In the triangle $A B C$ with an area of one unit, let $D$ and $E$ be the trisection points on side $A B$, and let $F$ be the midpoint of side $A C$. The line $F E$ intersects line $C B$ at $G$, and the line $F D$ intersects line $C B$ at $H$. Determine the area of triangle $F G H$.

Answer: \frac{3}{2}


Given an integer \( n \geq 2 \), let the non-negative real numbers \( a_1, a_2, \cdots, a_n \) satisfy
$$
a_1 \geq a_2 \geq \cdots \geq a_n, \quad a_1 + a_2 + \cdots + a_n = n.
$$

Find the minimum value of \( a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots + a_1 a_2 \cdots a_n \).

Answer: n


Machines A, B, and C operate independently and are supervised by a single worker, who cannot attend to two or more machines simultaneously. Given that the probabilities of these machines operating without needing supervision are 0.9, 0.8, and 0.85 respectively, calculate the probability that during a certain period,
(1) all three machines operate without needing supervision;
(2) the worker is unable to supervise effectively, leading to a shutdown.

Answer: 0.059


Calculate the determinant of the following matrix:

$$
\left|\begin{array}{ccc}
a^{2}+b^{2}-c^{2}-d^{2} & 2 b c-2 a d & 2 b d+2 a c \\
2 b c+2 a d & a^{2}-b^{2}+c^{2}-d^{2} & 2 c d-2 a b \\
2 b d-2 a c & 2 c d+2 a b & a^{2}-b^{2}-c^{2}+d^{2}
\end{array}\right|
$$

Answer: (^2+b^2+^2+^2)^3


Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays  $(\frac{1}{2})^L$  drachmas, where  $L$  is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not contain any of the strings Dedalo bought. Dedalo’s aim is to trap the Minotaur.
For instance, if Dedalo buys the strings  $00$  and  $11$  for a total of half a drachma, the Minotaur is able to escape using the infinite string  $01010101 \ldots$ .
On the other hand, Dedalo can trap the Minotaur by spending  $75$  cents of a drachma: he could for example buy the strings  $0$  and  $11$ , or the strings  $00, 11, 01$ .
Determine all positive integers  $c$  such that Dedalo can trap the Minotaur with an expense of at most  $c$  cents of a drachma.

Answer: 75


Segments \( AM \) and \( BH \) are the median and altitude, respectively, of an acute-angled triangle \( ABC \). It is known that \( AH = 1 \) and \( 2 \angle MAC = \angle MCA \). Find the side \( BC \).

Answer: 2


For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate
\[
\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.
\]

Answer: \frac{\pi^2}{16}


These two spinners are divided into thirds and quarters, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.

[asy]

size(5cm,5cm);

draw(Circle((0,0),1));

draw(Circle((3,0),1));

draw((0,0)--(0,1));

draw((0,0)--(-0.9,-0.47));

draw((0,0)--(0.9,-0.47));

draw((2,0)--(4,0));

draw((3,1)--(3,-1));

label("$3$",(-0.5,0.3));

label("$4$",(0.5,0.3));

label("$5$",(0,-0.5));

label("$5$",(2.6,-0.4));

label("$6$",(2.6,0.4));

label("$7$",(3.4,0.4));

label("$8$",(3.4,-0.4));

draw((0,0)--(0.2,0.8),Arrow);

draw((3,0)--(3.2,0.8),Arrow);

[/asy]

Answer: \frac{2}{3}


We denote with  $p_n(k)$  the number of permutations of the numbers  $1,2,...,n$  that have exactly  $k$  fixed points.
	a) Prove that  $\sum_{k=0}^n kp_n (k)=n!$ .
	b) If  $s$  is an arbitrary natural number, then: $\sum_{k=0}^n k^s p_n (k)=n!\sum_{i=1}^m R(s,i)$ ,
where with  $R(s,i)$  we denote the number of partitions of the set  $\{1,2,...,s\}$  into  $i$  non-empty non-intersecting subsets and  $m=min(s,n)$ .

Answer: n!\sum_{i=1}^R(,i)


On the same road, two trucks are driving in the same direction. Initially, Truck A is 4 kilometers ahead of Truck B. The speed of Truck A is 45 kilometers per hour, and the speed of Truck B is 60 kilometers per hour. How far apart are the two trucks 1 minute before Truck B catches up to Truck A, in meters?

Answer: 250


Let \( S = \{1, 2, \ldots, n\} \). \( A \) is an arithmetic sequence with at least two terms and a positive common difference, with all its terms in \( S \). Adding any other element from \( S \) to \( A \) does not form another arithmetic sequence with the same common difference as \( A \). Determine the number of such sequences \( A \). (An arithmetic sequence with only two terms is also considered an arithmetic sequence).

Answer: \lfloor\frac{n^2}{4}\rfloor


Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after "Gauss". Let $x\in\mathbb{R}$, use $[x]$ to represent the largest integer not exceeding $x$, and use $\{x\}=x-[x]$ to represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss function. It is known that the sequence $\{a_n\}$ satisfies: $$a_{1}= \sqrt {3}, a_{n+1}=[a_{n}]+ \frac {1}{\{a_{n}\}}, (n\in\mathbb{N}^{*})$$, then $a_{2017}=$ \_\_\_\_\_\_.

Answer: 3024+ \sqrt {3}


For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define \[ S_n = \sum | A \cap B | , \] where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3, \cdots , n \right\}$ with $|A| = |B|$. For example, $S_2 = 4$ because the sum is taken over the pairs of subsets \[ (A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} , \] giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$. Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by 1000.

Answer: 245


Please fold a long rope in half, then fold it in half again along the middle of the folded rope, and continue to fold it in half 5 times in total. Finally, cut the rope along the middle after it has been folded 5 times. At this point, the rope will be cut into ___ segments.

Answer: 33


At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the *product* of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are  $8^6=262144$  participants, and no two get the same score on every problem. Find the score of the participant whose rank was  $7^6 = 117649$ .

*Proposed by Yang Liu*

Answer: 1


Given distinct natural numbers \(a, b, c, d\) that satisfy the following conditions: \(a > d\), \(ab = cd\), and \(a + b + c + d = ac\). Find the sum of all four numbers.

Answer: 12


In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.6 times the probability of earning a B. The probability of earning a D is 0.3 times the probability of earning a B. Assuming that all grades are A, B, C, or D, how many B's will there be in a mathematics class of 50 students?

Answer: 14


Given the letters in the word $SUCCESS$, determine the number of distinguishable rearrangements where all the vowels are at the end.

Answer: 20


If \( a, b, c, d \) are four distinct positive integers such that \( a \times b \times c \times d = 2277 \), what is the maximum value of \( a + b + c + d \)?

Answer: 84


The vertices of a regular 120-gon are given as \(A_{1}, A_{2}, A_{3}, \ldots, A_{120}\). In how many ways can three vertices be chosen such that they form an obtuse triangle?

Answer: 205320


Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Compute $a+b+c$.

Answer: 141


Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice?

Answer: \frac{19}{40}


Each face of a fair six-sided die is marked with one of the numbers $1, 2, \cdots, 6$. When two such identical dice are rolled, the sum of the numbers on the top faces of these dice is the score for that roll. What is the probability that the product of the scores from three such rolls is divisible by 14? Express your answer as a simplified fraction.

Answer: 1/3


Given a parabola with vertex \( V \) and a focus \( F \), and points \( B \) and \( C \) on the parabola such that \( BF=25 \), \( BV=24 \), and \( CV=20 \), determine the sum of all possible values of the length \( FV \).

Answer: \frac{50}{3}


A regular hexagon with side length  $1$  is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is  $\sqrt7$ .

Notes: ''Using a ruler construct points  $\ldots$ '' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

Answer: \sqrt{7}


Answer: 8


Let  $1 = x_{1} < x_{2} < \dots < x_{k} = n$  denote the sequence of all divisors  $x_{1}, x_{2} \dots x_{k}$  of  $n$  in increasing order. Find the smallest possible value of  $n$  such that  $$ n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}. $$  *Proposed by Justin Lee*

Answer: 130


A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group?

Answer: 23


Libor divided a rectangle with a single line into two smaller rectangles. The perimeter of the large rectangle is \(76 \text{ cm}\), and the perimeters of the smaller rectangles are \(40 \text{ cm}\) and \(52 \text{ cm}\). Determine the dimensions of the large rectangle.

Answer: 30\,


An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of the smaller triangle?

Answer: 25\%


Given the function \( f(x) = \frac{1}{\sqrt[3]{1 - x^3}} \). Find \( f(f(f( \ldots f(19)) \ldots )) \), calculated 95 times.

Answer: \sqrt[3]{1 - \frac{1}{19^3}}


Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number)

Answer: 420


Let \( V \) be the set of vertices of a regular 21-gon. Given a non-empty subset \( U \) of \( V \), let \( m(U) \) be the number of distinct lengths that occur between two distinct vertices in \( U \). What is the maximum value of \( \frac{m(U)}{|U|} \) as \( U \) varies over all non-empty subsets of \( V \)?

Answer: 2


Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is:

Answer: 1


A three-digit number has a remainder of 2 when divided by 4, 5, and 6. If three digits are appended to this three-digit number to make it a six-digit number divisible by 4, 5, and 6, what is the smallest six-digit number that meets this condition?

Answer: 122040


The area of a quadrilateral formed by the midpoints of the bases and diagonals of a trapezoid is four times smaller than the area of the trapezoid itself. Find the ratio of the lengths of the bases of the trapezoid.

Answer: 3:1


A board for playing "Battleship" has a size of $10 \times 10$. What is the maximum number of ships with a size of $1 \times 4$ that can be placed on it?

Answer: 24


Lines \(AM\) and \(AN\) are symmetric with respect to the angle bisector of angle \(A\) in triangle \(ABC\) (points \(M\) and \(N\) lie on line \(BC\)). Prove that \(\frac{BM \cdot BN}{CM \cdot CN} = \frac{c^2}{b^2}\). In particular, if \(AS\) is the symmedian, then \(\frac{BS}{CS} = \frac{c^2}{b^2}\).

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


Consider a $5 \times 5$ grid, consisting of 25 white cells. Vincent wants to color some cells in red so that every $2 \times 3$ or $3 \times 2$ rectangle on the grid contains at least one colored cell. What is the minimum number of cells he needs to color?

Answer: 4


11 gardeners are planting trees. 2 of them are planting trees on Street A, and the remaining 9 are planting trees on Street B. In the evening, they return to their dormitory.

Gardener A says: "Although there are fewer of us, we spent the same amount of time as you did."
Gardener B says: "Although there are more of us, the length of our street is 5 times the length of your street."

If the intervals at which they plant the trees are the same and each gardener plants the same number of trees, planting only on one side of the road and including both ends of the street, how many trees did these 11 gardeners plant in total?

Answer: 44


Find the domain of the function $z(x) = \sqrt[3]{x - 1} + \sqrt[3]{8 - x}.$

Answer: (-\infty,\infty)


Answer: -57


A piece of paper is flipped upside down, and the digits 0, 1, and 8 remain unchanged when rotated by $180^{\circ}$. The digit 6 becomes 9, and 9 becomes 6. All other digits have no meaning when rotated by $180^{\circ}$. How many 7-digit numbers remain unchanged after being rotated by $180^{\circ}$? Among these, how many are divisible by 4? What is the total sum of these 7-digit numbers that remain unchanged after the rotation by $180^{\circ}$?

Answer: 1959460200


Given in the polar coordinate system, the equation of curve Ω is $\rho=6\cos\theta$. Taking the pole as the origin of the Cartesian coordinate system, with the polar axis as the positive half-axis of the x-axis, and using the same unit of length in both coordinate systems, establish a Cartesian coordinate system. The parametric equation of line $l$ is $$\begin{cases} x=4+t\cos\theta \\ y=-1+t\sin\theta \end{cases}$$ (t is the parameter, $\theta\in\mathbb{R}$).
(Ⅰ) Find the Cartesian coordinate equation of curve Ω and the general equation of line $l$;
(Ⅱ) Suppose line $l$ intersects curve Ω at points A and C, and a line $l_0$ passing through point (4, -1) and perpendicular to line $l$ intersects curve Ω at points B and D. Find the maximum value of the area of quadrilateral ABCD.

Answer: 16


In the convex quadrilateral \(ABCD\), it is known that \(\angle BCD = 80^\circ\), \(\angle ACB = 50^\circ\), and \(\angle ABD = 30^\circ\). Find the angle \(\angle ADB\).

Answer: 50


Answer: -243


Let $A_1A_2A_3\ldots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 19


Let \(ABC\) be a triangle with \(AB = 3\), \(BC = 4\), and \(CA = 5\). Let \(A_1\) and \(A_2\) be points on side \(BC\), \(B_1\) and \(B_2\) be points on side \(CA\), and \(C_1\) and \(C_2\) be points on side \(AB\). Suppose that there exists a point \(P\) such that \(PA_1A_2\), \(PB_1B_2\), and \(PC_1C_2\) are congruent equilateral triangles. Find the area of convex hexagon \(A_1A_2B_1B_2C_1C_2\).

Answer: \frac{12+22\sqrt{3}}{15}


Circle I is externally tangent to Circle II and passes through the center of Circle II. Given that the area of Circle I is increased to 16 square inches, determine the area of Circle II, in square inches.

Answer: 64


Answer: 9


There are 2009 non-negative integers arranged in a circle, each not exceeding 100. It is allowed to add 1 to two adjacent numbers, and this operation can be performed on any two adjacent numbers no more than $k$ times. What is the minimum $k$ such that all numbers can be made equal?

Answer: 100400


There are 36 criminal gangs operating in Chicago, some of which are hostile toward each other. Each gangster belongs to several gangs, with no two gangsters belonging to the same set of gangs. It is known that no gangster belongs to two gangs that are hostile toward each other. Additionally, it turns out that each gang, which a particular gangster does not belong to, is hostile to at least one gang that the gangster does belong to. What is the maximum number of gangsters that can be in Chicago?

Answer: 531441


A rectangular piece of paper is folded along its diagonal to form a non-convex pentagon that has an area of $\frac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle.

Answer: \sqrt{5}


In kite $ABCD$, $AB = AD$ and $BC = CD$. Two lines pass through the intersection point $O$ of $AC$ and $BD$, intersecting $AD$ at $E$, $BC$ at $F$, $AB$ at $G$, and $CD$ at $H$. Suppose $GF$ and $EH$ intersect $BD$ at $I$ and $J$ respectively. Prove that $IO = OJ$.

Answer: IO=OJ


The teacher gave each of her  $37$  students  $36$  pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.

Answer: 666


Answer: 2\sqrt{5}


Two circles with radii \( r_1 \) and \( r_2 \) touch the circle \( O \) with radius \( r \) externally (internally) at points such that the distance between these points is \( m \). Prove that the length \( t_{12} \) of the common external tangent of the first two circles is given by the formula:

\[
t_{12} = \frac{m}{r} \sqrt{(r \pm r_1)(r \pm r_2)}
\]

(The minus sign corresponds to external tangency).

Answer: t_{12}=\frac{}{r}\sqrt{(r\r_1)(r\r_2)}


Ilya takes a triplet of numbers and transforms it according to the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet on the 1989th step of applying this rule, given that the initial triplet of numbers was $\{70; 61; 20\}$? If the question allows multiple answers, list them without spaces in ascending order.

Answer: 50


Let  $M$  be a set consisting of  $n$  points in the plane, satisfying:
i) there exist  $7$  points in  $M$  which constitute the vertices of a convex heptagon;
ii) if for any  $5$  points in  $M$  which constitute the vertices of a convex pentagon, then there is a point in  $M$  which lies in the interior of the pentagon.
Find the minimum value of  $n$ .

*Leng Gangsong*

Answer: 11


A circle of 100 people is formed, each of whom is either a knight or a liar (liars always lie, and knights always tell the truth). Each person says, "I have a neighbor who is a liar." Find the minimum possible number of liars among these 100 people.

Answer: 34


What is the least positive integer  $k$  such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?

Answer: 249750


Vitya wrote down \( n \) distinct natural numbers in a notebook. For each pair of numbers from the notebook, he wrote their least common multiple on the board. Could it happen for some \( n > 100 \) that the \(\frac{n(n-1)}{2}\) numbers on the board form (in some order) consecutive terms of a non-constant arithmetic progression?

Answer: No


If \( f(t) = 2 - \frac{t}{3} \), and \( f(a) = -4 \), find \( a \).

If \( a + 9 = 12Q + r \), where \( Q, r \) are integers and \( 0 < r < 12 \), find \( r \).

\( x, y \) are real numbers. If \( x + y = r \) and \( M \) is the maximum value of \( xy \), find \( M \).

If \( w \) is a real number and \( 2^{2w} - 2^w - \frac{8}{9}M = 0 \), find \( w \).

Answer: 1


Let \( x \) and \( y \) be two non-zero numbers such that \( x^2 + xy + y^2 = 0 \) (where \( x \) and \( y \) are complex numbers, but that is not too important). Find the value of

\[ \left( \frac{x}{x+y} \right)^{2013} + \left( \frac{y}{x+y} \right)^{2013} \]

Answer: -2


On a plane, a point \( O \) is marked. Is it possible to position on the plane: a) five circles; b) four circles, not covering point \( O \), such that any ray starting at point \( O \) intersects at least two circles? ("Intersects" means having a common point.)

Answer: No


A heap of stones has a total weight of 100 kg, where the weight of each stone does not exceed 2 kg. Choose some stones from this heap in any combination and find the difference between the sum of these chosen stones' weights and 10 kg. Denote by $d$ the minimum value of the absolute value of these differences. Find the maximum value of $d$ among all heaps of stones satisfying the above conditions.

Answer: \frac{10}{11}


Let \(\mathcal{P}\) be a regular 17-gon. We draw in the \(\binom{17}{2}\) diagonals and sides of \(\mathcal{P}\) and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of \(\mathcal{P}\)) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?

Answer: 544


Let  $k$  be a positive integer with the following property: For every subset  $A$  of  $\{1,2,\ldots, 25\}$  with  $|A|=k$ , we can find distinct elements  $x$  and  $y$  of  $A$  such that  $\tfrac23\leq\tfrac xy\leq\tfrac 32$ .  Find the smallest possible value of  $k$ .

Answer: 7


Let  $n$  be a fixed positive integer. There are  $n \geq 1$  lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer  $i$  ( $1 \leq i \leq n$ ) and switch the state of the first  $i$  lamps from the left. Determine the smallest number  $k$  with the property that we can make all of the lamps be switched on using at most  $k$  moves, no matter what the initial configuration was.

*Proposed by Viktor Simjanoski and Nikola Velov*

Answer: n


What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal?

Answer: 3


In the rectangular coordinate system, circle \( C_1 \) has center \( C_1(-2,0) \) and radius 3. \( O \) is the origin, and \( A \) is \( (1,0) \). Prove that there exists a positive constant \( c \) such that for any point \( X \) outside \( C_1 \), \( |OX| - 1 \geqslant c \cdot \min \left\{|AX|,|AX|^2\right\} \). Find the minimum value of \( c \).

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


Construct a square \( A B C D \) with a side length of \( 6 \text{ cm} \) and label the intersection of its diagonals as \( S \). Construct a point \( K \) such that together with points \( S, B, \) and \( C \), they form a square \( B K C S \). Construct a point \( L \) such that together with points \( S, A, \) and \( D \), they form a square \( A S D L \). Construct the segment \( K L \), mark the intersection of segments \( K L \) and \( A D \) as \( X \), and mark the intersection of segments \( K L \) and \( B C \) as \( Y \).

Using the given data, calculate the length of the broken line \( K Y B A X L \).

Answer: 18


Determine the value of $$1 \cdot 2-2 \cdot 3+3 \cdot 4-4 \cdot 5+\cdots+2001 \cdot 2002$$

Answer: 2004002


For each positive integer \( n \), let \( a_n \) denote the number of \( n \)-digit integers formed by some or all of the digits 0, 1, 2, and 3 which contain neither a block of 12 nor a block of 21. Evaluate \( a_9 \).

Answer: 73368


Let $a, b$ be integers chosen independently and uniformly at random from the set $\{0,1,2, \ldots, 80\}$. Compute the expected value of the remainder when the binomial coefficient $\binom{a}{b}=\frac{a!}{b!(a-b)!}$ is divided by 3.

Answer: \frac{1816}{6561}


Prove that for any natural number \( n \):

\[
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{(n+1)(n+2)}=\frac{n+1}{n+2}
\]

and using this equality, solve the equation:

\[
(1+3+5+\ldots+(2 n+1)):\left(\frac{1}{2}+\frac{1}{6}+\ldots+\frac{1}{342}\right)=342
\]

Answer: 17


Given a regular square pyramid P-ABCD with its five vertices on the same sphere, if the volume of the pyramid is 1, find the minimum radius of the sphere.

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


In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 69


Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors and $(\overrightarrow{a} - 6\overrightarrow{b}) \perp \overrightarrow{a}$, $(2\overrightarrow{a} - 3\overrightarrow{b}) \perp \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: \frac{\pi}{3}


Solve the equation
\[
  \sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for  $x$  where  $0 < x < 90$ .

Answer: 9


On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles \(ABCD\) and \(EFGH\), whose sides are parallel to the edges of the sheet. It is known that \(AB = 9\), \(BC = 5\), \(EF = 3\), and \(FG = 10\). Find the area of the quadrilateral \(AFCH\).

Answer: 52.5


In a triangular pyramid, all the plane angles at vertex \( A \) are right angles, and edge \( AB \) is equal to the sum of the other two edges emanating from \( A \). Prove that the sum of the plane angles at vertex \( B \) is equal to \(\pi / 2\).

Answer: \frac{\pi}{2}


27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube?

Answer: 189


A particle is located on the coordinate plane at  $ (5,0)$ . Define a *move* for the particle as a counterclockwise rotation of  $ \pi/4$  radians about the origin followed by a translation of  $ 10$  units in the positive  $ x$ -direction. Given that the particle's position after  $ 150$  moves is  $ (p,q)$ , find the greatest integer less than or equal to  $ |p|\plus{}|q|$ .

Answer: 19


A ball is dropped from a height of $128$ meters, and each time it hits the ground, it bounces back to half of its original height. When it hits the ground for the $9$th time, the total distance it has traveled is ______ meters.

Answer: 383


In the "five rows by six columns" examination room, if two students A and B from the same school are to be seated with no two adjacent seats in any direction (front, back, left, right), calculate the number of different seating arrangements for students A and B.

Answer: 772


There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3\%$. Determine the initial percentage concentration of salt.

Answer: 27


How many ways are there to arrange the letters of the word $\text{B}_1\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{B}_2$, where three A's, two N's, and two B's are all considered different within each letter group but identical between groups?

Answer: 210


During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \ldots, z^{2012}$, in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^{2}$.

Answer: \frac{1005}{1006}


Given \( E(m, n) \) is a point inside the parabola
\[ y^{2} = 2px \quad (p>0) \]
Two lines with slopes \( k_1 \) and \( k_2 \) pass through \( E \) and intersect the parabola at points \( A, B, C, \) and \( D \). Points \( M \) and \( N \) are midpoints of segments \( AB \) and \( CD \), respectively.

1. When \( n=0 \) and \( k_1 k_2 = -1 \), find the minimum area of \(\triangle EMN\).
2. If \( k_1 + k_2 = \lambda \) (constant \(\lambda \neq 0\)), prove that line \( MN \) passes through a fixed point.

Answer: (-\frac{n}{\lambda},\frac{p}{\lambda})


An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the longer side of this frame?

Answer: 2:3


Find the number of integers $n$ with $1 \leq n \leq 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of 1001.

Answer: 99


Let \( A \) be any set of 7 real numbers. Define the set \( B = \{xy \mid x, y \in A \text{ and } x \neq y\} \). If \( |B| \) denotes the number of elements in the set \( B \), what is \( |B|_{\text{max}} + |B|_{\text{min}} \)?

Answer: 30


Answer: 2525


Define the function \( f(n) \) for \( n \in \mathbb{N} \) as follows:

Given the fraction:
$$
\frac{(2n)!}{n!(n + 1000)!} = \frac{A(n)}{B(n)},
$$
where \( A(n) \) and \( B(n) \) are coprime positive integers.

If \( B(n) = 1 \), then \( f(n) = 1 \). If \( B(n) \neq 1 \), then \( f(n) \) is the largest prime factor of \( B(n) \).

Prove that there are only finitely many different values of \( f(n) \), and find its maximum value.

(31st International Mathematical Olympiad, 1990)

Answer: 1999


On the extensions of the sides \(AB\), \(BC\), \(CD\) and \(DA\) of a convex quadrilateral \(ABCD\), points \(B_1\), \(C_1\), \(D_1\), and \(A_1\) are taken respectively, such that \(BB_1 = AB\), \(CC_1 = BC\), \(DD_1 = CD\), and \(AA_1 = DA\). By how many times is the area of quadrilateral \(ABCD\) smaller than the area of quadrilateral \(A_1 B_1 C_1 D_1\)? (10 points)

Answer: 5


Let \(k \geq 1\) be an integer.

We consider \(4k\) chips, \(2k\) of which are red and \(2k\) of which are blue. A sequence of those \(4k\) chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from \(r \underline{b b b r} \underline{r} b\) to \(r \underline{r r} b r \underline{b b b}\) where \(r\) denotes a red chip and \(b\) denotes a blue chip.

Determine the smallest number \(n\) (as a function of \(k\)) such that starting from any initial sequence of the \(4k\) chips, we need at most \(n\) moves to reach the state in which the first \(2k\) chips are red.

Answer: k


On a grid, there are three locations $A, B, C$, with each small square having a side length of 100 meters. If roads are to be built along the grid lines to connect the three locations, what is the minimum total length of the roads in meters?

Answer: 1000


If  $x$  is a real number, let  $\lfloor x \rfloor$  be the greatest integer that is less than or equal to  $x$ .  If  $n$  is a positive integer, let  $S(n)$  be defined by
\[
  S(n)
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor}
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.)  How many integers  $n$  from 1 to 2011 (inclusive) satisfy  $S(S(n)) = n$ ?

Answer: 108


The set $A$ consists of positive integers, and for any $x, y \in A$ with $x > y$, it holds that $x - y \geq \frac{xy}{25}$. What is the maximum number of elements that set $A$ can have?

Answer: 9


Solve
\[(x - 3)^4 + (x - 5)^4 = -8.\]Enter all the solutions, separated by commas.

Answer: 4 + i, 4 - i, 4 + i \sqrt{5}, 4 - i \sqrt{5}


Consider all  $6$ -digit numbers of the form  $abccba$  where  $b$  is odd. Determine the number of all such  $6$ -digit numbers that are divisible by  $7$ .

Answer: 70


Suppose that \(a, b, c\), and \(d\) are real numbers such that \(a+b+c+d=8\). Compute the minimum possible value of
$$
20\left(a^{2}+b^{2}+c^{2}+d^{2}\right)-\sum_{\text {sym }} a^{3} b,
$$
where the sum is over all 12 symmetric terms.

Answer: 112


The function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) has how many zeros in the interval \([0, \pi]\)?

Answer: 2017


A circle passes through the vertices \(A\) and \(C\) of triangle \(ABC\) and intersects its sides \(AB\) and \(BC\) at points \(K\) and \(T\) respectively, such that \(AK:KB = 3:2\) and \(BT:TC = 1:2\). Find \(AC\) if \(KT = \sqrt{6}\).

Answer: 3\sqrt{5}


Given vectors $\overrightarrow{m} = (a\cos x, \cos x)$ and $\overrightarrow{n} = (2\cos x, b\sin x)$, with a function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$ that satisfies $f(0) = 2$ and $f\left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}$,
(1) If $x \in \left[0, \frac{\pi}{2}\right]$, find the maximum and minimum values of $f(x)$;
(2) If $f\left(\frac{\theta}{2}\right) = \frac{3}{2}$, and $\theta$ is an internal angle of a triangle, find $\tan \theta$.

Answer: -\frac{4 + \sqrt{7}}{3}


If one person spends one minute waiting in line, we will say that one person-minute was wasted. There are eight people in line at the bank. Five of them plan simple operations that take 1 minute each, and the remaining three plan lengthy operations that take 5 minutes each. Find:

a) the minimum and maximum possible total number of wasted person-minutes;

b) the expected number of wasted person-minutes, assuming the customers join the line in a random order.

Answer: 70


Let  $P$  be an interior point of triangle  $ABC$ . Let  $a,b,c$  be the sidelengths of triangle  $ABC$  and let  $p$  be it's semiperimeter. Find the maximum possible value of  $$  \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$  taking into consideration all possible choices of triangle  $ABC$  and of point  $P$ .

by Elton Bojaxhiu, Albania

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


The product of three prime numbers. There is a number that is the product of three prime factors whose sum of squares is equal to 2331. There are 7560 numbers (including 1) less than this number and coprime with it. The sum of all the divisors of this number (including 1 and the number itself) is 10560. Find this number.

Answer: 8987


The intersection point of the altitudes of an isosceles triangle lies on the circle inscribed in it. Find the ratio of the sides of the triangle.

Answer: 3:3:4


Given that \( n! \) is evenly divisible by \( 1 + 2 + \cdots + n \), find the number of positive integers \( n \) less than or equal to 50.

Answer: 36


In a square $A B C D$ with side length 2, $E$ is the midpoint of $A B$. The square is folded along segments $E C$ and $E D$ such that segments $E A$ and $E B$ coincide (point $B$ coincides with point $A$), forming the tetrahedron $C D E A$. Find the volume of this tetrahedron.

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


For each integer  $n \ge 1$ , prove that there is a polynomial  $P_{n}(x)$  with rational coefficients such that  $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$ . Define the rational number  $a_{n}$  by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that  $a_{n}$  satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

Answer: |\pi-a_n|<\frac{1}{4^{5n-1}}


Find the minimum value of the expression \(\left\lfloor \frac{8(a+b)}{c} \right\rfloor + \left\lfloor \frac{8(a+c)}{b} \right\rfloor + \left\lfloor \frac{8(b+c)}{a} \right\rfloor\), where \(a\), \(b\), and \(c\) are arbitrary natural numbers.

Answer: 46


Given the hyperbola \( P : \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \) with left and right foci \( B \) and \( C \), point \( A \) lies on \( P \). \( I \) is the incenter of triangle \( ABC \) and the line \( AI \) passes through the point \( (1,0) \). If \( \overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C} \), then \( x+y \) equals ____.

Answer: \frac{3}{4}


Emir Ben Sidi Mohammed travels from his palace to the airport at a constant speed on a splendid highway through the desert. If his driver increased the average speed of the car by $20 \mathrm{km/h}$, the emir would save 2 minutes; if the driver drove 20 km/h slower, the emir would lose 3 minutes.

How far is the emir's palace from the airport?

Answer: 20\,


What is the value of $\frac12\cdot\frac41\cdot\frac18\cdot\frac{16}{1} \dotsm \frac{1}{512}\cdot\frac{1024}{1}$?

Answer: 32


Given that $α∈[\dfrac{π}{2}, \dfrac{3π}{2}]$, $β∈[-\dfrac{π}{2}, 0]$, and the equations $(α-\dfrac{π}{2})^{3}-\sin α-2=0$ and $8β^{3}+2\cos^{2}β+1=0$ hold, find the value of $\sin(\dfrac{α}{2}+β)$.

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


The function \( y = f(x) \) is defined on the set \( (0, +\infty) \) and takes positive values on it. It is known that for any points \( A \) and \( B \) on the graph of the function, the areas of the triangle \( AOB \) and the trapezoid \( ABH_BH_A \) are equal (\( H_A, H_B \) are the bases of the perpendiculars dropped from points \( A \) and \( B \) to the x-axis; \( O \) is the origin). Find all such functions. Given that \( f(1) = 4 \), find the value of \( f(4) \).

Answer: 1


Calculate $1010101_2 + 110011_2$ and express your answer in base $10$.

Answer: 136


Let  $BE$  and  $CF$  be altitudes in triangle  $ABC$ . If  $AE = 24$ ,  $EC = 60$ , and  $BF = 31$ , determine  $AF$ .

Answer: 32


Given the arithmetic sequence $\{a_n\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms.

Answer: 180


The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is

Answer: 2: 1


On the leg \(BC\) of the right triangle \(ABC\), a circle is constructed with \(BC\) as its diameter. This circle intersects the hypotenuse \(AB\) at point \(P\). Chord \(PQ\) is parallel to leg \(BC\). Line \(BQ\) intersects leg \(AC\) at point \(D\). Given that \(AC = b\) and \(DC = d\), find \(BC\).

Answer: \sqrt{bd}


In triangle \(ABC\) with \(\angle B = 120^\circ\), angle bisectors \(AA_1\), \(BB_1\), and \(CC_1\) are drawn. The segment \(A_1B_1\) intersects the angle bisector \(CC_1\) at point \(M\). Find the measure of angle \(\angle B_1MC_1\).

Answer: 60


A team of \( n \) students is participating in a game: each of them is given a hat of one of the \( k \) predetermined colors, and then, at the sound of a whistle, all students simultaneously choose one scarf. The team earns points based on how many of its members' hat colors match the colors of their scarves (there are enough scarves and hats of any color; during the game, each participant can see the hats of all the others, but cannot reveal any information before the whistle). What is the maximum number of points the team, having predetermined a plan of action for each member, can be guaranteed to get:

a) when \( n = k = 2 \)

b) for arbitrary fixed \( n \) and \( k \)?

Answer: \lfloor\frac{n}{k}\rfloor


Let $S$ be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in $S$ to the smallest area of any triangle in $S$.

Answer: 3+2 \sqrt{2}


Answer: (6,1)


We draw squares on the sides of a right-angled triangle. Calculate the area of the hexagon determined by the outer vertices (not identical with the vertices of the triangle) of the squares if we know the hypotenuse $(c)$ of our right-angled triangle, and the sum of the legs $(d)$.

Answer: ^2+^2


Answer: \frac{592}{969}


Given the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and the line $l: 4x - 5y + 40 = 0$. Is there a point on the ellipse for which the distance to line $l$ is minimal? If so, what is the minimal distance?

Answer: \frac{15}{\sqrt{41}}


Given a convex quadrilateral \(ABCD\). Point \(M\) is the midpoint of side \(BC\), and point \(N\) is the midpoint of side \(CD\). The segments \(AM\), \(AN\), and \(MN\) divide the quadrilateral into four triangles whose areas, in some order, are consecutive natural numbers. What is the largest possible area that triangle \(ABD\) can have?

Answer: 6


A plane intersects each segment of a spatial closed broken line \( P_{1} P_{2} \ldots P_{n} P_{1} \) at internal points \( Q_{i} \) on each segment \( P_{i} P_{i+1} \). Show that

\[
\frac{P_{1} Q_{1}}{Q_{1} P_{2}} \cdot \frac{P_{2} Q_{2}}{Q_{2} P_{3}} \cdot \cdots \cdot \frac{P_{n-1} Q_{n-1}}{Q_{n-1} P_{n}} \cdot \frac{P_{n} Q_{n}}{Q_{n} P_{1}} = 1
\]

Answer: 1


Let $O$ be the vertex of a parabola, $F$ be the focus, and $PQ$ be a chord passing through $F$. Given that $|OF|=a$ and $|PQ|=b$, find the area of $\triangle OPQ$.

Answer: \sqrt{}


The base of the pyramid \( SABC D \) is a parallelogram \( ABCD \). Let \( M \) be the midpoint of \( AB \) and \( N \) be the midpoint of \( SC \). In what ratio does the plane \( BSD \) divide the segment \( MN \)?

Answer: 1:1


Into how many parts can a plane be divided by four lines? Consider all possible cases and make a drawing for each case.

Answer: 11


In a flat plain at Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ lines such that no two lines are parallel and no three lines are concurrent. For each wall, Merlin colors one side red and the other side blue.

At the intersection of two walls, there are 4 corners: two corners where a red side and a blue side intersect, one corner where two red sides intersect, and one corner where two blue sides intersect. At this intersection, there exists a door connecting the two opposite corners where sides of different colors intersect.

After Merlin paints the walls, Morgana places knights in the labyrinth. The knights can walk through the doors but cannot pass through walls.

Let $k(\mathfrak{L})$ be the largest integer $k$ such that, no matter how Merlin colors the labyrinth $\mathfrak{L}$, Morgana can always place $k$ knights in such a way that no two of them can ever meet. For a fixed $n$, what are the possible values of $k(\mathfrak{L})$ for labyrinths $\mathfrak{L}$ consisting of $n$ lines?

Answer: n+1


There are $52$ people in a room. what is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 12$

Answer: \textbf{(D)}\5


Let  $x,y,z>0$  such that  $$ (x+y+z)\left(\frac1x+\frac1y+\frac1z\right)=\frac{91}{10} $$  Compute  $$ \left[(x^3+y^3+z^3)\left(\frac1{x^3}+\frac1{y^3}+\frac1{z^3}\right)\right] $$  where  $[.]$  represents the integer part.

*Proposed by Marian Cucoanoeş and Marius Drăgan*

Answer: 9


Calculate the value of the expression \[(5^{1003}+6^{1004})^2-(5^{1003}-6^{1004})^2\] and express it in the form of $k\cdot30^{1003}$ for some positive integer $k$.

Answer: 24


The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is:

Answer: $n(n+2)$


The sum of the distances from one vertex of a square with sides of length $2$ to the midpoints of each of the sides of the square is
$\textbf{(A) }2\sqrt{5}\qquad \textbf{(B) }2+\sqrt{3}\qquad \textbf{(C) }2+2\sqrt{3}\qquad \textbf{(D) }2+\sqrt{5}\qquad \textbf{(E) }2+2\sqrt{5}$

Answer: \textbf{(E)}2+2\sqrt{5}


Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola.

Answer: \frac{\sqrt{17}}{2}


A bag contains only green, yellow, and red marbles. The ratio of green marbles to yellow marbles to red marbles in the bag is \(3:4:2\). If 63 of the marbles in the bag are not red, the number of red marbles in the bag is:
(A) 14
(B) 18
(C) 27
(D) 36
(E) 81

Answer: 18


There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\mid z \mid = 1$. These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$, where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldots + \theta_{2n}$.

Answer: 840


A parallelogram is intersected by two sets of lines, each set being parallel to its sides. Each set consists of \( m \) lines.

How many parallelograms can be identified in the resulting grid?

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


Compute the sum of the squares of the roots of the equation \[x^{2018} +44x^{2015} + 3x^3 + 404 = 0.\]

Answer: 0


Given a device purchased at $40 less a 15% discount, and a goal of achieving a 25% selling profit based on the cost after allowing a 10% discount on the final sales price, determine the marked price of the device in dollars.

Answer: 47.22


A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
$\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad\textbf{(E) }\frac{1}{3}$

Answer: \textbf{(A)}\frac{5}{54}


Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$.

Answer: \frac{9}{4}


A circle is circumscribed around triangle \(ABC\). The median \(AD\) is extended to intersect the circle at point \(E\). It is known that \(AB + AD = DE\), \(\angle BAD = 60^\circ\), and \(AE = 6\). Find the area of triangle \(ABC\).

Answer: \frac{9\sqrt{3}}{4}


Answer: 1121_4


Let  $ABCD$  be a rectangle. We consider the points  $E\in CA,F\in AB,G\in BC$  such that  $DC\perp CA,EF\perp AB$  and  $EG\perp BC$ . Solve in the set of rational numbers the equation  $AC^x=EF^x+EG^x$ .

Answer: 2/3


I1.1 Let \( P \) be the unit digit of \( 3^{2003} \times 5^{2002} \times 7^{2001} \). Find the value of \( P \).

I1.2 If the equation \( \left(x^{2}-x-1\right)^{x+P-1}=1 \) has \( Q \) integral solutions, find the value of \( Q \).

I1.3 Let \( x, y \) be real numbers and \( x y=1 \).
If the minimum value of \( \frac{1}{x^{4}}+\frac{1}{Q y^{4}} \) is \( R \), find the value of \( R \).

11.4 Let \( x_R, x_{R+1}, \ldots, x_K (K>R) \) be \( K-R+1 \) distinct positive integers and \( x_R + x_{R+1} + \ldots + x_K = 2003 \). If \( S \) is the maximum possible value of \( K \), find the value of \( S \).

Answer: 62


Point \( M \) is located on the side \( BC \) of parallelogram \( ABCD \) such that \( BM: MC = 3:2 \). The segments \( AM \) and \( BD \) intersect at point \( K \). It is known that the area of the parallelogram is 1. Find the area of quadrilateral \( CMKD \).

Answer: \frac{31}{80}


In a cabinet, there are 3 pairs of different shoes. If 2 shoes are randomly taken out, let event A denote "the taken out shoes do not form a pair"; event B denote "both taken out shoes are for the same foot"; event C denote "one shoe is for the left foot and the other is for the right foot, but they do not form a pair".

(Ⅰ) Please list all the basic events;
(Ⅱ) Calculate the probabilities of events A, B, and C respectively.

Answer: \dfrac{2}{5}


Right $\triangle ABC$ has $AB=3$, $BC=4$, and $AC=5$.  Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $\overline{AC}$, $W$ on $\overline{AB}$, and $Z$ on $\overline{BC}$.  What is the side length of the square?

[asy]
pair A,B,C,W,X,Y,Z;
A=(-9,0); B=(0,12); C=(16,0);
W=(12A+25B)/37;
Z =(12C+25B)/37;
X=foot(W,A,C);
Y=foot(Z,A,C);
draw(A--B--C--cycle);
draw(X--W--Z--Y);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,E);
label("$W$",W,NW);
label("$X$",X,S);
label("$Y$",Y,S);
label("$Z$",Z,NE);
[/asy]

Answer: \frac{60}{37}


For every four points  $P_{1},P_{2},P_{3},P_{4}$  on the plane, find the minimum value of  $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$ .

Answer: 4 + 2\sqrt{2}


$p$ is a prime number such that the period of its decimal reciprocal is 200. That is,

$$
\frac{1}{p}=0.XXXX\ldots
$$

for some block of 200 digits $X$, but

$$
\frac{1}{p} \neq 0.YYYY\ldots
$$

for all blocks $Y$ with less than 200 digits. Find the 101st digit, counting from the left, of $X$.

Answer: 9


Let \(\triangle ABC\) be a triangle with \(AB=7\), \(BC=1\), and \(CA=4\sqrt{3}\). The angle trisectors of \(C\) intersect \(\overline{AB}\) at \(D\) and \(E\), and lines \(\overline{AC}\) and \(\overline{BC}\) intersect the circumcircle of \(\triangle CDE\) again at \(X\) and \(Y\), respectively. Find the length of \(XY\).

Answer: \frac{112}{65}


\(ABCD\) is a parallelogram with \(AB = 7\), \(BC = 2\), and \(\angle DAB = 120^\circ\). Parallelogram \(ECFA\) is contained within \(ABCD\) and is similar to it. Find the ratio of the area of \(ECFA\) to the area of \(ABCD\).

Answer: 39/67


Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Answer: 67


Let

$$
f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \quad \text{and} \quad g(x)=c_{n+1} x^{n+1}+c_{n} x^{n}+\cdots+c_{0}
$$

be non-zero polynomials with real coefficients such that $g(x) = (x + r) f(x)$ for some real number $r$. If $a = \max \left(\left|a_{n}\right|, \left|a_{n-1}\right|, \ldots, \left|a_{0}\right|\right)$ and $c = \max \left(\left|c_{n+1}\right|, \left|c_{n}\right|, \ldots, \left|c_{0}\right|\right)$, prove that $\frac{a}{c} \leq n+1$.

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


Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\), let \(A\) be the left vertex, \(F(c, 0)\) be the right focus, and \(2b\), \(a\), and \(c\) form a geometric progression. A line through the point \(F\) intersects the ellipse at points \(M\) and \(N\). The lines \(AM\) and \(AN\) intersect the right directrix at points \(P\) and \(Q\) respectively. Determine the size of the angle \(\angle PFQ\).

Answer: 90


Answer: 43/63


Given the decimal representation of $\frac{1}{30^{30}}$, determine how many zeros immediately follow the decimal point.

Answer: 44


Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral.

Answer: \frac{3200}{9}


Consider all ordered pairs  $(m, n)$  of positive integers satisfying  $59 m - 68 n = mn$ . Find the sum of all the possible values of  $n$  in these ordered pairs.

Answer: 237


The base of a pyramid inscribed in a cone is a quadrilateral with pairs of adjacent sides equal, and the angle between one pair of adjacent sides is $\alpha$. Find the ratio of the volume of the pyramid to the volume of the cone.

Answer: \frac{2\sin\alpha}{\pi}


If the odd function \( y=f(x) \) defined on \( \mathbf{R} \) is symmetrical about the line \( x=1 \), and when \( 0 < x \leqslant 1 \), \( f(x)=\log_{3}x \), find the sum of all real roots of the equation \( f(x)=-\frac{1}{3}+f(0) \) in the interval \( (0,10) \).

Answer: 30


A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2:1$, counting from the vertex. In what ratio does this line divide the sides of the triangle?

Answer: (\sqrt{6} + 2) : 1


29 boys and 15 girls came to the ball. Some of the boys danced with some of the girls (at most once with each person in the pair). After the ball, each individual told their parents how many times they danced. What is the maximum number of different numbers that the children could mention?

Answer: 29


Elector Georg has 100 coins, some of which are counterfeit (possibly all or none). Georg can show an expert between 10 and 20 coins, and the expert will tell him how many of them are counterfeit. The problem is that the only expert in the area, Baron Munchausen, always gives a result that is greater than the true number by some fixed (and unknown to Georg) natural number. The Baron is not bothered that he might, for example, say "thirteen" if given only twelve coins. Can Georg definitively determine which coins are counterfeit by consulting the expert fewer than 120 times?

Answer: Yes


Let the function be $$f(x)= \sqrt {3}\sin 2x+2\cos^{2}x+2$$.
(I) Find the smallest positive period and the range of $f(x)$;
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $$A= \frac {\pi }{3}$$ and the area of $\triangle ABC$ is $$\frac { \sqrt {3}}{2}$$, find $f(A)$ and the value of $a$.

Answer: \sqrt {3}


In the diagram below, triangle $ABC$ has been reflected over its median $\overline{AM}$ to produce triangle $AB'C'$.  If $AE = 6$, $EC =12$, and $BD = 10$, then find $AB$.

[asy]
size(250);
pair A,B,C,D,M,BB,CC,EE;
B = (0,0);
D = (10,0);
M = (15,0);
C=2*M;
A = D + (scale(1.2)*rotate(aCos((225-144-25)/120))*(M-D));
CC = D + D + D - A - A;
BB = reflect(A,M)*B;
EE = reflect(A,M)*D;
draw(M--A--BB--CC--A--B--C--A);
label("$M$",M,SE);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$C'$",CC,S);
label("$B'$",BB,E);
label("$D$",D,NW);
label("$E$",EE,N);
label("$12$",(EE+C)/2,N);
label("$6$",(A+EE)/2,S);
label("$10$",D/2,S);
[/asy]

Answer: 8\sqrt{3}


There are three balls of the same size but different colors in a pocket. One ball is drawn each time, the color is recorded, and then it is put back. The drawing stops when all three colors of balls have been drawn. If it stops after exactly 5 draws, the number of different ways to draw is \_\_\_\_\_\_\_.

Answer: 42


Find the sine of the angle of a rhombus if the opposite side is seen from the midpoint of one of its sides at an angle $\alpha$.

Answer: \frac{3}{4}\tan\alpha


Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After meeting, Person B also doubles their speed and turns back. When Person A returns to point $A$, Person B is still 120 meters away from point $A$. What is the distance between points $A$ and $B$ in meters?

Answer: 420


Harriet lives in a large family with 4 sisters and 6 brothers, and she has a cousin Jerry who lives with them. Determine the product of the number of sisters and brothers Jerry has in the house.

Answer: 24


A, B, C are positive reals with product 1. Prove that \((A - 1 + \frac{1}{B})(B - 1 + \frac{1}{C})(C - 1 + \frac{1}{A}) \le 1\).

Answer: 1


Given a triangle with side lengths \( l, m, n \), where \( l > m > n \) are integers, and it is known that:
\[
\left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\}
\]
where \( \{x\} = x - \lfloor x \rfloor \), and \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). Determine the minimum possible perimeter of such a triangle.

Answer: 3003


The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___

Answer: -1


There are $5$ different books to be distributed among three students, with each student receiving at least $1$ book and at most $2$ books. The number of different distribution methods is $\_\_\_\_\_\_\_\_$.

Answer: 90


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

Answer: 252


To any triangle with side lengths  $a,b,c$  and the corresponding angles  $\alpha, \beta, \gamma$  (measured in radians), the 6-tuple  $(a,b,c,\alpha, \beta, \gamma)$  is assigned. Find the minimum possible number  $n$  of distinct terms in the 6-tuple assigned to a scalene triangle.

Answer: 4


Show that the 4-digit number mnmn cannot be a cube in base 10. Find the smallest base \( b > 1 \) for which it can be a cube.

Answer: 7


If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.

(Ⅰ) Find the values of $\omega$ and $m$;
(Ⅱ) Find the sum of all zeros of the function $y = f(x)$ in the interval $x \in [0, 2\pi]$.

Answer: \frac{11\pi}{3}


Let \( P(x) \) be a polynomial of degree 2010. Suppose \( P(n) = \frac{n}{1+n} \) for all \( n = 0, 1, 2, \ldots, 2010 \). Find \( P(2012) \).

Answer: 0


Find the largest integer $n$ such that $3^{512}-1$ is divisible by $2^{n}$.

Answer: 11


Using the digits 1, 2, 3, and 4 (with digits allowed to be repeated), how many $n$-digit numbers ($n \geqslant 2$) can be formed that contain an odd number of 1's, an even number of 2's, and at least one 3?

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


From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained?

Answer: $\frac{1}{2}$


Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \).

Answer: \sqrt{3}


The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

Answer: \textbf{(A)}\12


Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$?

Answer: 8181


Given that the parabola $y^2=4x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a > 0, b > 0)$ have the same focus $F$, $O$ is the coordinate origin, points $A$ and $B$ are the intersection points of the two curves. If $(\overrightarrow{OA} + \overrightarrow{OB}) \cdot \overrightarrow{AF} = 0$, find the length of the real axis of the hyperbola.

Answer: 2\sqrt{2}-2


A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$.

Answer: 108


From point \( O \), three rays \( OA, OB, OC \) are drawn, with the angles between each pair of rays being \( 60^{\circ}, 90^{\circ}, \) and \( 120^{\circ} \) respectively. What is the minimum angle between the angle bisectors of these three angles?

Answer: 45


In a school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \).

Answer: 3721


The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?

Answer: 15


There are 3 boys and 4 girls, all lined up in a row. How many ways are there for the following situations?
- $(1)$ Person A is neither at the middle nor at the ends;
- $(2)$ Persons A and B must be at the two ends;
- $(3)$ Boys and girls alternate.

Answer: 144


Let $M \Theta N$ represent the remainder when the larger of $M$ and $N$ is divided by the smaller one. For example, $3 \Theta 10 = 1$. For non-zero natural numbers $A$ less than 40, given that $20 \Theta(A \bigodot 20) = 7$, find $A$.

Answer: 13


Let  $m$  and  $n$  be integers such that  $m + n$  and  $m - n$  are prime numbers less than  $100$ . Find the maximal possible value of  $mn$ .

Answer: 2350


If $V = gt + V_0$ and $S = \frac{1}{2}gt^2 + V_0t$, then $t$ equals:

Answer: \frac{2S}{V+V_0}


Given the function $f(x)$ that satisfies $f\left(\frac{\pi}{2} - x\right) + f(x) = 0$ and $f(\pi + x) = f(-x)$, calculate the value of $f\left(\frac{79\pi}{24}\right)$.

Answer: \frac{\sqrt{2} - \sqrt{6}}{4}


Define a positive integer \( n \) as a "factorial tail" if there exists a positive integer \( m \) such that the decimal representation of \( m! \) ends in exactly \( n \) zeros. How many positive integers less than 1992 are not "factorial tails"?

Answer: 396


If the digits \( a_{i} (i=1,2, \cdots, 9) \) satisfy
$$
a_{9} < a_{8} < \cdots < a_{5} \text{ and } a_{5} > a_{4} > \cdots > a_{1} \text{, }
$$

then the nine-digit positive integer \(\bar{a}_{9} a_{8} \cdots a_{1}\) is called a “nine-digit peak number”, for example, 134698752. How many nine-digit peak numbers are there?

Answer: 11875


Answer: 146


John rode his bicycle for 45 minutes at a speed of 20 mph, then walked for 120 minutes at a speed of 3 mph. Calculate the overall average speed in miles per hour.

Answer: 7.64


Let  $S$  be the sum of all real  $x$  such that  $4^x = x^4$ .  Find the nearest integer to  $S$ .

Answer: 5


In this version of SHORT BINGO, a $5\times5$ card is again filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. Now, the card is made by placing 5 distinct numbers from the set $1-15$ in the first column, 5 distinct numbers from $11-25$ in the second column, 4 distinct numbers from $21-35$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $31-45$ in the fourth column, and 5 distinct numbers from $41-55$ in the last column. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card?

Answer: 360360


Zhang Hua has to go through four traffic posts A, B, C, and D on his way to school. The probability of encountering a red light at posts A and B is $\frac{1}{2}$ each, and at posts C and D, it is $\frac{1}{3}$ each. Assuming that the events of encountering red lights at the four traffic posts are independent, let X represent the number of times he encounters red lights.
(1) If x≥3, he will be late. Find the probability that Zhang Hua is not late.
(2) Find $E(X)$, the expected number of times he encounters red lights.

Answer: \frac{5}{3}


Compute the nearest integer to
$$
100 \sum_{n=1}^{\infty} 3^{n} \sin ^{3}\left(\frac{\pi}{3^{n}}\right)
$$

Answer: 236


(Theorem of Helly) We consider four convex sets in the plane such that the intersection of any three of them is always non-empty.

- a) Show that the intersection of all four convex sets is non-empty.
- b) Does the theorem hold true if we replace 4 with $n \geqslant 4$?

Answer: Yes


A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum.

Answer: 10\pi \sqrt{61} + 68\pi


A factory produces a certain product for the Brazilian Olympic Games with an annual fixed cost of 2.5 million yuan. For every $x$ thousand units produced, an additional cost of $C(x)$ (in ten thousand yuan) is incurred. When the annual production is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$; when the annual production is not less than 80 thousand units, $C(x)=51x+\frac{10000}{x}-1450$. The selling price per product is 0.05 ten thousand yuan. Through market analysis, it is determined that all the products produced by the factory can be sold.

(1) Write the analytical expression of the annual profit $L$ (in ten thousand yuan) as a function of the annual production $x$ (in thousand units);

(2) At what annual production volume (in thousand units) does the factory maximize its profit from this product?

Answer: 100


Given that $F$ is the focus of the parabola $4y^{2}=x$, and points $A$ and $B$ are on the parabola and located on both sides of the $x$-axis. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = 15$ (where $O$ is the origin), determine the minimum value of the sum of the areas of $\triangle ABO$ and $\triangle AFO$.

Answer: \dfrac{ \sqrt{65}}{2}


Answer: \frac{1}{3}


There are  $13$  positive integers greater than  $\sqrt{15}$  and less than  $\sqrt[3]{B}$ . What is the smallest integer value of  $B$ ?

Answer: 4097


Answer: 2021


For any four points \( A, B, C, D \) on a plane, where the distance between any two points does not exceed 1, find the minimum radius of a circle that can cover these four points. Prove your answer.

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


A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\circ$ and $90^\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and $\theta$ varies from $0^\circ$ to $90^\circ$. As $\theta$ varies within this range, the highest points of the projectile paths trace out a curve. Calculate the area enclosed by this curve, which can be expressed as $c \cdot \frac{v^4}{g^2}$. Find the value of $c$.

Answer: \frac{\pi}{16}


Real numbers  $x, y$  satisfy the inequality  $x^2 + y^2 \le 2$ . Orove that  $xy + 3 \ge 2x + 2y$

Answer: xy+3\ge2x+2y


For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\sum_{i=1}^{216} x_i=1$ and $\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$. The maximum possible value of $x_2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 863


A *substring* of a number  $n$  is a number formed by removing some digits from the beginning and end of  $n$  (possibly a different number of digits is removed from each side). Find the sum of all prime numbers  $p$  that have the property that any substring of  $p$  is also prime.

Answer: 576


There are 55 points marked on a plane: the vertices of a regular 54-gon and its center. Petya wants to color a set of three marked points in red so that the colored points form the vertices of a regular triangle. In how many ways can Petya do this?

Answer: 72


In a right triangle \(ABC\) with a right angle at vertex \(B\), a segment \(BD\) is marked on the larger leg \(BC\), equal to the length of the smaller leg. Points \(K\) and \(N\) are the bases of the perpendiculars dropped onto the hypotenuse from points \(B\) and \(D\), respectively. Find the length of segment \(BK\), given that \(AK = ND = 2\).

Answer: 4


Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion. If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce 11 times before returning to $(2,0)$ for the first time. If $\cot \theta=a-\sqrt{b}$ for positive integers $a$ and $b$, compute $100 a+b$.

Answer: 403


Given that the positive integer \( m \) satisfies that \( m^{4} + 16m + 8 \) can be expressed as the product of two or more consecutive integers, determine the maximum value of \( m \).

Answer: 2


In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(110);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]

Answer: 12


A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi$, where n is a positive integer. Find $n$.

Answer: 216


Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$
$\textbf{(A) } 4444 \qquad\textbf{(B) } 4494 \qquad\textbf{(C) } 4944 \qquad\textbf{(D) } 9444 \qquad\textbf{(E) } 9944$

Answer: \mathrm{(C)\}4944


Given that the two trisection points on the minor axis of an ellipse and its two foci form a square, find the eccentricity of the ellipse.

Answer: \frac{\sqrt{10}}{10}


We will call a natural number interesting if all its digits, except for the first and the last, are less than the arithmetic mean of their two neighboring digits. Find the largest interesting number.

Answer: 96433469


The set  $X$  of  $N$  four-digit numbers formed from the digits  $1,2,3,4,5,6,7,8$  satisfies the following condition:

*for any two different digits from  $1,2,3,4,,6,7,8$  there exists a number in  $X$  which contains both of them.*

Determine the smallest possible value of  $N$ .

Answer: 6


Let  $ABC$  be a triangle with semiperimeter  $s$  and inradius  $r$ . The semicircles with diameters  $BC$ ,  $CA$ ,  $AB$  are drawn on the outside of the triangle  $ABC$ . The circle tangent to all of these three semicircles has radius  $t$ . Prove that
\[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \]
*Alternative formulation.* In a triangle  $ABC$ , construct circles with diameters  $BC$ ,  $CA$ , and  $AB$ , respectively. Construct a circle  $w$  externally tangent to these three circles. Let the radius of this circle  $w$  be  $t$ .
Prove:  $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$ , where  $r$  is the inradius and  $s$  is the semiperimeter of triangle  $ABC$ .

*Proposed by Dirk Laurie, South Africa*

Answer: \frac{}{2}<\le\frac{}{2}+(1-\frac{\sqrt{3}}{2})r


From the 8 English letters \( A, B, C, D, E, X, Y, Z \), select 5 letters (with repetition allowed) to form a "word." Arrange all possible "words" in "lexicographic order" (i.e., the order in which words appear in an English dictionary). Determine the number of "words" that lie between the "word" \(C Z Y E B\) and the "word" \(X C E D A\) (excluding these two words).

Answer: 9590


Find the minimum possible value of $\sqrt{58-42 x}+\sqrt{149-140 \sqrt{1-x^{2}}}$ where $-1 \leq x \leq 1$

Answer: \sqrt{109}


In a company of \(2n+1\) people, for any group of \(n\) people, there is another person who knows each of them. Prove that there is a person in this company who knows everyone.

Answer: 1


Let \( A = \{1, 2, \cdots, 10\} \). The mapping \( f \) from \( A \) to \( A \) satisfies the following two conditions:
1. For any \( x \in A \), \( f_{30}(x) = x \).
2. For every \( k \in \mathbf{N}_{+}, 1 \leq k \leq 29 \), there exists at least one \( a \in A \) such that \( f_{k}(a) \neq a \).

Determine the number of such mappings \( f \) (where it is given \( f_{1}(x) = f(x), f_{k+1}(x) = f(f_{k}(x)) \) for \( k = 1, 2, \cdots \)).

Answer: 120960


A line passing through the focus \( F \) of the parabola \( y^2 = 4x \) intersects the parabola at points \( M \) and \( N \). Let \( E(m, 0) \) be a point on the x-axis. The extensions of \( M E \) and \( N E \) intersect the parabola at points \( P \) and \( Q \). If the slopes \( k_1 \) of \( M N \) and \( k_2 \) of \( P Q \) satisfy \( k_1 = 3 k_2 \), then the value of the real number \( m \) is ______.

Answer: 3


The heights of a parallelogram are greater than 1. Is it necessarily possible to inscribe a unit square in it?

Answer: No


Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple?

Answer: \frac{1}{2}


In the final stage of a professional bowling tournament, the competition between the top five players is conducted as follows: the fifth and fourth place players compete first, the loser gets the 5th place prize; the winner competes with the third place player, the loser gets the 4th place prize; the winner competes with the second place player, the loser gets the 3rd place prize; the winner competes with the first place player, the loser gets the 2nd place prize, and the winner gets the 1st place prize. How many different possible outcomes of the prize distribution are there?

Answer: 16


Find the sum of all real roots of the equation \(3 \tan ^{2} x + 8 \tan x + 3 = 0\) in the range \(0 < x < 2\pi\).

Answer: 5\pi


Let \( m_{1}, m_{2}, \cdots, m_{r} \) (possibly with repetitions) and \( n_{1}, n_{2}, \cdots, n_{s} \) (possibly with repetitions) be two sets of positive integers. They satisfy the following condition: For any positive integer \( d > 1 \), the number of elements in the sequence \( m_{1}, m_{2}, \cdots, m_{r} \) that are divisible by \( d \) (including repetitions) is at least as many as the number of elements in the sequence \( n_{1}, n_{2}, \cdots, n_{s} \) that are divisible by \( d \) (including repetitions). Prove that \( \frac{m_{1} m_{2} \cdots m_{r}}{n_{1} n_{2} \cdots n_{s}} \) is an integer.

Answer: \frac{m_{1}m_{2}\cdotsm_{r}}{n_{1}n_{2}\cdotsn_{}}


Suppose  $x,y,z$ , and  $w$  are positive reals such that
\[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of  $(xy + wz)^2$ .

*Author: Alex Zhu*

Answer: 960


Answer: 166


Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019.

Answer: 1889


A wall made of mirrors has the shape of  $\triangle ABC$ , where  $AB = 13$ ,  $BC = 16$ , and  $CA = 9$ . A laser positioned at point  $A$  is fired at the midpoint  $M$  of  $BC$ . The shot reflects about  $BC$  and then strikes point  $P$  on  $AB$ . If  $\tfrac{AM}{MP} = \tfrac{m}{n}$  for relatively prime positive integers  $m, n$ , compute  $100m+n$ .

*Proposed by Michael Tang*

Answer: 2716


Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets.

Answer: 409510


Find all composite natural numbers \( n \) that possess the following property: every natural divisor of \( n \) (in particular, \( n \) itself), decreased by 1, is a square of an integer.

Answer: 10


In $\triangle ABC$, $\angle ABC = 60^\circ$, $O$ and $H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively. Points $D$ and $E$ lie on sides $BC$ and $AB$ respectively, such that $BD = BH$, $BE = BO$, and $BO = a$. Find the area of $\triangle BDE$.

Answer: \frac{\sqrt{3}}{4}^2


Choose three digits from the odd numbers 1, 3, 5, 7, 9 and two digits from the even numbers 2, 4, 6, 8 to form a five-digit number with no repeating digits, such that the odd and even digits alternate. How many such five-digit numbers can be formed?

Answer: 720


For how many integers \(n\) with \(1 \le n \le 2020\) is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero?

Answer: 337


Determine the greatest common divisor of all nine-digit integers formed by repeating a three-digit integer three times. For example, 256,256,256 or 691,691,691 are integers of this form.

Answer: 1001001


Find all values of the parameter  $m$  such that the equations  $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$  have only one root.

Answer: 0


Calculate:
$(1)-9+5-\left(-12\right)+\left(-3\right)$;
Calculate:
$(2)-(+1.5)-(-4\frac{1}{4})+3.75-(-8\frac{1}{2})$;
$(3)$Read the following solution process and answer the question:
Calculate:$\left(-15\right)\div (-\frac{1}{2}×\frac{25}{3}$）$÷\frac{1}{6}$
Solution: Original expression $=\left(-15\right)\div (-\frac{25}{6})\times 6(Step 1)$
$=\left(-15\right)\div \left(-25\right)(Step 2)$
$=\frac{3}{5}$(Step 3)
①The error in the solution process above starts from step ______, and the reason for the error is ______;
②Please write down the correct solution process.

Answer: \frac{108}{5}


The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is "$a=$ ______".

Answer: -2


Let \( x \) and \( y \) be positive real numbers, and \( x + y = 1 \). Find the minimum value of \( \frac{x^2}{x+2} + \frac{y^2}{y+1} \).

Answer: 1/4


Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age?

Answer: 14


There are 15 red points on a plane. Lines are drawn between some of these red points. A red point is marked with the number of lines connected to it. It is known that all points marked with the same number are not connected to each other. What is the maximum number of lines that can be drawn between these 15 red points?

Answer: 85


The average of the numbers $1, 2, 3,\dots, 148, 149,$ and $x$ is $50x$. What is $x$?

Answer: \frac{11175}{7499}


Given a triangle whose three sides are all positive integers, with only one side length equal to 5 and not the shortest side, find the number of such triangles.

Answer: 10


Let  $1=d_1<d_2<d_3<...<d_k=n$  be all different divisors of positive integer  $n$   written in ascending order. Determine all  $n$   such that   $$ d_7^2+d_{10}^2=(n/d_{22})^2. $$

Answer: 2040


Let \( n, k \in \mathbf{Z}_{+} \), and \( n \geqslant 2 \). It is known that there are \( n \) cities in a certain country, and there is a bus that can travel in both directions between every pair of cities. Prove that the number of ways to travel from city \( A \) to city \( B \) by taking exactly \( k \) buses is \( \frac{(n-1)^{k}-(-1)^{k}}{n} \).

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


Let \( a \) be the number of six-digit numbers divisible by 13 but not divisible by 17, and \( b \) be the number of six-digit numbers divisible by 17 but not divisible by 13.

Find \( a - b \).

Answer: 16290


Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$, find the maximum possible value of $a$.

Answer: \frac{50}{99}


Find the ratio \( m_{1} / m_{2} \) of the two hanging balls, given that the tension forces in the upper and lower threads differ by a factor of two.

Answer: 1


A projectile is launched with an initial velocity $u$ at an angle $\phi$ from the ground. The trajectory can be described by the parametric equations:
\[
x = ut \cos \phi, \quad y = ut \sin \phi - \frac{1}{2} g t^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and the angle $\phi$ varies over $0^\circ \le \phi \le 90^\circ$. The highest points of each trajectory are plotted as $\phi$ changes. Determine the area of the curve traced by these highest points, expressed as $d \cdot \frac{u^4}{g^2}$.

Answer: \frac{\pi}{8}


Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$

Answer: 8


Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. Determine the number of women in the luxury compartment if the train carries 300 passengers.

Answer: 32


Given three points $A, B, C$ on a line and a point $M$ outside the line. Perpendiculars are drawn at points $A, B, C$ to the lines $AM, BM, CM$ respectively, and let the intersections of these perpendiculars be $P, Q, R$. Show that the quadrilateral $PQRM$ is a cyclic quadrilateral.

Answer: PQRM


In the sequence \(\{a_n\}\), where \(a_n = a \cdot 2^n + b \cdot n - 80\) and \(a, b \in \mathbf{N}_{+}\), it is known that the sum of the first \(n\) terms, \(S_n\), attains its minimum value if and only if \(n = 6\), and \(a_{36}\) is divisible by 7. Find the value of \(\left|a_1\right|+\left|a_2\right|+\cdots+\left|a_{12}\right|\).

Answer: 8010


Given that point \( P \) is on the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), and \( F_1 \) is its left focus, \( Q \) lies on \( PF_1 \) and satisfies the condition
$$
\overrightarrow{O Q} = \frac{1}{2} \left( \overrightarrow{O P} + \overrightarrow{O F_1} \right), \quad |\overrightarrow{O Q}| = 3.
$$

Find the distance from point \( P \) to the left directrix of the ellipse.

Answer: 5


There are 7 volunteers, among which 3 people only speak Russian, and 4 people speak both Russian and English. From these, 4 people are to be selected to serve as translators for the opening ceremony of the "Belt and Road" summit, with 2 people serving as English translators and 2 people serving as Russian translators. There are a total of \_\_\_\_\_\_ different ways to select them.

Answer: 60


Given the fixed point $M(1,0)$, $A$ and $B$ are two moving points on the ellipse $\frac{x^2}{4}+y^2=1$, and $\overrightarrow{MA} \cdot \overrightarrow{MB}=0$, find the minimum value of $\overrightarrow{AM} \cdot \overrightarrow{AB}$.

Answer: \frac{2}{3}


In square \( ABCD \), diagonals \( AC \) and \( BD \) intersect at point \( O \). Point \( K \) is the midpoint of side \( AB \). Points \( M \) and \( N \) are chosen on sides \( AD \) and \( BC \) respectively, such that rays \( OK \), \( OM \), and \( ON \) divide the square into three regions of equal area. In what ratio does point \( M \) divide side \( AD \)?

Answer: 5:1


The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points.  All four points lie on a circle of radius $r.$  Find $r^2.$

Answer: \frac{13}{2}


One leg of a right triangle is 15 cm, and the projection of the other leg onto the hypotenuse is 16 cm. Find the radius of the circle inscribed in the triangle.

Answer: 5\,


What is the smallest $n$ for which there exists an $n$-gon that can be divided into a triangle, quadrilateral, ..., up to a 2006-gon?

Answer: 2006


Answer: \frac{3}{2}


Given $x \gt 0$, $y \gt 0$, and $x+y=1$, find the minimum value of $\frac{2{x}^{2}-x+1}{xy}$.

Answer: 2\sqrt{2}+1


An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball?

Answer: 51


Let  $a$  and  $b$  be positive integers not divisible by  $5$ . A sequence of integers is constructed as follows: the first term is  $5$ , and every consequent term is obtained by multiplying its precedent by  $a$  and adding  $b$ . (For example, if  $a = 2$  and  $b = 4$ , the first three terms are  $5,14,32$ .) What is the maximum possible number  of primes that can occur before encoutering the first composite term?

Answer: 5


What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)?

Answer: 12


What is the probability that $abc + ab + a$ is divisible by $4$, where positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 1024\}$.

Answer: \frac{7}{16}


Six musicians gathered at a chamber music festival. At each scheduled concert, some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that for every two musicians, each must play for the other in some concert?

Answer: 4


An electronic watch shows 6:20:25 at 6 hours, 20 minutes, and 25 seconds. Within the one-hour period from 5:00:00 to 5:59:59, how many instances are there where all five digits of the time displayed are different?

Answer: 840


Let \( S = \{1, 2, 3, \cdots, 280\} \). Find the smallest natural number \( n \) such that every subset of \( S \) with \( n \) elements contains 5 pairwise relatively prime numbers.

Answer: 217


There are  $2024$  people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were  $0,1, \ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly  $1$ . What is the minimal number of liars?

Answer: 1012


There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number?

Answer: 2601


Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|=5$, $|\overrightarrow {a}- \overrightarrow {b}|=6$, and $|\overrightarrow {a}+ \overrightarrow {b}|=4$, find the projection of vector $\overrightarrow {b}$ on vector $\overrightarrow {a}$.

Answer: -1


What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers?

Answer: -1000


Let \( A_{1}, A_{2}, \ldots, A_{2017} \) be the vertices of a regular polygon with 2017 sides. Prove that there exists a point \( P \) in the plane of the polygon such that the vector
$$
\sum_{k=1}^{2017} k \frac{\overrightarrow{P A_{k}}}{\left\|\overrightarrow{P A_{k}}\right\|^{5}}
$$
is the zero vector.
(The notation \( \|\overrightarrow{X Y}\| \) represents the length of the vector \( \overrightarrow{X Y} \).)

Answer: 0


Let  $p$  be a prime number and  $\mathbf{W}\subseteq \mathbb{F}_p[x]$  be the smallest set satisfying the following :
[list]
(a)  $x+1\in \mathbf{W}$  and  $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$ (b) For  $\gamma_1,\gamma_2$  in  $\mathbf{W}$ , we also have  $\gamma(x)\in \mathbf{W}$ , where  $\gamma(x)$  is the remainder  $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$ .[/list]
How many polynomials are in  $\mathbf{W}?$

Answer: p!


Find the volume of the solid formed by the revolution of the curve  $ x \equal{} a(1 \plus{} \cos \theta )\cos \theta ,\ y \equal{} a(1 \plus{} \cos \theta)\sin \theta\ (a > 0,\ \minus{} \pi < \theta \leq \pi)$  about the  $ x$  axis.

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


Let $P$ be a point inside equilateral $\triangle ABC$. Reflect $P$ over the lines $AB$, $BC$, and $CA$ to get points $C_1$, $A_1$, and $B_1$ respectively. Denote the areas of triangles $\triangle ABC$ and $\triangle A_1 B_1 C_1$ as $S$ and $S'$. Prove that $S' \leq S$.

Answer: S'\leS


A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of $n$.

Answer: 62


I bought a lottery ticket with a five-digit number such that the sum of its digits equals the age of my neighbor. Determine the number of this ticket, given that my neighbor easily solved this problem.

Answer: 99999


How many ordered pairs $(a, b)$ of positive integers satisfy the equation
\[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\]
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?
$\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}$

Answer: \textbf{(B)}~2


Answer: 1-\frac{1}{2003^{2003}}<\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{2003}}<1


In the Cartesian coordinate system, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \rho = -4\sqrt{3}\sin \theta$ intersects with circle $C$ at points $A$ and $B$.
(i) Find the polar equation of line $AB$.
(ii) If a line $C_2$ passing through point $C(2,0)$ with the parametric equations $\begin{cases} x=2+ \frac{\sqrt{3}}{2}t, \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter) intersects line $AB$ at point $D$ and intersects the $y$-axis at point $E$, calculate the ratio $|CD| : |CE|$.

Answer: 1 : 2


Misha has thought of a five-digit number, all of whose digits are distinct, and Igor is trying to guess it. In one turn, Igor can choose several digits of the number, and Misha will inform him of the digits at those places in random order. The order in which the digits are revealed is chosen by Misha. For example, if the number thought of is 67890, and Igor asks about the digits in positions 1 and 5, then Misha could answer "6 and 0" or "0 and 6". What is the minimum number of turns Igor will need to guarantee to find out the number?

Answer: 3


Find all polynomials with integer coefficients $P$ such that for all integers $a$ and $b$, the following holds:

$$
a + 2b \mid P(a) + 2P(b)
$$

Answer: P(x)=kx


Beatrix is going to place six rooks on a $6 \times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square.The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 371


Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set {1,2,3,4,5,6}.
2. No digit occurs more than once in the same integer.
3. The digits in each integer are in increasing order.
4. Any two integers have at least one digit in common (possibly at different positions).
5. There is no digit which appears in all the integers.

Answer: 32


Source: 1976 Euclid Part B Problem 1
-----
Triangle  $ABC$  has  $\angle{B}=30^{\circ}$ ,  $AB=150$ , and  $AC=50\sqrt{3}$ . Determine the length of  $BC$ .

Answer: 50\sqrt{3}


An equilateral triangle with a side length of 1 is cut along a line parallel to one of its sides, resulting in a trapezoid. Let $S = \frac{\text{(perimeter of the trapezoid)}^2}{\text{area of the trapezoid}}$. Find the minimum value of $S$.

Answer: \frac{32\sqrt{3}}{3}


There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.
$(1)$ How many unique three-digit numbers can be formed without repetition?
$(2)$ How many unique four-digit even numbers can be formed without repetition?

Answer: 2296


Suppose you have an equilateral triangle divided into 9 smaller equilateral triangles with the bottom side horizontal. Starting from the top corner labeled \( A \), you must walk to the bottom right corner labeled \( B \), and are only allowed to take steps along the edges down to the left, down to the right, or horizontally to the right. Determine the number of possible paths.

Answer: 22


Given the function \( f(x) = ax^2 - 1 \) where \( a \in \mathbb{R} \) and \( x \in \mathbb{R} \), and sets \( A = \{ x \mid f(x) = x \} \) and \( B = \{ x \mid f(f(x)) = x \} \), such that \( A = B \neq \emptyset \), find the range of the real number \( a \).

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


Define the function \( f(n) \left( n \in \mathbb{N}_{+} \right) \) as follows:
Let \(\frac{(2n)!}{n!(n+1002)!} = \frac{A(n)}{B(n)}\).
Here \( A(n) \) and \( B(n) \) are coprime positive integers:
If \( B(n) = 1 \), then \( f(n) = 1 \);
If \( B(n) \neq 1 \), then \( f(n) \) is the largest prime factor of \( B(n) \).
Prove that \( f(n) \) only takes finitely many different values, and determine its maximum value.

Answer: 2003


We write the equation on the board:

$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$

We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this?

Answer: 2016


Find the number of positive integers  $n$  for which there exists a sequence  $x_1, x_2, \cdots, x_n$  of integers with the following property: if indices  $1 \le i \le j \le n$  satisfy  $i+j \le n$  and  $x_i - x_j$  is divisible by  $3$ , then  $x_{i+j} + x_i + x_j + 1$  is divisible by  $3$ .

*Based on a proposal by Ivan Koswara*

Answer: 8


In a right triangle, the legs are 3 and 4. Find the distance between the centers of the inscribed and circumscribed circles. Write down the square of this distance.

Answer: 1.25


Leon has cards with digits from 1 to 7. How many ways are there to combine these cards into two three-digit numbers (one card will not be used) so that each of them is divisible by 9?

Answer: 36


Along the wall of a circular tower, two guards walk clockwise. The first guard is twice as fast as the second. In this wall, which has a length of 1, there are embrasures. A system of embrasures is called reliable if at any moment in time at least one of the guards is near an embrasure.

a) What is the minimum length an embrasure can have if a system consisting of just this embrasure is reliable?

b) Prove that the total length of embrasures in any reliable system is greater than $1 / 2$.

c) Prove that for any number $s > 1 / 2$, there exists a reliable system of embrasures with a total length less than $s$.

Answer: \frac{2}{3}


Consider all possible 100-digit numbers where each digit is either 1 or 2. For each number, compute the remainder when divided by 1024. How many different remainders are there?

Answer: 1024


Answer: 31


Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice roll. Each time the dice is rolled, the chess piece is moved from one vertex of the triangle to another vertex. If the number rolled on the dice is greater than 3, the movement is counterclockwise; if the number rolled is not greater than 3, the movement is clockwise. Let Pn(A), Pn(B), Pn(C) denote the probabilities of the chess piece being at points A, B, C after n dice rolls, respectively. Calculate the probability of the chess piece being at point A after 7 dice rolls.

Answer: \frac{21}{64}


Inequality

\[
A a(B b+C c)+B b(C c+A a)+C c(A a+B b)>\frac{1}{2}\left(A B c^{2}+B C a^{2}+C A b^{2}\right)
\]

where \(a>0, b>0, c>0\) are given numbers, holds for all \(A>0, B>0, C>0\). Can one form a triangle with the segments \(a, b, c\)?

Answer: Yes


Estimate the value of $\sqrt{90} - \sqrt{88}$.

Answer: 0.11


An \( n \times n \) board is colored in \( n \) colors such that the main diagonal (from top-left to bottom-right) is colored in the first color; the two adjacent diagonals are colored in the second color; the two next diagonals (one from above and one from below) are colored in the third color, etc.; the two corners (top-right and bottom-left) are colored in the \( n \)-th color. It happens that it is possible to place on the board \( n \) rooks, no two attacking each other and such that no two rooks stand on cells of the same color. Prove that \( n \equiv 0 \pmod{4} \) or \( n \equiv 1 \pmod{4} \).

Answer: n\equiv0\pmod{4}


Let \( a_{1}, a_{2}, \cdots, a_{21} \) be a permutation of \( 1, 2, \cdots, 21 \) that satisfies
\[ \left|a_{20} - a_{21}\right| \geq \left|a_{19} - a_{21}\right| \geq \left|a_{18} - a_{21}\right| \geq \cdots \geq \left|a_{1} - a_{21}\right|. \]
Determine the number of such permutations.

Answer: 3070


Maria baked 60 cakes, of which one-third contained strawberries, half contained blueberries, three-fifths contained raspberries, and one-tenth contained coconut flakes. What is the largest possible number of cakes that had none of these ingredients?

Answer: 24


Find all  $n$  such that there is a real polynomial  $f(x)$  of degree  $n$  such that  $f(x) \ge f'(x)$  for all real  $x$ .

Answer: n


Determine the coefficient of \(x^{29}\) in the expansion of \(\left(1 + x^{5} + x^{7} + x^{9}\right)^{16}\).

Answer: 65520


If the direction vector of line $l$ is $\overrightarrow{e}=(-1,\sqrt{3})$, calculate the inclination angle of line $l$.

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


Two tourists departed simultaneously from point A to point B, which is 40 km away from A. The first tourist walked at a speed of 6 km/h, while the second traveled by bicycle. When the second tourist passed the first by 5 km, the first tourist got into a passing car traveling at 24 km/h. Two hours after leaving A, the first tourist caught up with the second and arrived at B before him. Find the speed of the tourist traveling by bicycle.

Answer: 9\,


Define  $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$ . Let the sum of all  $H_n$  that are terminating in base 10 be  $S$ . If  $S = m/n$  where m and n are relatively prime positive integers, find  $100m+n$ .

*Proposed by Lewis Chen*

Answer: 9920


Let $X \sim B(4, p)$, and $P(X=2)=\frac{8}{27}$, find the probability of success in one trial.

Answer: \frac{2}{3}


Answer: 0