Given the function $f(x) = 2\cos(x)(\sin(x) + \cos(x))$ where $x \in \mathbb{R}$,
(Ⅰ) Find the smallest positive period of function $f(x)$;
(Ⅱ) Find the intervals on which function $f(x)$ is monotonically increasing;
(Ⅲ) Find the minimum and maximum values of function $f(x)$ on the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

Answer: \sqrt{2} + 1


Let \( p \) and \( q \) be positive integers such that \( \frac{5}{8} < \frac{p}{q} < \frac{7}{8} \). What is the smallest value of \( p \) so that \( p + q = 2005 \)?

Answer: 772


On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links?

Answer: 41


Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(3,3),$ $(5,5),$ $(7,7),$ and $(9,9),$ determine whether there is a point where the graphs of $y=h(3x)$ and $y=3j(x)$ intersect, and find the sum of the coordinates of that point if it exists.

Answer: 12


Answer: 8


How many pheasants did the clever gypsy actually catch if each of the three gatekeepers demanded "half of the stolen pheasants plus another half"?

(Note: The gypsy eventually gave the count to the lord without receiving any punishment, each gatekeeper received their desired amount, and the lord received three pheasants.)

Answer: 31


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

Answer: 18000


Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. What time does Leo think it is?

A) 11:30
B) 11:45
C) 12:00
D) 12:30
E) 12:45

Answer: 12:30


In a plane, there are 8 points. If any 4 of these points lie on a circle, it is called an "effective circle." Prove that it is possible to arrange these 8 points so that the number of "effective circles" is at least 12.

Answer: 12


In a winter camp, Vanya and Grisha share a room. Each night they draw lots to determine who will turn off the light before bed. The switch is near the door, so the loser has to walk back to bed in complete darkness, bumping into chairs.

Usually, Vanya and Grisha draw lots without any particular method, but this time Grisha has come up with a special way:

- "Let's flip a coin. If heads come up on an even-numbered throw, we stop flipping the coin: I win. If tails come up on an odd-numbered throw, you win."

a) What is the probability that Grisha wins?

b) Find the expected number of coin flips until the outcome is decided.

Answer: 2


Sixteen real numbers are arranged in a magic square of side  $4$  so that the sum of numbers in each row, column or main diagonal equals  $k$ . Prove that the sum of the numbers in the four corners of the square is also  $k$ .

Answer: k


The plate has the shape of a regular hexagon \(ABCDEF\) with side \(a\). \(O\) is the center of the hexagon. A square \(OMAN\) is cut out from the plate, with \(OA\) as one of its diagonals. Determine the distance from \(O\) to the center of gravity of the remaining plate.

Answer: \frac{3\sqrt{3}+1}{52}


Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly \(k\) chocolate chip cookies. Find the sum of the \(k\) for which Pearl is guaranteed to succeed regardless of how the cookies are arranged.

Answer: 71


In the real number $\alpha=\overline{0 . a_{1} a_{2} \cdots a_{n} \cdots}$, where $a_{n}=\left\{ \begin{array}{l}1, n \text { is a prime number;} \\ 0, n \text { is not a prime number} \end{array}(n=1,2, \cdots) \right.$.
Question: Is $\alpha$ a rational number? Prove your conclusion.

Answer: \alpha


What is the smallest positive integer  $k$  such that the number  $\textstyle\binom{2k}k$  ends in two zeros?

Answer: 13


The electricity price in a certain region last year was 0.80 yuan per kilowatt-hour, with an annual electricity consumption of \( a \) kilowatt-hours. This year, there is a plan to reduce the electricity price to between 0.55 yuan per kilowatt hour and 0.75 yuan per kilowatt hour, while the users expect the electricity price to be 0.40 yuan per kilowatt hour.

According to calculations, the increased electricity consumption after reducing the price is inversely proportional to the difference between the actual price and the users' expected price (the proportionality coefficient is \( k \)). The electricity cost in this region is 0.30 yuan per kilowatt hour.

(1) Write the functional relationship between the power department’s revenue \( y \) and the actual price \( x \) after reducing the price this year.
(2) Given that \( k = 0.2a \), what is the minimum price at which the power department’s revenue can still increase by at least 20% compared to last year?

Answer: 0.60


Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can guarantee a profit of at least \$n in one year. Find the greatest possible value of $n$.

Answer: 30000


In parallelogram ABCD, $\angle BAD=60^\circ$, $AB=1$, $AD=\sqrt{2}$, and P is a point inside the parallelogram such that $AP=\frac{\sqrt{2}}{2}$. If $\overrightarrow{AP}=\lambda\overrightarrow{AB}+\mu\overrightarrow{AD}$ ($\lambda,\mu\in\mathbb{R}$), then the maximum value of $\lambda+\sqrt{2}\mu$ is \_\_\_\_\_\_.

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


Express $0.6\overline{03}$ as a common fraction.

Answer: \frac{104}{165}


Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number).

Answer: 1080


Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains?

Answer: 5/84


Points \( K, L, M \) divide the sides of the convex quadrilateral \( A B C D \) in the ratio \( A K : K B = C L : L B = C M : M D = 1:2 \). The radius of the circumscribed circle of triangle \( K L M \) is \( \frac{5}{2} \), \( K L = 4 \), \( L M = 3 \). What is the area of the quadrilateral \( A B C D \) if it is known that \( K M < K L \)?

Answer: \frac{189}{25}


Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?

Answer: 8


650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

Answer: \frac{5}{2}


The circumcircle of the triangle \(A_1B_1C_1\) intersects the sides of the triangle \(ABC\) at segments of lengths \(\pm x\), \(\pm y\), and \(\pm z\) (signs are chosen according to the orientation of the triangle). Prove that \(\frac{x}{a^{n-1}}+\frac{y}{b^{n-1}}+\frac{z}{c^{n-1}}=0\), where \(a, b, c\) are the lengths of the sides of the triangle and \(A_1 B = b^n : c^n\).

Answer: \frac{x}{^{n-1}}+\frac{y}{b^{n-1}}+\frac{z}{^{n-1}}=0


The segment \( AB = 8 \) intersects the plane \( \alpha \) at an angle of \( 30^\circ \) and is divided by this plane in the ratio \( 1:3 \). Find the radius of the sphere passing through points \( A \) and \( B \) and intersecting the plane \( \alpha \) in a circle of the smallest radius.

Answer: 2\sqrt{7}


Convert the binary number \(11111011111_2\) to its decimal representation.

Answer: 2015


The teacher asked Xiao Ma to help calculate the class average score for the math exam. Xiao Ma mistakenly recorded his score of 91 as 19, resulting in an average score of 87. After discovering the mistake, the teacher recalculated and found the correct average score to be 90. How many students are in the class?

Answer: 24


At the beginning of school year in one of the first grade classes: $i)$  every student had exatly  $20$  acquaintances $ii)$  every two students knowing each other had exactly  $13$  mutual acquaintances $iii)$  every two students not knowing each other had exactly  $12$  mutual acquaintances
Find number of students in this class

Answer: 31


An equilateral triangle \( ABC \) is inscribed in the ellipse \( \frac{x^2}{p^2} + \frac{y^2}{q^2} = 1 \), such that vertex \( B \) is at \( (0, q) \), and \( \overline{AC} \) is parallel to the \( x \)-axis. The foci \( F_1 \) and \( F_2 \) of the ellipse lie on sides \( \overline{BC} \) and \( \overline{AB} \), respectively. Given \( F_1 F_2 = 2 \), find the ratio \( \frac{AB}{F_1 F_2} \).

Answer: \frac{8}{5}


The equation $\sqrt {x + 4} - \sqrt {x - 3} + 1 = 0$ has:
$\textbf{(A)}\ \text{no root} \qquad  \textbf{(B)}\ \text{one real root} \\ \textbf{(C)}\ \text{one real root and one imaginary root} \\ \textbf{(D)}\ \text{two imaginary roots} \qquad \qquad\textbf{(E)}\ \text{two real roots}$

Answer: \textbf{(A)}\


Among 51 consecutive odd numbers $1, 3, 5, \cdots, 101$, select $\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\mathrm{k}$?

Answer: 44


Two concentric circles have radii of 1 and 2 units, respectively. What is the minimum possible area of a cyclic quadrilateral inscribed in the larger circle that also contains the smaller circle?

Answer: 3\sqrt{3}


Expand $(1+0.1)^{500}$ by the binomial theorem and calculate each term as
\[{500 \choose k}(0.1)^k\] for $k = 0, 1, 2, \ldots, 500.$ Determine the value of $k$ for which the term is the largest.

Answer: 45


Call an integer $n>1$ radical if $2^{n}-1$ is prime. What is the 20th smallest radical number? If $A$ is your answer, and $S$ is the correct answer, you will get $\max \left(25\left(1-\frac{|A-S|}{S}\right), 0\right)$ points, rounded to the nearest integer.

Answer: 4423


Let $n$ be an odd integer with exactly 12 positive divisors. Find the number of positive divisors of $27n^3$.

Answer: 256


The value of \( a \) is chosen such that the number of roots of the first equation \( 4^{x} - 4^{-x} = 2 \cos a x \) is 2007. How many roots does the second equation \( 4^{x} + 4^{-x} = 2 \cos a x + 4 \) have for the same \( a \)?

Answer: 4014


Given that \( a_1, a_2, a_3, a_4 \) are 4 distinct numbers chosen from \( 1, 2, \cdots, 100 \), satisfying
$$
\left(a_1^2 + a_2^2 + a_3^2\right)\left(a_2^2 + a_3^2 + a_4^2\right) = \left(a_1 a_2 + a_2 a_3 + a_3 a_4\right)^2,
$$
determine the number of such ordered tuples \( (a_1, a_2, a_3, a_4) \).

Answer: 40


A person bequeathed an amount of money, slightly less than 1500 dollars, to be distributed as follows. His five children and the notary received amounts such that the square root of the eldest son's share, half of the second son's share, the third son's share minus 2 dollars, the fourth son's share plus 2 dollars, the daughter's share doubled, and the square of the notary's fee were all equal. All heirs and the notary received whole dollar amounts, and all the money was used to pay the shares and the notary's fee. What was the total amount left as inheritance?

Answer: 1464


In the triangular pyramid \( P-ABC \), given \( BC=3 \), \( CA=4 \), \( AB=5 \). If the dihedral angles between the three lateral faces and the base face are all \( 45^{\circ} \), find the volume of the triangular pyramid \( P-ABC \).

Answer: 2


$Z$ is a non-empty set of positive integers, satisfying the following conditions:
1. If $x \in Z$, then $4x \in Z$.
2. If $x \in Z$, then $\lfloor \sqrt{x} \rfloor \in Z$ (where $\lfloor a \rfloor$ denotes the greatest integer less than or equal to $a$).

Prove that $Z$ is the set of all positive integers.

Answer: Z


If five points are given on a plane, then by considering all possible triples of these points, 30 angles can be formed. Denote the smallest of these angles by $\alpha$. Find the maximum value of $\alpha$.

Answer: 36


Answer: -50


Given 5 points \( A, B, C, D, E \) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment?

Answer: 135


How many triangles with positive area can be formed where each vertex is at point $(i,j)$ in the coordinate grid, with integers $i$ and $j$ ranging from $1$ to $4$ inclusive?

Answer: 516


Given vector $\vec{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$, and $\vec{a}\cdot \vec{b}=\frac{1}{2}$, calculate the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$.

Answer: \frac{1}{2}


Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=-1$, and $a_{n}+a_{n+1}+4n+2=0 \ (n \in \mathbf{N}_{+})$, find the sum of the first $n$ terms of the subsequence formed by the odd-indexed terms of this sequence.

Answer: -2n^2+n


What digit can the positive integer \( n \geq 3 \) end with if \( n + n^{2} + \ldots + n^{2n-3} - 4 \) is a prime number?

Answer: 5


In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the total shaded area formed by the parts of the circle not included in the intersection with the rectangle. Assume the intersection forms a sector.
[Diagram not shown: Assume descriptive adequacy for the composition of the circle and the rectangle.]

Answer: 50\pi


Let $n$ represent the smallest integer that satisfies the following conditions:
$\frac n2$ is a perfect square.
$\frac n3$ is a perfect cube.
$\frac n5$ is a perfect fifth.
How many divisors does $n$ have that are not multiples of 10?

Answer: 242


Problem 4. Angel has a warehouse, which initially contains  $100$  piles of  $100$  pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.

However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.

What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse?

Answer: 199


On a board, the numbers $1, 2, \ldots, 1000$ are written. It is allowed to erase any two numbers $a$ and $b$ and write in their place the numbers $ab$ and $a^{2} + b^{2}$. Is it possible to achieve, through such operations, that at least 700 of the numbers written on the board are the same?

Answer: No


A dog from point $A$ chased a fox that was 30 meters away from the dog at point $B$. The dog's leap is 2 meters, and the fox's leap is 1 meter. The dog makes 2 leaps while the fox makes 3 leaps. At what distance from point $A$ will the dog catch the fox?

Answer: 120


How many terms of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \cdot n}{(2 n+1) 5^{n}}$ should be taken to calculate its sum with an accuracy of 0.01?

Answer: 2


In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$

Answer: 27


A true-false test has ten questions. If you answer five questions "true" and five "false," your score is guaranteed to be at least four. How many answer keys are there for which this is true?

Answer: 22


Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined?

Answer: 8\pi - 16


The picture shows several circles connected by segments. Tanya chooses a natural number \( n \) and places different natural numbers in the circles so that the following property holds for all these numbers:

If numbers \( a \) and \( b \) are not connected by a segment, then the sum \( a^2 + b^2 \) must be coprime with \( n \). If they are connected, then the numbers \( a^2 + b^2 \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?

Answer: 65


We want to set up an electric bell. The location of the bell is at a distance of $30 \mathrm{~m}$ from the ringing spot. The internal resistance of the bell is 2 ohms. We plan to use 2 Leclanché cells connected in series, each with an electromotive force of 1.5 Volts and an internal resistance of 1 ohm. What diameter of copper wire conductor can we use, if the bell will only operate reliably if at least 0.4 amperes flow through it? The resistance of 1 meter of copper wire with a cross-sectional area of $1 \mathrm{~mm}^{2}$ is $\frac{1}{55} \mathrm{ohm}$.

Answer: 0.63


The given expression $\boldsymbol{F}(n)$, where $n$ denotes a variable positive integer, should be assigned a plus sign when $n$ is even and a minus sign when $n$ is odd. How can this be expressed concisely in an algebraic form without providing any verbal explanation of the formula?

Answer: (-1)^nF(n)


Let  $S$  be a finite set of real numbers such that given any three distinct elements  $x,y,z\in\mathbb{S}$ , at least one of  $x+y$ ,  $x+z$ , or  $y+z$  is also contained in  $S$ . Find the largest possible number of elements that  $S$  could have.

Answer: 7


Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \in \mathbb{R}$).

$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.

$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$.

Answer: \dfrac{2}{3}


Koschei the Deathless captured 43 people and took them to an island. Prince Ivan set out with a two-person boat to rescue them. Koschei said to him:

- I'm tired of feeding these freeloaders. Let them sail on your boat willingly. Keep in mind: you can only sail from the island to the shore with two people, but one can manage the return trip alone. Before the crossing, I will tell each prisoner about at least 40 other prisoners, saying they are werewolves. You will choose whom to tell about whom. If a prisoner hears that someone is a werewolf, they will not sit in the boat with that person, but they can be on the shore together. I will enchant them so that they remain silent on land but talk to each other about all the werewolves they know while in the boat. As long as one prisoner remains on the island, you cannot sail with them. Only when all 43 are on the shore, one can sail back to fetch you. If you fail to arrange the crossing, you will remain with me forever.

Is there a way for Ivan to pass the test and bring the captives home? $\quad[8$ points]

Answer: Yes


Answer: .246


Laura typically performs her routine using two 10-pound dumbbells for 30 repetitions. If she decides to use two 8-pound dumbbells instead, how many repetitions must she complete to lift the same total weight as her usual routine?

Answer: 37.5


Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \( \frac{48}{\sqrt{K}} \). What is \( K \)?

Answer: \frac{36}{\pi^3}


In a box, there are 2 one-cent coins, 4 five-cent coins, and 6 ten-cent coins. Six coins are drawn in sequence without replacement, and each coin has an equal probability of being selected. What is the probability that the total value of the drawn coins is at least 50 cents?

Answer: $\frac{127}{924}$


In a speech competition, judges will score participants based on the content, delivery, and effectiveness of the speech, with weights of $4:4:2$ respectively. If a student receives scores of $91$, $94$, and $90$ in these three aspects, then the student's total score is ______ points.

Answer: 92


Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.

Answer: 600


Given the real number \( a \geqslant -2 \), with the sets defined as
\[
\begin{array}{l}
A=\{x \mid-2 \leqslant x \leqslant a\}, \\
B=\{y \mid y=2 x+3, x \in A\}, \\
C=\left\{z \mid z=x^{2}, x \in A\right\},
\end{array}
\]
if \( C \subseteq B \), then the range of values for \( a \) is

Answer: [\frac{1}{2},3]


Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which:
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$.
[/list]
[i]

Answer: c < \sqrt{2}


Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x-1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?

Answer: \frac{1}{4}


Find \( n > m \geq 1 \) such that the last two digits of the decimal representation of \( 42^{n} \) are the same as those of \( 42^{m} \), and such that the sum \( m+n \) is minimal.

Answer: 24


What is the maximum length of the segment intercepted by the sides of a triangle on the tangent to the inscribed circle, drawn parallel to the base, if the perimeter of the triangle is $2p$?

Answer: \frac{p}{4}


An ordinary domino tile can be identified as a pair  $(k,m)$  where numbers  $k$  and  $m$  can get values  $0, 1, 2, 3, 4, 5$  and  $6.$
Pairs  $(k,m)$  and  $(m, k)$  determine the same tile. In particular, the pair  $(k, k)$  determines one tile.
We say that two domino tiles *match*, if they have a common component.
*Generalized n-domino tiles*  $m$  and  $k$  can get values  $0, 1,... , n.$  What is the probability that two randomly chosen  $n$ -domino tiles match?

Answer: \frac{4(n+1)}{(n+2)(n+3)}


For a positive integer  $n$ , an *$n$ -branch*  $B$  is an ordered tuple  $(S_1, S_2, \dots, S_m)$  of nonempty sets (where  $m$  is any positive integer) satisfying  $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$ . An integer  $x$  is said to *appear* in  $B$  if it is an element of the last set  $S_m$ .  Define an *$n$ -plant* to be an (unordered) set of  $n$ -branches  $\{ B_1, B_2, \dots, B_k\}$ , and call it *perfect* if each of  $1$ ,  $2$ , \dots,  $n$  appears in exactly one of its branches.

Let  $T_n$  be the number of distinct perfect  $n$ -plants (where  $T_0=1$ ), and suppose that for some positive real number  $x$  we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If  $x = \tfrac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $m+n$ .

*Proposed by Yang Liu*

Answer: 76


Let \( m \) and \( n \) be positive integers such that \( m > n \). If the last three digits of \( 2012^m \) and \( 2012^n \) are identical, find the smallest possible value of \( m+n \).

Answer: 104


In the tetrahedron $ABCD$, $\triangle ABC$ is an equilateral triangle, $AD = BD = 2$, $AD \perp BD$, and $AD \perp CD$. Find the distance from point $D$ to the plane $ABC$.

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


Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is (    )

A 36      B 48       C 52       D 54

Answer: 48


Given a geometric progression \( b_1, b_2, \ldots, b_{3000} \) with all positive terms and a total sum \( S \). It is known that if every term with an index that is a multiple of 3 (i.e., \( b_3, b_6, \ldots, b_{3000} \)) is increased by 50 times, the sum \( S \) increases by 10 times. How will \( S \) change if every term in an even position (i.e., \( b_2, b_4, \ldots, b_{3000} \)) is increased by 2 times?

Answer: \frac{11}{8}


In \(\triangle ABC\), \(AB = 13\), \(BC = 14\), and \(CA = 15\). \(P\) is a point inside \(\triangle ABC\) such that \(\angle PAB = \angle PBC = \angle PCA\). Find \(\tan \angle PAB\).

Answer: \frac{168}{295}


On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz.

Answer: 13


In a triangle configuration, each row consists of increasing multiples of 3 unit rods. The number of connectors in a triangle always forms an additional row than the rods, with connectors enclosing each by doubling the requirements of connecting joints from the previous triangle. How many total pieces are required to build a four-row triangle?

Answer: 60


In an isosceles triangle \(ABC \) (\(AB = BC\)), the angle bisectors \(BD\) and \(AF\) intersect at point \(O\). The ratio of the area of triangle \(DOA\) to the area of triangle \(BOF\) is \(\frac{3}{8}\). Find the ratio \(\frac{AC}{AB}\).

Answer: \frac{1}{2}


Let $A B C D$ be an isosceles trapezoid with parallel bases $A B=1$ and $C D=2$ and height 1. Find the area of the region containing all points inside $A B C D$ whose projections onto the four sides of the trapezoid lie on the segments formed by $A B, B C, C D$ and $D A$.

Answer: \frac{5}{8}


Given the set $A={3,3^{2},3^{3},…,3^{n}}$ $(n\geqslant 3)$, choose three different numbers from it and arrange them in a certain order to form a geometric sequence. Denote the number of geometric sequences that satisfy this condition as $f(n)$.
(I) Find $f(5)=$ _______ ;
(II) If $f(n)=220$, find $n=$ _______ .

Answer: 22


Find the sum of the infinite series $$1+2\left(\frac{1}{1998}\right)+3\left(\frac{1}{1998}\right)^{2}+4\left(\frac{1}{1998}\right)^{3}+\ldots$$

Answer: \left(\frac{1998}{1997}\right)^{2} \text{ or } \frac{3992004}{3988009}


A turtle and a rabbit are racing on a 1000-meter track. The rabbit's speed is 15 times that of the turtle. However, the rabbit took a break during the race. When the rabbit woke up, the turtle had just reached the finish line, while the rabbit still had 100 meters left to go. How far did the turtle crawl while the rabbit was resting?

Answer: 940


In order to glaze 15 windows of various sizes and shapes, 15 panes of glass have been prepared exactly according to the windows (each window requires one pane). The glazier, not knowing that the panes have been matched, works as follows: he approaches the next window and sifts through the unused panes until he finds one that is large enough (either exactly fitting or one that can be cut to fit). If he cannot find such a pane, he moves on to the next window, and so on, until all the windows have been approached. It is not allowed to assemble a pane from multiple parts. What is the maximum number of windows that can remain unglazed?

Answer: 7


What is the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=21+11y-y^2$ that lies below the line $y=x-3$?

Answer: 8 \pi


It takes  $3$  rabbits  $5$  hours to dig  $9$  holes. It takes  $5$  beavers  $36$  minutes to build  $2$  dams. At this rate, how many more minutes does it take  $1$  rabbit to dig  $1$  hole than it takes  $1$  beaver to build  $1$  dam?

*2016 CCA Math Bonanza Team #1*

Answer: 10


The equation of the directrix of the parabola $y^{2}=6x$ is $x=\frac{3}{2}$.

Answer: -\dfrac{3}{2}


In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC=$

Answer: 30^\circ


Given the function $f\left(x\right)=\frac{2×202{3}^{x}}{202{3}^{x}+1}$, if the inequality $f(ae^{x})\geqslant 2-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______.

Answer: \frac{1}{e}


Two cars left point $A$ for point $B$ and from point $B$ to point $A$ simultaneously and met after 3 hours. The first car arrived at point $B$ 1.1 hours later than the second one arrived at point $A$. How many times greater is the speed of the second car compared to the speed of the first car?

Answer: \frac{6}{5}


A teacher wrote a sequence of consecutive odd numbers starting from 1 on the blackboard: $1, 3, 5, 7, 9, 11, \cdots$ After writing, the teacher erased two numbers, dividing the sequence into three segments. If the sums of the first two segments are 961 and 1001 respectively, what is the sum of the two erased odd numbers?

Answer: 154


Given \(3z \in \mathbf{C}, z^7 = 1 (z \neq 1)\), and the argument of \(z\) is \(\alpha\), find the value of \(\cos \alpha + \cos 2\alpha + \cos 4\alpha\).

Answer: -\frac{1}{2}


Determine, with proof, whether or not there exist distinct positive integers \( a_{1}, a_{2}, \ldots, a_{n} \) such that

\[
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}} = 2019.
\]

Answer: Yes


What is the largest positive integer with only even digits that is less than $10,000$ and is a multiple of $9$?

Answer: 8820


The angle that has the same terminal side as $- \frac{\pi}{3}$ is $\frac{\pi}{3}$.

Answer: \frac{5\pi}{3}


Find a number \( k \) (other than 1) such that both \( k \) and \( k^4 \) can each be expressed as the sum of the squares of two consecutive integers. Prove that there is only one such \( k \).

Answer: 13


The center of a hyperbola is at the origin, the foci are on the coordinate axes, and the distance from point \( P(-2,0) \) to its asymptote is \( \frac{2 \sqrt{6}}{3} \). If a line with slope \( \frac{\sqrt{2}}{2} \) passes through point \( P \) and intersects the hyperbola at points \( A \) and \( B \), intersects the y-axis at point \( M \), and \( P M \) is the geometric mean of \( P A \) and \( P B \), then the half focal length of the hyperbola is what?

Answer: \sqrt{3}


The chromatic number  $\chi$  of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.
Prove that  $4 \leq \chi \leq 7$ .

Answer: 4\le\chi\le7


Let \( x \) and \( y \) be positive real numbers, and let \( \theta \neq \frac{n \pi}{2} \) (where \( n \) is an integer). If \( \frac{\sin \theta}{x}=\frac{\cos \theta}{y} \), and \( \frac{\cos ^{4} \theta}{x^{4}}+\frac{\sin ^{4} \theta}{y^{4}}=\frac{97 \sin 2 \theta}{x^{3} y+y^{3} x} \), then find the value of \( \frac{y}{x}+\frac{x}{y} \).

Answer: 4


Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$

Answer: 135


Let $n$ be an integer and $$m=(n-1001)(n-2001)(n-2002)(n-3001)(n-3002)(n-3003)$$ Given that $m$ is positive, find the minimum number of digits of $m$.

Answer: 11


How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four?

Answer: 76


You are trying to cross a 6 foot wide river. You can jump at most 4 feet, but you have one stone you can throw into the river; after it is placed, you may jump to that stone and, if possible, from there to the other side of the river. However, you are not very accurate and the stone ends up landing uniformly at random in the river. What is the probability that you can get across?

Answer: \frac{1}{3}


The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 2007$ such that $a_i-i$ is even for $1\le i \le 10$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000.

Answer: 8


Out of two hundred ninth-grade students, $80\%$ received excellent grades on the first exam, $70\%$ on the second exam, and $59\%$ on the third exam. What is the minimum number of students who could have received excellent grades on all three exams?

Answer: 18


The sides  $AD$  and  $BC$  of a convex quadrilateral  $ABCD$  are extended to meet at  $E$ .  Let  $H$  and  $G$  be the midpoints of  $BD$  and  $AC$ , respectively.  Find the ratio of the area of the triangle  $EHG$  to that of the quadrilateral  $ABCD$ .

Answer: \frac{1}{4}


A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.

Hint: The equation of state for the gas is given by $\rho V=$ const, where $\rho$ is pressure and $V$ is volume.

Given:
$$
H=0.4 \text{ m}, \ h=0.2 \text{ m}, \ R=0.1 \text{ m}
$$

Answer: 900


On a circle of radius 1, a point $O$ is marked and from it, using a compass, a mark is made to the right with radius $l$. From the resulting point $O_{1}$, in the same direction, another mark is made with the same radius, and this process is repeated 1968 times. After this, the circle is cut at all 1968 marks, resulting in 1968 arcs. How many different arc lengths can be obtained in this process?

Answer: 3


Lucas wants to buy a book that costs $28.50. He has two $10 bills, five $1 bills, and six quarters in his wallet. What is the minimum number of nickels that must be in his wallet so he can afford the book?

Answer: 40


One side of the cards is painted in some color, and the other has a smiley face. There are four cards in front of you: the first is yellow, the second is black, the third has a happy smiley, and the fourth has a sad smiley. You need to check the statement: "If there is a happy smiley on one side of a card, then the other side is painted yellow."

What is the minimum number of cards you need to turn over to check the truth of this statement? Which ones? (7 points)

Answer: 2


On the hypotenuse \(BC\) of a right triangle \(ABC\), a point \(D\) is located such that \(AD \perp BC\). Find the hypotenuse \(BC\), given that \(AD = DC - BD = h\).

Answer: \sqrt{5}


A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo needs to complete to earn 30 study points.

Answer: 90


(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:
(1) If digits can be repeated, how many different five-digit even numbers can be formed?
(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not being 3, can be formed?
(3) If in the linear equation $ax + by = 0$, $a$ and $b$ can be any two different digits from the given six, how many different lines can be represented by the equation?

Answer: 20


Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.

[i]

Answer: f(x) = 2x


Given \( f(x) = x^2 + c \) and \( f(f(x)) = f(x^2 + 1) \),
1. Let \( g(x) = f(f(x)) \). Find the explicit expression for \( g(x) \).
2. Let \( \varphi(x) = g(x) - \lambda f(x) \). Determine the value of the real number \( \lambda \), such that \( \varphi(x) \) is a decreasing function on \( (-\infty, -1] \) and an increasing function on \( [-1, 0) \).

Answer: 4


Yura has unusual clocks with several minute hands, moving in different directions. Yura counted that in one hour, the minute hands coincided exactly 54 times in pairs. What is the maximum number of minute hands that Yura's clock can have?

Answer: 28


Find the number of natural numbers \( k \) not exceeding 353500 such that \( k^{2} + k \) is divisible by 505.

Answer: 2800


Suppose $a$ and $b$ are the points obtained by throwing a dice in order, and the function is $f(x)=\frac{1}{2}ax^{2}+bx+1$.

(1) Find the probability that $f(x)$ is a decreasing function in the interval $(-\infty,-1]$;

(2) Find the probability that the function $f(x)$ has zero points.

Answer: \frac{2}{3}


Let \(ABCD\) be a convex quadrilateral with \(\angle ABC = 90^\circ\), \(\angle BAD = \angle ADC = 80^\circ\). Let \(M\) and \(N\) be points on \([AD]\) and \([BC]\) such that \(\angle CDN = \angle ABM = 20^\circ\). Finally, assume \(MD = AB\). What is the measure of \(\angle MNB\)?

Answer: 70


What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$?

Answer: 970


A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection.

Answer: 40


The sequence $\left\{a_{n}\right\}$ is defined by the following conditions: $a_{1}=1, a_{n+1}=a_{n}+\frac{1}{a_{n}}$ for $n \geq 1$. Find the integer part of $a_{100}$.

Answer: 14


The area of an integer-sided triangle is also an integer and equals its semiperimeter. Show that this is possible only in the unique case when the sides of the triangle are $3, 4, 5$.

Answer: 3,4,5


In the convex quadrilateral \(ABCD\): \(AB = AC = AD = BD\) and \(\angle BAC = \angle CBD\). Find \(\angle ACD\).

Answer: 60


Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences.

Answer: 144


Let  $\triangle ABC$  be a triangle with  $AB=5, BC=6, CA=7$ . Suppose  $P$  is a point inside  $\triangle ABC$  such that  $\triangle BPA\sim \triangle APC$ . If  $AP$  intersects  $BC$  at  $X$ , find  $\frac{BX}{CX}$ .

[i]Proposed by Nathan Ramesh

Answer: 25/49


In a cube \( A B C D - A_{1} B_{1} C_{1} D_{1} \), let the points \( P \) and \( Q \) be the reflections of vertex \( A \) with respect to the plane \( C_{1} B D \) and the line \( B_{1} D \), respectively. Find the sine of the angle between the line \( P Q \) and the plane \( A_{1} B D \).

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


In parallelogram \( A B C D \), the height drawn from vertex \( B \) of the obtuse angle to side \( DA \) divides it in a ratio of 5:3, starting from vertex \( D \). Find the ratio \( AC:BD \) if \( AD:AB=2 \).

Answer: 2:1


Find the smallest prime number that can be expressed as the sum of five different prime numbers.

Answer: 43


Find the largest integer $n$ satisfying the following conditions:
 (i) $n^2$ can be expressed as the difference of two consecutive cubes;
 (ii) $2n + 79$ is a perfect square.

Answer: 181


Answer: 12500


A sequence of positive integers \(a_{1}, a_{2}, \ldots\) is such that for each \(m\) and \(n\) the following holds: if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_{m}\) is a divisor of \(a_{n}\) and \(a_{m} < a_{n}\). Find the least possible value of \(a_{2000}\).

Answer: 128


Today, Ivan the Confessor prefers continuous functions \(f:[0,1] \rightarrow \mathbb{R}\) satisfying \(f(x)+f(y) \geq|x-y|\) for all pairs \(x, y \in[0,1]\). Find the minimum of \(\int_{0}^{1} f\) over all preferred functions.

Answer: \frac{1}{4}


In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC$, $AF \perp BC$, and $BD=DC=FC=1$. Find $AC$.

Answer: \sqrt[3]{2}


The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radius of one circle is twice the radius of the other.

Answer: 2 \sqrt{3}


Point $O$ is the intersection point of the altitudes of an acute triangle $ABC$. Prove that the three circles passing through the points $O, A, B$; through the points $O, B, C$; and through the points $O, C, A$ are equal to each other.

Answer: 3


In a plane, there are $2n$ red lines and $n$ blue lines, none of which are parallel, and no three lines intersect at the same point. Prove that among the regions into which they divide the plane, there are at least $n$ regions bounded solely by red lines.

Answer: n


In a certain region of the planet, seismic activity was studied. 80 percent of all days were quiet. The predictions of the devices promised a calm environment in 64 out of 100 cases, and in 70 percent of all cases where the day was calm, the predictions of the devices came true. What percentage of days with increased seismic activity are those in which the predictions did not match reality?

Answer: 40


Given that $689\Box\Box\Box20312 \approx 69$ billion (rounded), find the number of ways to fill in the three-digit number.

Answer: 500


A local community club consists of four leaders and a number of regular members. Each year, the current leaders leave the club, and every regular member is responsible for recruiting three new members. At the end of the year, four new leaders are elected from outside the club. Initially, there are 20 people in total in the club. How many people will be in the club after four years?

Answer: 4100


For every  $a \in \mathbb N$  denote by  $M(a)$  the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{  is a divisor of } ab \}.\]
Find  $\max_{a\leq 1983} M(a).$

Answer: 121


The area of rectangle \(ABCD\) is 48, and the diagonal is 10. On the plane where the rectangle is located, a point \(O\) is chosen such that \(OB = OD = 13\). Find the distance from point \(O\) to the vertex of the rectangle farthest from it.

Answer: 17


Two circles with equal radii intersect as shown. The area of the shaded region equals the sum of the areas of the two unshaded regions. If the area of the shaded region is $216\pi$, what is the circumference of each circle?

Answer: 36\pi


James borrows $2000$ dollars from Alice, who charges an interest of $3\%$ per month (which compounds monthly). What is the least integer number of months after which James will owe more than three times as much as he borrowed?

Answer: 37


In a dumpling restaurant, dumplings can be ordered in portions of 6, 9, and 20 pieces. Therefore, not every number of dumplings can be ordered using these sets; for example, it is not possible to buy 1, 2, 3, 4, 5, 7, or 8 dumplings. What is the largest number of dumplings that cannot be ordered at the restaurant?

Answer: 43


Answer: y_1y_2y_3\gex_1x_2x_3


Consider the parabola \( C: y^{2} = 2px \) where \( p > 0 \). The line \( l \) intersects the parabola \( C \) at points \( A \) and \( B \). The line connecting point \( A \) and the vertex \( O \) of the parabola intersects the directrix at point \( B' \). The line \( BO \) intersects the directrix at point \( A' \). Both \( AA' \) and \( BB' \) are parallel to the \( x \)-axis.

(1) Prove that the line \( l \) passes through a fixed point.

(2) Find the minimum area of the quadrilateral \( ABB'A' \).

Answer: 2p^2


Optimus Prime in the shape of a robot travels from point $A$ to point $B$ and arrives on time. If he starts by transforming into a car, his speed increases by $\frac{1}{4}$ and he arrives at point $B$ 1 hour earlier. If he travels the first 150 kilometers as a robot and then transforms into a car, increasing his speed by $\frac{1}{5}$, he arrives 40 minutes earlier. Determine the distance between points $A$ and $B$ in kilometers.

Answer: 750


Real number  $C > 1$  is given. Sequence of positive real numbers  $a_1, a_2, a_3, \ldots$ , in which  $a_1=1$  and  $a_2=2$ , satisfy the conditions
\[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\]
for  $m, n = 1, 2, 3, \ldots$ . Prove that  $a_n = n$  for  $n=1, 2, 3, \ldots$ .

Answer: a_n=n


A circle with a radius equal to the height of an equilateral triangle rolls along one of its sides. Prove that the angular measure of the arc intercepted on the circle by the sides of the triangle is always $60^{\circ}$.

Answer: 60


The equation   $$ t^4+at^3+bt^2=(a+b)(2t-1) $$   has  $4$  positive real roots  $t_1<t_2<t_3<t_4$ . Show that  $t_1t_4>t_2t_3$ .

Answer: t_1t_4>t_2t_3


A five-digit number has one of its digits crossed out, and the resulting four-digit number is added to the original number. The sum is 54321. Find the original number.

Answer: 49383


The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, what is the value of $x$?

Answer: 11


Find the number of solutions in positive integers $(k ; a_{1}, a_{2}, \ldots, a_{k} ; b_{1}, b_{2}, \ldots, b_{k})$ to the equation $$a_{1}(b_{1})+a_{2}(b_{1}+b_{2})+\cdots+a_{k}(b_{1}+b_{2}+\cdots+b_{k})=7$$

Answer: 15


Answer: -3\cdot2^{2015}


A circle with radius $R$ is tangent to a line $l$ at point $A$. $AB$ is a diameter of this circle, and $BC$ is an arbitrary chord. Let $D$ be the foot of the perpendicular dropped from $C$ to $AB$. Point $E$ lies on the extension of $CD$ beyond point $D$, such that $|ED| = |BC|$. Tangents to the circle passing through $E$ intersect the line $l$ at points $K$ and $N$. Find the length of the segment $|KN|$.

Answer: 2R


Answer: \frac{1}{a_{2}^{2}}+\frac{1}{a_{3}^{2}}+\cdots+\frac{1}{a_{n}^{2}}\le\frac{n-1}{2}\cdot\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}


All three-digit numbers from 100 to 999 are written consecutively without spaces. Kostya underlined $k$ digits in a row in this sequence, and Andrey underlined another $k$ digits in a row in this sequence. It turned out that the $k$-digit numbers underlined by the boys are equal. For what is the maximum $k$ this could happen?

Answer: 5


The first term of an arithmetic series of consecutive integers is $k^2 + 1$. The sum of $2k + 1$ terms of this series may be expressed as:

Answer: $k^3 + (k + 1)^3$


Two players are playing a game. One of them thinks of a sequence of integers ($x_{1}, x_{2}, \ldots, x_{n}$), which are single-digit numbers, both positive and negative. The other player is allowed to ask for the value of the sum $a_{1} x_{1}+\ldots+a_{n} x_{n}$, where $(a_{1}, \ldots, a_{n})$ is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?

Answer: 1


A company has calculated that investing x million yuan in project A will yield an economic benefit y that satisfies the relationship: $y=f(x)=-\frac{1}{4}x^{2}+2x+12$. Similarly, the economic benefit y from investing in project B satisfies the relationship: $y=h(x)=-\frac{1}{3}x^{2}+4x+1$.
(1) If the company has 10 million yuan available for investment, how should the funds be allocated to maximize the total investment return?
(2) If the marginal effect function for investment is defined as $F(x)=f(x+1)-f(x)$, and investment is not recommended when the marginal value is less than 0, how should the investment be allocated?

Answer: 6.5


Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length 1. Find the difference between the sum of the areas of the squares and the sum of the areas of the circles.

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


Let $\gamma$ and $\gamma^{\prime}$ be two circles that intersect at points $A$ and $B$. Draw $t$, the common tangent to these two circles, which touches $\gamma$ at $T$ and $\gamma^{\prime}$ at $T^{\prime}$, such that $T A T^{\prime}$ contains $B$. Let $B^{*}$ be the reflection of $B$ with respect to $t$.

## Show that

1. $A, T, T^{\prime}, B^{*}$ are concyclic.
2. The circles $A T T^{\prime}$ and $B T T^{\prime}$ have the same radius.
3. Which exercise do we find in the special case where $A = B$?

Answer: 3


Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\text{eight}}.$

Answer: 585


Let \( k=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i} \). In the complex plane, the vertices of \(\triangle ABC\) correspond to the complex numbers \( z_{1}, z_{2}, z_{3} \) which satisfy the equation
\[ z_{1}+k z_{2}+k^{2}\left(2 z_{3}-z_{1}\right)=0 \text {. } \]

Find the radian measure of the smallest interior angle of this triangle.

Answer: \frac{\pi}{6}


There are \( n \) girls \( G_{1}, \ldots, G_{n} \) and \( n \) boys \( B_{1}, \ldots, B_{n} \). A pair \( \left(G_{i}, B_{j}\right) \) is called suitable if and only if girl \( G_{i} \) is willing to marry boy \( B_{j} \). Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?

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


The distance between the midpoints of the sides \(AB\) and \(CD\) of a convex quadrilateral \(ABCD\) is equal to the distance between the midpoints of its diagonals. Find the angle formed by the lines \(AD\) and \(BC\) at their intersection. Justify your answer.

Answer: 90


There are \( 2n + 1 \) cards. There are two cards with each integer from 1 to \( n \) and a joker. The cards are arranged in a line with the joker in the center position (with \( n \) cards on each side of it). For which \( n < 11 \) can we arrange the cards so that the two cards with the number \( k \) have just \( k-1 \) cards between them (for \( k = 1, 2, \ldots, n \))?

Answer: 3,4,7,8


In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can be repeated) so that the resulting 12-digit number is divisible by 12. How many ways can this be done?

Answer: 5184


Given that the four vertices of the quadrilateral $MNPQ$ are on the graph of the function $f(x)=\log_{\frac{1}{2}} \frac{ax+1}{x+b}$, and it satisfies $\overrightarrow{MN}= \overrightarrow{QP}$, where $M(3,-1)$, $N\left( \frac{5}{3},-2\right)$, then the area of the quadrilateral $MNPQ$ is \_\_\_\_\_\_.

Answer: \frac{26}{3}


Given functions $f(x)=xe^x$ and $g(x)=-\frac{lnx}{x}$, if $f(x_{1})=g(x_{2})=t\left( \gt 0\right)$, find the maximum value of $\frac{{x}_{1}}{{x}_{2}{e}^{t}}$.

Answer: \frac{1}{e}


The sequence of values \( S_{0}=0 \), \( S_{1}=\xi_{1} \), \( S_{2}=\xi_{1}+\xi_{2} \), \(\ldots\), \( S_{n}=\xi_{1}+\ldots+\xi_{n} \) can be considered as the trajectory of a random walk of a particle starting from zero and moving up or down by one unit at integer time moments.

Now, assume that the walk occurs at time moments \(\Delta, 2\Delta, \ldots, n\Delta\), where \(\Delta>0\), and at each step, the particle shifts up or down by \(\Delta x\). Instead of the probabilities \( P_{n}(k)=\mathrm{P}\{S_{n}=k\} \) introduced in the previous problem, consider the probabilities

\[ P_{n \Delta}(k \Delta x) = \mathrm{P}\{S_{n \Delta}=k \Delta x\} \]

By analogy with the recurrence relations (**), we obtain

\[ P_{(n+1) \Delta}(k \Delta x) - P_{n \Delta}(k \Delta x) = \frac{1}{2}\left[P_{n \Delta}((k+1) \Delta x) - 2 P_{n \Delta}(k \Delta x) + P_{n \Delta}((k-1) \Delta x)\right] \]

that is, the "discrete version of the first derivative with respect to time" matches up to a factor \(\frac{1}{2}\) with the "discrete version of the second derivative with respect to the spatial variable."

Let \(\Delta x = \sqrt{\Delta}\) and for \( t > 0 \) and \( x \in \mathbb{R} \) perform the limit transition as \( n \rightarrow \infty \) and \( k \rightarrow \infty \) such that \( n \Delta \rightarrow t \) and \( k \sqrt{\Delta} \rightarrow x \). Show that in such a limit transition

(a) the limit exists \( P_{t}(x)=\lim P_{n \Delta}(k \sqrt{\Delta}) / \sqrt{\Delta} \);

(b) the limiting functions \( P_{t}(x) \) satisfy the "diffusion" heat equation

\[ \frac{\partial P_{t}(x)}{\partial t}=\frac{1}{2} \frac{\partial^{2} P_{t}(x)}{\partial x^{2}} \quad \text{(Bachelier, Einstein).} \]

Answer: \frac{\partialP_{}(x)}{\partial}=\frac{1}{2}\frac{\partial^{2}P_{}(x)}{\partialx^{2}}


Given the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ with one of its foci at $F = (\sqrt{3}, 0)$, find a point $P = (p, 0)$ where $p > 0$ such that for any chord $\overline{AB}$ passing through $F$, the angles $\angle APF$ and $\angle BPF$ are equal.

Answer: \sqrt{3}


Given that the first term of a geometric sequence $\{a\_n\}$ is $\frac{3}{2}$, and the sum of the first $n$ terms is $S\_n$, where $n \in \mathbb{N}^*$. Also, $-2S\_2$, $S\_3$, and $4S\_4$ form an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. For a sequence $\{A\_n\}$, if there exists an interval $M$ such that $A\_i \in M$ for all $i = 1, 2, 3, ...$, then $M$ is called the "range interval" of sequence $\{A\_n\}$. Let $b\_n = S\_n + \frac{1}{S\_n}$, find the minimum length of the "range interval" of sequence $\{b\_n\}$.

Answer: \frac{1}{6}


Vivek is painting three doors. The doors are numbered 1, 2, and 3. Each door is to be painted either black or gold. In total, how many different ways can the three doors be painted?
(A) 8
(B) 7
(C) 5
(D) 4
(E) 3

Answer: 8


Answer: [2 \sqrt{2},4]


Let triangle \(ABC\) have \(AB=5\), \(BC=6\), and \(AC=7\), with circumcenter \(O\). Extend ray \(AB\) to point \(D\) such that \(BD=5\), and extend ray \(BC\) to point \(E\) such that \(OD=OE\). Find \(CE\).

Answer: \sqrt{59}-3


Find all sequences of positive integers  $\{a_n\}_{n=1}^{\infty}$ , for which  $a_4=4$  and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural  $n \geq 2$ .

*Peter Boyvalenkov*

Answer: a_n=n


Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,
\[x^2 + y^2 + z^3 + 1 \ge C(x + y + z).\]

Answer: \sqrt{2}


Given the natural number \( n \geq 2 \). Let \( M \) be a subset of the set \( \{(i, k) \mid i, k \in \mathbb{N}, i < k \leq n \} \). It is known that if the pair \( (i, k) \), where \( i < k \), belongs to the set \( M \), then any pair \( (k, m) \), where \( k < m \), does not belong to \( M \). Find the maximum number of elements in the set \( M \).

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


Given an isosceles trapezoid \(ABCD\) with bases \(AD > BC\), where a circle can be inscribed; \(CH\) is the height of the trapezoid, and \(AH = 7\). Find the lateral side of the trapezoid.

Answer: 7


The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$

Answer: 105


Given the real numbers \( x \) and \( y \) satisfy the equation \( 2x^2 + 3xy + 2y^2 = 1 \), find the minimum value of \( x + y + xy \).

Answer: -\frac{9}{8}


Solve the system of equations:
\[
\begin{cases}
2x + y + 8 \leq 0, \\
x^4 + 2x^2y^2 + y^4 + 9 - 10x^2 - 10y^2 = 8xy.
\end{cases}
\]

Answer: (-3,-2)


Find the value of \(a + b + c + d + e\) given the system of equations:
\[
\begin{array}{c}
3a + 2b + 4d = 10 \\
6a + 5b + 4c + 3d + 2e = 8 \\
a + b + 2c + 5e = 3 \\
2c + 3d + 3e = 4 \\
a + 2b + 3c + d = 7
\end{array}
\]

Answer: 4


It is known that, for all positive integers $k$,
$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$.
 Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $200$.

Answer: 112


Let the function $f(x)=a\ln x-x- \frac {1}{2}x^{2}$.
(I) For $a=2$, find the extreme values of the function $f(x)$.
(II) Discuss the monotonicity of the function $f(x)$.

Answer: -\frac{3}{2}


Let us consider a set  $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$ , satisfying the following properties:  $f(a_i) < 2003$  and  $f(a_i) = f(a_j) \quad \forall i, j$  from  $\{1, 2,\ldots , 2004\}$ , where  $f(a_i)$  denotes number of elements which are relatively prime with  $a_i$ . Find the least positive integer  $k$  for which in every  $k$ -subset of  $S$ , having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.

Answer: 1003


Answer: \frac{1}{2}


Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3$ cm from $P$?
$\textbf{(A)} \ 1 \qquad  \textbf{(B)} \ 2 \qquad  \textbf{(C)} \ 3 \qquad  \textbf{(D)} \ 4 \qquad  \textbf{(E)} \ 8$

Answer: \textbf{(B)}\2


Given two propositions:

Proposition \( p \): The function \( f(x)=\log_{a} x \) (\( x > 0 \)) is monotonically increasing;
Proposition \( q \): The function \( g(x) = x^2 + ax + 1 > 0 \) (\( x \in \mathbf{R} \)).

If \( p \vee q \) is a true proposition and \( p \wedge q \) is a false proposition, then the range of the real number \( a \) is...

Answer: (-2,1]\cup[2,+\infty)


Find the equations of the tangent plane and the normal to the surface given by the equation

$$
z = xy
$$

at the point $M(1,1)$.

Answer: \frac{x-1}{1}=\frac{y-1}{1}=\frac{z-1}{-1}


An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs (supporting chief A and supporting chief B), each of whom had 2 equal, inferior officers. If the tribe at one point had 10 members, what is the number of different ways to choose the leadership of the tribe?  That is, in how many ways can we choose a chief, 2 supporting chiefs, and two inferior officers reporting to each supporting chief?

Answer: 151200


Find the sum of all integer values of \(a\) belonging to the interval \([-10, 10]\), for each of which the inequality \(5 \leq x \leq 10\) implies the inequality \(a x + 3a^{2} - 12a + 12 > a^{2} \sqrt{x-1}\).

Answer: -47


In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of the line $l$ is given by $\rho \sin \left( \theta -\dfrac{\pi }{4} \right)=\sqrt{2}$.

(1) Find the Cartesian equation of $C$ and the angle of inclination of $l$;

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

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


Let n be the number of real values of $p$ for which the roots of
$x^2-px+p=0$
are equal. Then n equals:
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{a finite number greater than 2}\qquad \textbf{(E)}\ \infty$

Answer: \textbf{(C)}\2


There are 94 rectangular bricks of dimensions \(4 \times 10 \times 19\). We need to build a tower using these 94 bricks (one brick stacked on top of another). How many different possible heights can the tower have?

Answer: 465


In triangle \( ABC \), \( AB = 86 \) and \( AC = 97 \). A circle with center at point \( A \) and radius \( AB \) intersects side \( BC \) at points \( B \) and \( X \). Given that both \( BX \) and \( CX \) have integer lengths, what is the length of \( BC \)?

Answer: 61


In the plane Cartesian coordinate system \( xOy \), an ellipse \( C \) : \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \( (a>b>0) \) has left and right foci \( F_{1} \) and \( F_{2} \) respectively. Chords \( ST \) and \( UV \) are parallel to the \( x \)-axis and \( y \)-axis respectively, intersecting at point \( P \). Given the lengths of segments \( PU \), \( PS \), \( PV \), and \( PT \) are \(1, 2, 3,\) and \( 6 \) respectively, find the area of \( \triangle P F_{1} F_{2} \).

Answer: \sqrt{15}


Given the function $f(x)=x^3+ax^2+bx+1$, its derivative $f'(x)$ satisfies $f'(1)=2a$, $f'(2)=-b$, where $a$ and $b$ are constants.

(I) Find the values of $a$ and $b$;

(II) Let $g(x)=f'(x)e^{-x}$, find the extreme values of the function $g(x)$.

Answer: 15e^{-3}


Find the range of real values of \( a \) such that the inequality \( \sin^{6} x + \cos^{6} x + 2a \sin x \cos x \geq 0 \) holds for all real numbers \( x \).

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


How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?

Answer: 616


To enter a park, a group of two men, four women, and two children paid 226 reais, while a group of three men, three women, and one child paid 207 reais.

a) How much would a group of 8 men, 10 women, and 4 children pay to enter the park?

b) If the ticket prices are all natural numbers, how many possible prices are there for the tickets?

Answer: 21


Given a regular quadrangular pyramid \( S-ABCD \) with a side edge length of 4, and \( \angle ASB = 30^\circ \). Points \( E \), \( F \), and \( G \) are taken on side edges \( SB \), \( SC \), and \( SD \), respectively. Find the minimum value of \( AE + EF + FG + GA \).

Answer: 4\sqrt{3}


Ezekiel has a rectangular piece of paper with an area of 40. The width of the paper is more than twice the height. He folds the bottom left and top right corners at $45^{\circ}$ and creates a parallelogram with an area of 24. What is the perimeter of the original rectangle?

Answer: 28


Given the set
$$
M=\{1,2, \cdots, 2020\},
$$
for any non-empty subset $A$ of $M$, let $\lambda_{A}$ be the sum of the maximum and minimum numbers in the subset $A$. What is the arithmetic mean of all such $\lambda_{A}$?

Answer: 2021


let  ${a_{n}}$  be a sequence of integers, $a_{1}$  is odd,and for any positive integer  $n$ ,we have $n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3$ ,in addition,we have  $2010$  divides  $a_{2009}$ find the smallest  $n\ge\ 2$ ,so that  $2010$  divides  $a_{n}$

Answer: 671


In the rhombus \(ABCD\), the measure of angle \(B\) is \(40^\circ\). \(E\) is the midpoint of \(BC\), and \(F\) is the foot of the perpendicular dropped from \(A\) to \(DE\). Find the measure of angle \(DFC\).

Answer: 110


Determine $m$ so that $4x^2-6x+m$ is divisible by $x-3$. The obtained value, $m$, is an exact divisor of:

$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 64$

Answer: \textbf{(C)}36


Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?

Answer: 360


Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.

Answer: 476


Let $0 \le a,$ $b,$ $c \le 1.$  Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\]

Answer: 1


The base of a quadrangular pyramid is a rhombus \(ABCD\) in which \(\angle BAD = 60^\circ\). It is known that \(SD = SB\) and \(SA = SC = AB\). Point \(E\) is taken on edge \(DC\) such that the area of triangle \(BSE\) is the smallest among all sections of the pyramid containing segment \(BS\) and intersecting segment \(DC\). Find the ratio \(DE : EC\).

Answer: 2:5


Let $m \geq 2$ be an integer, $A$ a finite set of integers, and $B_{1}, B_{2}, \ldots, B_{m}$ subsets of $A$. Suppose that for each $k=1,2, \ldots, m$, the sum of the elements of $B_{k}$ equals $m^{k}$. Prove that $A$ contains at least $\frac{m}{2}$ elements.

Answer: \frac{}{2}


Answer: \frac{\sqrt[8]{2}}{\sqrt{5}}


The side of a square is increased by $20\%$. To keep the area of the square unchanged, what percentage must the other side be reduced?

Answer: 16.67\%


$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals:
$\textbf{(A) }2\sqrt{3}\qquad\textbf{(B) }3\sqrt{2}\qquad\textbf{(C) }3\sqrt{3}\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }2$

Answer: \textbf{(B)}


In an alphabet with $n>1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Find the maximum possible number of letters in a "good" word.

Answer: 2n+1


Let  $n\ge2$  be an integer and let  $x_1,x_2,\ldots,x_n$  be real numbers. Consider  $N=\binom n2$  sums  $x_i+x_j$ ,  $1\le i<j\le n$ , and denote them by  $y_1,y_2,\ldots,y_N$  (in an arbitrary order). For which  $n$  are the numbers  $x_1,x_2,\ldots,x_n$  uniquely determined by the numbers  $y_1,y_2,\ldots,y_N$ ?

Answer: n\ge3


An object moves $8$ cm in a straight [line](https://artofproblemsolving.com/wiki/index.php/Line) from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that $AC < 7$?
$\mathrm{(A)}\ \frac{1}{6} \qquad\mathrm{(B)}\ \frac{1}{5} \qquad\mathrm{(C)}\ \frac{1}{4} \qquad\mathrm{(D)}\ \frac{1}{3} \qquad\mathrm{(E)}\ \frac{1}{2}$

Answer: \textbf{(D)}\frac{1}{3}


A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
$\textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{12} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ \frac{5}{18}$

Answer: \textbf{(B)}\\frac{1}{12}


On hypotenuse $AB$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $BC=1$, $AC=b$, and $AD=2$, then $BD$ equals:

Answer: \sqrt{b^2-3}


Petya wants to color some cells of a $6 \times 6$ square so that there are as many vertices as possible that belong to exactly three colored squares. What is the maximum number of such vertices he can achieve?

Answer: 25


Let \(a, b, c, d\) be real numbers such that \(b-d \geq 5\) and all zeros \(x_{1}, x_{2}, x_{3},\) and \(x_{4}\) of the polynomial \(P(x) = x^{4} + a x^{3} + b x^{2} + c x + d\) are real. Find the smallest value the product \(\left(x_{1}^{2} + 1\right)\left(x_{2}^{2} + 1\right)\left(x_{3}^{2} + 1\right)\left(x_{4}^{2} + 1\right)\) can take.

Answer: 16


Find a number \( N \) with five digits, all different and none zero, which equals the sum of all distinct three-digit numbers whose digits are all different and are all digits of \( N \).

Answer: 35964


Given that \(\alpha = \frac{2\pi}{1999}\), find the value of \(\cos \alpha \cos 2 \alpha \cos 3 \alpha \cdots \cos 999 \alpha\).

Answer: \frac{1}{2^{999}}


Given the following system of equations:  $$ \begin{cases} R I +G +SP = 50   R I +T + M = 63  G +T +SP = 25   SP + M = 13    M +R I = 48  N = 1 \end{cases} $$
Find the value of L that makes  $LMT +SPR I NG = 2023$  true.

Answer: \frac{341}{40}


An isosceles trapezoid \(ABCD\) is circumscribed around a circle. The lateral sides \(AB\) and \(CD\) are tangent to the circle at points \(M\) and \(N\), respectively, and \(K\) is the midpoint of \(AD\). In what ratio does the line \(BK\) divide the segment \(MN\)?

Answer: 1:3


Inside a square, point $P$ has distances $a, b, c$ from vertices $A, B, C$ respectively. What is the area of the square? What is it in the case when $a = b = c$?

Answer: 2a^2


Given the function $f(x)=\sin \left( \omega x- \frac{\pi }{6} \right)+\sin \left( \omega x- \frac{\pi }{2} \right)$, where $0 < \omega < 3$. It is known that $f\left( \frac{\pi }{6} \right)=0$.

(1) Find $\omega$;

(2) Stretch the horizontal coordinates of each point on the graph of the function $y=f(x)$ to twice its original length (the vertical coordinates remain unchanged), then shift the resulting graph to the left by $\frac{\pi }{4}$ units to obtain the graph of the function $y=g(x)$. Find the minimum value of $g(x)$ on $\left[ -\frac{\pi }{4},\frac{3\pi }{4} \right]$.

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


Given the function \( f(x) = 4 \sin x \cdot \sin^{2}\left(\frac{\pi}{4} + \frac{x}{2}\right) + \cos 2x \):
1. Suppose \( \omega > 0 \) is a constant. If the function \( y = f(\omega x) \) is increasing on the interval \(\left[-\frac{\pi}{2}, \frac{2\pi}{3}\right]\), find the range of values for \(\omega\).
2. Let the set \( A = \left\{ x \left|\, \frac{\pi}{6} \leqslant x \leqslant \frac{2\pi}{3} \right. \right\} \) and \( B = \left\{ x \big|| f(x) - m \big| < 2 \right\} \). If \( A \cup B = B \), determine the range of the real number \( m \).

Answer: (1,4)


Given real numbers \( x_{1}, x_{2}, \cdots, x_{2011} \) satisfying
\[
\left|x_{1}\right|=99 \text{ and } \left|x_{n}\right|=\left|x_{n-1}+1\right|
\]
for \( n = 2, 3, \cdots, 2014 \), find the minimum value of \( x_{1} + x_{2} + \cdots + x_{2014} \).

Answer: -5907


Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]

Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \).

Answer: 1/16


On the plane  $S$  in a space, given are unit circle  $C$  with radius 1 and the line  $L$ . Find the volume of the solid bounded by the curved surface formed by the point  $P$  satifying the following condition  $(a),\ (b)$ . $(a)$  The point of intersection  $Q$  of the line passing through  $P$  and perpendicular to  $S$  are on the perimeter or the inside of  $C$ . $(b)$  If  $A,\ B$  are the points of intersection of the line passing through  $Q$  and pararell to  $L$ , then  $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$ .

Answer: \pi


Cara is sitting at a circular table with her five friends as shown below. How many different possible pairs of people could Cara be sitting between?

[asy]
draw(circle((0,0),1));
label("$\_$",1.5dir(0));
label("$\_$",1.5dir(60));
label("Cara",1.5dir(120));
label("$\_$",1.5dir(180));
label("$\_$",1.5dir(240));
label("$\_$",1.5dir(300));
[/asy]

Answer: 10


We have 1000 cards, numbered from 000 to 999. Additionally, we have 100 boxes, numbered from 00 to 99. A card can only be placed in a box if its number can be obtained by removing one digit from the card's number. Show that the cards can be placed into 50 boxes, but not fewer.

Answer: 50


Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?

Answer: 78


A capricious mathematician writes a book with pages numbered from  $2$  to  $400$ . The pages are to be read in the following order. Take the last unread page ( $400$ ), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be  $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$ . What is the last page to be read?

Answer: 397


For positive integers  $m$  and  $n$ , find the smalles possible value of  $|2011^m-45^n|$ .

*(Swiss Mathematical Olympiad, Final round, problem 3)*

Answer: 14


A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?

Answer: 10


How many five-digit positive integers have the product of their digits equal to 1000?

Answer: 40


The network of bus routes in the city of Lisse is arranged such that: a) each route has three stops; b) any two routes either do not share any stops or share exactly one common stop. What is the maximum number of routes that can exist in this city, given that there are a total of nine different stops?

Answer: 12


Let  $T = TNFTPP$ .   $x$  and  $y$  are nonzero real numbers such that \[18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0.\] The smallest possible value of  $\tfrac{y}{x}$  is equal to  $\tfrac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, here's the value of T$T=6$.

Answer: 7


In the polar coordinate system, the equation of curve C is $\rho^2= \frac{3}{1+2\sin^2\theta}$. Point R is at $(2\sqrt{2}, \frac{\pi}{4})$.
P is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and minimum values of the perimeter of rectangle PQRS and the polar angle of point P when these values occur.

Answer: \frac{\pi}{6}


What is the largest possible rational root of the equation  $ax^2 + bx + c = 0{}$  where  $a, b$  and  $c{}$  are positive integers that do not exceed  $100{}$ ?

Answer: -\frac{1}{99}


Compute the  $100^{\text{th}}$  smallest positive integer  $n$  that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4},  \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4},  \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] Here  $\lfloor \cdot \rfloor$  denotes the greatest integer function.

*Proposed by Michael Tang*

Answer: 6491


The parabola \(C_{1}: x^{2}=2 py\) has its focus at \(F\). The hyperbola \(C_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) has its left and right foci at \(F_{1}\) and \(F_{2}\). Point \(P\) is a common point of the two curves in the first quadrant. If points \(P\), \(F\), and \(F_{1}\) are collinear and \(C_{1}\) and \(C_{2}\) share a common tangent at \(P\), find the eccentricity of \(C_{2}\).

Answer: \sqrt{2}


Vojta began writing the number of the current school year 2019202020192020... continuously. When he wrote 2020 digits, he got bored.

How many twos did he write?

Hint: How many twos would Vojta write if he wrote only 20 digits?

Answer: 757


If $3 \in \{a, a^2 - 2a\}$, then the value of the real number $a$ is ______.

Answer: -1


Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that:
\[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]

Answer: \frac{1}{x^2}


Let \( n \) be an integer. Determine the largest possible constant \( C \) such that for all \( a_1, a_2, \ldots, a_n \geq 0 \) we have \(\left(\sum a_i\right)^{2} \geqslant C \sum_{i<j} a_i a_j\).

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


In a three-dimensional space, there is a convex polyhedron whose vertices all lie on lattice points (each coordinate of every vertex is an integer). Additionally, there are no other lattice points inside the polyhedron, on its faces, or on its edges. Prove that the number of vertices of the polyhedron does not exceed 8.

(40th Moscow Mathematical Olympiad, 1977)

Answer: 8


Triangle \( ABC \) has a right angle at \( B \). Point \( D \) lies on side \( BC \) such that \( 3 \angle BAD = \angle BAC \). Given \( AC = 2 \) and \( CD = 1 \), compute \( BD \).

Answer: \frac{3}{8}


In the triangular pyramid \(ABCD\), it is known that \(DC = 9\), \(DB = AD\), and the edge \(AC\) is perpendicular to the face \(ABD\). A sphere of radius 2 touches the face \(ABC\), the edge \(DC\), and also touches the face \(DAB\) at the intersection point of its medians. Find the volume of the pyramid.

Answer: 36


Given a positive integer $n$ that is no less than 2, find the maximum value of $\frac{\left(\sum_{i=1}^{n} i a_{i}\right)\left(\sum_{i=1}^{n} \frac{a_{i}}{i}\right)}{\left(\sum_{i=1}^{n} a_{i}\right)^{2}}$ for all non-negative sequences $a_{i} (i = 1, 2, \cdots, n)$.

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


Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.

Answer: 2 \text{ and } 3


Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value?

Answer: 0.9232


Let $O$ be the origin, the parabola $C_{1}$: $y^{2}=2px\left(p \gt 0\right)$ and the hyperbola $C_{2}$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ have a common focus $F$. The line passing through $F$ and perpendicular to the $x$-axis intersects $C_{1}$ at points $A$ and $B$, and intersects $C_{2}$ in the first quadrant at point $M$. If $\overrightarrow{OM}=m\overrightarrow{OA}+n\overrightarrow{OB}\left(m,n\in R\right)$ and $mn=\frac{1}{8}$, find the eccentricity of the hyperbola $C_{2}$.

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


Prove that for any integer \( n \), among the numbers \( n, n+1, n+2, n+3, n+4 \), there is at least one number that is relatively prime to the other four of these numbers.

Answer: True


Kelvin the frog currently sits at \((0,0)\) in the coordinate plane. If Kelvin is at \((x, y)\), either he can walk to any of \((x, y+1)\), \((x+1, y)\), or \((x+1, y+1)\), or he can jump to any of \((x, y+2)\), \((x+2, y)\) or \((x+1, y+1)\). Walking and jumping from \((x, y)\) to \((x+1, y+1)\) are considered distinct actions. Compute the number of ways Kelvin can reach \((6,8)\).

Answer: 1831830


An electric company installed a total of 402 poles along both sides of the road, with a distance of 20 meters between each adjacent pole. Later, all poles were replaced, and only 202 poles were installed. The distance between each adjacent pole after the replacement is $\qquad$ meters.

Answer: 40


Given \( x, y, z \) are 3 real numbers greater than or equal to 1, find the sum of the numerator and denominator of the simplest fraction representing the minimum value of

$$\left(\frac{\sqrt{x^{2}(y-1)^{2}+y^{2}}}{x y}+\frac{\sqrt{y^{2}(z-1)^{2}+z^{2}}}{y z}+\frac{\sqrt{z^{2}(x-1)^{2}+x^{2}}}{z x}\right)^{2}$$.

Answer: 11


Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial.

Answer: 8\sqrt{6}


$TRN$ and $HAM$ are congruent equilateral triangles. Point $T$ is the centroid of triangle $HAM$ and point $R$ lies on the ray $TA$.

What is the ratio of the areas of the parts of triangle $TRN$ that are inside and outside of triangle $HAM$?

Hint: The centroid is the intersection of the medians.

Answer: 1:5


Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\frac{\pi}{6}$, and the distance from origin to this line is $\frac{\sqrt{3}}{2}$.

(1) Find the equation of the ellipse.
(2) Suppose a line with a positive slope passes through point D($-1, 0$) and intersects the ellipse at points E and F. If $\overrightarrow{ED} = 2\overrightarrow{DF}$, find the equation of line EF.
(3) Is there a real number $k$ such that the line $y = kx + 2$ intersects the ellipse at points P and Q, and the circle with diameter PQ passes through point D($-1, 0$)? If it exists, find the value of $k$; if not, explain why.

Answer: k = \frac{7}{6}


Two numbers need to be inserted between $4$ and $16$ such that the first three numbers are in arithmetic progression and the last three numbers are in geometric progression. What is the sum of those two numbers?
A) $6\sqrt{3} + 8$
B) $10\sqrt{3} + 6$
C) $6 + 10\sqrt{3}$
D) $16\sqrt{3}$

Answer: 6\sqrt{3} + 8


A digit was removed from a five-digit number, and the resulting four-digit number was added to the original number. The sum turned out to be 54321. Find the original number.

Answer: 49383


In a right triangle $ABC$ with a right angle at $A$, a circle is constructed on the altitude $AD$ as its diameter. This circle intersects the side $AB$ at point $K$ and the side $AC$ at point $M$. The segments $AD$ and $KM$ intersect at point $L$. Find the acute angles of triangle $ABC$ given that $AK: AL = AL: AM$.

Answer: 75


In a large 15 by 20 rectangular region, one quarter area of the rectangle is shaded. If the shaded quarter region itself represents one fourth of its quarter area, calculate the fraction of the total area that is shaded.
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{4}$
D) $\frac{3}{20}$
E) $\frac{1}{5}$

Answer: \frac{1}{16}


The difference between the maximal and the minimal diagonals of the regular  $n$ -gon equals to its side (  $n > 5$  ).  Find  $n$ .

Answer: 9


Numbers  $u_{n,k} \ (1\leq k \leq n)$  are defined as follows
\[u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.\]
(the empty sum is defined to be equal to zero). Prove that  $n \mid u_{n,k}$  for every natural number  $n$  and for every  $k \ (1 \leq k \leq n).$

Answer: n\midu_{n,k}


The set of quadruples  $(a,b,c,d)$  where each of  $a,b,c,d$  is either  $0$  or  $1$  is *called vertices of the four dimensional unit cube* or *4-cube* for short. Two vertices are called *adjacent*, if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from  $(0,0,0,0)$  and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to  $(0,0,0,0)$  after  $4042$  turns? Note that it is <u>NOT</u> forbidden for the robot to pass through  $(0,0,0,0)$  before the  $4042$ -nd turn.

Answer: 2^{4041}+2^{8081}


In equality  $$ 1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023 $$  Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?

Answer: 2


Given \( f(x) = \frac{bx + 1}{2x + a} \), where \( a \) and \( b \) are constants, and \( ab \neq 2 \). If \( f(x) \cdot f\left(\frac{1}{x}\right) = k \) is a constant, then \( k \) is equal to what value?

Answer: \frac{1}{4}


A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of $40 \mathrm{~km} / \mathrm{h}$ and the minivan travels at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass between the time at which the minivan arrives in Betatown and the time at which the car arrives in Betatown?

Answer: 2


Suppose the estimated €25 billion (Euros) cost to send a person to the planet Mars is shared equally by the 300 million people in a consortium of countries. Given the exchange rate of 1 Euro = 1.2 dollars, calculate each person's share in dollars.

Answer: 100


Let \( p = 2^{24036583} - 1 \), the largest prime currently known. For how many positive integers \( c \) do the quadratics \( \pm x^2 \pm p x \pm c \) all have rational roots?

Answer: 0


In a convex quadrilateral \( ABCD \), two equal circles are inscribed such that they touch each other. The center of the first circle lies on the segment connecting vertex \( D \) and the midpoint \( E \) of side \( AB \), while the center of the second circle lies on segment \( CE \). The first circle touches sides \( AB \), \( AD \), and \( CD \), and the second circle touches sides \( AB \), \( BC \), and \( CD \). Find the sine of the angle between the diagonals of quadrilateral \( ABCD \).

Answer: \frac{4}{5}


Find all positive integers  $d$  for which there exist polynomials  $P(x)$  and  $Q(x)$  with real coefficients such that degree of  $P$  equals  $d$  and  $$ P(x)^2+1=(x^2+1)Q(x)^2. $$

Answer: d


Let the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0) \) pass through the point \( (0, \sqrt{3}) \), with eccentricity \( \frac{1}{2} \). The line \( l \) passes through the right focus \( F \) of the ellipse \( C \) and intersects the ellipse \( C \) at points \( A \) and \( B \). The projections of \( A \), \( F \), and \( B \) on the line \( x=4 \) are denoted as \( D \), \( K \), and \( E \) respectively.

1. Find the equation of the ellipse \( C \).
2. Connect \( AE \) and \( BD \). When the angle of inclination of line \( l \) changes, do lines \( AE \) and \( BD \) intersect at a fixed point? If so, find the coordinates of the fixed point and provide a proof; if not, explain why.

Answer: (\frac{5}{2},0)


The numbers 1, 9, 8, 1 are the first four terms of a sequence, where each subsequent term is equal to the last digit of the sum of the four previous terms. Can the numbers 1, 2, 3, 4 appear consecutively in this sequence?

Answer: No


In triangle \(ABC\), the side \(AC\) is longer than \(AB\). The line \(l\) is the bisector of the external angle at \(C\). A line passing through the midpoint of \(AB\) and parallel to \(l\) intersects \(AC\) at point \(E\). Find \(CE\) if \(AC = 7\) and \(CB = 4\). (The external angle of a triangle is the angle adjacent to the internal angle at a given vertex.)

Answer: \frac{11}{2}


How many positive integers \( 2 \leq a \leq 101 \) have the property that there exists a positive integer \( N \) for which the last two digits in the decimal representation of \( a^{2^{n}} \) are the same for all \( n \geq N \)?

Answer: 36


$N$  students are seated at desks in an  $m \times  n$  array, where  $m, n \ge  3$ . Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are  $1020 $ handshakes, what is  $N$ ?

Answer: 280


The absolute value of -1.2 is ____, and its reciprocal is ____.

Answer: -\frac{5}{6}


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

\[
A = \sin(x - y) + \sin(y - z) + \sin(z - x)
\]

Answer: 2


Fifteen students numbered $1, 2, 3, \ldots, 15$ stand in a circle facing inward. On the first turn, the student with number 1 turns around to face outward. On the second turn, the students numbered 2 and 3 turn around. On the third turn, the students numbered $4, 5, 6$ turn around, and so on. On the 15th turn, all students turn around. After the 12th turn, how many students are still facing outward?

Answer: 12


A parallelogram $ABCD$ has a base $\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by parallelogram $ABCD$ and triangle $ACE$.

Answer: 18 + 3\sqrt{5}


A regular octagon has sides alternating in length between 1 and $\sqrt{2}$. What is the area of the octagon?

Answer: 7


Given that the product of Kiana's age and the ages of her two older siblings is 256, and that they have distinct ages, determine the sum of their ages.

Answer: 38


As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum?

Answer: 15


Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each side of the triangle.

Answer: 300


Given two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=a$ ($a>0$), $b_1-a_1=1$, $b_2-a_2=2$, and $b_3-a_3=3$.
(1) If $a=1$, find the general formula for the sequence $\{a_n\}$.
(2) If the sequence $\{a_n\}$ is unique, find the value of $a$.

Answer: \frac{1}{3}


In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.

Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round to whole degrees. What is the maximum possible deviation of the obtained value from the original temperature in Celsius degrees?

Answer: 13/18


Find all natural numbers \( n \) such that the product of their decimal digits is \( n^2 - 10n - 22 \).

Answer: 12


Given a parabolic curve $y^{2}=2x$ with a point $P$ on it, points $B$ and $C$ are on the y-axis. A circle described by $(x-1)^{2}+y^{2}=1$ is inscribed within the triangle $PBC$. Find the minimum area of triangle $PBC$.

Answer: 8


Let  $ P(x)$  be a nonzero polynomial such that, for all real numbers  $ x$ ,  $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$ . Determine the maximum possible number of real roots of  $ P(x)$ .

Answer: 4


Alice's favorite number has the following properties:

- It has 8 distinct digits.
- The digits are decreasing when read from left to right.
- It is divisible by 180.

What is Alice's favorite number?

*Author: Anderson Wang*

Answer: 97654320


What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square) if:

1) No pawn can be placed on the e4 square;
2) No two pawns can be placed on squares that are symmetric with respect to the e4 square?

Answer: 39


The number \(abcde\) has five distinct digits, each different from zero. When this number is multiplied by 4, the result is a five-digit number \(edcba\), which is the reverse of \(abcde\). What is the value of \(a + b + c + d + e\)?

Answer: 27


Given a trapezoid \( MNPQ \) with bases \( MQ \) and \( NP \). A line parallel to the bases intersects the lateral side \( MN \) at point \( A \), and the lateral side \( PQ \) at point \( B \). The ratio of the areas of the trapezoids \( ANPB \) and \( MABQ \) is \( \frac{2}{7} \). Find \( AB \) if \( NP = 4 \) and \( MQ = 6 \).

Answer: \frac{2\sqrt{46}}{3}


The roots of the quadratic equation \( ax^2 + bx + c = 0 \), which has integer coefficients, are distinct and both lie between 0 and 1. Show that if \( a \) is positive, it is at least 5.

Answer: 5


How many pairs of integers  $a$  and  $b$  are there such that  $a$  and  $b$  are between  $1$  and  $42$  and  $a^9 = b^7 \mod 43$ ?

Answer: 42


Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:

$\bullet$  Carolyn always has the first turn.

$\bullet$  Carolyn and Paul alternate turns.

$\bullet$  On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.

$\bullet$  On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed, and he can optionally choose one additional number that is a multiple of any divisor he is removing.

$\bullet$  If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.

For example, if $n=8,$ a possible sequence of moves could be considered.

Suppose that $n=8$ and Carolyn removes the integer $4$ on her first turn. Determine the sum of the numbers that Carolyn removes.

Answer: 12


The cafe "Buratino" operates 6 days a week and is closed on Mondays. Kolya made two statements: "from April 1 to April 20, the cafe was open for 18 days" and "from April 10 to April 30, the cafe was also open for 18 days." It is known that he made a mistake once. How many days was the cafe open from April 1 to April 27?

Answer: 23


The price of the jacket was increased and then decreased by a certain percent, and then a 10% discount was applied, resulting in a final price that is 75% of the original price. Determine the percent by which the price was increased and then decreased.

Answer: 40.82


Given complex numbers \( z_{1} \) and \( z_{2} \) such that \( \left|z_{1}\right| = 2 \) and \( \left|z_{2}\right| = 3 \). If the angle between their corresponding vectors is \( 60^{\circ} \), then find the value of \( \left|\frac{z_{1} + z_{2}}{z_{1} - z_{2}}\right| \).

Answer: \sqrt{\frac{19}{7}}


Answer: 900


Find all continuous solutions of Cauchy's equation on \(\mathbb{R}\).

Answer: f(x)=ax


In a $4 \times 4$ grid, place 4 pieces such that there is exactly one piece in each row, each column, and each diagonal. How many ways are there to do this?

Answer: 8


In a certain high school, there are 300 freshmen students, including 180 boys and 120 girls. In order to understand the height information of the freshmen students, a stratified random sampling method is used to select samples according to the proportion of the sample size. It is found that the average height of the boys' sample is $\overline{x}=170$ (unit: $cm$) with a variance of ${s}_{1}^{2}=14$, and the average height of the girls' sample is $\overline{y}=160$ (unit: $cm$) with a variance of ${s}_{2}^{2}=24$. Based on the above data, estimate the average height and variance of the freshmen students in this high school.

Answer: 42


For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $c$ denote the number of positive integers $n \leq 1000$ with $S(n)$ odd, and let $d$ denote the number of positive integers $n \leq 1000$ with $S(n)$ even. Find $|c-d|.$

Answer: 33


If three numbers are chosen simultaneously from the integers $1,2,\dots,14$ in such a way that the absolute value of the difference between any two numbers is not less than 3, how many different ways can this be done?

Answer: 120


Find the possible value of \(x+y\) if it is known that \(x^{3}-6x^{2}+15x=12\) and \(y^{3}-6y^{2}+15y=16\).

Answer: 4


On a  $3 \times 3$  board the numbers from  $1$  to  $9$  are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.

Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.

Answer: 720


Given \(a > 0\), the function \(f(x) = x^3 - a x\) is a monotonic function on \([1, +\infty)\).
1. Find the range of values for \(a\).
2. If \(x_{0} \geq 1\) and satisfies \(f(x_{0}) \geq 1\) and \(f(f(x_{0})) = x_{0}\), prove that \(f(x_{0}) = x_{0}\).

Answer: f(x_0)=x_0


A hostess baked a pie for guests. At the table, there can be either \( p \) people or \( q \) people, where \( p \) and \( q \) are relatively prime. Into what minimum number of pieces (not necessarily equal) should the pie be cut in advance so that it can be evenly distributed in either case?

Answer: p+1


In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $m=(\sin \frac{A}{2},\cos \frac{A}{2})$ and $n=(\cos \frac{A}{2},-\cos \frac{A}{2})$, and $2m\cdot n+|m|=\frac{ \sqrt{2}}{2}$, find $\angle A=$____.

Answer: \frac{5\pi }{12}


The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)?

Answer: 30


A state issues car license plates consisting of 6 digits (each digit ranging from $0$ to $9$), with the condition that any two license plate numbers must differ in at least two places. (For example, license numbers 027592 and 020592 cannot both be used). Determine the maximum number of license plate numbers possible under this condition. Provide a proof.

Answer: 100000


Given that the eccentricity of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\frac{\sqrt{2}}{2}$, and it passes through the point $P(\sqrt{2},1)$. The line $y=\frac{\sqrt{2}}{2}x+m$ intersects the ellipse at two points $A$ and $B$.
(1) Find the equation of the ellipse $C$;   (2) Find the maximum area of $\triangle PAB$.

Answer: \sqrt{2}


Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 43


(1) For which positive integers \( n \) does there exist a finite set of \( n \) distinct positive integers \( S_n \) such that the geometric mean of any subset of \( S_n \) is an integer?

(2) Is there an infinite set of distinct positive integers \( S \) such that the geometric mean of any finite subset of \( S \) is an integer?

Answer: No


Six points are placed inside a rectangle of dimensions $4 \times 3$. Show that among these points, there are two whose distance is less than or equal to $\sqrt{5}$.

Answer: \sqrt{5}


$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF} = \stackrel{\frown}{CD},\ \stackrel{\frown}{AE} = \stackrel{\frown}{BD}$. The length of $BD$ can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

Answer: 14


Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different shape or all three of the card have the same shape.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
How many different complementary three-card sets are there?

Answer: 117


Find the smallest number in which all digits are different and the sum of all digits equals 32.

Answer: 26789


A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$

Answer: 373


You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits.

Answer: 11.125


Given $a$, $b$, $c \in \{1, 2, 3, 4, 5, 6\}$, if the lengths $a$, $b$, and $c$ can form an isosceles (including equilateral) triangle, then there are \_\_\_\_\_\_ such triangles.

Answer: 27