Given the function
\[ f(x) = a \cos x + b \cos 2x + c \cos 3x, \]

and \( f(x) \geq -1 \) always holds true. Find the maximum value of \( a - b + c \).

Answer: 1


A fishing boat fishes in the territorial waters of a foreign country without permission. Each time the net is cast, it causes an equivalent-valued loss in the country's fish resources. The probability that the boat will be detained by the foreign coast guard during any net casting is $1/k$, where $k$ is a natural number specific to the country. Assume that the events of being detained or not during each net casting are independent of previous fishing activities. If the boat is detained by the coast guard, all previously caught fish are confiscated, and future fishing in those waters is prohibited. The captain plans to leave the foreign territorial waters after casting the net $n$ times.

Given that there is always a possibility of being detained by the foreign coast guard, making the fishing profit a random variable, determine the value of $n$ that maximizes the expected value of the fishing profit.

Answer: k-1


Given the function $f(x)= \frac {1}{3}x^{3}+x^{2}+ax+1$, and the slope of the tangent line to the curve $y=f(x)$ at the point $(0,1)$ is $-3$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Find the extrema of $f(x)$.

Answer: -\frac{2}{3}


As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Emerson?
$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16$

Answer: \textbf{(D)}\15


Read the following material and complete the corresponding tasks.

平衡多项式
定义:对于一组多项式$x+a$,$x+b$,$x+c$,$x+d(a$,$b$,$c$,$d$是常数）,当其中两个多项式的乘积与另外两个多项式乘积的差是一个常数$p$时,称这样的四个多项式是一组平衡多项式,$p$的绝对值是这组平衡多项式的平衡因子.
例如:对于多项式$x+1$,$x+2$,$x+5$,$x+6$,因为$(x+1)(x+6)-(x+2)(x+5)=(x^{2}+7x+6)-(x^{2}+7x+10)=-4$,所以多项式$x+1$,$x+2$,$x+5$,$x+6$是一组平衡多项式,其平衡因子为$|-4|=4$.

任务:
$(1)$小明发现多项式$x+3$,$x+4$,$x+6$,$x+7$是一组平衡多项式,在求其平衡因子时,列式如下:$(x+3)(x+7)-(x+4)(x+6)$,根据他的思路求该组平衡多项式的平衡因子；
$A$.判断多项式$x-1$,$x-2$,$x-4$,$x-5$是否为一组平衡多项式,若是,求出其平衡因子；若不是,说明理由.
$B$.若多项式$x+2$,$x-4$,$x+1$,$x+m(m$是常数）是一组平衡多项式,求$m$的值.

Answer: -5


A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite.

Answer: 216


In an NBA basketball competition, the teams Warriors, Nuggets, Jazz, Trail Blazers, and Rockets won the top five spots. Five individuals, A, B, C, D, and E, made guesses about the specific rankings:

A: 1st place is Warriors, 3rd place is Nuggets;
B: 3rd place is Warriors, 5th place is Jazz;
C: 4th place is Rockets, 2nd place is Warriors;
D: 2nd place is Nuggets, 5th place is Trail Blazers;
E: 3rd place is Trail Blazers, 5th place is Rockets.

Each person guessed half correctly, and each place was guessed correctly by someone. What place did the Trail Blazers secure?

Answer: 3


On each of the lines \( x=0 \) and \( x=2 \), there are 62 points marked with ordinate values \( 1, 2, 3, \ldots, 62 \). In how many ways can you choose three points from the marked 124 such that they form the vertices of a right triangle?

Answer: 7908


A circle touches the sides of an angle at points $A$ and $B$. The distance from point $C$, which lies on the circle, to the line segment $AB$ is 8. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 30 less than the other.

Answer: 34


Zan has created this iterative rule for generating sequences of whole numbers:

1) If a number is 25 or less, double the number.

2) If a number is greater than 25, subtract 12 from it.

Let $F$ be the first number in a sequence generated by the rule above. $F$ is a "sweet number" if 16 is not a term in the sequence that starts with $F$. How many of the whole numbers 1 through 50 are "sweet numbers"?

Answer: 16


A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of the circle.

Answer: 4:1


There are 999 numbers placed on a circle, each equal to either 1 or -1, and not all numbers are the same. Consider all products of 10 consecutive numbers and sum them up.

a) What is the minimum sum that can be obtained?

b) What is the maximum sum that can be obtained?

Answer: 995


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

Answer: 10


The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is

Answer: 10.5


Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\frac{1}{3}$, find the eccentricity of the hyperbola $\frac{y^2}{m^2}-\frac{x^2}{n^2}=1$.

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


Place four different balls - red, black, blue, and yellow - into three different boxes, with at least one ball in each box. The red and blue balls cannot be in the same box. How many different arrangements are there?

Answer: 30


A bus ticket number is a six-digit number. If the sum of the digits of the first three numbers equals the sum of the digits of the last three numbers, the ticket is called "lucky." Prove that the sum of all lucky ticket numbers is divisible by 13.
(5th All-Russian Mathematical Olympiad, 1965)

Answer: 13


Ivan has received $3n$ guests for his birthday. He has $3n$ top hats with the letters A, B, and C written on them, with exactly $n$ hats of each type. Ivan wants to organize a ball where guests wear the hats and form one or more circles such that the length of each circle is divisible by 3, and when viewed from above, each circle reads A B C A B C...A B C in a clockwise direction. Prove that Ivan can organize the ball in exactly $(3n)!$ different ways. (Hats with the same letter are indistinguishable; all guests are unique.)

Answer: (3n)!


Let the set \( M = \{1, 2, 3, \cdots, 2002\} \). For any non-empty subset \( A \) of \( M \), define \( x_{A} \) as the sum of the maximum and minimum numbers in \( A \). Find the arithmetic mean of all such \( x_{A} \).

Answer: 2003


The sides of an \( N \)-sided regular polygon are extended to form a "star". If the angle at each point of that "star" is \( 108^{\circ} \), find \( N \).

Answer: 10


In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.

Answer: 90


Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\triangle GEM$.

Answer: 25


A cylinder with a base diameter of 12 is cut by a plane that forms a $30^{\circ}$ angle with the base, forming an ellipse $S$. What is the eccentricity of the ellipse $S$?

Answer: \frac{1}{2}


Solve the inequality

$$
\sqrt{10x-21} - \sqrt{5x^2 - 21x + 21} \geqslant 5x^2 - 31x + 42
$$

In the answer, provide the sum of all integer values of \( x \) that satisfy the inequality.

Answer: 7


Given that a rancher bought 600 sheep at an unknown cost price, then sold 550 sheep for the cost price of 600 sheep, and the remaining 50 sheep at the same price per head as the first 550, calculate the percent gain on the entire transaction.

Answer: 9.09\%


In a club with 30 members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of 10 members such that no one in the group has received a hat from another one in the group.

Answer: 10


How many consecutive tickets must be purchased at a bus cashier to be sure to get a lucky one? A ticket is called lucky if the sum of its first three digits equals the sum of its last three digits. The number of tickets at the cashier is unlimited.

Answer: 1001


Given \( a > 0 \), a line passing through the point \( M(a, 0) \) intersects the parabola \( y^2 = 2px \) \( (p > 0) \) at points \( P \) and \( Q \). If \(\frac{1}{|MP|^2} + \frac{1}{|MQ|^2}\) is constant, find the value of \( a \).

Answer: p


Find the total number of different integer values the function
$$
f(x)=[x]+[2x]+\left[\frac{5x}{3}\right]+[3x]+[4x]
$$
takes for real numbers $x$ with \(0 \leq x \leq 100\).

Note: \([t]\) is the largest integer that does not exceed \(t\).

Answer: 734


Given the sequence \( x, y, z, w \) satisfying \( xy + 2yz + zw \leq A(x^2 + y^2 + z^2 + w^2) \), find the minimum value of the real number \( A \).

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


Answer: 51


Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$?

Answer: 248


There are two ingots of different alloys of copper and tin weighing 6 kg and 12 kg, respectively. Identical pieces were cut from each of them, and the first piece was fused with the remnants of the second ingot, while the second piece was fused with the remnants of the first ingot. As a result, the ratio of copper to tin in the two new ingots was the same. Find the weight of each of the cut pieces.

Answer: 4


According to the latest revision of the "Regulations on the Application and Use of Motor Vehicle Driving Licenses" by the Ministry of Public Security: each driving license applicant must pass the "Subject One" (theoretical subject) and "Comprehensive Subject" (combination of driving skills and some theoretical knowledge of "Subject One") exams. It is known that Mr. Li has passed the "Subject One" exam, and the score of "Subject One" is not affected by the "Comprehensive Subject". The "Comprehensive Subject" offers 5 chances to take the exam within three years. Once an exam is passed, the driving license will be issued, and the applicant will no longer participate in subsequent exams; otherwise, the exams will continue until the 5th attempt. The probabilities of Mr. Li passing the "Comprehensive Subject" in each attempt are 0.5, 0.6, 0.7, 0.8, and 0.9, respectively.
(1) Calculate the distribution and mathematical expectation of the number of times ξ that Mr. Li takes the driving license test within three years.
(2) Calculate the probability of Mr. Li obtaining a driving license within three years.

Answer: 0.9988


Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and  places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and Julián do the same, emptying the ballot box. Pedro informs the teacher that he has picked three consecutive numbers whose product is $5$
times the sum. Ana informs that she has no prime number, but two consecutive and that the product of these three numbers is $4$ times the sum of them. What numbers did Julian remove?

Answer: 2, 6, 7


A natural number \( N \) is divisible by 2020. In its decimal representation, all digits are different, and if any two of them are swapped, the resulting number is not divisible by 2020. For what number of digits in the decimal representation of the number \( N \) is this possible?

Answer: 6


Let  $a \geq 0$  be a natural number. Determine all rational  $x$ , so that
\[\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}\]
All occurring square roots, are not negative.**Note.** It seems the set of natural numbers =  $\mathbb N = \{0,1,2,\ldots\}$  in this problem.

Answer: 0,1


At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time?

Answer: 9:36


Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \).

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


The number \( N \) has the smallest positive divisor 1, the second largest positive divisor \( k \), and the third largest positive divisor \( m \). Moreover, \( k^k + m^m = N \). What is \( N \)?

Answer: 260


A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen if Biff and Jacob must serve together or not at all, and Alice and Jane refuse to serve with each other?

Answer: 41


Let $f(x)$ be a function defined on $\mathbb{R}$ with a minimum positive period of $3\pi$, and its expression in the interval $(-\pi,2\pi]$ is $f(x)= \begin{cases} \sin x & (0\leqslant x\leqslant 2\pi) \\ \cos x & (-\pi < x < 0) \end{cases}$. Evaluate the expression $f(- \frac {308\pi}{3})+f( \frac {601\pi}{6})$.

Answer: -1


Enter all the solutions to
\[ \sqrt{4x-3}+\frac{10}{\sqrt{4x-3}}=7,\]separated by commas.

Answer: \frac 74,7


Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2$=
$\text{(A) } 1369\quad \text{(B) } 1679\quad \text{(C) } 1748\quad \text{(D) } 2109\quad \text{(E) } 8825$

Answer: 1679


Some natural numbers are marked. It is known that on every segment of the number line of length 1999, there is a marked number.

Prove that there exists a pair of marked numbers, one of which is divisible by the other.

Answer: 1


Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Report the sum $p+q$.

Answer: 101


Given a convex $n$-gon with no two sides being parallel, prove that the number of distinct triangles mentioned in problem $\underline{22.8}$ is at least $n-2$.

Answer: n-2


The numbers \( a, b, c, d \) belong to the interval \([-5, 5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \).

Answer: 110


All natural numbers from 1 to $N$ (where $N \geq 2$) are written in some order in a circle. For any pair of neighboring numbers, there is at least one digit that appears in the decimal representation of both. Find the smallest possible value of $N$.

Answer: 29


Let the quadratic function \( f(x) = ax^2 + bx + c \) (where \(a, b, c \in \mathbf{R}\) and \(a \neq 0\) ) satisfy the following conditions:
1. For \( x \in \mathbf{R} \), \( f(x-4) = f(2-x) \), and \( f(x) \geq x \);
2. For \( x \in (0, 2) \), \( f(x) \leq \left( \frac{x+1}{2} \right)^2 \);
3. The minimum value of \( f(x) \) on \( \mathbf{R} \) is 0.

Find the largest \( m \) (where \( m > 1 \)) such that there exists \( t \in \mathbf{R} \) satisfying \( f(x+t) \leq x \) for all \( x \in [1, m] \).

Answer: 9


Let the function \( f(x) \) be defined on \(\mathbf{R}\). For any \( x \in \mathbf{R} \), the equations \( f(1+4x)=f(3-2x) \) and \( f(2+3x)=-f(7-6x) \) hold. Find the value of \(\sum_{k=1}^{100}[f(2k-1)+f(4k-2)]\).

Answer: 0


Let \( x, y, z \) be positive numbers satisfying the system of equations:

\[ \left\{\begin{array}{l}
x^{2}+xy+y^{2}=75 \\
y^{2}+yz+z^{2}=4 \\
z^{2}+xz+x^{2}=79
\end{array}\right. \]

Find the value of the expression \( xy + yz + xz \).

Answer: 20


A certain school arranges $5$ senior high school teachers to visit $3$ schools for exchange and learning. One school has $1$ teacher, one school has $2$ teachers, and one school has $2$ teachers. The total number of different arrangements is _______.

Answer: 90


Given the function $f(x)=\sin(\omega x+\varphi)$, which is monotonically increasing on the interval ($\frac{\pi}{6}$,$\frac{{2\pi}}{3}$), and the lines $x=\frac{\pi}{6}$ and $x=\frac{{2\pi}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, find $f(-\frac{{5\pi}}{{12}})$.

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


A certain store sells a batch of helmets for $80 each. It can sell 200 helmets per month. During the "Creating a Civilized City" period, the store plans to reduce the price of the helmets for sale. After investigation, it was found that for every $1 decrease in price, an additional 20 helmets are sold per month. It is known that the cost price of the helmets is $50 each.
$(1)$ If the price of each helmet is reduced by $10, the store can sell ______ helmets per month, and the monthly profit from sales is ______ dollars.
$(2)$ If the store plans to reduce the price of these helmets to reduce inventory while ensuring a monthly profit of $7500, find the selling price of the helmets.

Answer: 65


What is the minimum number of planes determined by  $6$  points in space which are not all coplanar, and among which no three are collinear?

Answer: 11


A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

Answer: 7


A large batch of tires contains $1.5\%$ defects. What should be the sample size for the probability of finding at least one defective tire in the sample to be more than $0.92 ?$

Answer: 168


Answer: 250


Given \(0 \le x_0 < 1\), let
\[x_n = \left\{ \begin{array}{ll}
3x_{n-1} & \text{if } 3x_{n-1} < 1 \\
3x_{n-1} - 1 & \text{if } 1 \le 3x_{n-1} < 2 \\
3x_{n-1} - 2 & \text{if } 3x_{n-1} \ge 2
\end{array}\right.\]
for all integers \(n > 0\), determine the number of values of \(x_0\) for which \(x_0 = x_6\).

Answer: 729


In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes

Answer: 265 - 132\sqrt{3}


Answer: 32


O is a point inside the convex quadrilateral \(ABCD\). The line through O parallel to \(AB\) meets the side \(BC\) at \(L\) and the line through O parallel to \(BC\) meets the side \(AB\) at \(K\). The line through O parallel to \(AD\) meets the side \(CD\) at \(M\) and the line through O parallel to \(CD\) meets the side \(DA\) at \(N\). The area of \(ABCD\) is \(k\), the area of \(AKON\) is \(k_1\), and the area of \(LOMC\) is \(k_2\). Show that \((k_1)^{1/2} \geq (k_1)^{1/2} + (k_2)^{1/2}\).

Answer: \sqrt{k}\ge\sqrt{k_1}+\sqrt{k_2}


Let  $ AB$  be the diameter of a circle with a center  $ O$  and radius  $ 1$ . Let  $ C$  and  $ D$  be two points on the circle such that  $ AC$  and  $ BD$  intersect at a point  $ Q$  situated inside of the circle, and  $ \angle AQB\equal{} 2 \angle COD$ . Let  $ P$  be a point that intersects the tangents to the circle that pass through the points  $ C$  and  $ D$ .

Determine the length of segment  $ OP$ .

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


For a polygon \(A_{1} A_{2} \ldots A_{n}\), the area of the polygon formed by the points dividing the sides \(A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n} A_{1}\) in the same ratio \(\lambda\) is minimized at what value of \(\lambda\)?

Answer: \frac{1}{2}


Inside a cube \( ABCD-A_1B_1C_1D_1 \) with side length 1, there is a small sphere that is tangent to the diagonal \( AC_1 \) of the cube. Determine the maximum radius of the sphere.

Answer: \frac{4-\sqrt{6}}{5}


Let \(a, b,\) and \(c\) denote the sides of a triangle, and let \(S\) be the area of the same triangle. Prove that

$$
a^{2}+b^{2}+c^{2} \geq 4 S \sqrt{3}
$$

In what case does equality hold?

Answer: ^{2}+b^{2}+^{2}\ge4S\sqrt{3}


The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begin with either G or K, end with T, cannot contain S, and have no letters that repeat?

Answer: 1008


Compute the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{2}+5\right)\left(n^{4}+2\right)}-\sqrt{n^{6}-3 n^{3}+5}}{n}
$$

Answer: \frac{5}{2}


Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a,\frac{b}{a},1\}=\{a^{2},a+b,0\}$, determine the value of $a^{2023}+b^{2024}$.

Answer: -1


Find the ordered pair $(a,b)$ of integers such that
\[\sqrt{9 - 8 \sin 50^\circ} = a + b \csc 50^\circ.\]

Answer: (3,-1)


The integer sequence \(a_1, a_2, a_3, \dots\) is defined as follows: \(a_1 = 1\). For \(n \geq 1\), \(a_{n+1}\) is the smallest integer greater than \(a_n\) such that for all \(i, j, k \in \{1, 2, \dots, n+1\}\), the condition \(a_i + a_j \neq 3a_k\) is satisfied. Find the value of \(a_{22006}\).

Answer: 66016


Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.

Answer: (1, 3, 2, 1)


Given a set of sample data with $8$ numbers, the average is $8$, and the variance is $12$. Two unknown numbers are added to this set of sample data to form a new set of sample data. It is known that the average of the new sample data is $9$. Find the minimum value of the variance of the new sample data.

Answer: 13.6


It is known that for certain natural numbers \(a, b\), the number \(N=\frac{a^{2}+b^{2}}{a b-1}\) is also a natural number. Find all possible values of \(N\).

Answer: 5


Find the minimum value of the expression:
\[(15 - x)(8 - x)(15 + x)(8 + x).\]

Answer: -6480.25


The nonzero complex numbers \( z_{i} (i=1, 2, 3, 4, 5) \) have equal magnitudes, and \( \sum_{i=1}^{5} z_{i} = \sum_{i=1}^{5} z_{i}^{2} = 0 \). Prove that \( z_{1}, z_{2}, z_{3}, z_{4}, z_{5} \) are the vertices of a regular pentagon in the complex plane.

Answer: z_1,z_2,z_3,z_4,z_5


In a volleyball tournament, every team played against every other team exactly once. We know that three teams "beat each other in a cycle," meaning one team defeated the second, the second defeated the third, and the third defeated the first. Prove that there were two teams that had the same number of victories.

Answer: 2


Find all positive integers  $n$  that satisfy the following inequalities:  $$ -46 \leq \frac{2023}{46-n} \leq 46-n $$

Answer: 90


A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\%$ chance of getting a small rabbit toy.

$(1)$ Individuals Jia, Yi, and Bing each buy $1$ set of blind box set $B$. Let the random variable $\xi$ represent the number of small rabbit toys among them. Find the distribution table and mathematical expectation of $\xi$.

$(2)$ A consumer bought $1$ set of blind box set $A$ and $1$ set of blind box set $B$ on the first and second days of sale, respectively. After mixing all $6$ items together and randomly selecting $1$ to open, a small rabbit toy was found. Find the probability that this small rabbit toy came from blind box set $B$.

Answer: \frac{1}{3}


In $\triangle ABC, \overline{CA} = \overline{CB}$. On $CB$ square $BCDE$ is constructed away from the triangle. If $x$ is the number of degrees in $\angle DAB$, then
$\textbf{(A)}\ x\text{ depends upon }\triangle ABC \qquad \textbf{(B)}\ x\text{ is independent of the triangle} \\ \textbf{(C)}\ x\text{ may equal }\angle CAD \qquad \\ \textbf{(D)}\ x\text{ can never equal }\angle CAB \\ \textbf{(E)}\ x\text{ is greater than }45^{\circ}\text{ but less than }90^{\circ}$

Answer: \textbf{B}


Celine has 240 feet of fencing. She needs to enclose a rectangular area such that the area is eight times the perimeter of the rectangle. If she uses up all her fencing material, how many feet is the largest side of the enclosure?

Answer: 101


In trapezoid \(ABCD\), points \(K\) and \(M\) are the midpoints of the bases \(AB\) and \(CD\), respectively. It is known that \(AM\) is perpendicular to \(DK\) and \(CK\) is perpendicular to \(BM\), and the angle \(CKD\) is equal to \(60^\circ\). Find the area of the trapezoid if its height is 1.

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


Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Answer: 11


Compute the sum:
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \le a < b < c.$)

Answer: \frac{1}{21216}


22. Let the function  $f : \mathbb{Z} \to \mathbb{Z}$  take only integer inputs and have integer outputs. For any integers  $x$  and  $y$ , f satisfies  $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$  If  $f(2016) = 6102$  and  $f(6102) = 2016$ , what is  $f(1)$ ?

23. Let  $d$  be a randomly chosen divisor of  $2016$ . Find the expected value of  $$ \frac{d^2}{d^2 + 2016} $$  24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of  $N$  colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of  $N$ ?

Answer: 8117


For a sequence  $(a_{n})_{n\geq 1}$  of real numbers it is known that  $a_{n}=a_{n-1}+a_{n+2}$  for  $n\geq 2$ .
What is the largest number of its consecutive elements that can all be positive?

Answer: 5


Given that \(w\) and \(z\) are complex numbers such that \(|w+z|=1\) and \(\left|w^{2}+z^{2}\right|=14\), find the smallest possible value of \(\left|w^{3}+z^{3}\right|\). Here, \(|\cdot|\) denotes the absolute value of a complex number, given by \(|a+bi|=\sqrt{a^{2}+b^{2}}\) whenever \(a\) and \(b\) are real numbers.

Answer: \frac{41}{2}


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

\[ f(f(x) + x f(y)) = x + f(x) y \]

where \(\mathbb{Q}\) is the set of rational numbers.

Answer: f(x)=x


A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?

[asy]
draw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));
draw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cycle,linewidth(1));

dot((0,0));
dot((2,0));
dot((4,0));
dot((1,sqrt(3)));
dot((3,sqrt(3)));
dot((2,2sqrt(3)));

label("Row 2",(-1,1));
label("Row 1",(0,2.5));

draw((3.5,2sqrt(3))--(2.2,2sqrt(3)),Arrow);
draw((4,2.5)--(2.8,2.5),Arrow);

label("connector",(5,2sqrt(3)));
label("unit rod",(5.5,2.5));
[/asy]

Answer: 153


A full can of milk weighs 34 kg, and a half-filled can weighs 17.5 kg. How much does the empty can weigh?

Answer: 1\,


On a board, the numbers $2,3,5, \ldots, 2003,2011,2017$ are written, i.e., all prime numbers not exceeding 2020. In one operation, you can replace two numbers $a, b$ with the maximum prime number not exceeding $\sqrt{a^2 - ab + b^2}$. After several operations, one number remains on the board. What is the maximum value it can take?

Answer: 2011


The diagram shows a shaded semicircle of diameter 4, from which a smaller semicircle has been removed. The two semicircles touch at exactly three points. What fraction of the larger semicircle is shaded?

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


Let \(\mathcal{P}\) be a convex polygon. Prove that there is a convex hexagon that is contained in \(\mathcal{P}\) and that occupies at least 75 percent of the area of \(\mathcal{P}\).

Answer: 75


Determine the number of juggling sequences of length \( n \) with exactly 1 ball.

Answer: 2^n-1


How many different positions can appear on a chessboard if both players, starting from the initial position, make just one move each?

Answer: 400


Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2 &=& c^2+d^2 &=& 2016, \\
ac &=& bd &=& 1024.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$.

Answer: 127


For a positive number  $n$ ,  we write  $d (n)$  for the number of positive divisors of  $n$ .
Determine all positive integers  $k$  for which exist positive integers  $a$  and  $b$   with the property  $k = d (a) = d (b) = d (2a + 3b)$ .

Answer: allevenpositiveintegers


An equilateral triangle has a side length of 20. If a square has the same perimeter as this triangle, what is the area of the square?
(A) 25
(B) 400
(C) 225
(D) 60
(E) 100

Answer: 225


Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'?

Answer: \frac{11}{36}


An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always at least two connected by a bridge. What is the maximum possible value of $N$?

Answer: 36


In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle?

Answer: 1 : \sqrt{2}


Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$,
$$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$

Answer: \frac{n-1 + \sqrt{n-1}}{\sqrt{n}}


Among all proper fractions whose numerator and denominator are two-digit numbers, find the smallest fraction that is greater than \(\frac{4}{9}\). Provide the numerator of this fraction in your answer.

Answer: 41


Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$.
 ~ pi_is_3.14

Answer: 4


Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of  $5$  random keystrokes; at each stroke, each key is equally likely to be pressed.  The calculator then evaluates the entire expression, yielding a result of  $E$ .  Find the expected value of  $E$ .
(Note: Negative numbers are permitted, so 13-22 gives  $E = -9$ . Any excess operators are parsed as signs, so -2-+3 gives  $E=-5$  and -+-31 gives  $E = 31$ .  Trailing operators are discarded, so 2++-+ gives  $E=2$ .  A string consisting only of operators, such as -++-+, gives  $E=0$ .)

*Proposed by Lewis Chen*

Answer: 1866


Determine the number of zeros in the quotient $Q = R_{30}/R_6$, where $R_k$ is a number consisting of $k$ repeated digits of 1 in base-ten.

Answer: 25


Let \( S \) be the unit circle on the complex plane (i.e., the set of complex numbers with modulus equal to 1), and \( f \) be a mapping from \( S \) to \( S \). For any \( z \in S \), define

If \( c \in S \) and a natural number \( n \) such that
$$
f^{(1)}(c) \neq c, f^{(2)}(c) \neq c, \cdots, f^{(n-1)}(c) \neq c, f^{(n)}(c)=c
$$

we say \( c \) is an \( n \)-periodic point of \( f \).
Let \( m \) be a natural number greater than 1, and \( f \) is defined as follows:
$$
f(z)=z^{m}, z \in S
$$

Calculate the total number of 1989-periodic points of \( f \).

Answer: ^{1989}-^{663}-^{153}-^{117}+^{51}+^{39}+^{9}-^{3}


Draw the circle tangent to side \(AB\) at \(B\) and passing through \(C\), the circle tangent to side \(BC\) at \(C\) and passing through \(A\), and the circle tangent to side \(CA\) at \(A\) and passing through \(B\) for triangle \(ABC\). Prove that the three circles pass through a common point.

Answer: P


Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the numbers on the three faces adjacent to that vertex. If the sum of the numbers on the vertices is $1512$, and the sum of the numbers on one pair of opposite faces is $8$, what is the sum of the numbers on all the faces?

Answer: 38


Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps.

Answer: 91


Given the function \( f(x)=\{\begin{array}{ll}x+\frac{1}{2} & 0 \leqslant x \leqslant \frac{1}{2}, \\ 2(1-x) & \frac{1}{2}<x \leqslant 1,\end{array} \), define \( f_{n}(x)=\underbrace{f(f(\cdots f}_{n \uparrow 1}(x) \cdots)), n \in \mathbf{N}^{*} \). Find the value of \( f_{2006}\left(\frac{2}{15}\right) \).

Answer: \frac{19}{30}


A father wants to distribute his property among his children: first giving 1000 yuan and $\frac{1}{10}$ of the remaining property to the eldest child, then giving 2000 yuan and $\frac{1}{10}$ of the remaining property to the second child, and then giving 3000 yuan and $\frac{1}{10}$ of the remaining property to the third child, and so on. It turns out that each child receives the same amount of property. The father has $\qquad$ children.

Answer: 9


Given the set \( S = \{1, 2, \cdots, 1990\} \). A subset of 31 elements from \( S \) is called a good subset if the sum of its elements is divisible by 5. Find the number of good subsets of \( S \).

Answer: \frac{\binom{1990}{31}}{5}


Given \( a > 0 \) and \( a \neq 1 \), and the function \( f(x) = \log_a (6ax^2 - 2x + 3) \) is monotonically increasing on the interval \([ \frac{3}{2}, 2 ]\), determine the range of values for \( a \).

Answer: (\frac{1}{24},\frac{1}{12}]\cup(1,+\infty)


The altitudes of the acute-angled triangle $ABC$ intersect at point $O$. A circle with radius $R$ and centered at point $O$ passes through vertex $B$, touches side $AC$, and intersects side $AB$ at point $K$ such that $BK: AK = 5:1$.

Find the length of side $BC$.

Answer: 3R


Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24, q(0)=30$, and $p(q(x))=q(p(x))$ for all real numbers $x$. Find the ordered pair $(p(3), q(6))$.

Answer: (3,-24)


Two players, A and B, take turns shooting baskets. The probability of A making a basket on each shot is $\frac{1}{2}$, while the probability of B making a basket is $\frac{1}{3}$. The rules are as follows: A goes first, and if A makes a basket, A continues to shoot; otherwise, B shoots. If B makes a basket, B continues to shoot; otherwise, A shoots. They continue to shoot according to these rules. What is the probability that the fifth shot is taken by player A?

Answer: \frac{247}{432}


Given real numbers \( x, y \in (1,+\infty) \) such that \( xy - 2x - y + 1 = 0 \), find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \).

Answer: 15


Two places A and B are 66 kilometers apart. Person A and Person C start walking from A to B, and Person B starts walking from B to A. Person A walks 12 km per hour, Person B walks 10 km per hour, and Person C walks 8 km per hour. If all three start at the same time, after how many hours will Person B be exactly at the midpoint of the distance between Person A and Person C?

Answer: 3.3


Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\left\lfloor\frac{N}{3}\right\rfloor$.

Answer: 29


Mr. Canada chooses a positive real \( a \) uniformly at random from \((0,1]\), chooses a positive real \( b \) uniformly at random from \((0,1]\), and then sets \( c = \frac{a}{a+b} \). What is the probability that \( c \) lies between \(\frac{1}{4}\) and \(\frac{3}{4}\)?

Answer: \frac{2}{3}


Given the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions.

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


Given the function $f(x)=a\sin x - \sqrt{3}\cos x$, one of its graphs has an axis of symmetry at $x=-\frac{\pi}{6}$, and $f(x_1) - f(x_2) = -4$, calculate the minimum value of $|x_1+x_2|$.

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


Let \( f(x) = \left\{ \begin{array}{ll} x - [x], & x \leq 0, \\ f(x - 1), & x > 0, \end{array} \right. \)
where \([x]\) represents the greatest integer less than or equal to \( x \). If the equation \( f(x) = kx + k \) (with \( k > 0 \)) has three distinct real roots, determine the range of the real number \( k \).

Answer: [\frac{1}{4},\frac{1}{3})


Answer: \frac{-1}{2007}


In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Answer: 16


Let \( s(n) \) denote the sum of the digits of a natural number \( n \). How many zeros are at the end of the number equal to the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \)?

Answer: 19


Scheherazade tells the Shah mathematical stories. A mathematical story is an equation of the form \( a x + b y = c \), where \( a, b, c \) are natural numbers ranging from 1 to 17 (inclusive). A story is sad if the corresponding equation has no solutions in integers (for example, the story \( 2 x + 4 y = 7 \) is sad, while the story \( 3 x + 5 y = 2 \) is not, as the equation has a solution \( x = -1, y = 1 \) ). Can Scheherazade come up with 1001 different sad stories?

Answer: Yes


Let \( ABC \) be an equilateral triangle with side length \( a \), and \( M \) be a point on the plane at a distance \( d \) from the center of triangle \( ABC \). Prove that the area of a triangle whose sides are the segments \( |MA|, |MB|, |MC| \) is given by the formula \( S = \frac{\sqrt{3}}{12} \left| a^2 - 3d^2 \right| \).

Answer: \frac{\sqrt{3}}{12}|^2-3d^2|


What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)} ~0\qquad\textbf{(B)} ~\frac{1}{4}\qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~1 \qquad\textbf{(E)} ~2$

Answer: \textbf{(D)}~1


Solve the inequality \(\sqrt{x^{2}-x-56}-\sqrt{x^{2}-25x+136}<8\sqrt{\frac{x+7}{x-8}}\) and find the sum of its integer solutions that belong to the interval \([-25, 25]\).

Answer: -285


Let  $ABC$  be a triangle with circumradius  $17$ , inradius  $4$ , circumcircle  $\Gamma$  and  $A$ -excircle  $\Omega$ . Suppose the reflection of  $\Omega$  over line  $BC$  is internally tangent to  $\Gamma$ .  Compute the area of  $\triangle ABC$ .

Answer: 128


In a cycling competition with  $14$  stages, one each day, and  $100$  participants, a competitor was characterized by finishing  $93^{\text{rd}}$  each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)

Answer: 2


Divide the natural numbers \(2, 3, 4, \ldots, n\) into two groups such that (1) the product of any two numbers in the same group is not in that group; (2) any number and its square are not in the same group. Find the maximum \(n\).

Answer: 31


A certain scenic area has two attractions that require tickets for visiting. The three ticket purchase options presented at the ticket office are as follows:
Option 1: Visit attraction A only, $30$ yuan per person;
Option 2: Visit attraction B only, $50$ yuan per person;
Option 3: Combined ticket for attractions A and B, $70$ yuan per person.
It is predicted that in April, $20,000$ people will choose option 1, $10,000$ people will choose option 2, and $10,000$ people will choose option 3. In order to increase revenue, the ticket prices are adjusted. It is found that when the prices of options 1 and 2 remain unchanged, for every $1$ yuan decrease in the price of the combined ticket (option 3), $400$ people who originally planned to buy tickets for attraction A only and $600$ people who originally planned to buy tickets for attraction B only will switch to buying the combined ticket.
$(1)$ If the price of the combined ticket decreases by $5$ yuan, the number of people buying tickets for option 1 will be _______ thousand people, the number of people buying tickets for option 2 will be _______ thousand people, the number of people buying tickets for option 3 will be _______ thousand people; and calculate how many tens of thousands of yuan the total ticket revenue will be?
$(2)$ When the price of the combined ticket decreases by $x$ (yuan), find the functional relationship between the total ticket revenue $w$ (in tens of thousands of yuan) in April and $x$ (yuan), and determine at what price the combined ticket should be to maximize the total ticket revenue in April. What is the maximum value in tens of thousands of yuan?

Answer: 188.1


On the bases \(AD\) and \(BC\) of trapezoid \(ABCD\), squares \(ADEF\) and \(BCGH\) are constructed outside the trapezoid. The diagonals of the trapezoid intersect at point \(O\). Find the length of segment \(AD\) if \(BC = 2\), \(GO = 7\), and \(GF = 18\).

Answer: \frac{22}{7}


Are there integers $m$ and $n$ such that
\[5m^2 - 6mn + 7n^2 = 1985 \ ?\]

Answer: \text{No}


The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again?

Answer: 2022


In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=1+\dfrac{\sqrt{2}}{2}t \\ y=2+\dfrac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the non-negative half-axis of $x$ as the polar axis), the equation of circle $C$ is $\rho=6\sin\theta$.
- (I) Find the standard equation of circle $C$ in Cartesian coordinates;
- (II) If point $P(l,2)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA| + |PB|$.

Answer: 2\sqrt{7}


A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \).

Answer: 35


The horizontal and vertical distances between adjacent points equal 1 unit.  What is the area of triangle $ABC$?

Answer: \frac{1}{2}


Find all functions  $ f: \mathbb{R} \to \mathbb{R}$  satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all  $ x \neq y$ .

Answer: f(x)=x


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

\[ 3|x+3a|+\left|x+a^{2}\right|+2x=a \]

has no solution.

Answer: (-\infty,0)\cup(10,\infty)


The hares are cutting the log again, but now both ends of the log are fixed. Ten middle logs fell, and the two end ones remained fixed. How many cuts did the hares make?

Answer: 11


Consider the set  $M=\{1,2,3,...,2020\}.$  Find the smallest positive integer  $k$  such that for any subset  $A$  of  $M$  with  $k$  elements, there exist  $3$  distinct numbers  $a,b,c$  from  $M$  such that  $a+b, b+c$  and  $c+a$  are all in  $A.$

Answer: 1011


Let  $R$  be the region in the Cartesian plane of points  $(x,y)$  satisfying  $x\geq 0$ ,  $y\geq 0$ , and  $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$ .  Determine the area of  $R$ .

Answer: \frac{9}{2}


Let  $X=\{1,2,3,...,10\}$ . Find the number of pairs of  $\{A,B\}$  such that  $A\subseteq X, B\subseteq X, A\ne B$  and  $A\cap B=\{5,7,8\}$ .

Answer: 2186


Given a triangle $ABC$ with an area of 1, points $D$ and $E$ are located on sides $AB$ and $AC$ respectively. Line segments $BE$ and $CD$ intersect at point $P$. Points $D$ and $E$ move along $AB$ and $AC$ such that the area of quadrilateral $BCED$ is twice the area of triangle $PBC$. Find the maximum area of triangle $PDE$.

Answer: 5\sqrt{2}-7


Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $\text{Area}(ABC)=3\sqrt{5}/8$, calculate $|AB|$.

Answer: \frac{9}{8}


Two equilateral triangles with perimeters of 12 and 15 are positioned such that their sides are respectively parallel. Find the perimeter of the resulting hexagon.

Answer: 27


Consider  $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each  $i = 1, ... ,4$  we have  $ (x_i -a)^2 +(y_i -b)^2 \le  c^2$ . Prove that  $a^2 + b^2 < c^2$ .
Restate this fact in the form of geometric result in plane geometry.

Answer: ^2+b^2<^2


Compute $\tan 40^\circ + 4 \sin 40^\circ.$

Answer: \sqrt{3}


Answer: \frac{2b}{3}


A standard deck of cards, excluding jokers, has 4 suits with 52 cards in total. Each suit has 13 cards, with face values from 1 to 13. If Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs from this deck, and the sum of the face values of the spades is 11 times the sum of the face values of the hearts, while the sum of the face values of the clubs is 45 more than the sum of the face values of the diamonds, what is the sum of the face values of these 14 cards?

Answer: 101


Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, $D$, $M$, and $N$, respectively. If $\overrightarrow{CN} = \overrightarrow{MD}$, find the value of $k$ and calculate the chord length $|MN|$.

Answer: \frac{\sqrt{42}}{2}


Concatenate the natural number $N$ to the right of each natural number. If the resulting new number is always divisible by $N$, then $N$ is called a magic number. How many magic numbers are there among the natural numbers less than 130?

Answer: 9


Find all odd numbers \( n \) that satisfy \( n \mid 3^n + 1 \).

Answer: 1


Given two circles $\odot K$ and $\odot O$ with radii $r$ and $R$, respectively, where $\odot K$ is contained within $\odot O$ and the distance between their centers is $d$, prove that there exists a quadrilateral that is both inscribed in $\odot O$ and circumscribed about $\odot K$ if and only if the following equation holds:
$$
\frac{1}{(R+d)^{2}}+\frac{1}{(R-d)^{2}}=\frac{1}{r^{2}}
$$

Answer: \frac{1}{(R+)^2}+\frac{1}{(R-)^2}=\frac{1}{r^2}


The Binomial Expansion is valid for exponents that are not integers.  That is, for all real numbers  $ x, y,$  and  $ r$  with  $ |x| > |y|,$
\[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots
\]
What are the first three digits to the right of the decimal point in the decimal representation of  $ \left(10^{2002} \plus{} 1\right)^{10/7}?$

Answer: 428


Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that \[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\]

Answer: 16


For any positive integers \( m \) and \( n \), define \( r(m, n) \) as the remainder of \( m \div n \) (for example, \( r(8,3) \) represents the remainder of \( 8 \div 3 \), so \( r(8,3)=2 \)). What is the smallest positive integer solution satisfying the equation \( r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4 \)?

Answer: 120


Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair.

Answer: \frac{8}{9}


Find the value of \( a \) such that the equation \( \sin 4x \cdot \sin 2x - \sin x \cdot \sin 3x = a \) has a unique solution in the interval \([0, \pi)\).

Answer: 1


If the number 79777 has the digit 9 crossed out, the result is the number 7777. How many different five-digit numbers exist from which 7777 can be obtained by crossing out one digit?

Answer: 45


How many pairs of integers $(a, b)$, with $1 \leq a \leq b \leq 60$, have the property that $b$ is divisible by $a$ and $b+1$ is divisible by $a+1$?

Answer: 106


A bag contains 999 balls, marked with the 999 numbers \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1000}\) respectively. Each ball is either red or blue, and the number of red balls is a positive even number. Let \(S\) denote the product of the numbers on all red balls. Find the sum of all possible values of \(S\).

Answer: \frac{498501}{2000}


suppose that polynomial  $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$  does not have a real root. what is the maximum number of coefficients to be  $-1$ ?(14 points)

Answer: 1005


An equilateral triangle with side length 10 cm is inscribed in a circle. The triangle's side is the diameter of the circle. In two opposite sectors of the circle, there are small shaded regions. If the sum of the areas of the two small shaded regions is in the form \(a\pi - b\sqrt{c}\), calculate \(a+b+c\).

Answer: 28


In the region $x > 0$, $y > 0$, $xy = 1$, find the range of the function
$$
f(x, y) = \frac{x + y}{\lfloor x \rfloor \lfloor y \rfloor + \lfloor x \rfloor + \lfloor y \rfloor + 1}.
$$

Answer: {\frac{1}{2}}\cup[\frac{5}{6},\frac{5}{4})


Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \).

Answer: \frac{1}{9}


A circle passes through vertices \( A \) and \( B \) of the rectangle \( ABCD \) and touches side \( CD \) at its midpoint. A line is drawn through vertex \( D \) that touches the same circle at point \( E \) and then intersects the extension of side \( AB \) at point \( K \).

Find the area of the trapezoid \( BCDK \) if \( AB = 10 \) and \( KE : KA = 3 : 2 \).

Answer: 210


In the diagram, $\triangle PQR$ is isosceles with $PQ = PR = 39$ and $\triangle SQR$ is equilateral with side length 30. The area of $\triangle PQS$ is closest to:

Answer: 75


In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 28


Find the maximum number of elements in the set \( S \) that satisfy the following conditions:
1. Each element of \( S \) is a positive integer not exceeding 100.
2. For any two different elements \( a \) and \( b \) in \( S \), there exists an element \( c \) in \( S \) such that the greatest common divisor of \( a \) and \( c \) is equal to 1, and the greatest common divisor of \( b \) and \( c \) is also equal to 1.
3. For any two different elements \( a \) and \( b \) in \( S \), there exists an element \( d \) in \( S \) different from \( a \) and \( b \) such that the greatest common divisor of \( a \) and \( d \) is greater than 1, and the greatest common divisor of \( b \) and \( d \) is also greater than 1.

Answer: 72


A regular hexagon \( K L M N O P \) is inscribed in an equilateral triangle \( A B C \) such that the points \( K, M, O \) lie at the midpoints of the sides \( A B, B C, \) and \( A C \), respectively. Calculate the area of the hexagon \( K L M N O P \) given that the area of triangle \( A B C \) is \( 60 \text{ cm}^2 \).

Answer: 30


In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?

Answer: $12$


A rectangular sheet of paper is lying on the table. Sasha cuts it in a straight line into two parts and places the parts on the table. Then he takes one of the parts, cuts it again in a straight line into two parts and puts these parts back on the table. He then takes another part from the table, cuts it again in a straight line into two parts, and so on. What is the minimum number of cuts that Sasha needs to make in order to have at least 252 eleven-sided polygons on the table?

Answer: 2015


Let the function $f(x)= \sqrt{3}\cos^2\omega x+\sin \omega x\cos \omega x+a$ where $\omega > 0$, $a\in\mathbb{R}$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $\dfrac{\pi}{6}$.

(Ⅰ) Find the smallest positive period of $f(x)$; (Ⅱ) If the minimum value of $f(x)$ in the interval $\left[-\dfrac{\pi}{3}, \dfrac{5\pi}{6}\right]$ is $\sqrt{3}$, find the value of $a$.

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


Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers defined by $a_{0}=21, a_{1}=35$, and $a_{n+2}=4 a_{n+1}-4 a_{n}+n^{2}$ for $n \geq 2$. Compute the remainder obtained when $a_{2006}$ is divided by 100.

Answer: 0


For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$.

Answer: 5120


How many positive odd integers greater than 1 and less than $150$ are square-free?

Answer: 59


The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.

[asy]
unitsize(0.8 cm);

int i;

real func (real x) {
  return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);
}

draw(graph(func,-4.5,4.5));
draw((-4.5,0)--(4.5,0));
draw((0,-5.5)--(0,5.5));

for (i = -4; i <= 4; ++i) {
  draw((i,-0.1)--(i,0.1));
}

for (i = -5; i <= 5; ++i) {
  draw((-0.1,i)--(0.1,i));
}

label("$-3$", (-3,-0.1), S);
label("$3$", (3,-0.1), S);
label("$10$", (-0.1,5), W);
label("$-10$", (-0.1,-5), W);

limits((-4.5,-5.5),(4.5,5.5),Crop);
[/asy]

Which of the following is the smallest?

A. $P(-1)$
B. The product of the zeros of $P$
C. The product of the non-real zeros of $P$
D. The sum of the coefficients of $P$
E. The sum of the real zeros of $P$

Answer: \text{C}


For each positive integer \( n \), define \(\varphi(n)\) to be the number of positive divisors of \( n \). For example, \(\varphi(10)=4\), since 10 has 4 positive divisors, namely \(\{1,2,5,10\}\). Suppose \( n \) is a positive integer such that \(\varphi(2n)=6\). Determine the minimum possible value of \(\varphi(6n)\).

Answer: 8


$2018$  people (call them  $A, B, C, \ldots$ ) stand in a line with each permutation equally likely. Given that  $A$  stands before  $B$ , what is the probability that  $C$  stands after  $B$ ?

Answer: 1/3


A circle with center $O$, inscribed in triangle $ABC$, touches its sides $AB$ and $AC$ at points $M$ and $N$. A circle with center $Q$ is inscribed in triangle $AMN$. Find $OQ$, given $AB=13$, $BC=15$, and $AC=14$.

Answer: 4


Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions.

Answer: 12


For  $n$  people, if it is known that

(a) there exist two people knowing each other among any three people, and

(b) there exist two people not knowing each other among any four people.

Find the maximum of  $n$ .

Here, we assume that if  $A$  knows  $B$ , then  $B$  knows  $A$ .

Answer: 8


Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n$.

Answer: 118


Let \(ABC\) be a triangle that satisfies \(AB=13, BC=14, AC=15\). Given a point \(P\) in the plane, let \(P_{A}, P_{B}, P_{C}\) be the reflections of \(A, B, C\) across \(P\). Call \(P\) good if the circumcircle of \(P_{A}P_{B}P_{C}\) intersects the circumcircle of \(ABC\) at exactly one point. The locus of good points \(P\) encloses a region \(\mathcal{S}\). Find the area of \(\mathcal{S}\).

Answer: \frac{4225}{64}\pi


Rectangles $R_1$ and $R_2,$ and squares $S_1, S_2,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

Answer: 651


Define
\[ A' = \frac{1}{1^2} + \frac{1}{7^2} - \frac{1}{11^2} - \frac{1}{13^2} + \frac{1}{19^2} + \frac{1}{23^2} - \dotsb, \]
which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and
\[ B' = \frac{1}{5^2} - \frac{1}{25^2} + \frac{1}{35^2} - \frac{1}{55^2} + \frac{1}{65^2} - \frac{1}{85^2} + \dotsb, \]
which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5.

Determine $\frac{A'}{B'}.$

Answer: 26


Positive integers \(a_{1}, a_{2}, \cdots, a_{18}\) satisfy the following two conditions:
1. \(a_{1} < a_{2} < a_{3} < \cdots < a_{18}\)
2. \(a_{1} + a_{2} + \cdots + a_{18} = 2001\)

What is the maximum value of \(a_{9}\)?

Answer: 192


The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8$

Answer: \textbf{(D)}\7


There are exactly 120 ways to color five cells in a $5 \times 5$ grid so that there is exactly one colored cell in each row and each column.

There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that there is exactly one colored cell in each row and each column.

How many ways are there to color five cells in a $5 \times 5$ grid without two corner cells so that there is exactly one colored cell in each row and each column?

Answer: 78


In the quadrilateral \( ABCD \), \( AB = a \), and the ratio of the measures of the four angles \( A, B, C, D \) is \( 3:7:4:10 \). Additionally, \(\angle CDB = 30^{\circ}\). Find the length of \( BC \).

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


Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then find $\tan B.$

Answer: \frac{4 \sqrt{3}}{11}


Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$

Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$.

Answer: 2010


Given the function $g(x)=x-1$, and the function $f(x)$ satisfies $f(x+1)=-2f(x)-1$. When $x \in (0,1]$, $f(x)=x^{2}-x$. For any $x_1 \in (1,2]$ and $x_2 \in R$, determine the minimum value of $(x_1-x_2)^2+(f(x_1)-g(x_2))^2$.

Answer: \frac{49}{128}


Let \(AA_1B_1B\) be the axial section of a cylinder, and let \(C\) be a point on the circumference of the base. Given \(AA_1 = 1\), \(AB = 4\), and \(\angle BAC = 60^\circ\), find the dihedral angle between plane \(A_1CB_1\) and the base of the cylinder \(AB\).

Answer: 30


The diagonals of an isosceles trapezoid are perpendicular. Find the area of the trapezoid if its midsegment is equal to 5.

Answer: 25


In a regular \( n \)-gon, \( A_{1} A_{2} A_{3} \cdots A_{n} \), where \( n > 6 \), sides \( A_{1} A_{2} \) and \( A_{5} A_{4} \) are extended to meet at point \( P \). If \( \angle A_{2} P A_{4}=120^\circ \), determine the value of \( n \).

Answer: 18


Find all integers \( n \) such that \( n^{4} + 6 n^{3} + 11 n^{2} + 3 n + 31 \) is a perfect square.

Answer: 10


Is it possible? Can three people cover a distance of 60 km in 3 hours if they have a two-seater motorcycle at their disposal? The speed of the motorcycle is 50 km/h, and the speed of a pedestrian is 5 km/h.

Answer: Yes


Class one and class two each select 4 players who will compete in a table tennis singles match in a predetermined order. The first players from both classes compete against each other; the loser is eliminated, and the winner plays against the next player from the losing side. This continues until all players from one side are eliminated, and the remaining side is declared the winner. The sequence of matches from the first to the last, including the outcome of each match, is called a match process. How many different possible match processes can occur in total?

Answer: 70


What is the least integer whose square is 36 more than three times its value?

Answer: -6


A circle $\omega$ with center at point \( I \) is inscribed in a convex quadrilateral \( ABCD \) and touches side \( AB \) at point \( M \), and side \( CD \) at point \( N \), with \(\angle BAD + \angle ADC < 180^\circ\). On the line \( MN \), a point \( K \neq M \) is chosen such that \( AK = AM \). In what ratio can the line \( DI \) divide the segment \( KN \)? Provide all possible answers and prove that there are no other solutions.

Answer: 1:1


Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) represent the sum of the largest and smallest numbers in \( X \). What is the arithmetic mean of all such \( a_X \)?

Answer: 1001


Let  $n>3$  be a positive integer. Equilateral triangle ABC is divided into  $n^2$  smaller congruent equilateral triangles (with sides parallel to its sides). Let  $m$  be the number of rhombuses that contain two small equilateral triangles and  $d$  the number of rhombuses that contain eight small equilateral triangles. Find the difference  $m-d$  in terms of  $n$ .

Answer: 6n-9


Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn.

Answer: 2.8


Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, calculate his age in 2013.

Answer: 18


Two heights of a parallelogram, drawn from the vertex of an obtuse angle, are $h_{1}$ and $h_{2}$, and the angle between them is $\alpha$. Find the longer diagonal of the parallelogram.

Answer: \frac{\sqrt{h_{1}^{2}+h_{2}^{2}+2h_{1}h_{2}\cos\alpha}}{\sin\alpha}


The graph of $y=f(x)$ for $-3\le x\le 3$ is shown below. On this interval, what is the range of $f(x)-x$? Express your answer in interval notation.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
rr_cartesian_axes(-3,3,-3,3);
draw((-3,-3)--(-2,-3),red+1);
draw((-2,-2)--(-1,-2),red+1);
draw((-1,-1)--(0,-1),red+1);
draw((0,0)--(1,0),red+1);
draw((1,1)--(2,1),red+1);
draw((2,2)--(3,2),red+1);
dot((-3,-3),red); dot((-2,-2),red); dot((-1,-1),red); dot((0,0),red); dot((1,1),red); dot((2,2),red); dot((3,3),red);
dot((-2,-3),red,UnFill); dot((-1,-2),red,UnFill); dot((0,-1),red,UnFill); dot((1,0),red,UnFill); dot((2,1),red,UnFill); dot((3,2),red,UnFill);
[/asy]

Answer: (-1,0]


If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chairs does Person B originally have?

Answer: 20


Answer: 1


The plane is divided by parallel lines that are a distance of $2a$ apart. A needle of length $2l$ $(l<a)$ is randomly thrown onto the plane. Find the probability that the needle will intersect any of the lines.

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


In Mexico, environmentalists achieved the passage of a law that requires each car to be off the road at least one day a week (the owner reports the car number and the designated "day off" to the police). In a certain family, all adults wish to drive daily (each for their own activities!). How many cars (at a minimum) must the family have if there are

a) 5 adults? b) 8 adults?

Answer: 10


Given \( x, y, z \in \mathbb{Z}_{+} \) with \( x \leq y \leq z \), how many sets of solutions satisfy the equation \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}\) ?

Answer: 10


Find the maximum value of the function $y=\sin^2x+3\sin x\cos x+4\cos^2x$ for $0 \leqslant x \leqslant \frac{\pi}{2}$ and the corresponding value of $x$.

Answer: \frac{\pi}{8}


Find the minimum value of the real number \(\lambda\), such that for any integer \(n \geq 2\) and positive real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) satisfying \(\sum_{i=1}^{n} a_{i}=n\), the following inequality always holds:
$$
\sum_{i=1}^{n} \frac{1}{a_{i}}-\lambda \prod_{i=1}^{n} \frac{1}{a_{i}} \leq n-\lambda .
$$

Answer: e


Given a natural number  $n \geq 2$ , consider all the fractions of the form  $\frac{1}{ab}$ , where  $a$  and  $b$  are natural numbers, relative primes and such that: $a < b \leq n$ ,  $a+b>n$ .
Show that for each  $n$ , the sum of all this fractions are  $\frac12$ .

Answer: \frac{1}{2}


The sum of all of the digits of the integers from 1 to 2008 is to be calculated.

Answer: 28054


An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction.

Answer: \frac{5}{2}


A finite sequence of integers  $a_{0}, a_{1}, \cdots, a_{n}$  is called quadratic if for each  $i \in \{1,2,\cdots,n \}$  we have the equality  $\vert a_{i}-a_{i-1} \vert = i^2$ . [list=a] [*] Prove that for any two integers  $b$  and  $c$ , there exists a natural number  $n$  and a quadratic sequence with  $a_{0}=b$  and  $a_{n}=c$ . [*] Find the smallest natural number  $n$  for which there exists a quadratic sequence with  $a_{0}=0$  and  $a_{n}=1996$ . [/list]

Answer: 19


Let \( p_{1}, \ldots, p_{n} \) be odd prime numbers. Show that \( 2^{p_{1} p_{2} \ldots p_{n}} + 1 \) has at least \( 2^{2^{n-1}} \) divisors.

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


Angle PQR is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square?

[asy]
draw((0,0)--(12,0)--(0,5)--cycle);
dot((0,0));
dot((12,0));
dot((0,5));
draw((0,0)--(0,5)--(-5,5)--(-5,0)--cycle);
draw((0,0)--(0,-12)--(12,-12)--(12,0));
draw((0,5)--(5,17)--(17,12)--(12,0)--cycle);
label("$P$",(0,5),NW);
label("$Q$",(0,0),SE);
label("$R$",(12,0),E);

[/asy]

Answer: 169


On one line, there are 10 points marked, and on a parallel line, there are 11 points marked.

How many a) triangles; b) quadrilaterals can be formed with vertices at these points?

Answer: 2475


Find \(g(2022)\) if for any real numbers \(x\) and \(y\) the following equation holds:
$$
g(x-y)=2022(g(x)+g(y))-2021 x y .
$$

Answer: 2043231


It is known that there exists a natural number \( N \) such that

$$
(\sqrt{3}-1)^{N}=4817152-2781184 \cdot \sqrt{3}
$$

Find \( N \).

Answer: 16


If the inequality
$$
\left|a x^{2}+b x+a\right| \leqslant x
$$

holds for $x \in [1,2]$, then the maximum value of $3a + b$ is ___.

Answer: 3


Points $M$ and $N$ are taken on the sides $AB$ and $AD$ of parallelogram $ABCD$ such that the lines $MC$ and $NC$ divide the parallelogram into three equal areas. Find $MN$, if $BD = d$.

Answer: \frac{}{3}


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


A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z?

Answer: 6


Find the triangle. The sides and the height of a certain triangle are expressed by four consecutive integers. What is the area of this triangle?

Answer: 84


Compute the number of even positive integers $n \leq 2024$ such that $1,2, \ldots, n$ can be split into $\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of 3.

Answer: 675


Rewrite the system in the form

$$
\left\{\begin{array}{l}
|x-1|-y=1-a^{4}-a^{4}(x-1)^{4} \\
(x-1)^{2}+y^{2}=1
\end{array}\right.
$$

and make the substitution $\mathrm{t}=\mathrm{x}-1$.

We get

$$
\left\{\begin{array}{l}
|t|-y=1-a^{4}-a^{4} t^{4} \\
t^{2}+y^{2}=1
\end{array}\right.
$$

If (t, y) is a solution of the system, then (-t, y) is also a solution. If the solution is unique, then $t=0$, and

$$
\left\{\begin{array}{l}
y=a^{4}-1 \\
y^{2}=1
\end{array}\right.
$$

For $\mathrm{y}=-1$, $\mathrm{a}=0$, and for $\mathrm{y}=1$, $\mathrm{a}^{4}=2$.

Let $\mathrm{a}=0$, then the system becomes

$$
\left\{\begin{array}{l}
|t|-y=1 \\
t^{2}+y^{2}=1
\end{array}\right.
$$

There are more than one solution, for example, $(0,-1)$, $(1,0)$ (where the first value in the pair is t).

Let $\mathrm{a}^{4}=2$, then the system becomes

$$
\left\{\begin{array}{l}
|t|-y=-1-2 t^{4} \\
t^{2}+y^{2}=1,
\end{array},<=>\left\{\begin{array}{l}
y=1+|t|+2 t^{4} \\
y^{2}=1-t^{2} .
\end{array}\right.\right.
$$

From the first equation, $\mathrm{y} \geq 1$, with $\mathrm{y}=1$ only when $\mathrm{t}=0$.

In the second equation, for $\mathrm{t} \neq 1$, $\mathrm{y}<1$. Thus, for $\mathrm{t} \neq 0$, there is no solution, and $(0,1)$ is the unique solution when $\mathrm{a}^{4}=2$.


A thief on a bus gets off at a bus stop and walks in the direction opposite to the bus’s travel direction. The bus continues its journey, and a passenger realizes they have been robbed. The passenger gets off at the next stop and starts chasing the thief. If the passenger's speed is twice that of the thief, the bus's speed is ten times the speed of the thief, and the bus takes 40 seconds to travel between two stops, how many seconds will it take for the passenger to catch up with the thief after getting off the bus?

Answer: 440


A four-digit number $\overline{\mathrm{abcd}}$ and its reverse $\overline{\mathrm{dcba}}$ are both multiples of 65. Find this number.

Answer: 5005


Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}.
$$

Answer: 1/7


You have $7$ red shirts, $8$ green shirts, $10$ pairs of pants, $10$ blue hats, $10$ red hats, and $5$ scarves (each distinct). How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one scarf, without having the same color of shirts and hats?

Answer: 7500


Each valve $A$, $B$, and $C$, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. The number of hours required with only valves $A$ and $B$ open is
 $\textbf{(A)}\ 1.1\qquad\textbf{(B)}\ 1.15\qquad\textbf{(C)}\ 1.2\qquad\textbf{(D)}\ 1.25\qquad\textbf{(E)}\ 1.75$

Answer: \textbf{(C)}\1.2


In the addition below, identical letters represent the same digit and different letters represent different digits. Find the number $ABCDE$.

\[ \begin{array}{cccccc}
    & & & & A & B & C & D & E \\
    + & & & & B & C & D & E \\
    + & & & & & C & D & E \\
    + & & & & & & D & E \\
    + & & & & & & & E \\
    \hline
    & & A & A & A & A & A \\
\end{array} \]

Answer: 52487


For each value of \( n \in \mathbf{N} \), find the largest value of \( k \in \mathbf{Z}^{+} \) for which the number \(\left\lfloor (3+\sqrt{11})^{2n-1} \right\rfloor\) is divisible by \( 2^k \).

Answer: n


The numbers \(a\) and \(n\) are coprime. Prove that for some natural number \(m\), the number \(a^{m} - 1\) is divisible by \(n\).

Answer: \varphi(n)


Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2, 2^2, 2^3, ..., 2^{25}\}$. What is the probability that $\log_a b$ is an integer?

Answer: \frac{31}{300}


Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\]
Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals:

Answer: f(n)


Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said?

Answer: 100


Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50 \%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and $2^{n}$ face each other in the last round? Express your answer in terms of $n$.

Answer: \frac{2^{n}-1}{8^{n}}


There are \( n \) people, and it is known that any two of them can call each other at most once. Among them, the total number of calls between any group of \( n-2 \) people is the same and is \( 3^{k} \) times, where \( k \) is a natural number. Find all possible values of \( n \).

Answer: 5


A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle.  What is the area of the union of the two triangles?

Answer: \frac{32}{3}


A triangle has sides of lengths 6 cm and 8 cm that create a 45-degree angle between them. Calculate the length of the third side.

Answer: 5.67


Answer: \frac{1}{3}


Given triangle  $ ABC$  and  $ AB\equal{}c$ ,  $ AC\equal{}b$  and  $ BC\equal{}a$  satisfying  $ a \geq b \geq c$ ,  $ BE$  and  $ CF$  are two interior angle bisectors.  $ P$  is a point inside triangle  $ AEF$ .  $ R$  and  $ Q$  are the projections of  $ P$  on sides  $ AB$  and  $ AC$ .

Prove that  $ PR \plus{} PQ \plus{} RQ < b$ .

Answer: PR+PQ+RQ<b


A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline.

Answer: 1/8


Circle \( \odot O \) is internally tangent to circle \( \odot O' \) at point \( A \). The radii of the two circles are \( r \) and \( r' \) respectively. Line \( BC \) is perpendicular to line \( OO' \) and intersects the two circles at points \( B \) and \( C \) respectively. The radius of the circumcircle of triangle \( ABC \) is \( R \). Prove that \( R^2 = rr' \).

Answer: R^2=rr'


Given that \(11 \cdot 14 n\) is a non-negative integer and \(f\) is defined by \(f(0)=0\), \(f(1)=1\), and \(f(n)=f\left(\left\lfloor \frac{n}{2} \right\rfloor \right)+n-2\left\lfloor \frac{n}{2} \right\rfloor\), find the maximum value of \(f(n)\) for \(0 \leq n \leq 1991\). Here, \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).

Answer: 10


Determine the number of subsets \( S \) of \(\{1,2,3, \ldots, 10\}\) with the following property: there exist integers \(a < b < c\) with \(a \in S, b \notin S, c \in S\).

Answer: 968


Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).

Answer: 501


In a certain middle school, there are 180 students in both the eighth and ninth grades. To understand the physical health of students in these two grades, a sampling survey was conducted as follows:
$(1)$ Data Collection:
Twenty students were randomly selected from each of the eighth and ninth grades for physical health tests. The test scores (in percentage) are as follows:

| Grade | Scores |
|-------|--------|
| Eighth | 78, 86, 74, 81, 75, 76, 87, 70, 75, 90, 75, 79, 81, 76, 74, 80, 86, 69, 83, 77 |
| Ninth | 93, 73, 88, 81, 72, 81, 94, 83, 77, 83, 80, 81, 70, 81, 73, 78, 82, 80, 70, 40 |

$(2)$ Data Organization and Description:
Organize and describe the two sets of sample data into the following score ranges:

| Score $x$ | $40\leqslant x\leqslant 49$ | $50\leqslant x\leqslant 59$ | $60\leqslant x\leqslant 69$ | $70\leqslant x\leqslant 79$ | $80\leqslant x\leqslant 89$ | $90\leqslant x\leqslant 100$ |
|-----------|-----------------------------|-----------------------------|-----------------------------|-----------------------------|-----------------------------|------------------------------|
| Eighth Grade Students | 0 | 0 | 1 | 11 | 7 | 1 |
| Ninth Grade Students | 1 | 0 | 0 | 7 | 10 | 2 |

(Note: A score of 80 or above indicates excellent physical health, 70-79 indicates good physical health, 60-69 indicates qualified physical health, and below 60 indicates poor physical health.)

$(3)$ Data Analysis:
The mean, median, and mode of the two sets of sample data are shown in the table below. Please complete the table:

|          | Mean | Median | Mode |
|----------|------|--------|------|
| Eighth Grade | 78.3 | 77.5 | ①______ |
| Ninth Grade | 78 | ②______ | 81 |

$(4)$ Conclusion:
① Estimate the number of students in the ninth grade who have excellent physical health to be ______.
② It can be inferred that ______ students in each grade have better physical health. The reason is ______. (Provide at least two different perspectives to explain the reasonableness of the inference.)

Answer: 108


Let \(n\) be a positive integer. Alice writes \(n\) real numbers \(a_{1}, a_{2}, \ldots, a_{n}\) in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors \(\left(a_{n}\) is not a neighbor of \(a_{1}\), nor vice versa\). A number changes sign if it changes from being nonnegative to negative or vice versa. In terms of \(n\), determine the maximum number of times that \(a_{1}\) can change sign, across all possible values of \(a_{1}, a_{2}, \ldots, a_{n}\) and all possible sequences of moves Alice may make.

Answer: n-1


Given $\triangle ABC$, find a point $P$ such that $PA + PB + PC$ is minimized.

Answer: P


In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?
$\text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9$

Answer: (C)\7


On a certain island, only knights, who always tell the truth, and liars, who always lie, live. One time, 1001 inhabitants of this island stood in a circle, and each one said: "All ten people following me in a clockwise direction are liars." How many knights could there be among those standing in the circle?

Answer: 91


Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[ N = a + (a+1) +(a+2) + \cdots + (a+k-1) \] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?

Answer: 16


Two players take turns placing Xs and Os in the cells of a $9 \times 9$ square (the first player places Xs, and their opponent places Os). At the end of the game, the number of rows and columns where there are more Xs than Os are counted as points for the first player. The number of rows and columns where there are more Os than Xs are counted as points for the second player. How can the first player win (score more points)?

Answer: 10


Let \( n \) be a natural number with the following property: from the numbers \( 1, 2, 3, \cdots, n \), any selection of 51 different numbers must include two numbers whose sum is equal to 101. What is the maximum value of \( n \) with this property?

Answer: 100


Solve the equation \(\frac{3}{\log_{2} x} = 4x - 5\).

Answer: 2


Let the tangent line passing through a point  $A$   outside the circle with center  $O$  touches the circle at  $B$  and  $C$ . Let  $[BD]$  be the diameter of the circle. Let the lines  $CD$  and  $AB$  meet at  $E$ . If the lines  $AD$  and  $OE$  meet at  $F$ , find  $|AF|/|FD|$ .

Answer: \frac{1}{2}


Find all natural numbers \( n \) such that \(\frac{1}{n}=\frac{1}{p}+\frac{1}{q}+\frac{1}{p q}\) for some prime numbers \( p \) and \( q \).

Answer: 1


Let $S$ be a subset with four elements chosen from \{1,2, \ldots, 10\}$. Michael notes that there is a way to label the vertices of a square with elements from $S$ such that no two vertices have the same label, and the labels adjacent to any side of the square differ by at least 4 . How many possibilities are there for the subset $S$ ?

Answer: 36


A solid has a triangular base with sides of lengths $s$, $s$, $s \sqrt{2}$. Two opposite vertices of the triangle extend vertically upward by a height $h$ where $h = 3s$. Given $s = 2\sqrt{2}$, what is the volume of this solid?

Answer: 24\sqrt{2}


How many 100-digit even numbers exist, where each digit in the decimal representation is one of the digits 0, 1, or 3?

Answer: 2\cdot3^{98}


A grasshopper jumps in a straight line. The first time it jumps 1 cm in some direction, the second time it jumps 2 cm, and so on.

Prove that after 1985 jumps, it cannot end up where it started.

Answer: 0


Find the smallest positive integer $n$ such that the divisors of $n$ can be partitioned into three sets with equal sums.

Answer: 120


Given that the center of circle $M$ lies on the $y$-axis, the radius is $1$, and the chord intercepted by line $l: y = 2x + 2$ on circle $M$ has a length of $\frac{4\sqrt{5}}{5}$. Additionally, the circle center $M$ is located below line $l$.

(1) Find the equation of circle $M$;

(2) Let $A(t, 0), B(t + 5, 0) \, (-4 \leqslant t \leqslant -1)$, if $AC, BC$ are tangent lines to circle $M$, find the minimum value of the area of $\triangle ABC$.

Answer: \frac{125}{21}


Let \(O_1\) be the center of the inscribed circle in triangle \(ABC\) with radius \(R\), and \(O_2\) be the center of the inscribed circle in triangle \(ADE\) with radius \(r\). Let \(p_1\) be the semiperimeter of triangle \(ABC\), and \(p_2\) be the semiperimeter of triangle \(ADE\). Then \(AK = p_2\), and \(AK = p_1 - BC\).

Using properties of tangents to circles:

\[
\begin{gathered}
2 AK = AK + AH = AD + DK + EH + AE = AD + (DM + ME) + AE = 2p_2 \\
2 AK = (AB - BK) + (AC - CH) = AB + AC + (BF + CF) - 2(BF + CF) = 2(p_1 - BC)
\end{gathered}
\]

Then \(r = \frac{S_{ADE}}{p_2} = \frac{S_{ADE}}{AK} = \frac{1}{6}\).

Since a circle can be circumscribed around quadrilateral \(BDEC\), \(\angle ABC + \angle DEC = 180^\circ\), and \(\angle AED = \angle ABC\). Therefore, triangles \(ADE\) and \(ACB\) are similar, and \(\frac{r}{R} = \frac{p_2}{p_1}\), \(R = \frac{r(AK + BC)}{AK} = \frac{3 + 15}{6 \cdot 3} = 1\).

Let \(\angle BAC = \alpha\).

Then, \(\operatorname{tg}(\alpha / 2) = \frac{R}{AK} = \frac{1}{3}\), and \(\operatorname{tg} \alpha = \frac{2 \operatorname{tg}(\alpha / 2)}{(1 - \operatorname{tg}^2(\alpha / 2))} = \frac{3}{4}\).

Answer: \frac{3}{4}


Maria wants to purchase a book which costs \$35.50. She checks her purse and discovers she has two \$20 bills, and twelve quarters, and a bunch of nickels. Determine the minimum number of nickels that Maria needs with her to buy the book.

Answer: 150


Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together?

$\textbf{(A) }70 \qquad \textbf{(B) }84 \qquad \textbf{(C) }100 \qquad \textbf{(D) }106 \qquad \textbf{(E) }120$

Answer: \textbf{(E)}\120


Construct regular triangles outwardly on sides $AB$ and $BC$ of parallelogram $ABCD$. The third vertices of these triangles are $E$ and $F$, respectively. Show that the sum of angles $CED$ and $AFD$ is $60^{\circ}$.

Answer: 60


Let  $\Gamma$  be the maximum possible value of  $a+3b+9c$  among all triples  $(a,b,c)$  of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If  $\Gamma = \frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers, then find  $p+q$ .

Answer: 16


In triangle \( ABC \), where angle \( A \) is \( 120^\circ \), the angle bisectors \( AF \), \( BG \), and \( CH \) are drawn. Prove that angle \( GFH \) is a right angle.

Answer: 90


How many  $5$ -digit numbers  $N$  (in base  $10$ ) contain no digits greater than  $3$  and satisfy the equality  $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of  $N$  cannot be zero.)

*Based on a proposal by Yannick Yao*

Answer: 256


In a right triangle ABC with sides 9, 12, and 15, a small circle with center Q and radius 2 rolls around the inside of the triangle, always remaining tangent to at least one side of the triangle. When Q first returns to its original position, through what distance has Q traveled?

Answer: 24


In a regular triangular prism, a plane passes through a side of the lower base and the opposite vertex of the upper base, forming a $45^\circ$ angle with the plane of the lower base. The area of the cross-section is $S$. Find the volume of the prism.

Answer: \frac{S\sqrt{S}\cdot\sqrt[4]{6}}{2}


Find the number of solutions to:
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval $(0,50 \pi)$.

Answer: 50


In a tetrahedron $A B C D$ where all edges are of equal length, points $E$ and $F$ are the midpoints of edges $A D$ and $B C$ respectively. Connect $A$ to $F$ and $C$ to $E$. Find the angle between the skew lines $A F$ and $C E$.

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


Given an integer \( n \geq 3 \), let \( A_{1}, A_{2}, \ldots, A_{2n} \) be distinct non-empty subsets of the set \(\{1, 2, \ldots, n\}\). Define \( A_{2n+1} = A_{1} \). Find the maximum value of \( \sum_{i=1}^{2n} \frac{\left| A_{i} \cap A_{i+1} \right|}{\left| A_{i} \right| \cdot \left| A_{i+1} \right|} \).

Answer: n


A certain type of alloy steel production company needs to ensure that the carbon content percentage of the alloy steel is within a specified range. Under the same test conditions, the inspector randomly samples $10$ times each day and measures the carbon content (unit: $\%$). It is known that the carbon content of their product follows a normal distribution $N(\mu, \sigma^2)$.

$(1)$ Assuming the production is normal, let $X$ represent the number of times in a day of $10$ samples where the carbon content percentage is outside of $(\mu - 3\sigma, \mu + 3\sigma)$. Find $P(X \geq 1)$ and the expected value of $X$.

$(2)$ In a day of inspections, if there is at least one detection of carbon content outside of $(\mu - 3\sigma, \mu + 3\sigma)$, it is considered that there may be an abnormality in the production process that day, and the production process of that day needs to be inspected. The following are the measurement results of carbon content (unit: $\%$) obtained by the inspector in $10$ tests in a day:

| Number | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|--------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|
| Carbon Content $\left(\%\right)$ | $0.31$ | $0.32$ | $0.34$ | $0.31$ | $0.30$ | $0.31$ | $0.32$ | $0.31$ | $0.33$ | $0.32$ |

After calculation, it is found that $\overline{x} = \frac{1}{10}\sum_{i=1}^{10}x^{i} = 0.317$ and $s = \sqrt{\frac{1}{10}\sum_{i=1}^{10}({x}_{i}-\overline{x})^{2}} = 0.011$, where $x_{i}$ is the carbon content percentage of the $i$-th sample $(i=1,2,\cdots,10)$.

$(i)$ Using the sample mean $\overline{x}$ as the estimate $\hat{\mu}$ of $\mu$ and the sample standard deviation $s$ as the estimate $\hat{\sigma}$ of $\sigma$, determine if it is necessary to inspect the production process of that day.

$(ii)$ If $x_{1}$ is removed, and the mean and standard deviation of the remaining data are denoted as $\mu_{1}$ and $\sigma_{1}$, respectively, write the formula for $\sigma_{1}$ in terms of $\overline{x}$, $s$, $x_{1}$, and $\mu_{1}$.

Note: If a random variable $Z$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - 3\sigma < Z < \mu + 3\sigma) = 0.9974$. $0.9974^{10} \approx 0.9743$.

Answer: 0.026


A moth starts at vertex \(A\) of a certain cube and is trying to get to vertex \(B\), which is opposite \(A\), in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches \(B\). How many ways can the moth achieve its objective?

Answer: 48


Find all real solutions $(x, y)$ of the system $x^{2}+y=12=y^{2}+x$.

Answer: (3,3),(-4,-4),\left(\frac{1+3 \sqrt{5}}{2}, \frac{1-3 \sqrt{5}}{2}\right),\left(\frac{1-3 \sqrt{5}}{2}, \frac{1+3 \sqrt{5}}{2}\right)


Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inward. Calculate the ratio of the shaded area formed between the semicircles and the quarter-circle to the area of a circle with a radius equal to the radius of the quarter circle.

Answer: \frac{3}{4}


The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence as number 0, determine the time (in minutes) until they coincide for the 21st time. Round your answer to hundredths.

Answer: 1374.55


Seven thousand twenty-two can be expressed as the sum of a two-digit number and a four-digit number.

Answer: 7022


Answer: 2


Call a positive integer $N$ a 7-10 double if the digits of the base-$7$ representation of $N$ form a base-$10$ number that is twice $N$. For example, $51$ is a 7-10 double because its base-$7$ representation is $102$. What is the largest 7-10 double?

Answer: 315


Amelia has a coin that lands heads with a probability of $\frac{3}{7}$, and Blaine has a coin that lands on heads with a probability of $\frac{1}{4}$. Initially, they simultaneously toss their coins once. If both get heads, they stop; otherwise, if either gets a head and the other a tail, that person wins. If both get tails, they start tossing their coins alternately as in the original scenario until one gets a head and wins. Amelia goes first in the alternate rounds. What is the probability that Amelia wins the game?
A) $\frac{5}{14}$
B) $\frac{9}{14}$
C) $\frac{11}{14}$
D) $\frac{13}{14}$
E) $\frac{1}{2}$

Answer: \frac{9}{14}


A certain country has \( n(n \geqslant 2) \) cities, and there is a direct bidirectional flight route between each pair of cities. The aviation authority authorizes certain airlines to use these routes under the following conditions:
1. Each flight route is used by exactly one airline.
2. For each airline, passengers can use its flight routes to travel from one city to any other city.

Find the maximum number of airlines that the aviation authority can authorize.

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


Using 12 different animal patterns to make some animal cards, with each card containing 4 different animal patterns, such that any two cards have exactly one animal pattern in common. What is the maximum number of cards that can be made?

Answer: 9


In the expression \( x_{1}: x_{2}: \ldots: x_{n} \), brackets are placed to indicate the order of operations, and the result is written as a fraction:

\[
\frac{x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}}{x_{i_{1}} x_{l_{2}} \ldots x_{i_{n-k}}}
\]

(where each of the variables \( x_{1}, x_{2}, \ldots, x_{n} \) is either in the numerator or the denominator). How many distinct expressions can be obtained in this way for all possible ways of placing the brackets?

Answer: 2^{n-2}


Kolya and Vitya are playing the following game. There is a pile of 100 stones on the table. The boys take turns making moves, with Kolya going first. On a turn, the player divides a pile that contains more than one stone into two smaller piles. The winner is the one who, after making their move, leaves only piles with one stone each. Can Kolya ensure that he wins regardless of Vitya's moves?

Answer: Yes


In a tetrahedron \(ABCD\), \(AD = \sqrt{2}\) and all other edge lengths are 1. Find the shortest path distance from the midpoint \(M\) of edge \(AB\) to the midpoint \(N\) of edge \(CD\) along the surface of the tetrahedron.

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


It is known that the only solution to the equation

$$
\pi / 4 = \operatorname{arcctg} 2 + \operatorname{arcctg} 5 + \operatorname{arcctg} 13 + \operatorname{arcctg} 34 + \operatorname{arcctg} 89 + \operatorname{arcctg}(x / 14)
$$

is a natural number. Find it.

Answer: 2016


Let  $x, y, z, t$  be real numbers such that  $(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2$ .
Prove that  $x^2 + y^2 > 1$ .

Answer: x^2+y^2>1


Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\sin^2\left(\angle CAD\right)$.
A) $\frac{3}{4}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $\frac{\sqrt{3}}{2}$

Answer: \frac{1}{2}


Given:
$$
\begin{array}{l}
A \cup B \cup C=\{a, b, c, d, e, f\}, \\
A \cap B=\{a, b, c, d\}, \\
c \in A \cap B \cap C .
\end{array}
$$

How many sets $\{A, B, C\}$ satisfy the given conditions?

Answer: 200


There are 183 different cards with the numbers $1, 2, 11, 2^{2}, 11^{2}, \ldots, 2^{91}, 11^{91}$ (each card has exactly one number, and each number occurs exactly once). In how many ways can you choose 3 cards such that the product of the numbers on the selected cards is a square of an integer and divisible by 22?

Answer: 184275


2002 is a palindromic year, meaning it reads the same backward and forward. The previous palindromic year was 11 years ago (1991). What is the maximum number of non-palindromic years that can occur consecutively (between the years 1000 and 9999)?

Answer: 109


Consider all possible quadratic polynomials $x^2 + px + q$ with a positive discriminant, where the coefficients $p$ and $q$ are integers divisible by 5. Find the largest natural number $n$ such that for any polynomial with the described properties, the sum of the hundredth powers of the roots is an integer divisible by $5^n$.

Answer: 50


A journalist wants to take a photo of 5 volunteers and 2 elderly individuals they are assisting, requiring them to line up in a row. The 2 elderly individuals must stand next to each other but not at the ends. The total number of different arrangements is ____.

Answer: 960


The quadratic polynomial \( y = ax^2 + bx + c \) has no roots and \( a + b + c > 0 \). Determine the sign of the coefficient \( c \).

Answer: 0


Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 4$, and $\vec{a}\cdot \vec{b}=2$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

Answer: \dfrac{\pi }{3}


Let \( XYZ \) be a triangle with \( \angle X = 60^\circ \) and \( \angle Y = 45^\circ \). A circle with center \( P \) passes through points \( A \) and \( B \) on side \( XY \), \( C \) and \( D \) on side \( YZ \), and \( E \) and \( F \) on side \( ZX \). Suppose \( AB = CD = EF \). Find \( \angle XPY \) in degrees.

Answer: 120


From point \( M \), located outside a circle at a distance \( \sqrt{7} \) from the center, a secant is drawn, where the internal part is twice as short as the external part and is equal to the radius of the circle.

Find the radius of the circle.

Answer: 1


In triangle \(ABC\), there is an inscribed circle with center \(O\). The line \(BO\) intersects this circle at points \(M\) and \(N\), and the line segments \(AO\) and \(CO\) intersect the circle at points \(P\) and \(Q\) respectively. Find the ratio of the areas of triangles \(MNP\) and \(MQN\) given angles \(\angle A = \alpha\) and \(\angle C = \gamma\).

Answer: \frac{\cos\frac{\gamma}{2}}{\cos\frac{\alpha}{2}}


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.\]What is the smallest possible value of $n$?

Answer: 20


The unknown real numbers $x, y, z$ satisfy the equations $$\frac{x+y}{1+z}=\frac{1-z+z^{2}}{x^{2}-x y+y^{2}} ; \quad \frac{x-y}{3-z}=\frac{9+3 z+z^{2}}{x^{2}+x y+y^{2}}$$ Find $x$.

Answer: \sqrt[3]{14}


On segment \( AB \) of length 10, a circle \(\omega\) is constructed as its diameter. A tangent line to \(\omega\) is drawn through point \( A \), and a point \( K \) is chosen on this tangent. A line passing through point \( K \), different from \( AK \), touches the circle \(\omega\) at point \( C \). The altitude \( CH \) of triangle \( ABC \) intersects segment \( BK \) at point \( L \). Find the area of triangle \( CKL \), given that \( BH : AH = 1 : 4 \).

Answer: 8