For nonnegative integers $a$ and $b$ with  $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where  $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.

Answer: 564


In $\triangle ABC$, $AB \neq AC$, $AD \perp BC$, and $D$ is the foot of the perpendicular. The line passing through the incenter $O_1$ of right $\triangle ABD$ and the incenter $O_2$ of right $\triangle ACD$ intersects $AB$ at $K$ and $AC$ at $L$. If $AK = AL$, then $\angle BAC = 90^{\circ}$.

Answer: 90


Little Rabbit and Little Turtle start from point $A$ to the Forest Amusement Park simultaneously. Little Rabbit jumps forward 36 meters in 1 minute and rests after every 3 minutes of jumping. The first rest period is 0.5 minutes, the second rest period is 1 minute, the third rest period is 1.5 minutes, and so on, with the $k$th rest period being $0.5k$ minutes. Little Turtle does not rest or play on the way. It is known that Little Turtle reaches the Forest Amusement Park 3 minutes and 20 seconds earlier than Little Rabbit. The distance from point $A$ to the Forest Amusement Park is 2640 meters. How many meters does Little Turtle crawl in 1 minute?

Answer: 12


In each cell of a $100 \times 100$ square, a certain natural number is written. We call a rectangle, whose sides follow the grid lines, good if the sum of the numbers in all its cells is divisible by 17. It is allowed to paint all the cells in some good rectangle simultaneously. It is prohibited to paint any cell twice. What is the greatest $d$ for which it is possible to paint at least $d$ cells regardless of the arrangement of numbers?

Answer: 9744


Given triangle \( ABC \), where \( 2h_c = AB \) and \( \angle A = 75^\circ \). Find the measure of angle \( C \).

Answer: 75


$ABCDE$  is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments  $AC$  and  $BD$ ?

Answer: 72


The set of vertices of a polygon consists of points with coordinates $(x, y)$ where $x$ and $y$ are positive integers, and also satisfy $x \mid 2y + 1$ and $y \mid 2x + 1$. What is the maximum possible area of the polygon?

Answer: 20


Square $ABCD$ is inscribed in circle $\omega$. An arbitrary point $M$ is chosen on the minor arc $CD$ of circle $\omega$. Inside the square, points $K$ and $L$ are marked such that $KLMD$ is a square. Find $\angle AKD$.

Answer: 135


Around a circular table, there are exactly 60 chairs. $N$ people sit at the table according to the following rule: every new person must sit next to someone who is already seated. What is the smallest possible value of $N$?

(A) 15
(B) 20
(C) 30
(D) 40
(E) 58

Answer: 20


Some positive integers are initially written on a board, where each  $2$  of them are different.
Each time we can do the following moves:
(1) If there are 2 numbers (written in the board) in the form  $n, n+1$  we can erase them and write down  $n-2$ (2) If there are 2 numbers (written in the board) in the form  $n, n+4$  we can erase them and write down  $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer  $c$  such that:
Independetly of the starting numbers, each number which appears in any move is greater or equal to  $c$

Answer: -3


A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (pieces) as follows:
$$
P= \begin{cases}
\frac {1}{96-x} & (1\leq x\leq 94, x\in \mathbb{N}) \\
\frac {2}{3} & (x>94, x\in \mathbb{N})
\end{cases}
$$
It is known that for every qualified instrument produced, a profit of $A$ yuan can be made, but for every defective product produced, a loss of $\frac {A}{2}$ yuan will be incurred. The factory wishes to determine an appropriate daily output.
(1) Determine whether producing this instrument can be profitable when the daily output (pieces) exceeds 94 pieces, and explain the reason;
(2) When the daily output $x$ pieces does not exceed 94 pieces, try to express the daily profit $T$ (yuan) of producing this instrument as a function of the daily output $x$ (pieces);
(3) To obtain the maximum profit, how many pieces should the daily output $x$ be?

Answer: 84


Given a triangle \(ABC\). It is required to divide it into the smallest number of parts so that, by flipping these parts to the other side, the same triangle \(ABC\) can be formed.

Answer: 3


Given the circle \( \odot O: x^{2}+y^{2}=5 \) and the parabola \( C: y^{2}=2px \) (with \( p > 0 \)), they intersect at point \( A(x_{0}, 2) \). \( AB \) is a diameter of \( \odot O \). A line through point \( B \) intersects the parabola \( C \) at two points \( D \) and \( E \). Find the product of the slopes of the lines \( AD \) and \( AE \).

Answer: 2


Let $A M O L$ be a quadrilateral with $A M=10, M O=11$, and $O L=12$. Given that the perpendicular bisectors of sides $A M$ and $O L$ intersect at the midpoint of segment $A O$, find the length of side LA.

Answer: $\sqrt{77}$


From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible?

Answer: 42


Given that for the angles of triangle \(ABC\), \(\sin A + \cos B = \sqrt{2}\) and \(\cos A + \sin B = \sqrt{2}\), find the measure of angle \(C\).

Answer: 90


Specify the smallest number that ends in 37, has a digit sum of 37, and is divisible by 37.

Answer: 99937


The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$.

Answer: 13


Answer: \sqrt{2007}


How many numbers with less than four digits (from 0 to 9999) are neither divisible by 3, nor by 5, nor by 7?

Answer: 4571


Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?

Answer: p \geq 7


We play on a chessboard with sides of length 100. A move consists of choosing a rectangle formed by squares on the chessboard and inverting the colors. What is the smallest number of moves needed to make the entire chessboard completely black?

Answer: 100


Construct a rational parameterization of the circle \( x^2 + y^2 = 1 \) by drawing lines through the point \((1,0)\).

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


Find the ratio of the volume of a regular hexagonal pyramid to the volume of a regular triangular pyramid, given that the sides of their bases are equal and their slant heights are twice the length of the sides of the base.

Answer: \frac{6 \sqrt{1833}}{47}


Point $A$ has two tangents drawn to a circle, touching the circle at points $M$ and $N$, respectively. A secant from the same point $A$ intersects the circle at points $B$ and $C$, and intersects the chord $MN$ at point $P$. Given that $AB:BC = 2:3$, find the ratio $AP:PC$.

Answer: 4:3


Circles with centers \( O_1 \) and \( O_2 \) and equal radii are inscribed in the angles at \( B \) and \( C \) of triangle \( ABC \) respectively. Point \( O \) is the center of the incircle of triangle \( ABC \). These circles touch side \( BC \) at points \( K_1, K_2, \) and \( K \) respectively, with \( BK_1 = 4, CK_2 = 8, \) and \( BC = 18 \).

a) Find the length of segment \( CK \).

b) Suppose the circle with center \( O_1 \) touches side \( AB \) at point \( K_3 \). Find the angle \( ABC \), given that point \( O_1 \) is the center of the circumscribed circle around triangle \( OK_1K_3 \).

Answer: 60


The circles \(\odot O_{1}\) and \(\odot O_{2}\) intersect at points \(P\) and \(Q\). The chord \(PA\) of \(\odot O_{1}\) is tangent to \(\odot O_{2}\), and the chord \(PB\) of \(\odot O_{2}\) is tangent to \(\odot O_{1}\). Let the circumcenter of \(\triangle PAB\) be \(O\). Prove that \(OQ \perp PQ\).

Answer: OQ\perpPQ


Morteza marked six points on a plane and calculated the areas of all 20 triangles with vertices at these points. Can all of these areas be integers, and can their sum be equal to 2019?

Answer: No


Find the angle $C$ of triangle $ABC$ if vertex $A$ is equidistant from the centers of the excircles that touch sides $AB$ and $BC$.

Answer: 90


Two identical two-digit numbers were written on the board. One of them had 100 added to the left, and the other had 1 added to the right, resulting in the first number becoming 37 times larger than the second. What were the numbers written on the board?

Answer: 27


Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]

[i]Razvan Gelca.[/i]

Answer: (2, 4, 6)


Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.

Answer: 34


Inside an isosceles triangle \( ABC \), a point \( K \) is marked such that \( CK = AB = BC \) and \(\angle KAC = 30^\circ\). Find the angle \( AKB \).

Answer: 150


In the right triangle \(ABC\) with the right angle at vertex \(C\), the angle bisector \(CL\) and the median \(CM\) are drawn.

Find the area of triangle \(ABC\) if \(LM = a\) and \(CM = b\).

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


In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ).

Answer: 19


The ecology club at a school has 30 members: 12 boys and 18 girls.  A 4-person committee is to be chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?

Answer: \dfrac{530}{609}


What is the maximum value of $\frac{(3^t - 5t)t}{9^t}$ for real values of $t$?

Answer: \frac{1}{20}


One side of a rectangle was tripled, and the other side was halved to form a square. What is the side length of the square if the area of the rectangle is $54 \ \mathrm{m}^{2}$?

Answer: 9\,


For which smallest natural number \( k \) does the expression \( 2016 \cdot 2017 \cdot 2018 \cdot 2019 + k \) become a square of a natural number?

Answer: 1


Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?

Answer: 120


Let \( f(x) = |\ln x| \). If the function \( g(x) = f(x) - a x \) has three zero points on the interval \( (0, 4) \), determine the range of values for the real number \( a \).

Answer: (\frac{\ln2}{2},\frac{1}{e})


How many positive integers \( k \) are there such that
\[
\frac{k}{2013}(a+b) = \mathrm{lcm}(a, b)
\]
has a solution in positive integers \( (a, b) \)?

Answer: 1006


Given a semicircle with diameter \( AB \), center \( O \), a line intersects the semicircle at points \( C \) and \( D \), and intersects the line \( AB \) at point \( M \) (\(MB < MA, MD < MC\)). Let the second intersection point of the circumcircles of \( \triangle AOC \) and \( \triangle DOB \) be \( K \). Prove that \( \angle M K O \) is a right angle.

Answer: \angleMKO=90


Mom decides to take Xiaohua on a driving tour to 10 cities during the holidays. After checking the map, Xiaohua is surprised to find that among any three cities, either all three pairs of cities are connected by highways, or exactly one pair of cities is not connected by a highway. How many highways must have been constructed between these 10 cities? (Note: There can be at most one highway between any two cities.)

Answer: 40


What is the maximum possible area of a triangle if the sides \(a, b, c\) satisfy the following inequalities:

$$
0 < a \leq 1 \leq b \leq 2 \leq c \leq 3
$$

Answer: 1


In triangle $PQR$, let the side lengths be $PQ = 7,$ $PR = 8,$ and $QR = 5$. Calculate:
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\]

Answer: \frac{16}{7}


For how many positive integers \( n \) less than 200 is \( n^n \) a cube and \( (n+1)^{n+1} \) a square?

Answer: 40


Let  $ABC$  be an acute triangle. Let $H$  and  $D$  be points on  $[AC]$  and  $[BC]$ , respectively, such that  $BH \perp AC$  and  $HD \perp BC$ . Let  $O_1$  be the circumcenter of  $\triangle ABH$ , and  $O_2$  be the circumcenter of  $\triangle BHD$ , and  $O_3$  be the circumcenter of  $\triangle HDC$ . Find the ratio of area of  $\triangle O_1O_2O_3$  and  $\triangle ABH$ .

Answer: 1/4


1. The focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $Ax+By+C=0$ is $d= \frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}$. By analogy, the distance between the point $(0,1,3)$ and the plane $x+2y+3z+3=0$ is \_\_\_\_\_\_\_\_\_\_\_\_
4. If point $A$ has coordinates $(1,1)$, $F_{1}$ is the lower focus of the ellipse $5y^{2}+9x^{2}=45$, and $P$ is a moving point on the ellipse, then the maximum value of $|PA|+|PF_{1}|$ is $M$, the minimum value is $N$, so $M-N=$ \_\_\_\_\_\_\_\_\_\_\_\_

Answer: 2\sqrt{2}


Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$.

Answer: -4


On the side \( AC \) of triangle \( ABC \), a point \( E \) is marked. It is known that the perimeter of triangle \( ABC \) is 25 cm, the perimeter of triangle \( ABE \) is 15 cm, and the perimeter of triangle \( BCE \) is 17 cm. Find the length of segment \( BE \).

Answer: 3.5\,


Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest square?

Answer: \frac{500}{3}


Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
    \item $\frac{a}{2} \le x \le 2a$
    \item $\frac{a}{2} \le y \le 2a$
    \item $x+y \ge a$
    \item $x+a \ge y$
    \item $y+a \ge x$
\end{enumerate}
The boundary of set $S$ is a polygon with

Answer: 6


If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals

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


\(\{a_{n}\}\) is an arithmetic sequence with \(a_{1} = 1\) and common difference \(d\). \(\{b_{n}\}\) is a geometric sequence with common ratio \(q\). The first three terms of the sequence \(\{a_{n} + b_{n}\}\) are 3, 12, 23. Find \(d + q\).

Answer: 9


Let  $ n\ge 3$  be a natural number. Find all nonconstant polynomials with real coeficcietns  $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$ , for which
\[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad  1\le k\le n,\]
for every real  $ x$  (with  $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$  and  $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$ ).

Answer: f_k(x)=x^2


A natural number  $k > 1$  is called *good* if there exist natural numbers  $$ a_1 < a_2 < \cdots < a_k $$  such that  $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots +  \dfrac{1}{\sqrt{a_k}} = 1 $$  .

Let  $f(n)$  be the sum of the first  $n$  *[good* numbers,  $n \geq$  1. Find the sum of all values of  $n$  for which  $f(n+5)/f(n)$  is an integer.

Answer: 18


Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:

(i) \( f(1) = 0 \),

(ii) \( f(p) = 1 \) for all prime numbers \( p \),

(iii) \( f(x y) = y f(x) + x f(y) \) for all \( x, y \) in \( \mathbb{Z}_{>0} \).

Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n) = n \).

Answer: 3125


In the quadrilateral  $ABCD$ ,  $AB = BC = CD$  and  $\angle BMC = 90^\circ$ , where  $M$  is the midpoint of  $AD$ . Determine the acute angle between the lines  $AC$  and  $BD$ .

Answer: 30


In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have?

Answer: 13


Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\%$ per annum. The interest earned each month is added to the account balance, and the interest for the following month is calculated on the new balance. After how many months will there be enough money in the account to buy the car?

Answer: 29


Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy]

Answer: 23


Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 71


A student's final score on a 150-point test is directly proportional to the time spent studying multiplied by a difficulty factor for the test. The student scored 90 points on a test with a difficulty factor of 1.5 after studying for 2 hours. What score would the student receive on a second test of the same format if they studied for 5 hours and the test has a difficulty factor of 2?

Answer: 300


Answer: 90


In a regular tetrahedron \( S-ABC \) with side length \( a \), \( E \) and \( F \) are the midpoints of \( SA \) and \( BC \) respectively. Find the angle between the skew lines \( BE \) and \( SF \).

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


The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).

Answer: 1705


Answer: \frac{2}{3}


Compute
\[\cos^6 0^\circ + \cos^6 1^\circ + \cos^6 2^\circ + \dots + \cos^6 90^\circ.\]

Answer: \frac{229}{8}


Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$.

Answer: 20


In bag A, there are 3 white balls and 2 red balls, while in bag B, there are 2 white balls and 4 red balls. If a bag is randomly chosen first, and then 2 balls are randomly drawn from that bag, the probability that the second ball drawn is white given that the first ball drawn is red is ______.

Answer: \frac{17}{32}


In $\triangle ABC$, the sides opposite to angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $a= \sqrt{2}$, $b=2$, and $\sin B - \cos B = \sqrt{2}$, find the measure of angle $A$.

Answer: \frac{\pi}{6}


The positive reals  $ x,y$  and  $ z$  are satisfying the relation  $ x \plus{} y \plus{} z\geq 1$ . Prove that:
 $ \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2}$

*Proposer*:**Baltag Valeriu**

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


Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_.

Answer: 12\pi


Let \(ABCD\) be a quadrilateral with \(BC = CD = DA = 1\), \(\angle DAB = 135^\circ\), and \(\angle ABC = 75^\circ\). Find \(AB\).

Answer: \frac{\sqrt{6}-\sqrt{2}}{2}


In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} \overset{x=t}{y=1+t}\end{cases}$$ (t is the parameter), line m is parallel to line l and passes through the coordinate origin, and the parametric equation of circle C is $$\begin{cases} \overset{x=1+cos\phi }{y=2+sin\phi }\end{cases}$$ (φ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis.

1. Find the polar coordinate equations of line m and circle C.
2. Suppose line m and circle C intersect at points A and B. Find the perimeter of △ABC.

Answer: 2+ \sqrt {2}


The real number sequence \(a_0, a_1, a_2, \cdots, a_n, \cdots\) satisfies the following equation: \(a_0 = a\), where \(a\) is a real number,
\[ a_n = \frac{a_{n-1} \sqrt{3} + 1}{\sqrt{3} - a_{n-1}}, \quad n \in \mathbf{N} \]
Find \( a_{1994} \).

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


In the quadrilateral \(ABCD\), \(AB\) is parallel to \(CD\) and \(AD\) is parallel to \(BC\). Prove that there exists a point from which the distances to the lines containing the sides of the quadrilateral are proportional to these sides.

Answer: 0


The vertices of a regular hexagon are labeled $\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real $\theta$ ), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge?

Answer: 14


Find all surjective functions  $f:\mathbb{N}\to\mathbb{N}$  such that for all positive integers  $a$  and  $b$ , exactly one of the following equations is true:
\begin{align*}
f(a)&=f(b), <br />
 f(a+b)&=\min\{f(a),f(b)\}.
\end{align*}
*Remarks:*  $\mathbb{N}$  denotes the set of all positive integers. A function  $f:X\to Y$  is said to be surjective if for every  $y\in Y$  there exists  $x\in X$  such that  $f(x)=y$ .

Answer: f(n)=\nu_2(n)+1


The function $y=f(x)$ is an even function with the smallest positive period of $4$, and when $x \in [-2,0]$, $f(x)=2x+1$. If there exist $x\_1$, $x\_2$, $…x\_n$ satisfying $0 \leqslant x\_1 < x\_2 < … < x\_n$, and $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_1)|+…+|f(x\_{n-1}-f(x\_n))|=2016$, then the minimum value of $n+x\_n$ is \_\_\_\_\_\_.

Answer: 1513


A rectangular piece of paper has integer side lengths. The paper is folded so that a pair of diagonally opposite vertices coincide, and it is found that the crease is of length 65. Find a possible value of the perimeter of the paper.

Answer: 408


If, in the expansion of \((a + 2b)^n\), there exist three consecutive terms whose binomial coefficients form an arithmetic progression, then the largest three-digit positive integer \(n\) is _____.

Answer: 959


Define a function \( D(n) \) on the set of positive integers such that:
(i) \( D(1)=0 \),
(ii) \( D(p)=1 \) when \( p \) is a prime number,
(iii) For any two positive integers \( u \) and \( v \),
\[ D(uv) = uD(v) + vD(u). \]

1. Prove that these three conditions are consistent (i.e., not contradictory) and uniquely determine the function \( D(n) \). Derive a formula for \( \frac{D(n)}{n} \) assuming \( n=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}} \), where \( p_{1}, p_{2}, \cdots, p_{k} \) are distinct primes.
2. Find \( n \) such that \( D(n)=n \).
3. Define \( D^{1}(n)=D(n) \) and \( D^{k+1}(n)=D\left(D^{k}(n)\right) \) for \( k=1,2, \cdots \). Find the limit of \( D^{m}(63) \) as \( m \rightarrow \infty \).

(Note: This is from the 10th Putnam Mathematics Competition, 1950)

Answer: +\infty


Quadrilateral \(ABCD\) is inscribed in a circle with diameter \(AC\). Points \(K\) and \(M\) are the projections of vertices \(A\) and \(C\) respectively onto the line \(BD\). Through point \(K\) a line parallel to \(BC\) is drawn, intersecting \(AC\) at point \(P\). Prove that the angle \(KPM\) is a right angle.

Answer: \angleKPM=90


Find the sum of all positive integers  $B$  such that  $(111)_B=(aabbcc)_6$ , where  $a,b,c$  represent distinct base  $6$  digits,  $a\neq 0$ .

Answer: 237


Given the function $f(x)=\cos (x- \frac {π}{4})-\sin (x- \frac {π}{4}).$
(I) Determine the evenness or oddness of the function $f(x)$ and provide a proof;
(II) If $θ$ is an angle in the first quadrant and $f(θ+ \frac {π}{3})= \frac { \sqrt {2}}{3}$, find the value of $\cos (2θ+ \frac {π}{6})$.

Answer: \frac {4 \sqrt {2}}{9}


Three of the roots of the equation  $x^4 -px^3 +qx^2 -rx+s = 0$  are  $\tan A, \tan B$ , and  $\tan C$ , where  $A, B$ , and  $C$  are angles of a triangle. Determine the fourth root as a function only of  $p, q, r$ , and  $s.$

Answer: \frac{r-p}{-1}


Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$.

Answer: -4


Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 100 residents of this island lined up, and each of them said one of the following phrases:

- "To the left of me there are as many liars as knights."
- "To the left of me there is 1 more liar than knights."
- "To the left of me there are 2 more liars than knights."
- "To the left of me there are 99 more liars than knights."

It is known that each phrase was said by exactly one person. What is the minimum number of liars that can be among these 100 residents?

Answer: 50


Suppose that $m$ and $n$ are positive integers with $m<n$ such that the interval $[m, n)$ contains more multiples of 2021 than multiples of 2000. Compute the maximum possible value of $n-m$.

Answer: 191999


In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the number \( M \) have? List all possible answers.

Answer: 29


Let positive integers $a$, $b$, $c$, and $d$ be randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2000\}$. Find the probability that the expression $ab+bc+cd+d+1$ is divisible by $4$.

Answer: \frac{1}{4}


In a chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.
(1) If there are 4 people per row, how many different arrangements are possible?
(2) If the lead singer stands in the front row and the male singers stand in the back row, with again 4 people per row, how many different arrangements are possible?

Answer: 5760


Let positive real numbers \(a, b, c\) satisfy \(a+b+c=1\). Prove that \(10\left(a^{3}+b^{3}+c^{3}\right)-9\left(a^{5}+b^{5}+c^{5}\right) \geqslant 1\).

Answer: 1


The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
$\textbf{(A)}\; 20 \qquad\textbf{(B)}\; \dfrac{360}{17} \qquad\textbf{(C)}\; \dfrac{107}{5} \qquad\textbf{(D)}\; \dfrac{43}{2} \qquad\textbf{(E)}\; \dfrac{281}{13}$

Answer: \textbf{(E)}\;\frac{281}{13}


The cells of a 2001×2001 chessboard are colored in a checkerboard pattern in black and white, such that the corner cells are black. For each pair of differently colored cells, a vector is drawn from the center of the black cell to the center of the white cell. Prove that the sum of the drawn vectors equals 0.

Answer: 0


Let the sequence  $a_{1}, a_{2}, \cdots$  be defined recursively as follows:  $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of  $a_{1}$  can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

Answer: 121


Answer: 35


Given a parallelogram  $ABCD$ , let  $\mathcal{P}$  be a plane such that the distance from vertex  $A$  to  $\mathcal{P}$  is  $49$ , the distance from vertex  $B$  to  $\mathcal{P}$  is  $25$ , and the distance from vertex  $C$  to  $\mathcal{P}$  is  $36$ . Find the sum of all possible distances from vertex  $D$  to  $\mathcal{P}$ .

*Proposed by **HrishiP***

Answer: 220


How many ways are there to insert plus signs (+) between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30?

Answer: 2002


Let the integer part and decimal part of $2+\sqrt{6}$ be $x$ and $y$ respectively. Find the values of $x$, $y$, and the square root of $x-1$.

Answer: \sqrt{3}


In a circle, two perpendicular chords $KM$ and $LN$ are drawn. It is known that lines $KL$ and $MN$ are parallel, and two sides of the quadrilateral $KLMN$ are equal to 2. Find the radius of the circle.

Answer: \sqrt{2}


All old Mother Hubbard had in her cupboard was a Giant Bear chocolate bar. She gave each of her children one-twelfth of the chocolate bar. One third of the bar was left. How many children did she have?
A) 6
B) 8
C) 12
D) 15
E) 18

Answer: 8


Find the maximum of  $x+y$  given that  $x$  and  $y$  are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]

Answer: 12


Answer: \frac{\sqrt{15}}{9}


Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division?

Answer: \frac{1}{3}


Write the decomposition of the vector $x$ in terms of the vectors $p, q, r$:

$x = \{8 ; 1 ; 12\}$

$p = \{1 ; 2 ; -1\}$

$q = \{3 ; 0 ; 2\}$

$r = \{-1 ; 1 ; 1\}$

Answer: -p+4q+3r


A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , 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  $a^2 + b^2 + c^2$ .

Answer: 422


An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle.

Answer: \frac{3}{2}


What is the maximum number of equal non-convex polygons into which a square can be divided so that all sides of the polygons are parallel to the sides of the square and none of these polygons can be obtained from one another by parallel translation?

Answer: 8


Let $\alpha$ and $\beta$ be real numbers.  Find the minimum value of
\[(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.\]

Answer: 100


Given the random variable $\xi$ follows a binomial distribution $B(5,0.5)$, and $\eta=5\xi$, calculate the values of $E\eta$ and $D\eta$ respectively.

Answer: \frac{125}{4}


Xiao Jun is playing a dice game. He starts at the starting square. If he rolls a 1 to 5, he moves forward by the number of spaces shown on the dice. If he rolls a 6 or moves beyond the final square at any time, he must immediately return to the starting square. How many possible ways are there for Xiao Jun to roll the dice three times and exactly reach the ending square?

Answer: 19


Let $f(x)=(x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left |\sum_{k=1}^n\log_{10}f(k)\right|=1.\]

Answer: 21


Given the set of integers $\{1, 2, 3, \dots, 9\}$, from which three distinct numbers are arbitrarily selected as the coefficients of the quadratic function $f_{(x)} = ax^2 + bx + c$, determine the total number of functions $f_{(x)}$ that satisfy $\frac{f(1)}{2} \in \mathbb{Z}$.

Answer: 264


A gardener is preparing to plant a row of 20 trees, with the choice between two types of trees: maple trees and sycamore trees. The number of trees between any two maple trees (excluding these two maple trees) cannot be equal to 3. What is the maximum number of maple trees that can be planted in these 20 trees?

Answer: 12


**H**orizontal parallel segments  $AB=10$  and  $CD=15$  are the bases of trapezoid  $ABCD$ . Circle  $\gamma$  of radius  $6$  has center within the trapezoid and is tangent to sides  $AB$ ,  $BC$ , and  $DA$ . If side  $CD$  cuts out an arc of  $\gamma$  measuring  $120^{\circ}$ , find the area of  $ABCD$ .

Answer: \frac{225}{2}


A circle is divided into two segments by a chord equal to the side of a regular inscribed triangle. Determine the ratio of the areas of these segments.

Answer: \frac{4\pi - 3\sqrt{3}}{8\pi + 3\sqrt{3}}


In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$ with $a > b$, $a > c$. The radius of the circumcircle of $\triangle ABC$ is $1$, and the area of $\triangle ABC$ is $S= \sqrt{3} \sin B\sin C$.

$(1)$ Find the size of angle $A$;

$(2)$ If a point $D$ on side $BC$ satisfies $BD=2DC$, and $AB\perp AD$, find the area of $\triangle ABC$.

Answer: \dfrac{ \sqrt{3}}{4}


Find the number of square units in the area of the shaded region. Consider a rectangular plot with vertices at points (0,0), (40,0), (40,20), and (0,20). Within this rectangle, a shaded polygon is formed with vertices at (0,0), (20,0), (40,10), (40,20), and (10,20). Determine the area of this shaded region.

Answer: 600


Find the remainder when $3^{3^{3^3}}$ is divided by 1000.

Answer: 387


Given a moving point \( P \) on the \( x \)-axis, \( M \) and \( N \) lie on the circles \((x-1)^{2}+(y-2)^{2}=1\) and \((x-3)^{2}+(y-4)^{2}=3\) respectively. Find the minimum value of \( |PM| + |PN| \).

Answer: 2\sqrt{10} - \sqrt{3} - 1


Find the functions $f: \mathbb{N} \longmapsto \mathbb{N}$ such that $f(f(n)) < f(n+1)$.

Answer: f(n)=n


Rachelle picks a positive integer \(a\) and writes it next to itself to obtain a new positive integer \(b\). For instance, if \(a=17\), then \(b=1717\). To her surprise, she finds that \(b\) is a multiple of \(a^{2}\). Find the product of all the possible values of \(\frac{b}{a^{2}}\).

Answer: 77


The side of the base \(ABC\) of a regular prism \(ABCA_1B_1C_1\) is 1, and each of the lateral edges has a length of \(\frac{1}{\sqrt{3}}\). A right circular cylinder is positioned such that the point \(A_1\) and the midpoint \(M\) of the edge \(CC_1\) lie on its lateral surface, and the axis of the cylinder is parallel to the line \(AB_1\) and is at a distance of \(\frac{1}{4}\) from it. Find the radius of the cylinder.

Answer: \frac{\sqrt{7}}{4}


Which of the following integers is equal to a perfect square: $2^{3}$, $3^{5}$, $4^{7}$, $5^{9}$, $6^{11}$?

Answer: 4^{7}


Answer: 4


Given $\triangle ABC$ with circumcenter $O$ and incenter $I$, the incircle touches sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$ respectively. The line $DE$ intersects $AB$ at point $Q$, and $DF$ intersects $AC$ at point $P$. The midpoints of line segments $PE$ and $QF$ are $M$ and $N$ respectively. Prove that $MN \perp IO$.

Answer: MN\perpIO


A circle that has its center on the line \( y = b \) intersects the parabola \( y = \frac{4}{3} x^2 \) at least at three points; one of these points is the origin, and two of the remaining intersection points lie on the line \( y = \frac{4}{3} x + b \). Find all possible values of \( b \) for which this configuration is possible.

Answer: \frac{25}{12}


A positive integer is called *cool* if it can be expressed in the form  $a!\cdot b!+315$  where  $a,b$  are positive integers. For example,  $1!\cdot 1!+315=316$  is a cool number. Find the sum of all cool numbers that are also prime numbers.

[i]Proposed by Evan Fang

Answer: 317


In trapezoid \(ABCD\), the side \(AB\) is perpendicular to the base \(BC\). A circle passes through points \(C\) and \(D\) and is tangent to line \(AB\) at point \(E\).

Find the distance from point \(E\) to line \(CD\), if \(AD = 4\) and \(BC = 3\).

Answer: 2\sqrt{3}


How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)?

Answer: 334000


Through the vertex \(C\) of the base of a regular triangular pyramid \(SABC\), a plane is drawn perpendicular to the lateral edge \(SA\). This plane forms an angle with the base plane, the cosine of which is \( \frac{2}{3} \). Find the cosine of the angle between two lateral faces.

Answer: \frac{1}{7}


In a senior high school class, there are two study groups, Group A and Group B, each with 10 students. Group A has 4 female students and 6 male students; Group B has 6 female students and 4 male students. Now, stratified sampling is used to randomly select 2 students from each group for a study situation survey. Calculate:

(1) The probability of exactly one female student being selected from Group A;

(2) The probability of exactly two male students being selected from the 4 students.

Answer: \dfrac{31}{75}


Prove that the alternating series

$$
\sum_{i=1}^{\infty}(-1)^{i+1} \ln \left(1+\frac{1}{i}\right)
$$

is conditionally convergent, and find its sum.

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


A function $f(x) = a \cos ωx + b \sin ωx (ω > 0)$ has a minimum positive period of $\frac{π}{2}$. The function reaches its maximum value of $4$ at $x = \frac{π}{6}$.

1. Find the values of $a$, $b$, and $ω$.
2. If $\frac{π}{4} < x < \frac{3π}{4}$ and $f(x + \frac{π}{6}) = \frac{4}{3}$, find the value of $f(\frac{x}{2} + \frac{π}{6})$.

Answer: -\frac{4\sqrt{6}}{3}


Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
$\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$

Answer: \textbf{(B)}244


Let be a group  $ G $  of order  $ 1+p, $  where  $ p $  is and odd prime. Show that if  $ p $  divides the number of automorphisms of  $ G, $  then  $ p\equiv 3\pmod 4. $

Answer: p\equiv3\pmod{4}


Let \(\vec{a}\) and \(\vec{b}\) be two perpendicular plane vectors, and \( |\vec{a}| = 2|\vec{b}| = 10 \). For \( 0 \leqslant t \leqslant 1 \), let the vectors \(t \vec{a}+(1-t) \vec{b}\) and \(\left(t-\frac{1}{5}\right) \vec{a}+(1-t) \vec{b}\) have a maximum angle \(\theta\). Then, \(\cos \theta =\) ____.

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


In $\bigtriangleup ABC$, $AB = 75$, and $AC = 100$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. It is known that $\overline{BX}$ and $\overline{CX}$ have integer lengths. Calculate the length of segment $BC$.

Answer: 125


Find the number of squares in the sequence given by  $ a_0\equal{}91$  and  $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$  for  $ n \ge 0.$

Answer: 0


Answer: 51


In $\triangle ABC$, it is known that $\cos A= \frac{1}{7}$, $\cos (A-B)= \frac{13}{14}$, and $0 < B < A < \frac{\pi}{2}$. Find the measure of angle $B$.

Answer: \frac{\pi}{3}


Different numbers \(a\), \(b\), and \(c\) are such that the equations \(x^{2}+a x+1=0\) and \(x^{2}+b x+c=0\) have a common real root. In addition, the equations \(x^{2}+x+a=0\) and \(x^{2}+c x+b=0\) also have a common real root. Find the sum \(a+b+c\).

Answer: -3


Select three digits from 1, 3, 5, 7, 9, and two digits from 0, 2, 4, 6, 8 to form a five-digit number without any repeating digits. How many such numbers can be formed?

Answer: 11040


Let  $m,n$  be co-prime integers, such that  $m$  is even and  $n$  is odd. Prove that the following expression does not depend on the values of  $m$  and  $n$ :
\[ \frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} . \]

*Bogdan Enescu*

Answer: \frac{1}{2}


A right triangle  $ABC$  is inscribed in the circular base of a cone. If two of the side lengths of  $ABC$  are  $3$  and  $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is  $3$ , then the minimum possible volume of the cone can be written as  $\frac{m\pi\sqrt{n}}{p}$ , where  $m$ ,  $n$ , and  $p$  are positive integers,  $m$  and  $p$  are relatively prime, and  $n$  is squarefree. Find  $m + n + p$ .

Answer: 60


Given: $\because 4 \lt 7 \lt 9$, $\therefore 2 \lt \sqrt{7} \lt 3$, $\therefore$ the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$. The integer part of $\sqrt{51}$ is ______, and the decimal part of $9-\sqrt{51}$ is ______.

Answer: 8-\sqrt{51}


In a plane, there is a square \(ABCD\) with side length 1 and a point \(X\). It is known that \(XA = \sqrt{5}\) and \(XC = \sqrt{7}\). What is the length of \(XB\)?

Answer: \sqrt{6-\sqrt{10}}


Let \( n \geq 2 \) be a fixed integer. Find the smallest constant \( C \) such that for all non-negative reals \( x_1, x_2, \ldots, x_n \):
\[ \sum_{i < j} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4. \]
Determine when equality occurs.

Answer: \frac{1}{8}


The sequences  $(a_{n})$ ,  $(b_{n})$  are deﬁned by  $a_{1} = \alpha$ ,  $b_{1} = \beta$ ,  $a_{n+1} = \alpha a_{n} - \beta b_{n}$ ,   $b_{n+1} = \beta a_{n} + \alpha b_{n}$  for all  $n > 0.$  How many pairs  $(\alpha, \beta)$  of real numbers are there such that  $a_{1997} = b_{1}$  and  $b_{1997} = a_{1}$ ?

Answer: 1999


For the subset \( S \) of the set \(\{1,2, \cdots, 15\}\), if a positive integer \( n \) and \( n+|S| \) are both elements of \( S \), then \( n \) is called a "good number" of \( S \). If a subset \( S \) has at least one "good number", then \( S \) is called a "good set". Suppose 7 is a "good number" of a "good set" \( X \). How many such subsets \( X \) are there?

Answer: 4096


Given the coordinates of the three vertices of $\triangle P_{1}P_{2}P_{3}$ are $P_{1}(1,2)$, $P_{2}(4,3)$, and $P_{3}(3,-1)$, the length of the longest edge is ________, and the length of the shortest edge is ________.

Answer: \sqrt {10}


A store arranges a decorative tower of balls where the top level has 2 balls and each lower level has 3 more balls than the level above. The display uses 225 balls. What is the number of levels in the tower?

Answer: 12


A store purchased a batch of New Year cards at a price of 21 cents each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how many cents did the store earn in total?

Answer: 470


How many 12 step paths are there from point $A$ to point $C$ which pass through point $B$ on a grid, where $A$ is at the top left corner, $B$ is 5 steps to the right and 2 steps down from $A$, and $C$ is 7 steps to the right and 4 steps down from $A$?

Answer: 126


Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be two circles with centers $O_{1}$ and $O_{2}$ intersecting at $P$ and $Q$. Let $U$ be on $\mathcal{C}_{1}$ and $V$ be on $\mathcal{C}_{2}$ such that $U$, $P$, and $V$ are collinear. Show that $\widehat{U Q V}=\widehat{O_{1} Q O_{2}}$.

Answer: \widehat{UQV}=\widehat{O_{1}QO_{2}}


A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere, where \( A B C \) is the base and \( A A_{1}, B B_{1}, C C_{1} \) are the lateral edges. The segment \( C D \) is the diameter of this sphere, and point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism if \( C K = 2 \sqrt{3} \) and \( D K = 2 \sqrt{2} \).

Answer: 9\sqrt{2}


How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?

*Proposed by Milan Haiman.*

Answer: 8


On a circle, there are some white and black points (at least 12 points in total), such that for each point, among the 10 neighboring points (5 on each side), there are exactly half white points and half black points. Prove that the number of points is a multiple of 4.

Answer: 4


Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it also holds that \( x y z \mid (x+y+z)^{n} \).

Answer: 13


In an acute-angled triangle \( ABC \), the altitude \( AA_1 \) is drawn. \( H \) is the orthocenter of triangle \( ABC \). It is known that \( AH = 3 \), \( A_1H = 2 \), and the radius of the circumcircle of triangle \( ABC \) is 4. Find the distance from the center of this circumcircle to \( H \).

Answer: 2


Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$.

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


Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. The minimum height of this regular tetrahedron is:

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


In a  $25 \times n$  grid, each square is colored with a color chosen among  $8$  different colors. Let  $n$  be as minimal as possible such that, independently from the coloration used, it is always possible to select  $4$  coloumns and  $4$  rows such that the  $16$  squares of the interesections are all of the same color. Find the remainder when  $n$  is divided by  $1000$ .

*Proposed by **FedeX333X***

Answer: 601


Let $F(0)=0, F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2} F(n-1)-F(n-2)$ for $n \geq 2$. Determine whether or not $\sum_{n=0}^{\infty} \frac{1}{F\left(2^{n}\right)}$ is a rational number.

Answer: 1


For a set  $ P$  of five points in the plane, no three of them being collinear, let  $ s(P)$  be the numbers of acute triangles formed by vertices in  $ P$ .
Find the maximum value of  $ s(P)$  over all such sets  $ P$ .

Answer: 7


Answer: 4


The perimeter of triangle \(ABC\) is 4. Points \(X\) and \(Y\) are marked on rays \(AB\) and \(AC\) such that \(AX = AY = 1\). Segments \(BC\) and \(XY\) intersect at point \(M\). Prove that the perimeter of one of the triangles \(ABM\) or \(ACM\) is 2.

Answer: 2


Answer: \frac{3}{4}


The number 5.6 may be expressed uniquely (ignoring order) as a product \( \underline{a} \cdot \underline{b} \times \underline{c} . \underline{d} \) for digits \( a, b, c, d \) all nonzero. Compute \( \underline{a} . \underline{b} + \underline{c} . \underline{d} \).

Answer: 5.1


A meal at a diner includes a burger weighing 150 grams, of which 40 grams are filler. What percent of the burger is not filler?

Answer: 73.33\%


For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.

Answer: 736


Among the three-digit numbers formed by the digits 1, 2, 3, 4, 5, 6 with repetition allowed, how many three-digit numbers have exactly two different even digits (for example: 124, 224, 464, …)? (Answer with a number).

Answer: 72


A cube $A B C D - A' B' C' D'$ with edge length 12 is intersected by a plane $\alpha$ passing through points $A$, $E$ (on $B B'$), and $F$ (on $D D'$). Given that $D E = D F = 9$, find the area of the section of the cube cut by the plane $\alpha$.

Answer: 28\sqrt{34}


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

Answer: 108


Given the equation \( x e^{-2 x} + k = 0 \) has exactly two real roots in the interval \((-2, 2)\), find the range of values for \( k \).

Answer: (-\frac{1}{2e},-\frac{2}{e^{4}})


Given that
\begin{eqnarray*}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\     &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\     &(3)& z=|x-y|. \end{eqnarray*}
How many distinct values of $z$ are possible?

Answer: 9


Given that \(a, b\) are real numbers and \(a < b\), find the range of values for \(b - a\) such that the difference between the maximum and minimum values of the function \(y = \sin x\) on the closed interval \([a, b]\) is 1.

Answer: [\frac{\pi}{3},\pi]


When simplified and expressed with negative exponents, the expression $(x + y)^{ - 1}(x^{ - 1} + y^{ - 1})$ is equal to:

Answer: x^{ - 1}y^{ - 1}


For the inequality system about $y$ $\left\{\begin{array}{l}{2y-6≤3(y-1)}\\{\frac{1}{2}a-3y＞0}\end{array}\right.$, if it has exactly $4$ integer solutions, then the product of all integer values of $a$ that satisfy the conditions is ______.

Answer: 720


On a table, there are 2 candles, each 20 cm long, but of different diameters. The candles burn evenly, with the thin candle burning completely in 4 hours and the thick candle in 5 hours. After how much time will the thin candle become twice as short as the thick candle if they are lit simultaneously?

Answer: 20/3


During an earthquake, the epicenter emits both primary (P) waves and secondary (S) waves in all directions simultaneously. The propagation speeds are 5.94 km/s for P-waves and 3.87 km/s for S-waves. In a certain earthquake, a seismic monitoring station receives the P-wave first and then receives the S-wave 11.5 seconds later. What is the distance from the epicenter to the seismic monitoring station, in kilometers (round to the nearest integer)?

Answer: 128


Given a sequence $\{a_n\}$ where each term is a positive number and satisfies the relationship $a_{n+1}^2 = ta_n^2 +(t-1)a_na_{n+1}$, where $n\in \mathbb{N}^*$.

(1) If $a_2 - a_1 = 8$, $a_3 = a$, and the sequence $\{a_n\}$ is unique:
   ① Find the value of $a$.
   ② Let another sequence $\{b_n\}$ satisfy $b_n = \frac{na_n}{4(2n+1)2^n}$. Is there a positive integer $m, n$ ($1 < m < n$) such that $b_1, b_m, b_n$ form a geometric sequence? If it exists, find all possible values of $m$ and $n$; if it does not exist, explain why.

(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, with $k \in \mathbb{N}^*$, determine the minimum value of $a_{2k+1} + a_{2k+2} + \ldots + a_{3k}$.

Answer: 32


Given that $-6 \leq x \leq -3$ and $1 \leq y \leq 5$, what is the largest possible value of $\frac{x+y}{x}$?

Answer: \frac{1}{6}


Let \( A B C \) be a triangle with orthocenter \( H \) and let \( P \) be the second intersection of the circumcircle of triangle \( A H C \) with the internal bisector of \( \angle B A C \). Let \( X \) be the circumcenter of triangle \( A P B \) and let \( Y \) be the orthocenter of triangle \( A P C \). Prove that the length of segment \( X Y \) is equal to the circumradius of triangle \( A B C \).

Answer: XY=R


Consider an arbitrary triangle $A B C$. Let $A_{1}, B_{1}, C_{1}$ be three points on the lines $B C$, $C A$, and $A B$, respectively. Define the following:

$$
\begin{aligned}
R & =\frac{\left|A C_{1}\right|}{\left|C_{1} B\right|} \cdot \frac{\left|B A_{1}\right|}{\left|A_{1} C\right|} \cdot \frac{\left|C B_{1}\right|}{\left|B_{1} A\right|} \\
R^{*} & =\frac{\sin \widehat{A C C_{1}}}{\sin \widehat{C_{1} C B}} \cdot \frac{\sin \widehat{B A A_{1}}}{\sin \widehat{A_{1} A C}} \cdot \frac{\sin \widehat{C B B_{1}}}{\sin \widehat{B_{1} B A}}
\end{aligned}
$$

Prove that $R=R^{*}$.

Answer: R^*


Calculate the double integral \(\iint_{\Omega} \sqrt{\left|x^{2}-y^{2}\right|} \, dx \, dy\), where \(\Omega=\{(x, y): |x| + |y| \leq a\}\).

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


For the function $f(x) = x - 2 - \ln x$, we know that $f(3) = 1 - \ln 3 < 0$, $f(4) = 2 - \ln 4 > 0$. Using the bisection method to find the approximate value of the root of $f(x)$ within the interval $(3, 4)$, we first calculate the function value $f(3.5)$. Given that $\ln 3.5 = 1.25$, the next function value we need to find is $f(\quad)$.

Answer: 3.25


Pablo has 27 solid $1 \times 1 \times 1$ cubes that he assembles in a larger $3 \times 3 \times 3$ cube. If 10 of the smaller cubes are red, 9 are blue, and 8 are yellow, what is the smallest possible surface area of the larger cube that is red?

Answer: 12


Answer: -1011


A line $c$ is given by the equation $y = 2x$. Points $A$ and $B$ have coordinates $A(2, 2)$ and $B(6, 2)$. On the line $c$, find point $C$ from which the segment $AB$ is seen at the largest angle.

Answer: (2,4)


Let \( H \) be the orthocenter of \( \triangle ABC \), and suppose
\[ 3 \overrightarrow{HA} + 4 \overrightarrow{HB} + 5 \overrightarrow{HC} = \mathbf{0} \]
Then, \(\cos \angle AHB =\)

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


Find all values of the parameter \(a\) for which the equation \(x^{2} + 2x + 2|x + 1| = a\) has exactly two roots.

Answer: -1


The minimum value of the function $f(x) = \cos^2 x + \sin x$ is given by $\frac{-1 + \sqrt{2}}{2}$.

Answer: -1


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

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

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


On a sphere, there are four points A, B, C, and D satisfying $AB=1$, $BC=\sqrt{3}$, $AC=2$. If the maximum volume of tetrahedron D-ABC is $\frac{\sqrt{3}}{2}$, then the surface area of this sphere is _______.

Answer: \frac{100\pi}{9}


Two individuals, A and B, are undergoing special training on the same elliptical track. They start from the same point simultaneously and run in opposite directions. Each person, upon completing the first lap and reaching the starting point, immediately turns around and accelerates for the second lap. During the first lap, B's speed is $\frac{2}{3}$ of A's speed. During the second lap, A's speed increases by $\frac{1}{3}$ compared to the first lap, and B's speed increases by $\frac{1}{5}$. Given that the second meeting point between A and B is 190 meters from the first meeting point, what is the length of the elliptical track in meters?

Answer: 400


Among all integers that alternate between 1 and 0, starting and ending with 1 (e.g., 101, 10101, 10101…), how many are prime numbers? Why? And list all the prime numbers.

Answer: 101


Marisa has a collection of \(2^{8}-1=255\) distinct nonempty subsets of \(\{1,2,3,4,5,6,7,8\}\). For each step, she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process \(2^{8}-2=254\) times until there is only one set left in the collection. What is the expected size of this set?

Answer: \frac{1024}{255}


In front of Vasya, there is a stack of 15 red, 15 blue, and 15 yellow cards. Vasya needs to choose 15 out of all 45 cards to earn the maximum number of points. Points are awarded as follows: for each red card, Vasya earns one point. For each blue card, Vasya earns points equal to twice the number of red cards chosen, and for each yellow card, Vasya earns points equal to three times the number of blue cards chosen. What is the maximum number of points Vasya can earn?

Answer: 168


Given the hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (\(a>0, b>0\)) with left and right foci \( F_{1}, F_{2} \) and eccentricity \(\frac{5}{3}\), a line \( l \) passing through \( F_{1} \) is tangent to the circle \( x^{2}+y^{2}=a^{2} \) at point \( T \). The line \( l \) also intersects the right branch of the hyperbola \( C \) at point \( P \). Find \(\frac{\left|\overrightarrow{F_{1} P}\right|}{\left|\overrightarrow{F_{1} T}\right|} \).

Answer: 4


Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that for any integers $x, y$, we have $f\left(x^{2}-3 y^{2}\right)+f\left(x^{2}+y^{2}\right)=2(x+y) f(x-y)$. Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \cdot f(2016)$ is a perfect square. Find the minimum possible value of $f(1)+f(2)$.

Answer: 246


Inside triangle \( ABC \) with angles \(\angle A = 50^\circ\), \(\angle B = 60^\circ\), \(\angle C = 70^\circ\), a point \( M \) is chosen such that \(\angle AMB = 110^\circ\) and \(\angle BMC = 130^\circ\). Find \(\angle MBC\).

Answer: 20


There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012.  Let $x$ be one of these numbers.  Find the maximum value of $x + \frac{1}{x}.$

Answer: \frac{8045}{2012}


Consider all natural numbers whose decimal representations do not contain the digit zero. Prove that the sum of the reciprocals of any number of these numbers does not exceed a certain constant \( C \).

Answer: 90


Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score?
$\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$

Answer: \textbf{(A)}\40


In the tetrahedron \( A B C D \),
$$
\begin{array}{l}
AB=1, BC=2\sqrt{6}, CD=5, \\
DA=7, AC=5, BD=7.
\end{array}
$$

Find its volume.

Answer: \frac{\sqrt{66}}{2}


In a school fundraising campaign, $25\%$ of the money donated came from parents. The rest of the money was donated by teachers and students. The ratio of the amount of money donated by teachers to the amount donated by students was $2:3$. What is the ratio of the amount of money donated by parents to the amount donated by students?

Answer: 5:9


Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral \(ABED\).

Answer: 28/3


Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \dfrac{\sqrt{2}}{2}$, the length of the minor axis is $2$.
$(1)$ Find the equation of the ellipse;

$(2)$ Point $A$ is a moving point on the ellipse (not the endpoints of the major axis), the extension line of $AF_2$ intersects the ellipse at point $B$, and the extension line of $AO$ intersects the ellipse at point $C$. Find the maximum value of the area of $\triangle ABC$.

Answer: \sqrt{2}


A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?

Answer: 338350


Each square of a  $(2^n-1) \times (2^n-1)$  board contains either  $1$  or  $-1$ .  Such an arrangement is called *successful* if each number is the product of its neighbors.  Find the number of successful arrangements.

Answer: 1


Given a square grid of size $2023 \times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six cells. What is the smallest $n$ for which this configuration is possible?

Answer: 338


Inside the circle $\omega$ are two intersecting circles $\omega_{1}$ and $\omega_{2}$, which intersect at points $K$ and $L$, and are tangent to the circle $\omega$ at points $M$ and $N$. It turns out that the points $K$, $M$, and $N$ are collinear. Find the radius of the circle $\omega$, given that the radii of the circles $\omega_{1}$ and $\omega_{2}$ are 3 and 5, respectively.

Answer: 8


Let \( A = (2, 0) \) and \( B = (8, 6) \). Let \( P \) be a point on the circle \( x^2 + y^2 = 8x \). Find the smallest possible value of \( AP + BP \).

Answer: 6\sqrt{2}


The internal angle bisector of a triangle from vertex $A$ is parallel to the line $OM$, but not identical to it, where $O$ is the circumcenter and $M$ is the orthocenter. What is the angle at vertex $A$?

Answer: 120


If \( x^{*} y = xy + 1 \) and \( a = (2 * 4) * 2 \), find \( a \).

If the \( b \)th prime number is \( a \), find \( b \).

If \( c = \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right) \cdots \left(1 - \frac{1}{50}\right) \), find \( c \) in the simplest fractional form.

If \( d \) is the area of a square inscribed in a circle of radius 10, find \( d \).

Answer: 200


Simplify the expression \(\frac{\operatorname{tg}\left(\frac{5}{4} \pi - 4 \alpha\right) \sin^{2}\left(\frac{5}{4} \pi + 4 \alpha\right)}{1 - 2 \cos^{2} 4 \alpha}\).

Answer: -\frac{1}{2}


Given an isosceles right triangle \(ABC\) with a right angle at \(A\), there is a square \(KLMN\) positioned such that points \(K, L, N\) lie on the sides \(AB, BC, AC\) respectively, and point \(M\) is located inside the triangle \(ABC\).

Find the length of segment \(AC\), given that \(AK = 7\) and \(AN = 3\).

Answer: 17


If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is
$\textbf {(A) } -2 \qquad \textbf {(B) } -\frac{2}{5} \qquad \textbf {(C) } -\frac{1}{4} \qquad \textbf {(D) } \frac{2}{5} \qquad \textbf {(E) } 2$

Answer: \textbf{E}


Find the smallest natural number, which divides  $2^{n}+15$  for some natural number  $n$  and can be expressed in the form  $3x^2-4xy+3y^2$  for some integers  $x$  and  $y$ .

Answer: 23


In a table tennis match between player A and player B, the "best of five sets" rule is applied, which means the first player to win three sets wins the match. If the probability of player A winning a set is $\dfrac{2}{3}$, and the probability of player B winning a set is $\dfrac{1}{3}$, then the probability of the match ending with player A winning three sets and losing one set is ______.

Answer: \dfrac{8}{27}


Let vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{1}{2}$, and $(\overrightarrow{a}- \overrightarrow{c})\cdot(\overrightarrow{b}- \overrightarrow{c})=0$. Then, calculate the maximum value of $|\overrightarrow{c}|$.

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


Let  $\{\epsilon_n\}^\infty_{n=1}$  be a sequence of positive reals with  $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$ . Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right)  \]

Answer: -1


Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey?

Answer: 55.38


2011 warehouses are connected by roads such that each warehouse can be reached from any other, possibly by traveling through several roads. Each warehouse contains $x_{1}, \ldots, x_{2011}$ kg of cement, respectively. In one trip, any amount of cement can be transported from one warehouse to another along a connecting road. Ultimately, the plan is to have $y_{1}, \ldots, y_{2011}$ kg of cement at the warehouses, respectively, with the condition that $x_{1} + x_{2} + \ldots + x_{2011} = y_{1} + y_{2} + \ldots + y_{2011}$. What is the minimum number of trips required to execute this plan for any values of $x_{i}$ and $y_{i}$ and any road scheme?

Answer: 2010


Solve the inequality in integers:

$$
\frac{1}{\sqrt{x-2y+z+1}} + \frac{2}{\sqrt{2x-y+3z-1}} + \frac{3}{\sqrt{3y-3x-4z+3}} > x^2 - 4x + 3
$$

Answer: (3,1,-1)


On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings for the first time when the situation is reversed: the first and third glasses are empty, and the second and fourth glasses are full.

Answer: 6


We regularly transport goods from city $A$ to city $B$, which is $183 \mathrm{~km}$ away. City $A$ is $33 \mathrm{~km}$ from the river, while city $B$ is built on the riverbank. The cost of transportation per kilometer is half as much on the river as on land. Where should we build the road to minimize transportation costs?

Answer: 11\sqrt{3}


The positive integer \( N \) has six digits in increasing order. For example, 124689 is such a number. However, unlike 124689, three of the digits of \( N \) are 3, 4, and 5, and \( N \) is a multiple of 6. How many possible six-digit integers \( N \) are there?

Answer: 3


The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at  $(5, 1)$  and put poison in two pipes, one along the line  $y=x$  and one along the line  $x=7$ . However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?

Answer: 4\sqrt{5}


Each face of a regular icosahedron has a non-negative integer written on it such that the sum of all 20 numbers is 39. Prove that there exist two different faces sharing a common vertex with the same number written on them.

An icosahedron is a convex polyhedron with 12 vertices, 20 faces, and 30 edges, where all faces are congruent equilateral triangles. Five edges meet at each vertex of the icosahedron.

Answer: 2


The sides of a triangle are 1 and 2, and the angle between them is $60^{\circ}$. A circle is drawn through the center of the inscribed circle of this triangle and the ends of the third side. Find its radius.

Answer: 1


Let \( m \) be a positive real number such that for any positive real numbers \( x \) and \( y \), the three line segments with lengths \( x + y \), \( \sqrt{x^2 + y^2 + xy} \), and \( m \sqrt{xy} \) can form the three sides of a triangle. Determine the range of values for \( m \).

Answer: (2-\sqrt{3},2+\sqrt{3})


What is the area enclosed by the geoboard quadrilateral below?
[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2;  for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b)  {   dot((a,b));  };  draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]
$\textbf{(A)}\ 15\qquad \textbf{(B)}\ 18\frac12 \qquad \textbf{(C)}\ 22\frac12 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 41$

Answer: \textbf{(C)}\22\frac{1}{2}


Given the vertices of a regular 120-gon \( A_{1}, A_{2}, A_{3}, \ldots, A_{120} \), in how many ways can three vertices be selected to form an obtuse triangle?

Answer: 205320


Given that \(0\) is the circumcenter of \(\triangle ABC\) and it satisfies \(\overrightarrow{CO} = t \cdot \overrightarrow{CA} + \left(\frac{1}{2} - \frac{3t}{4}\right) \cdot \overrightarrow{CB}\) for \(t \in \mathbb{R} \setminus \{0\}\), and that \(|AB| = 3\), find the maximum area of \(\triangle ABC\).

Answer: 9


Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all \( x, y, z \in \mathbf{R} \), the inequality \( f(x+y) + f(y+z) + f(z+x) \geq 3 f(x + 2y + 3z) \) is satisfied.

Answer: f(x)=


A sequence  $(a_1, a_2,\ldots, a_k)$  consisting of pairwise distinct squares of an  $n\times n$  chessboard is called a *cycle* if  $k\geq 4$  and squares  $a_i$  and  $a_{i+1}$  have a common side for all  $i=1,2,\ldots, k$ , where  $a_{k+1}=a_1$ . Subset  $X$  of this chessboard's squares is *mischievous* if each cycle on it contains at least one square in  $X$ .

Determine all real numbers  $C$  with the following property: for each integer  $n\geq 2$ , on an  $n\times n$  chessboard there exists a mischievous subset consisting of at most  $Cn^2$  squares.

Answer: \frac{1}{3}


Solve the inequality \(\operatorname{tg} \arccos x \leqslant \sin \operatorname{arctg} x\).

Answer: [-\frac{1}{\sqrt[4]{2}},0)\cup[\frac{1}{\sqrt[4]{2}},1]


Given $|m|=4$, $|n|=3$.

(1) When $m$ and $n$ have the same sign, find the value of $m-n$.

(2) When $m$ and $n$ have opposite signs, find the value of $m+n$.

Answer: -1


In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is:
$\textbf{(A)}\ 11\qquad  \textbf{(B)}\ 33\qquad  \textbf{(C)}\ 55\qquad  \textbf{(D)}\ 66\qquad  \textbf{(E)}\ 132$

Answer: \textbf{D}


A rectangle is divided by two straight lines into four smaller rectangles with perimeters of 6, 10, and 12 (the rectangle with a perimeter of 10 shares sides with the rectangles with perimeters of 6 and 12). Find the perimeter of the fourth rectangle.

Answer: 8


Given $0\leqslant x\_0 < 1$, for all integers $n > 0$, let $x\_n= \begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\ 2x_{n-1}-1, & 2x_{n-1} \geqslant 1 \end{cases}$. Find the number of $x\_0$ that makes $x\_0=x\_6$ true.

Answer: 64


Calculate the limit

$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$

Answer: \frac{2}{5}


Find all solutions to  $aabb=n^4-6n^3$ , where  $a$  and  $b$  are non-zero digits, and  $n$  is an integer.  ( $a$  and  $b$  are not necessarily distinct.)

Answer: 6655


Given that $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\frac{2}{a - 1} + \frac{a}{b}$.

Answer: 4(1 + \sqrt{2})


In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the nearest 0.01.

Answer: 29.32


There exists a scalar $c$ so that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) = c \mathbf{v}\]for all vectors $\mathbf{v}.$  Find $c.$

Answer: 2


There are 100 people in a room with ages $1,2, \ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?

Answer: 43


Given a positive integer $n \geq 2$, and a set $S$ containing $n$ pairwise distinct real numbers $a_{1}, a_{2}, \ldots, a_{n}$, define $k(S)$ as the number of distinct numbers of the form $a_{i} + 2^{j}$ for $i, j = 1, 2, \ldots, n$. Find the minimum possible value of $k(S)$ for all possible sets $S$.

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


Given the parabola C: y² = 3x with focus F, and a line l with slope $\frac{3}{2}$ intersecting C at points A and B, and the x-axis at point P.
(1) If |AF| + |BF| = 4, find the equation of line l;
(2) If $\overrightarrow{AP}$ = 3$\overrightarrow{PB}$, find |AB|.

Answer: \frac{4\sqrt{13}}{3}


Find the minimum of \( |\sin x + \cos x + \tan x + \cot x + \sec x + \csc x| \) for real \( x \).

Answer: 2\sqrt{2}-1


For all \( m \) and \( n \) satisfying \( 1 \leq n \leq m \leq 5 \), the polar equation
$$
\rho = \frac{1}{1 - C_{m}^{n} \cos \theta}
$$
represents how many different hyperbolas?

Answer: 10


Let \(a, b, c\) be numbers such that \(0 < c \leq b \leq a\). Prove that

$$
\frac{a^{2}-b^{2}}{c}+\frac{c^{2}-b^{2}}{a}+\frac{a^{2}-c^{2}}{b} \geq 3a-4b+c
$$

Answer: \frac{^2-b^2}{}+\frac{^2-b^2}{}+\frac{^2-^2}{b}\ge3a-4b+


Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$

Find all possible values of $a_{2021}$.

Answer: 2021


Given that \(\theta\) is an angle in the third quadrant, and \(\sin^{4}\theta + \cos^{4}\theta = \frac{5}{9}\), determine the value of \(\sin 2\theta\).

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


Given that a floor is tiled in a similar pattern with a $4 \times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are preserved.

Answer: \frac{1}{2}


Consider the following algorithm.

Step 0. Set \( n = m \).

Step 1. If \( n \) is even, divide \( n \) by two. If \( n \) is odd, increase \( n \) by one.

Step 2. If \( n > 1 \), go to Step 1. If \( n = 1 \), end the algorithm. How many natural numbers \( m \) exist such that Step 1 will be performed exactly 15 times when this algorithm is executed?

Answer: 610


Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together with Mouse can pull up the Turnip, but without Mouse they can't. How many Mice are needed so that they can pull up the Turnip on their own?

Answer: 1237


Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)

Answer: 255


On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted.

Answer: 21


Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate--in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to $p$?
$\textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$

Answer: \textbf{(C)}\\frac{1}{11}


Given a sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, $a_{1}=3$, $\frac{{S}_{n+1}}{{S}_{n}}=\frac{{3}^{n+1}-1}{{3}^{n}-1}$, $n\in N^{*}$.
$(1)$ Find $S_{2}$, $S_{3}$, and the general formula for $\{a_{n}\}$;
$(2)$ Let $b_n=\frac{a_{n+1}}{(a_n-1)(a_{n+1}-1)}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is denoted as $T_{n}$. If $T_{n}\leqslant \lambda (a_{n}-1)$ holds for all $n\in N^{*}$, find the minimum value of $\lambda$.

Answer: \frac{9}{32}


There are a total of 10040 numbers written on a blackboard, including 2006 of the number 1, 2007 of the number 2, 2008 of the number 3, 2009 of the number 4, and 2010 of the number 5. Each operation consists of erasing 4 different numbers and writing the 5th number (for example, erasing one 1, one 2, one 3, and one 4, and writing one 5; or erasing one 2, one 3, one 4, and one 5, and writing one 1, etc.). If after a finite number of operations, there are exactly two numbers left on the blackboard, what is the product of these two numbers?

Answer: 8


Let $\mathrm{P}$ be a point on the side $\mathrm{BC}$ of a triangle $\mathrm{ABC}$. The parallel through $\mathrm{P}$ to $\mathrm{AB}$ intersects side $\mathrm{AC}$ at point $\mathrm{Q}$, and the parallel through $\mathrm{P}$ to $\mathrm{AC}$ intersects side $\mathrm{AB}$ at point $\mathrm{R}$. The ratio of the areas of triangles $\mathrm{RBP}$ and $\mathrm{QPC}$ is $\mathrm{k}^{2}$.

Determine the ratio of the areas of triangles $\mathrm{ARQ}$ and $\mathrm{ABC}$.

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


The integers from 1 to \( k \) are concatenated to form the integer \( N = 123456789101112 \ldots \). Determine the smallest integer value of \( k > 2019 \) such that \( N \) is divisible by 9.

Answer: 2024


Answer: -1,0,1


Answer: \frac{1}{4}


Line  $\ell$  passes through  $A$  and into the interior of the equilateral triangle  $ABC$ .   $D$  and  $E$  are the orthogonal projections of  $B$  and  $C$  onto  $\ell$  respectively.  If  $DE=1$  and  $2BD=CE$ , then the area of  $ABC$  can be expressed as  $m\sqrt n$ , where  $m$  and  $n$  are positive integers and  $n$  is not divisible by the square of any prime.  Determine  $m+n$ .

[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label(" $A$ ",A,S);
label(" $B$ ",B,S);
label(" $C$ ",C,N);
label(" $D$ ",D,dir(B--D));
label(" $E$ ",E,dir(C--E));
[/asy]

Answer: 10


Given the function $f(x) = x^2 - (2t + 1)x + t \ln x$ where $t \in \mathbb{R}$,

(1) If $t = 1$, find the extreme values of $f(x)$.

(2) Let $g(x) = (1 - t)x$, and suppose there exists an $x_0 \in [1, e]$ such that $f(x_0) \geq g(x_0)$ holds. Find the maximum value of the real number $t$.

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


A point is randomly dropped on the interval $[5 ; 7]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3 k-4\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.

Answer: 1/3


Let ABC be an acute, non-isosceles triangle with orthocenter H, with M being the midpoint of side AB, and w being the angle bisector of angle ACB. Let S be the intersection point of the perpendicular bisector of side AB with w, and F be the foot of the perpendicular dropped from H to w.

Prove that segments MS and MF are of equal length.

(Karl Czakler)

Answer: MS=MF


What is the maximum number of non-intersecting diagonals that can be drawn in a convex $n$-gon (diagonals that share a common vertex are allowed)?

Answer: n-3


Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?

Answer: 20


Given \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots
\]

A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy
\[
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \quad a_{1} a_{2} \cdots a_{10} > 0
\]

Find the sum of the lengths of all intervals on the number line that represent all such "good numbers".

Answer: 61/210


Prove that if $m$ and $n$ are integers such that

$$
(\sqrt{2}-1)^{n} \geq \sqrt{m}-\sqrt{m-1}
$$

then $m \geq n^{2}$.

Answer: \gen^{2}


Answer: 108


Answer: 504


Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.

Answer: 330


For a positive integer  $n$  we denote by  $u(n)$  the largest prime number less than or equal to  $n$ , and with  $v(n)$  the smallest prime number larger than  $n$ . Prove that  \[ \frac 1 {u(2)v(2)} + \frac 1{u(3)v(3)} + \cdots + \frac 1{ u(2010)v(2010)} = \frac 12 - \frac 1{2011}. \]

Answer: \frac{1}{2}-\frac{1}{2011}


Given the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1\), the right focus is \(F\), and \(P\) is a point on the ellipse. Point \(A\) is at \((0, 2 \sqrt{3})\). When the perimeter of \(\triangle APF\) is maximized, what is the area of \(\triangle APF\)?

Answer: \frac{21\sqrt{3}}{4}


Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ________.

Answer: 330


In triangle \(ABC\), angle \(B\) is a right angle, and the medians \(AD\) and \(BE\) are mutually perpendicular. Find angle \(C\).

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


Two spheres with equal mass and charge are initially at a distance of \( l \) from each other when they are released without an initial velocity. After \( t \) seconds, the distance between them doubles. How much time will it take for the distance between the spheres to double if they are initially released from a distance of \( 3l \)? Air resistance can be neglected.

Answer: 3\sqrt{3}


The letter T is formed by placing a $2\:\text{inch} \times 6\:\text{inch}$ rectangle vertically and a $2\:\text{inch} \times 4\:\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches?

Answer: 24


Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.

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


Consider three collinear points \(B\), \(C\), and \(D\) such that \(C\) is between \(B\) and \(D\). Let \(A\) be a point not on the line \(BD\) such that \(AB = AC = CD\).

(a) If \(\angle BAC = 36^\circ\), then verify that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]

(b) Now, suppose that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]

Verify that \(\angle BAC = 36^\circ\).

Answer: 36


Five numbers form an increasing arithmetic progression. The sum of their cubes is zero, and the sum of their squares is 70. Find the smallest of these numbers.

Answer: -2\sqrt{7}


Compute the number of nonempty subsets  $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$   that satisfy   $$ |S| +\ min(S) \cdot \max (S) = 0. $$

Answer: 335


Let \( S(n) \) be the sum of the digits in the decimal representation of the number \( n \). Find \( S\left(S\left(S\left(S\left(2017^{2017}\right)\right)\right)\right) \).

Answer: 1


If $x=1$ is an extremum point of the function $f(x)=(x^{2}+ax-1)e^{x-1}$, determine the maximum value of $f(x)$.

Answer: 5e^{-3}


Let $F_{1}$ and $F_{2}$ be the two foci of an ellipse $C$. Let $AB$ be a chord of the ellipse passing through the point $F_{2}$. In the triangle $\triangle F_{1}AB$, the lengths are given as follows:
$$
|F_{1}A| = 3, \; |AB| = 4, \; |BF_{1}| = 5.
$$

Find $\tan \angle F_{2}F_{1}B$.

Answer: \frac{1}{7}


The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$.

Answer: 125 - 5\sqrt{2}\pi


A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___.

Answer: 600


Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this?

Answer: 16


A boy, standing on a horizontal surface of the ground at a distance $L=5$ m from a vertical wall of a house, kicked a ball lying in front of him on the ground. The ball flew at an angle $\alpha=45^{\circ}$ to the horizontal and, after an elastic collision with the wall, fell back to the same place where it initially lay.
1) Find the flight time of the ball.
2) At what height from the ground did the ball bounce off the wall?

Assume the acceleration due to gravity is $10 \mathrm{~m} / \mathrm{c}^{2}$.

Answer: 2.5


The circles OAB, OBC, and OCA have equal radius \( r \). Show that the circle ABC also has radius \( r \).

Answer: r


The operation $*$ is defined by
\[a * b = \frac{a - b}{1 - ab}.\]Compute
\[1 * (2 * (3 * (\dotsb (999 * 1000) \dotsb))).\]

Answer: 1


Given \( x^{2} + y^{2} = 25 \), find the maximum value of the function \( f(x, y) = \sqrt{8 y - 6 x + 50} + \sqrt{8 y + 6 x + 50} \).

Answer: 6\sqrt{10}


Given the sequence $\left\{a_{n}\right\}$ with its sum of the first $n$ terms $S_{n}$ satisfying $2 S_{n}-n a_{n}=n$ for $n \in \mathbf{N}^{*}$, and $a_{2}=3$:
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=\frac{1}{a_{n} \sqrt{a_{n+1}}+a_{n+1} \sqrt{a_{n}}}$ and $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. Determine the smallest positive integer $n$ such that $T_{n}>\frac{9}{20}$.

Answer: 50


If  $a$  and  $b$  are nonzero real numbers such that  $\left| a \right| \ne \left| b \right|$ , compute the value of the expression
\[
  \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
  \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
  \left(
    \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
      - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
  \right).
\]

Answer: -8


In the election for the class president, Petya and Vasya competed. Over three hours, 27 students voted for one of the two candidates. During the first two hours, Petya received 9 more votes than Vasya. In the last two hours, Vasya received 9 more votes than Petya. In the end, Petya won. By the largest possible margin, how many more votes could Petya have received than Vasya?

Answer: 9


Define polynomials  $f_{n}(x)$  for  $n \geq 0$  by  $f_{0}(x)=1, f_{n}(0)=0$  for  $n \geq 1,$  and  $$ \frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1) $$  for  $n \geq 0 .$  Find, with proof, the explicit factorization of  $f_{100}(1)$  into powers of distinct primes.

Answer: 101^{99}


A cycle of three conferences had a constant attendance, meaning that each session had the same number of participants. However, half of those who attended the first session did not return; a third of those who attended the second conference only attended that one, and a quarter of those who attended the third conference did not attend either the first or the second. Knowing that there were 300 registered participants and that each one attended at least one conference, determine:

a) How many people attended each conference?

b) How many people attended all three conferences?

Answer: 37


A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinders, in square inches? Express your answer in terms of $\pi$.

Answer: 16\pi


Let  $n$  be a positive integer. Compute, in terms of  $n$ , the number of sequences  $(x_1,\ldots,x_{2n})$  with each  $x_i\in\{0,1,2,3,4\}$  such that  $x_1^2+\dots+x_{2n}^2$  is divisible by  $5$ .

*2020 CCA Math Bonanza Individual Round #13*

Answer: 5^{2n-1}+4\cdot5^{n-1}


Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).

Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of \( g(3) \). Find \( m \times t \).

Answer: \frac{6019}{3}


Consider the game of "tic-tac-toe" on a 3D cubic grid $8 \times 8 \times 8$. How many straight lines can be indicated on which there are 8 symbols in a row?

Answer: 244


At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.

[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.

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


\(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of the area of quadrilateral \(EFPH\) to the area of parallelogram \(ABCD\).

Answer: 37/72


Consider a sequence $\{a_n\}$ with property P: if $a_p = a_q$ for $p, q \in \mathbb{N}^{*}$, then it must hold that $a_{p+1} = a_{q+1}$. Suppose the sequence $\{a_n\}$ has property P, and it is given that $a_1=1$, $a_2=2$, $a_3=3$, $a_5=2$, and $a_6+a_7+a_8=21$. Determine the value of $a_{2017}$.

Answer: 16


On the extension of side \(AC\) of the equilateral triangle \(ABC\) beyond point \(A\), a point \(M\) is taken, and circles are circumscribed around the triangles \(ABM\) and \(MBC\). Point \(A\) divides the arc \(MAB\) in the ratio \(MA:AB = n\). In what ratio does point \(C\) divide the arc \(MCB\)?

Answer: 2n+1


Find all natural numbers   $k$  such that there exist natural numbers  $a_1,a_2,...,a_{k+1}$  with  $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider  $0$  to be a natural number.

Answer: 3


Find all real \( x \) such that \( 0 < x < \pi \) and \(\frac{8}{3 \sin x - \sin 3x} + 3 \sin^2 x \le 5\).

Answer: \frac{\pi}{2}


There are 11 quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this
$$
\star x^{2}+\star x+\star=0 \text {. }
$$

Two players are playing a game making alternating moves. In one move each of them replaces one star with a real nonzero number.
The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible.
What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players.

Answer: 6


Let \(ABCD\) be a rectangle inscribed in circle \(\Gamma\), and let \(P\) be a point on minor arc \(AB\) of \(\Gamma\). Suppose that \(PA \cdot PB = 2\), \(PC \cdot PD = 18\), and \(PB \cdot PC = 9\). The area of rectangle \(ABCD\) can be expressed as \(\frac{a \sqrt{b}}{c}\), where \(a\) and \(c\) are relatively prime positive integers and \(b\) is a squarefree positive integer. Compute \(100a + 10b + c\).

Answer: 21055


What is the base $2$ representation of $84_{10}$?

Answer: 1010100_2


Given an equilateral triangle ABC. Point D is chosen on the extension of side AB beyond point A, point E on the extension of BC beyond point C, and point F on the extension of AC beyond point C such that CF = AD and AC + EF = DE. Find the angle BDE.

Answer: 60


Three circles  $\mathcal{K}_1$ ,  $\mathcal{K}_2$ ,  $\mathcal{K}_3$  of radii  $R_1,R_2,R_3$  respectively, pass through the point  $O$  and intersect two by two in  $A,B,C$ . The point  $O$  lies inside the triangle  $ABC$ .

Let  $A_1,B_1,C_1$  be the intersection points of the lines  $AO,BO,CO$  with the sides  $BC,CA,AB$  of the triangle  $ABC$ . Let  $ \alpha = \frac {OA_1}{AA_1} $ ,  $ \beta= \frac {OB_1}{BB_1} $  and  $ \gamma = \frac {OC_1}{CC_1} $  and let  $R$  be the circumradius of the triangle  $ABC$ . Prove that
\[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

Answer: \alphaR_1+\betaR_2+\gammaR_3\geR


Given a square pyramid \(M-ABCD\) with a square base such that \(MA = MD\), \(MA \perp AB\), and the area of \(\triangle AMD\) is 1, find the radius of the largest sphere that can fit into this square pyramid.

Answer: \sqrt{2} - 1


Prove that if \( A, B, C, \) and \( D \) are arbitrary points on a plane, then four circles, each passing through three points: the midpoints of the segments \( AB, AC \) and \( AD; BA, BC \) and \( BD; CA, CB \) and \( CD; DA, DB \) and \( DC \), have a common point.

Answer: 4


A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)} \ 6 \qquad\textbf{(D)} \ 3 + 2\sqrt{3} \qquad\textbf{(E)} \ 6 + \frac{\sqrt{3}}{3}$

Answer: \textbf{(D)}\3+2\sqrt{3}


 $ABCD$  - quadrilateral inscribed in circle, and  $AB=BC,AD=3DC$  . Point  $R$  is on the  $BD$  and  $DR=2RB$ . Point  $Q$  is on  $AR$  and  $\angle ADQ = \angle BDQ$ .  Also  $\angle ABQ + \angle CBD = \angle QBD$  .  $AB$  intersect line  $DQ$  in point  $P$ .
Find  $\angle APD$

Answer: 90


The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George and Alex?

Answer: 3795


How many 8-digit numbers begin with 1 , end with 3 , and have the property that each successive digit is either one more or two more than the previous digit, considering 0 to be one more than 9 ?

Answer: 21


A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).

Answer: 43


In an \( n \times n \) grid, each cell is filled with one of the numbers from \( 1 \) to \( n^2 \). If, no matter how the grid is filled, there are always two adjacent cells whose numbers differ by at least 1011, find the smallest possible value of \( n \).

Answer: 2020


Let points \( C \) and \( D \) be the trisection points of \( AB \). At 8:00, person \( A \) starts walking from \( A \) to \( B \) at a constant speed. At 8:12, person \( B \) starts walking from \( B \) to \( A \) at a constant speed. After a few more minutes, person \( C \) starts walking from \( B \) to \( A \) at a constant speed. When \( A \) and \( B \) meet at point \( C \), person \( C \) reaches point \( D \). When \( A \) and \( C \) meet at 8:30, person \( B \) reaches \( A \) exactly at that time. At what time did person \( C \) start walking?

Answer: 8:16


A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on Avani's cards?
A 6
B 9
C 10
D 11
E 12

Answer: 12


Let  $M$  be a set of  $99$  different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of  $M$  inside in. What is the maximum number of obtuse angles formed by two rays in  $M$ ?

Answer: 3267


In triangle  $ABC$ ,  $AB = 28$ ,  $AC = 36$ , and  $BC = 32$ . Let  $D$  be the point on segment  $BC$  satisfying  $\angle BAD = \angle DAC$ , and let  $E$  be the unique point such that  $DE \parallel AB$  and line  $AE$  is tangent to the circumcircle of  $ABC$ . Find the length of segment  $AE$ .

*Ray Li*

Answer: 18


When point P moves on the circle $C: x^2 - 4x + y^2 = 0$, there exist two fixed points $A(1, 0)$ and $B(a, 0)$, such that $|PB| = 2|PA|$, then $a = \ $.

Answer: -2


It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{1}{3}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^2 y \).

Answer: \frac{4}{5}


Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row.

Answer: 3430


Ben "One Hunna Dolla" Franklin is flying a kite KITE such that $I E$ is the perpendicular bisector of $K T$. Let $I E$ meet $K T$ at $R$. The midpoints of $K I, I T, T E, E K$ are $A, N, M, D$, respectively. Given that $[M A K E]=18, I T=10,[R A I N]=4$, find $[D I M E]$.

Answer: 16


What is the smallest base-10 integer that can be represented as $XX_6$ and $YY_8$, where $X$ and $Y$ are valid digits in their respective bases?

Answer: 63


Determine the largest natural number \( n \) such that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.

Answer: 2004


Given that the domain of the function \( f \) is
$$
\{(x, y) \mid x, y \in \mathbf{R}, xy \neq 0\},
$$
and it takes values in the set of positive real numbers, satisfying the following conditions:
(i) For any \( x, y \neq 0 \), we have
$$
f(xy, z) = f(x, z) f(y, z);
$$
(ii) For any \( x, y \neq 0 \), we have
$$
f(x, yz) = f(x, y) f(x, z);
$$
(iii) For any \( x \neq 0, 1 \), we have
$$
f(x, 1-x) = 1.
$$

Prove:
(1) For all \( x \neq 0 \), we have
$$
f(x, x) = f(x, -x) = 1;
$$
(2) For all \( x, y \neq 0 \), we have
$$
f(x, y) f(y, x) = 1.
$$

Answer: f(x,y)f(y,x)=1


In quadrilateral $ABCD$, there exists a point $E$ on segment $AD$ such that $\frac{AE}{ED}=\frac{1}{9}$ and $\angle BEC$ is a right angle. Additionally, the area of triangle $CED$ is 27 times more than the area of triangle $AEB$. If $\angle EBC=\angle EAB, \angle ECB=\angle EDC$, and $BC=6$, compute the value of $AD^{2}$.

Answer: 320


For each integer  $n\geqslant2$ , determine the largest real constant  $C_n$  such that for all positive real numbers  $a_1, \ldots, a_n$  we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]

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

Answer: \frac{1}{2n}


How many non-negative integers with not more than 1993 decimal digits have non-decreasing digits? [For example, 55677 is acceptable, 54 is not.]

Answer: \binom{2002}{9}


Four roads branch off a highway to four villages A, B, C, and D sequentially. It is known that the distance along the road/highway/road from A to B is 9 km, from A to C is 13 km, from B to C is 8 km, and from B to D is 14 km. Find the length of the path along the road/highway/road from A to D. Explain your answer.

Answer: 19\,


The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\{a_n\}$.

Answer: 3240