: 5


In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid.

Answer: 56


Let \( AB \) and \( AC \) be two distinct rays not lying on the same line, and let \( \omega \) be a circle with center \( O \) that is tangent to ray \( AC \) at \( E \) and ray \( AB \) at \( F \). Let \( R \) be a point on segment \( EF \). The line through \( O \) parallel to \( EF \) intersects line \( AB \) at \( P \). Let \( N \) be the intersection of lines \( PR \) and \( AC \), and let \( M \) be the intersection of line \( AB \) and the line through \( R \) parallel to \( AC \). Prove that line \( MN \) is tangent to \( \omega \).

Answer: MNistangentto\omega


Answer: \frac{1}{^5+b^5+^5}=\frac{1}{^5}+\frac{1}{b^5}+\frac{1}{^5}


Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed?

Answer: 14


Three cubes with volumes 1, 8, and 27 are glued together at their faces. If \(a\) is the smallest possible surface area of the resulting polyhedron, find the value of \(a\).

Answer: 72


Given an integer  $n\geq 2$ , determine the maximum value the sum  $x_1+\cdots+x_n$  may achieve, as the  $x_i$  run through the positive integers, subject to  $x_1\leq x_2\leq \cdots \leq x_n$  and  $x_1+\cdots+x_n=x_1 x_2\cdots x_n$ .

Answer: 2n


Let $A B C$ be an isosceles triangle with $A B=A C$. Let $D$ and $E$ be the midpoints of segments $A B$ and $A C$, respectively. Suppose that there exists a point $F$ on ray $\overrightarrow{D E}$ outside of $A B C$ such that triangle $B F A$ is similar to triangle $A B C$. Compute $\frac{A B}{B C}$.

Answer: \sqrt{2}


There are two ribbons, one is 28 cm long and the other is 16 cm long. Now, we want to cut them into shorter ribbons of the same length without any leftovers. What is the maximum length of each short ribbon in centimeters? And how many such short ribbons can be cut?

Answer: 11


The coach of the math training team needs to photocopy a set of materials for 23 team members. The on-campus copy shop charges 1.5 yuan per page for the first 300 pages and 1 yuan per page for any additional pages. The cost of photocopying these 23 sets of materials together is exactly 20 times the cost of photocopying a single set. How many pages are in this set of photocopy materials?

Answer: 950


Let \( A_1 \), \( B_1 \), \( C_1 \) be the feet of the altitudes of an acute-angled triangle \( ABC \). The circle inscribed in triangle \( A_1B_1C_1 \) is tangent to the sides \( A_1B_1 \), \( A_1C_1 \), and \( B_1C_1 \) at points \( C_2 \), \( B_2 \), and \( A_2 \), respectively. Prove that the lines \( AA_2 \), \( BB_2 \), \( CC_2 \) intersect at a single point that lies on the Euler line of triangle \( ABC \).

Answer: 0


The line \( y = kx + 1 \) intersects the left branch of the hyperbola \( x^2 - y^2 = 1 \) at points \( A \) and \( B \). The line \( l \) passes through the point \((-2, 0)\) and the midpoint of segment \( AB \). Find the range of values for the y-intercept \( b \) of line \( l \).

Answer: (-\infty,-2-\sqrt{2})\cup(2,\infty)


Let  $r$  be a positive integer. Show that if a graph  $G$  has no cycles of length at most  $2r$ , then it has at most  $|V|^{2016}$  cycles of length exactly  $2016r$ , where  $|V|$  denotes the number of vertices in the graph  $G$ .

Answer: |V|^{2016}


Let  $ABCD$  be a cyclic quadrilateral with circumradius  $100\sqrt{3}$  and  $AC=300$ . If  $\angle DBC = 15^{\circ}$ , then find  $AD^2$ .

*Proposed by Anand Iyer*

Answer: 60000


In the cells of a \(3 \times 3\) square are the numbers \(1, 2, 3, \ldots, 9\). It is known that any two consecutive numbers are located in neighboring (side-adjacent) cells. What number can be in the center cell if the sum of the numbers in the corner cells is \(18\)?

Answer: 7


Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ and $3 x+1 \in S$. Determine the number of non-negative integers in $S$ less than 2008.

Answer: 128