The convex quadrilateral \(ABCD\) has an area \(T\), and \(P\) is a point inside it. A line parallel to \(BC\) passing through \(P\) intersects \(BA\) at point \(E\), a line parallel to \(AD\) intersecting \(CD\) at \(G\), a line parallel to \(AB\) intersecting \(BC\) at \(F\), and a line parallel to \(CD\) intersecting \(AD\) at \(H\). Denote the area of quadrilateral \(AEPH\) by \(t_{1}\) and the area of quadrilateral \(PFCG\) by \(t_{2}\). Prove that \(\sqrt{t_{1}} + \sqrt{t_{2}} \leq \sqrt{T}\).

Answer: \sqrt{t_{1}}+\sqrt{t_{2}}\le\sqrt{T}


An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
$\textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16}$

Answer: \textbf{(C)}\\frac{17}{8}


Let \( ABCD \) be a quadrilateral with an inscribed circle \( \omega \) that has center \( I \). If \( IA = 5 \), \( IB = 7 \), \( IC = 4 \), and \( ID = 9 \), find the value of \( \frac{AB}{CD} \).

Answer: \frac{35}{36}


Let \( P(x) \) be a polynomial of degree at most 3 such that \( P(x) = \frac{1}{1 + x + x^2} \) for \( x = 1, 2, 3, 4 \). What is \( P(5) \)?

Answer: \frac{-3}{91}


Calculate the limit of the function:

$$\lim _{x \rightarrow \pi} \frac{\sin ^{2} x-\operatorname{tg}^{2} x}{(x-\pi)^{4}}$$

Answer: -1


Alice wants to color the integers between 2 and 31 (inclusive) using $k$ colors. She wants that, if $m$ and $n$ are integers between 2 and 31 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ should be of different colors. Determine the smallest integer $k$ for which Alice can color the integers $2,3, \ldots, 31$ using $k$ colors.

Answer: 4


Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube.

Answer: \frac{1}{4}


Find the angle $B$ of triangle $ABC$ if the length of the altitude $CH$ is half the length of side $AB$, and $\angle BAC = 75^\circ$.

Answer: 30


Given a positive integer  $m$ . Let  $$ A_l = (4l+1)(4l+2)...(4(5^m+1)l) $$  for any positive integer  $l$ . Prove that there exist infinite number of positive integer  $l$  which  $$ 5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l $$  and find the minimum value of  $l$  satisfying the above condition.

Answer: \frac{5^{+1}-1}{4}


I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I decide to highlight exactly two books (not necessarily different)?

Answer: 26460


For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$?

Answer: 30


All natural numbers are divided into "good" and "bad" based on the following rules:

a) From any "bad" number, you can subtract some natural number that does not exceed half of it in such a way that the resulting difference becomes "good".

b) From a "good" number, you cannot subtract any number that does not exceed half of it in such a way that it remains "good".

It is known that the number 1 is "good". Find the closest "good" number to 2015.

Answer: 2047


Given that the odd function $f(x)$ is defined on the domain $\mathbb{R}$ and satisfies $f(x) = f(2 - x)$ for any real number $x$. When $x \in (0, 1)$, $f(x) = 2^x + 1$. Find the value of $f(\log_{\frac{1}{2}} \frac{1}{15})$.

Answer: -\frac{31}{15}


The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$.

Answer: -1


Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\), find the measure of \(\angle BAM\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:

- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- \(\angle MAK\) is the maximum possible.

How many degrees is \(\angle BAM\)?

Answer: 44


When three different numbers from the set $\{ -3, -2, -1, 4, 5 \}$ are multiplied, the largest possible product is
$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 60$

Answer: C


Each cell of a $10 \times 10$ table contains a natural number. Then, a cell is shaded if the number in that cell is less than one of its neighbors but greater than another neighbor. (Two cells are considered neighbors if they share a side.) As a result, only two cells remain unshaded. What is the minimal possible sum of the numbers in these two unshaded cells?

Answer: 20


There are a set of tickets numbered from 1 to 30 (numbers may repeat). Each student drew one ticket. The teacher can perform the following operation: read out a list of several (possibly just one) numbers and ask their holders to raise their hands. How many times does the teacher need to perform this operation to find out the number of each student? (There may be fewer than 30 students.)

Answer: 5


Convert the binary number $101101_2$ to an octal number. The result is

Answer: 55_8


Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Find $N$.

Answer: 772


Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.

Answer: 33


Answer: -7


Given that \( a_{k} \) is the number of integer terms in \( \log_{2} k, \log_{3} k, \cdots, \log_{2018} k \). Calculate \( \sum_{k=1}^{2018} a_{k} \).

Answer: 4102


Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $15$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

Answer: 49


Given that $a$, $b$, and $c$ are nonzero real numbers, find all possible values of the expression
\[\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}.\]Enter all possible values, separated by commas.

Answer: 4, 0, -4


If the bisector of the acute angle $B$ in the right triangle $ABC$ divides the leg $AC$ in such a way that the segments are in the ratio $2:\sqrt{3}$, then the angle $C$ of the triangle is $60^{\circ}$.

Answer: 60


A showroom has 150 lights, all of which are initially turned on. Each light has an individual switch, numbered from 1 to 150. A student first toggles all switches that are multiples of 3, and then toggles all switches that are multiples of 5. How many lights remain on in the showroom?

Answer: 80


Answer: uU^2+vV^2+wW^2=UVW+4uvw


Find the greatest integer  $A$  for which in any permutation of the numbers  $1, 2, \ldots , 100$  there exist ten consecutive numbers whose sum is at least  $A$ .

Answer: 505


For a positive integer \( n \), let \( \varphi(n) \) denote the number of positive integers less than and relatively prime to \( n \). Let \( S_{k} = \sum_{n} \frac{\varphi(n)}{n} \), where \( n \) runs through all positive divisors of \( 42^{k} \). Find the largest positive integer \( k < 1000 \) such that \( S_{k} \) is an integer.

Answer: 996


In triangle \(ABC\), it is known that \(AC = 12\) and \(AB = BC = 3\sqrt{10}\). Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(C\), respectively, and are located on opposite sides of this plane. The distance between the centers of these spheres is 15. The center of a third sphere is at point \(B\), and this sphere touches the two given spheres externally. Find the radius of the third sphere.

Answer: 6


There are positive integers $x$ and $y$ that satisfy the system of equations\begin{align*} \log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) &= 60\\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*}Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $3m+2n$.

Answer: 880


An infinite sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the recurrence $$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$ for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\cdots+a_{100}$.

Answer: 3


Prove that all numbers in the $n$-th row of the probability triangle do not exceed $\frac{1}{\sqrt{n}}$.

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


If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \_\_\_\_\_\_ .

Answer: -\frac{2}{5}


Answer: \sum_{i=1}^{n}\frac{1}{1+x_i}\ge\sum_{i=1}^{n}\frac{x_i}{1+x_i^2}


Given integers $a$ and $b$ whose sum is $20$, determine the number of distinct integers that can be written as the sum of two, not necessarily different, fractions $\frac{a}{b}$.

Answer: 14


Let  $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$ . For all points  $(a, b)$ , let  $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$ . Kathy constructs a set  $T$  by adding  $n$  distinct points from  $S$  to  $T$  at random. If the expected value of  $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$  is 4, then compute  $n$ .

*Proposed by Lewis Chen*

Answer: 2013


Given: A hyperbola \( C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{2 a^{2}}=1(a>0) \), and a parabola \( C_{2} \) with its vertex at the origin \( O \). The focus of \( C_{1} \) is also the left focus \( F_{1} \) of \( C_{2} \). Determine if there exists a chord \( A B \) of \( C_{2} \) passing through \( F_{1} \) such that the area \( S_{\triangle A O B} \) is maximized or minimized. If it exists, find the equation of the line containing \( A B \) and the extreme value. If it doesn't exist, provide an explanation.

Answer: 6a^2


The sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers satisfies the recurrence $a_{n+1}=\frac{a_{n}^{2}-a_{n-1}+2 a_{n}}{a_{n-1}+1}$. Given that $a_{1}=1$ and $a_{9}=7$, find $a_{5}$.

Answer: 3


Given \( x, y, z \in \mathbf{Z} \) such that \(\left\{\begin{array}{l}x+ y + z = 3 \\ x^3 + y^3 + z^3 = 3\end{array}\right.\), determine the set of all possible values of \( x^2 + y^2 + z^2 \).

Answer: {3,57}


A straight line $l$ passes through a vertex and a focus of an ellipse. If the distance from the center of the ellipse to $l$ is one quarter of its minor axis length, calculate the eccentricity of the ellipse.

Answer: \dfrac{1}{2}


The number of triangles with vertices' coordinates $(x, y)$ that satisfy $1 \leqslant x \leqslant 4, 1 \leqslant y \leqslant 4$, and where $x$ and $y$ are integers is $\qquad$ .

Answer: 516


Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers?

Answer: 11


Positive numbers \( a, b, c \) satisfy \( a^2 b + b^2 c + c^2 a = 3 \). Find the minimum value of the expression:

\[ A = \frac{\sqrt{a^6 + b^4 c^6}}{b} + \frac{\sqrt{b^6 + c^4 a^6}}{c} + \frac{\sqrt{c^6 + a^4 b^6}}{a}. \]

Answer: 3\sqrt{2}


Find the sum of all integer solutions \( x \in [-3, 13] \) that satisfy the inequality

$$
\left(1-\operatorname{ctg}^{2} \frac{\pi x}{12}\right)\left(1-3 \operatorname{ctg}^{2} \frac{\pi x}{12}\right)\left(1-\operatorname{tg} \frac{\pi x}{6} \cdot \operatorname{ctg} \frac{\pi x}{4}\right) \leqslant 16
$$

Answer: 28


Let  $ABC$  be a triangle with sides  $51, 52, 53$ . Let  $\Omega$  denote the incircle of  $\bigtriangleup ABC$ . Draw tangents to  $\Omega$  which are parallel to the sides of  $ABC$ . Let  $r_1, r_2, r_3$  be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed  $r_1 + r_2 + r_3$ .

Answer: 15


Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?
$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72\qquad\textbf{(E)}\ 80$

Answer: \textbf{(C)}\64


Show that √7, √(7 - √7), √(7 - √(7 + √7)), √(7 - √(7 + √(7 - √7))), ... converges and find its limit.

Answer: 2


A rectangular piece of paper with a length of 20 cm and a width of 12 cm is folded along its diagonal (refer to the diagram). What is the perimeter of the shaded region formed?

Answer: 64


Given the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), the line \( x = -\sqrt{2} b \) intersects the ellipse at only one point. Let \( P \) be any point on the ellipse \( C \), with \( P_{1}(-1,0) \) and \( P_{2}(1,0) \), and \(\overrightarrow{P P_{1}} \cdot \overrightarrow{P P_{2}} \) has a minimum value of \( \frac{a}{2} \).
(1) Find the equation of the ellipse \( C \);
(2) Let the line \( l: y=k x+m \) intersect the ellipse \( C \) at points \( A \) and \( B \), with \( O \) being the origin, and \(\overrightarrow{O M}=\frac{1}{2}(\overrightarrow{O A}+\overrightarrow{O B})\). When the area of \(\triangle A O B\) is maximized, find the range of \( T=\frac{1}{\left|M P_{1}\right|^{2}}-2\left|M P_{2}\right|.

Answer: [3-4\sqrt{2},1)


A certain state issues license plates consisting of six digits (from 0 to 9).  The state requires that any two license plates differ in at least two places.  (For instance, the numbers 027592 and 020592 cannot both be used.)  Determine, with proof, the maximum number of distinct license plates that the state can use.

Answer: 100000


Given the ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ with its left and right foci being F1 and F2, and its top vertex being B. If the perimeter of $\triangle BF_{1}F_{2}$ is 6, and the distance from point F1 to the line BF2 is $b$.
(1) Find the equation of ellipse C;
(2) Let A1 and A2 be the two endpoints of the major axis of ellipse C, and point P is any point on ellipse C different from A1 and A2. The line A1P intersects the line $x=m$ at point M. If the circle with MP as its diameter passes through point A2, find the value of the real number $m$.

Answer: 14


Determine all positive integers \( a \) for which the equation

$$
\left(1+\frac{1}{x}\right) \cdot\left(1+\frac{1}{x+1}\right) \cdots\left(1+\frac{1}{x+a}\right) = a - x
$$

has at least one integer solution for \( x \).

For each such \( a \), provide the corresponding solutions.

Answer: 7


In a rhombus \(ABCD\), angle \(BAD\) is acute. A circle inscribed in this rhombus touches sides \(AB\) and \(CD\) at points \(M\) and \(N\) respectively. It intersects segment \(CM\) at point \(P\) and segment \(BN\) at point \(Q\). Find the ratio \(BQ\) to \(QN\) if \(CP: PM = 9:16\).

Answer: 1:9


The surface area of a sphere inscribed in a regular truncated triangular pyramid is related to the total surface area of the pyramid as \(\pi : 6 \sqrt{3}\). Find the angle between the lateral face and the base plane of the pyramid.

Answer: \arctan2


During training shooting, each of the soldiers shot 10 times. One of them successfully completed the task and scored 90 points. How many times did he score 9 points if he scored four 10s, and the results of the other shots were 7s, 8s, and 9s, with no misses?

Answer: 3


A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of "ten thousands" as the unit, it becomes ____ ten thousands.

Answer: 603.4


The set \( S = \{1, 2, \cdots, 1990\} \). If some 31-element subset of \( S \) has a sum that is divisible by 5, it is called a "good subset" of \( S \). Find the number of "good subsets" of \( S \).

Answer: \frac{1}{5}\binom{1990}{31}


Harvard has recently built a new house for its students consisting of \( n \) levels, where the \( k \)-th level from the top can be modeled as a 1-meter-tall cylinder with radius \( k \) meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is 35 percent of the total surface area of the building (including the bottom), compute \( n \).

Answer: 13


If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is
$\textbf{(A) }8\qquad \textbf{(B) }9\qquad \textbf{(C) }12\qquad \textbf{(D) }18\qquad \textbf{(E) }24$

Answer: \textbf{(D)}18


Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?

Answer: 194


In ancient China, Confucianism required students to master six basic skills (Six Arts): rites, music, archery, charioteering, calligraphy, and mathematics. A school's traditional Chinese culture society held a "Six Arts" lecture activity on weekends, with six sessions in a day, one for each art. Calculate the total number of different scheduling methods between "archery" and "mathematics" with at most one art in between.

Answer: 432


Given the numbers 2 and 8, find the product of three numbers that form a geometric sequence with these two numbers.

Answer: 64


Let \( x \) be a real number. How many of the first 1000 positive integers can be expressed in the form \( [2x] + [4x] + [6x] + [8x] \)?

Answer: 600


Express the number one million using numbers that contain only the digit 9 and the algebraic operations of addition, subtraction, multiplication, division, powers, and roots. Determine at least three different solutions.

Answer: 1000000


In the rhombus \(ABCD\), the angle \(\angle ABC = 60^{\circ}\). A circle is tangent to the line \(AD\) at point \(A\), and the center of the circle lies inside the rhombus. Tangents to the circle, drawn from point \(C\), are perpendicular. Find the ratio of the perimeter of the rhombus to the circumference of the circle.

Answer: \frac{\sqrt{3} + \sqrt{7}}{\pi}


Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface),
if $x + y + z = s$, and $0 \leq x \leq y \leq z \leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\_\_\_\_\_\_\_\_\_\_$.

Answer: \frac{1}{6}


Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, determine the minimum value of $m$.

Answer: \dfrac{e^{2}-1}{e^{2}+1}


Find the sum of all \( x \) such that \( 0 \leq x \leq 360 \) and \( \cos 12 x^{\circ} = 5 \sin 3 x^{\circ} + 9 \tan^2 x^{\circ} + \cot^2 x^{\circ} \).

Answer: 540


How many four-digit numbers whose digits add up to $9$ are divisible by $11$?

Answer: 0


Given the function
$$
f(x)=\sin \left(\omega x+\frac{\pi}{2}\right) \cdot \sin \left(\omega x+\frac{\pi}{4}\right)(\omega>0)
$$
where the smallest positive period of \( f(x) \) is \( \pi \). Find the range of \( f(x) \) over the interval \(\left[0, \frac{\pi}{2}\right]\).

Answer: [0,\frac{2+\sqrt{2}}{4}]


Try to divide the set $\{1,2,\cdots, 1989\}$ into 117 mutually disjoint subsets $A_{i}, i = 1,2,\cdots, 117$, such that
(1) Each $A_{i}$ contains 17 elements;
(2) The sum of the elements in each $A_{i}$ is the same.

Answer: 16915


The numbers from 1 to 200, inclusive, are placed in a bag. A number is randomly selected from the bag. What is the probability that it is neither a perfect square, a perfect cube, nor a multiple of 7? Express your answer as a common fraction.

Answer: \frac{39}{50}


In an acute-angled triangle \( ABC \), the altitude \( BH \) and the median \( AM \) are drawn. It is known that the angle \( MCA \) is twice the angle \( MAC \), and \( BC = 10 \).

Find \( AH \).

Answer: 5


Given the function $f(x)=e^{-x}+ \frac {nx}{mx+n}$.
$(1)$ If $m=0$, $n=1$, find the minimum value of the function $f(x)$.
$(2)$ If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac {m}{n}$.

Answer: \frac {1}{2}


Let the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ intersect the positive direction of the $y$-axis at point $B$. Find the number of right-angled isosceles triangles inscribed in the ellipse with vertex $B$.

Answer: 3


For how many values of $k$ is $18^{18}$ the least common multiple of the positive integers $6^9$, $9^9$, and $k$?

Answer: 19


In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$
$\mathrm{(A)}\ 12\qquad \mathrm{(B)}\ 17\qquad \mathrm{(C)}\ 20\qquad \mathrm{(D)}\ 25\qquad \mathrm{(E)}\ 35$

Answer: \textbf{(B)}\17


Given rectangle $ABCD$. We construct the circle $k_{1}$ which passes through $A$ and $B$, and is tangent to line $CD$, as well as the circle $k_{2}$ which passes through $A$ and $D$ and is tangent to line $BC$. Denote the radii of $k_{1}$ and $k_{2}$ by $r_{1}$ and $r_{2}$ respectively, and the sides of the rectangle by $a$ and $b$. Prove that:

$$
r_{1}+r_{2} \geqq \frac{5}{8}(a+b)
$$

Answer: r_1+r_2\ge\frac{5}{8}(b)


Four numbers form a geometric progression. If the second number is increased by 4 and the fourth number is decreased by 28, an arithmetic progression is obtained. Determine these four numbers.

Answer: 2,6,18,54


Given points \( A(4,0) \) and \( B(2,2) \) are inside the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \), and \( M \) is a point on the ellipse, find the maximum value of \( |MA| + |MB| \).

Answer: 10 + 2\sqrt{10}


In the Cartesian coordinate plane \( xOy \), the number of integer points (points with both coordinates being integers) not passed through by the line \( x \cos \theta + y \sin \theta = 1 \) is \( n = \) $\qquad$.

Answer: 1


The areas of triangles \(ABC\), \(A_1B_1C_1\), and \(A_2B_2C_2\) are \(S\), \(S_1\), and \(S_2\) respectively, with the conditions \(AB = A_1B_1 + A_2B_2\), \(AC = A_1C_1 + A_2C_2\), and \(BC = B_1C_1 + B_2C_2\). Prove that \(S \leq 4 \sqrt{S_1 S_2}\).

Answer: S\le4\sqrt{S_1S_2}


R is the reals. X is the square \([0, 1] \times [0, 1]\). \(f : X \rightarrow R\) is continuous. If \(\int_{Y} f(x, y) \, dx \, dy = 0\) for all squares \(Y\) such that:
1. \(Y \subseteq X\),
2. \(Y\) has sides parallel to those of \(X\),
3. at least one of \(Y\)'s sides is contained in the boundary of \(X\),

is it true that \(f(x, y) = 0\) for all \(x, y\)?

Answer: f(x,y)=0forall(x,y)\inX


A person has a probability of $\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value)

Answer: \frac{3}{16}


The difference between the maximum and minimum values of the function $f(x)= \frac{2}{x-1}$ on the interval $[-2,0]$ is $\boxed{\frac{8}{3}}$.

Answer: \frac{4}{3}


Dean is playing a game with calculators. The 42 participants (including Dean) sit in a circle, and Dean holds 3 calculators. One calculator reads 1, another 0, and the last one -1. Dean starts by pressing the cube button on the calculator that shows 1, pressing the square button on the one that shows 0, and on the calculator that shows -1, he presses the negation button. After this, he passes all of the calculators to the next person in the circle. Each person presses the same buttons on the same calculators that Dean pressed and then passes them to the next person. Once the calculators have all gone around the circle and return to Dean so that everyone has had one turn, Dean adds up the numbers showing on the calculators. What is the sum he ends up with?

Answer: 0


A worm is called an *adult* if its length is one meter. In one operation, it is possible to cut an adult worm into two (possibly unequal) parts, each of which immediately becomes a worm and begins to grow at a speed of one meter per hour and stops growing once it reaches one meter in length. What is the smallest amount of time in which it is possible to get  $n{}$  adult worms starting with one adult worm? Note that it is possible to cut several adult worms at the same time.

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


Define $a\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\clubsuit y=y\clubsuit x$?
$\textbf{(A)}\ \text{a finite set of points}\\ \qquad\textbf{(B)}\ \text{one line}\\ \qquad\textbf{(C)}\ \text{two parallel lines}\\ \qquad\textbf{(D)}\ \text{two intersecting lines}\\ \qquad\textbf{(E)}\ \text{three lines}$

Answer: \textbf{(E)}\


On the side \( BC \) of triangle \( ABC \) a circle is constructed with a radius of 20 cm, using \( BC \) as the diameter. This circle intersects sides \( AB \) and \( AC \) at points \( X \) and \( Y \) respectively. Find \( BX \cdot AB + CY \cdot AC \).

Answer: 1600


A circle with center $O$ has a diameter $AB$. From a point $C$ on the circumference, which is distinct from points $A$ and $B$, a perpendicular is drawn to $AB$ intersecting $AB$ at point $D$. From $O$, a perpendicular is drawn to $BC$, intersecting $BC$ at point $M$. Determine the measure of angle $ABC$ given that $DB = 3 \cdot OM$.

Answer: 30


Given 2020 points in the plane, no three of which are collinear, demonstrate that it is possible to construct 505 pairwise disjoint quadrilaterals, not necessarily convex, with vertices among the given points.

Answer: 505


Answer: 1296


Select 5 people from 4 boys and 5 girls to participate in a math extracurricular group. How many different ways are there to select under the following conditions?

 (1) Select 2 boys and 3 girls, and girl A must be selected;

 (2) Select at most 4 girls, and boy A and girl B cannot be selected at the same time.

Answer: 90


Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combinations need to be checked to guarantee guessing Vasya's password?

Answer: 504


If \(a > b > c > d > 0\) are integers such that \(ad = bc\), show that \((a - d)^2 \geq 4d + 8\).

Answer: ()^2\ge4d+8


The sides of  $\triangle ABC$  measure 11,20, and 21 units. We fold it along  $PQ,QR,RP$  where  $P,Q,R$  are the midpoints of its sides until  $A,B,C$  coincide. What is the volume of the resulting tetrahedron?

Answer: 45


In parallelogram \(ABCD\) with angle \(A\) equal to \(60^\circ\), the bisector of angle \(B\) intersects side \(CD\) at point \(E\). A circle with radius \(R\) is inscribed in triangle \(ECB\). Another circle is inscribed in trapezoid \(ABED\). Find the distance between the centers of these circles.

Answer: R\sqrt{7}


Let \( a, b, c \) be the side lengths of a triangle with a perimeter not exceeding \( 2\pi \). Prove that \( \sin a, \sin b, \sin c \) can form the side lengths of a triangle.

Answer: \sin,\sinb,\sin


Let  ${f : \Bbb{R} \rightarrow \Bbb{R}}$  be a continuous and strictly increasing function for which
\[ \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) \]
for all  ${x,y \in \Bbb{R}} ({f^{-1}}$  denotes the inverse of  ${f})$ . Prove that there exist real constants  ${a \neq 0}$  and  ${b}$  such that  ${f(x)=ax+b}$  for all  ${x \in \Bbb{R}}.$ *Proposed by Zoltán Daróczy*

Answer: f(x)=ax+b


\( y = f(x) \) is a real-valued solution (for all real \( x \)) of the differential equation \( y'' - 2y' + y = 2e^x \) which is positive for all \( x \). Is \( f'(x) \) necessarily positive for all \( x \)?

\( y = g(x) \) is another real-valued solution which satisfies \( g'(x) > 0 \) for all real \( x \). Is \( g(x) \) necessarily positive for all \( x \)?

Answer: Yes


Let \( N=2^{\left(2^{2}\right)} \) and \( x \) be a real number such that \( N N^{\left(N^{N}\right)}=2^{\left(2^{x}\right)} \). Find \( x \).

Answer: 66


Let \( a \) and \( b \) be real numbers satisfying \( a > b > 0 \). Evaluate
\[
\int_{0}^{2\pi} \frac{1}{a + b \cos(\theta)} \, d\theta
\]
Express your answer in terms of \( a \) and \( b \).

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


Jerry has nine cards with digits from 1 to 9. He arranges them in a row to form a nine-digit number. Tom writes on the board all 8 two-digit numbers formed by consecutive digits (for example, for the number 789456123, these numbers are 78, 89, 94, 45, 56, 61, 12, 23). For each two-digit number that is divisible by 9, Tom gives Jerry a piece of cheese. What is the maximum number of pieces of cheese Jerry can receive?

Answer: 4


Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\).

Answer: 10


Given the fundamental set of solutions $y_{1}=\ln x, y_{2}=x$ of the corresponding homogeneous equation, find a particular solution to the equation

$$
x^{2}(1-\ln x) y^{\prime \prime}+x y^{\prime}-y=\frac{(1-\ln x)^{2}}{x}
$$

that satisfies the condition $\lim _{x \rightarrow+\infty} y=0$.

Answer: \frac{1-2\ln{x}}{4x}


On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ to $B$ were the same as on the section from $B$ to $C$, the entire journey from $A$ to $C$ would take 5.5 hours. How much time would the boat take to travel upstream from $C$ to $A$ under these conditions? The boat's own speed remains unchanged in all cases.

Answer: 7.7


Let \( A(a, b, c) \) be the area of a triangle with sides \( a, b, c \). Let \( f(a, b, c) = \sqrt{A(a, b, c)} \). Prove that for any two triangles with sides \( a, b, c \) and \( a', b', c' \) we have \( f(a, b, c) + f(a', b', c') \leq f(a + a', b + b', c + c') \). When do we have equality?

Answer: f(,b,)+f(',b',')\lef(',b',')


Answer: $6^7$


Given the function \( f(x) = x^2 + ax + b \), where \( a \) and \( b \) are real numbers, let \( M(a, b) \) be the maximum value of \( |f(x)| \) on the interval \([-1,1]\). When \( M(a, b) \leq 2 \), find the maximum value of \( |a| + |b| \).

Answer: 3


Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and $a_{12} = 123456789101112$. For $1 \le k \le 150$, how many $a_k$ are divisible by both 3 and 5?

Answer: 10


Answer: 4\midn


Consider all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(f(x)+2 x+20)=15$$ Call an integer $n$ good if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$.

Answer: -35


Buses. At the stop near Absent-minded Scientist's house, two bus routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly followed, but the schedules of these two routes are not coordinated and are independent of each other. At a completely random moment, the Scientist arrives at the stop and gets on the first bus that comes to reach the metro. What is the probability that the Scientist will board bus No. 251?

Answer: \frac{5}{14}


**p1.** What is the maximum possible value of  $m$  such that there exist  $m$  integers  $a_1, a_2, ..., a_m$  where all the decimal representations of  $a_1!, a_2!, ..., a_m!$  end with the same amount of zeros?**p2.** Let  $f : R \to R$  be a function such that  $f(x) + f(y^2) = f(x^2 + y)$ , for all  $x, y \in R$ . Find the sum of all possible  $f(-2017)$ .**p3.** What is the sum of prime factors of  $1000027$ ?**p4.** Let   $$ \frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m}, $$   where  $n, m$  are relatively prime. Find  $(m - n)$ .**p5.** Determine the number of ordered pairs of real numbers  $(x, y)$  such that  $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$ **p6.** Triangle  $\vartriangle ABC$  has  $\angle B = 120^o$ ,  $AB = 1$ . Find the largest real number  $x$  such that  $CA - CB > x$  for all possible triangles  $\vartriangle ABC$ .**p7.**Jung and Remy are playing a game with an unfair coin. The coin has a probability of  $p$  where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round  $ 1$ . Whoever gets heads first wins the game. Given that Jung has the probability of  $8/15$  , what is the value of  $p$ ?**p8.** Consider a circle with  $7$  equally spaced points marked on it. Each point is  $ 1$  unit distance away from its neighbors and labelled  $0,1,2,...,6$  in that order counterclockwise. Feng is to jump around the circle, starting at the point  $0$  and making six jumps counterclockwise with distinct lengths  $a_1, a_2, ..., a_6$  in a way such that he will land on all other six nonzero points afterwards. Let  $s$  denote the maximum value of  $a_i$ . What is the minimum possible value of  $s$ ?**p9.**Justin has a  $4 \times 4 \times 4$  colorless cube that is made of  $64$  unit-cubes. He then colors  $m$  unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of  $m$ ?**p10.** Yikai wants to know Liang’s secret code which is a  $6$ -digit integer  $x$ . Furthermore, let  $d(n)$  denote the digital sum of a positive integer  $n$ . For instance,  $d(14) = 5$  and  $d(3) = 3$ . It is given that   $$ x + d(x) + d(d(x)) + d(d(d(x))) = 999868. $$   Please find  $x$ .

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 5


Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 157


Answer: \frac{3}{4}


Simplify the fraction \(\frac{\sqrt{-x}-\sqrt{-3 y}}{x+3 y+2 \sqrt{3 x y}}\).

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


Lunasa, Merlin, and Lyrica each have a distinct hat. Every day, two of these three people, selected randomly, switch their hats. What is the probability that, after 2017 days, every person has their own hat back?

Answer: 0


Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who is 172 cm tall.

Answer: 60.316


Two thieves stole an open chain with  $2k$  white beads and  $2m$  black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets  $k$  white beads and  $m$  black beads. What is the minimal number of cuts that is always sufficient?

Answer: 2


As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respectively, find the area of \(\triangle ABC\).

Answer: 63


Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\frac{x}{\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$.

Answer: 4\sqrt{e}-2


We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*

Answer: 21600


Answer: 7789


Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$

Answer: 742


The specific heat capacity of a body with mass \( m = 3 \) kg depends on the temperature in the following way: \( c = c_{0}(1 + \alpha t) \), where \( c_{0} = 200 \) J/kg·°C is the specific heat capacity at \( 0^{\circ} \mathrm{C} \), \( \alpha = 0.05 \,^{\circ} \mathrm{C}^{-1} \) is the temperature coefficient, and \( t \) is the temperature in degrees Celsius. Determine the amount of heat that needs to be transferred to this body to heat it from \( 30^{\circ} \mathrm{C} \) to \( 80^{\circ} \mathrm{C} \).

Answer: 112.5


A merchant placed on display some dresses, each with a marked price. He then posted a sign "$1/3$ off on these dresses." The cost of the dresses was $3/4$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was:

Answer: \frac{1}{2}


If \( x \) and \( y \) are both greater than 0, and \( x + 2y = 1 \), then what is the minimum value of \( u = \left( x + \frac{1}{x} \right) \left( y + \frac{1}{4y} \right) \)?

Answer: \frac{25}{8}


Given the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) with general terms \( a_{n}=2^{n} \) and \( b_{n}=3n+2 \), respectively, arrange the common terms of \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) in ascending order to form a new sequence \( \left\{c_{n}\right\} \). Determine the general term of \( \left\{c_{n}\right\} \).

Answer: c_n=2^{2n+1}


Find the smallest positive integer \( n \) such that the mean of the squares of the first \( n \) natural numbers (\( n > 1 \)) is an integer.

(Note: The mean of the squares of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is given by \( \sqrt{\frac{a_{1}^2 + a_{2}^2 + \cdots + a_{n}^2}{n}} \).)

(Note: Fifteenth American Mathematical Olympiad, 1986)

Answer: 337


A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement?

Answer: 11


Given that $17^{-1} \equiv 11 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)

Answer: 42


Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games did Innokentiy play?

Answer: 14


Let the three sides of a triangle be integers \( l \), \( m \), and \( n \) with \( l > m > n \). It is known that \( \left\{\frac{3^l}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\} \), where \( \{x\} \) denotes the fractional part of \( x \). Determine the minimum value of the perimeter of the triangle.

Answer: 3003


The center of a circle with a radius of 5, circumscribed around an isosceles trapezoid, lies on the longer base, and the shorter base is equal to 6. Find the area of the trapezoid.

Answer: 32


Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of paper has the original sheet been cut into?

A) 27
B) 30
C) 37
D) 40
E) 47

Answer: 37


Given the triangle \( \triangle ABC \) with a semiperimeter \( p \) and an incircle of radius \( r \), semicircles are constructed on the sides \( BC \), \( CA \), and \( AB \) of the triangle externally. A circle \(\tau\) that is tangent to all three semicircles has a radius \( t \). Prove that:

\[ \frac{p}{2} < t \leq \frac{p}{2} + \left( 1 - \frac{\sqrt{3}}{2} \right)r. \]

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


What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?

Answer: 16.67\%


In the Cartesian coordinate system \(xOy\), given the ellipse \(C: \frac{x^{2}}{27}+\frac{y^{2}}{18}=1\) with right focus \(F\), a line \(l\) passing through point \(F\) intersects the ellipse \(C\) at points \(A\) and \(B\). Is there a fixed point \(P\) on the \(x\)-axis such that, as the line \(l\) rotates about point \(F\), \(\overrightarrow{PA} \cdot \overrightarrow{PB}\) remains constant?

Answer: (4,0)


What is the smallest positive value of $m$ so that the equation $18x^2 - mx + 252 = 0$ has integral solutions?

Answer: 162


Let  $ABCD$  be a cyclic quadrilateral, and suppose that  $BC = CD = 2$ . Let  $I$  be the incenter of triangle  $ABD$ . If  $AI = 2$  as well, find the minimum value of the length of diagonal  $BD$ .

Answer: 2\sqrt{3}


Given a parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), point \( X \) is chosen on edge \( A_1 D_1 \) and point \( Y \) is chosen on edge \( BC \). It is known that \( A_1 X = 5 \), \( B Y = 3 \), and \( B_1 C_1 = 14 \). The plane \( C_1 X Y \) intersects the ray \( DA \) at point \( Z \). Find \( DZ \).

Answer: 20


Andrei traveled by bus to his club and back home, spending 115 rubles in total. In January, he did not have his pass and had to buy tickets from the driver at 15 rubles each on some days and at 11 rubles each on other days. Every time he returned from his club, he walked home. Andrei calculated that he spent exactly 115 rubles in January.

Determine how many times Andrei went to his club in January.

Answer: 9


Each letter in the table represents a different digit, and different letters represent different digits. The leading digit of each number cannot be zero. The three numbers in each row from left to right form an arithmetic sequence, and the three numbers in each column from top to bottom also form an arithmetic sequence. What is the five-digit number $\overline{\mathrm{CDEFG}}$?

\begin{tabular}{|c|c|c|}
\hline
$A$ & $BA$ & $AA$ \\
\hline
$AB$ & $CA$ & $EF$ \\
\hline
$CD$ & $GA$ & $BDC$ \\
\hline
\end{tabular}

Answer: 40637


A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\circ$?

Answer: 48


In a massive school which has  $m$  students, and each student took at least one subject. Let  $p$  be an odd prime. Given that:

(i) each student took at most  $p+1$  subjects.
(ii) each subject is taken by at most  $p$  students.
(iii) any pair of students has at least  $1$  subject in common.

Find the maximum possible value of  $m$ .

Answer: p^2


Given the function f(x) = sinωx + cosωx, if there exists a real number x₁ such that for any real number x, f(x₁) ≤ f(x) ≤ f(x₁ + 2018) holds true, find the minimum positive value of ω.

Answer: \frac{\pi}{2018}


Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

Answer: \frac{3}{16}


Let  $r_1,r_2,r$ , with  $r_1 < r_2 < r$ , be the radii of three circles  $\Gamma_1,\Gamma_2,\Gamma$ , respectively. The circles  $\Gamma_1,\Gamma_2$  are internally tangent to  $\Gamma$  at two distinct points  $A,B$  and intersect in two distinct points. Prove that the segment  $AB$  contains an intersection point of  $\Gamma_1$  and  $\Gamma_2$  if and only if  $r_1 +r_2 = r$ .

Answer: r_1+r_2=r


Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b} < \frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Answer: 36


A number consists of 600 digits of 6 and a certain number of 0s. Can it be a perfect square? (36th Moscow Mathematical Olympiad, 1973)

Answer: No


Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair  $(x, y)$  of positive integers such that  $x \leq 20$  and  $y \leq 23$ . (Philena knows that Nathan’s pair must satisfy  $x \leq 20$  and  $y \leq 23$ .) The game then proceeds in rounds; in every round, Philena chooses an ordered pair  $(a, b)$  of positive integers and tells it to Nathan; Nathan says YES if  $x \leq a$  and  $y \leq b$ , and NO otherwise. Find, with proof, the smallest positive integer  $N$  for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most  $N$  rounds.

Answer: 9


We write the equation on the board:

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

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

Answer: 2016


Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red?

Answer: \frac{9}{26}


Consider all words containing only letters  $A$  and  $B$ . For any positive integer  $n$ ,  $p(n)$  denotes the number of all  $n$ -letter words without four consecutive  $A$ 's or three consecutive  $B$ 's. Find the value of the expression
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]

Answer: 2


Among the two-digit numbers less than 20, the largest prime number is ____, and the largest composite number is ____.

Answer: 18


Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry?

Answer: 1956


Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Answer: 489


Let \( N \) be a three-digit integer such that the difference between any two positive integer factors of \( N \) is divisible by 3. Let \( d(N) \) denote the number of positive integers which divide \( N \). Find the maximum possible value of \( N \cdot d(N) \).

Answer: 5586


Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \dots, a_n$.

Answer: 3


In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than  $12$  employees have a more difficult work;
(ii) At least  $30$  employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.

Answer: 42


Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying
\[n+k |\binom{2n}{n}\]

Answer: k \neq 1


Given $f(x)=\frac{1}{2}\cos^{2}x-\frac{1}{2}\sin^{2}x+1-\sqrt{3}\sin x \cos x$.
$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing.
$(2)$ Find the minimum value of $f(x)$ on $[0,\frac{\pi}{2}]$ and the corresponding set of independent variables.

Answer: \left\{\frac{\pi}{3}\right\}


In trapezoid \(ABCD\), the angles \(A\) and \(D\) at the base \(AD\) are \(60^{\circ}\) and \(30^{\circ}\) respectively. Point \(N\) lies on the base \(BC\) such that \(BN : NC = 2\). Point \(M\) lies on the base \(AD\), the line \(MN\) is perpendicular to the bases of the trapezoid and divides its area in half. Find the ratio \(AM : MD\).

Answer: 3:4


The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Determine the maximum average storage period for the product without incurring a loss.

Answer: 56


Consider a $5 \times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do not form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?

Answer: 12


In \( \triangle ABC \), if \( |\overrightarrow{AB}| = 2 \), \( |\overrightarrow{BC}| = 3 \), and \( |\overrightarrow{CA}| = 4 \), find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \).

Answer: -\frac{29}{2}


In the quadrilateral  $ABCD$ , the angles  $B$  and  $D$  are right . The diagonal  $AC$  forms with the side  $AB$  the angle of  $40^o$ , as well with side  $AD$  an angle of  $30^o$ . Find the acute angle between the  diagonals  $AC$  and  $BD$ .

Answer: 80


The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid.

Answer: 96


Let $\alpha \in \mathbb{Q}^{+}$. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ such that for all $x, y \in \mathbb{Q}^{+}$, it holds that

$$
f\left(\frac{x}{y}+y\right)=\frac{f(x)}{f(y)}+f(y)+\alpha x.
$$

Here, $\mathbb{Q}^{+}$ denotes the set of positive rational numbers.

Answer: f(x)=x^2


Find the least value of  $k$  such that for all  $a,b,c,d \in \mathbb{R}$  the inequality
\[ \begin{array}  c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)}   \  \ge 2( ab+bc+cd+da+ac+bd)-k  \end{array}\]
holds.

Answer: 4


Suppose \( t = \left(\frac{1}{2}\right)^x + \left(\frac{2}{3}\right)^x + \left(\frac{5}{6}\right)^x \). Find the sum of all real solutions \( x \) of the equation \( (t-1)(t-2)(t-3)=0 \).

Answer: 4


Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same.

$\square + \square \square = \square \square \square$.

What is the largest possible three-digit number in the equation?

Answer: 105


Determine all polynomials  $P(x)$  with degree  $n\geq 1$  and integer coefficients so that for every real number  $x$  the following condition is satisfied   $$ P(x)=(x-P(0))(x-P(1))(x-P(2))\cdots (x-P(n-1)) $$

Answer: P(x)=x


Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), \( M \) is the center of the face \( A B B_{1} A_{1} \), \( N \) is a point on the edge \( B_{1} C_{1} \), \( L \) is the midpoint of \( A_{1} B_{1} \); \( K \) is the foot of the perpendicular dropped from \( N \) to \( BC_{1} \). In what ratio does point \( N \) divide the edge \( B_{1} C_{1} \) if \( \widehat{L M K} = \widehat{M K N} \)?

Answer: \sqrt{2} + 1


Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{2014}}{3}\right]$.

Answer: 15


Given the convex quadrilateral \(ABCD\) with an area of 1 unit. Reflect vertex \(A\) over \(B\), \(B\) over \(C\), \(C\) over \(D\), and \(D\) over \(A\). What is the area of the quadrilateral formed by the reflected points?

Answer: 5


The base of the pyramid is a right triangle with an acute angle $\alpha$. This triangle is inscribed in the base of a cone. The vertex of the pyramid coincides with the midpoint of one of the generatrices of the cone. Find the ratio of the volume of the cone to the volume of the prism.

Answer: \frac{2\pi}{\sin2\alpha}


For an upcoming holiday, the weather forecast indicates a probability of $30\%$ chance of rain on Monday and a $60\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\%$ chance that the rain will continue into the next day without interruption. Calculate the probability that it rains on at least one day during the holiday period. Express your answer as a percentage.

Answer: 72\%


Xiaoming saw a tractor pulling a rope slowly on the road and decided to measure the length of the rope. If Xiaoming walks in the direction the tractor is moving, it takes him a total of 140 steps to walk from one end of the rope to the other. If Xiaoming walks in the opposite direction to the tractor, it takes him 20 steps to walk from one end of the rope to the other. The speeds of both the tractor and Xiaoming remain constant, and Xiaoming covers 1 meter with each step. What is the length of the rope in meters?

Answer: 35


Let  $f:S\to\mathbb{R}$  be the function from the set of all right triangles into the set of real numbers, defined by  $f(\Delta ABC)=\frac{h}{r}$ , where  $h$  is the height with respect to the hypotenuse and  $r$  is the inscribed circle's radius. Find the image,  $Im(f)$ , of the function.

Answer: (2,1+\sqrt{2}]


Find the smallest natural number such that when multiplied by 9, the resulting number consists of the same digits but in some different order.

Answer: 1089


Let $\left\{a_n\right\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, with $S_{11}= \frac{11}{3}\pi$. Let $\left\{b_n\right\}$ be a geometric sequence, and $b_4, b_8$ be the two roots of the equation $4x^2+100x+{\pi}^2=0$. Find the value of $\sin \left(a_6+b_6\right)$.

Answer: -\frac{1}{2}


For positive integers $a$ and $N$, let $r(a, N) \in\{0,1, \ldots, N-1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \leq 1000000$ for which $r(n, 1000)>r(n, 1001)$.

Answer: 499500


Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady's $2010$th step, considering the procedure implies hexadecimal (base 16) operations rather than quinary (base 5)?

Answer: 30


Given \( x, y, z \in (0,1] \), find the minimum value of the expression

\[
A = \frac{(x+2y) \sqrt{x+y-xy} + (y+2z) \sqrt{y+z-yz} + (z+2x) \sqrt{z+x-zx}}{xy + yz + zx}
\]

Answer: 3


A certain high school encourages students to spontaneously organize various sports competitions. Two students, A and B, play table tennis matches in their spare time. The competition adopts a "best of seven games" format (i.e., the first player to win four games wins the match and the competition ends). Assuming that the probability of A winning each game is $\frac{1}{3}$.
$(1)$ Find the probability that exactly $5$ games are played when the match ends;
$(2)$ If A leads with a score of $3:1$, let $X$ represent the number of games needed to finish the match. Find the probability distribution and expectation of $X$.

Answer: \frac{19}{9}


Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1.

Answer: 2010


Consider the sequence  $$ 1,7,8,49,50,56,57,343\ldots $$  which consists of sums of distinct powers of $7$ , that is,  $7^0$ ,  $7^1$ ,  $7^0+7^1$ ,  $7^2$ , $\ldots$  in increasing order. At what position will  $16856$  occur in this sequence?

Answer: 36


Let $\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\mathcal{S}_{11}$.

Answer: 927


Suppose that by omitting a finite number of positive elements from the set of natural numbers, we obtained a set $S$ that is closed under addition. Let $k$ be one of the elements of $S$. How many elements of $S$ are there such that subtracting $k$ from them results in a number that does not belong to $S$?

Answer: k


In $\triangle A B C$, the incircle centered at $I$ touches sides $A B$ and $B C$ at $X$ and $Y$, respectively. Additionally, the area of quadrilateral $B X I Y$ is $\frac{2}{5}$ of the area of $A B C$. Let $p$ be the smallest possible perimeter of a $\triangle A B C$ that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter $p$.

Answer: 2 \sqrt{5}


"That" and "this," and half of "that" and "this" - how many percent is this of three-quarters of "that" and "this"?

Answer: 200


In how many ways can the number 1500 be represented as the product of three natural numbers (variations where the factors are the same but in different orders are considered identical)?

Answer: 32


The area of the base of a pyramid is 3, and the volume of the pyramid is also 3. Two planes parallel to the base of the pyramid are drawn. The areas of the resulting cross-sections are 1 and 2. Find the volume of the part of the pyramid located between the planes.

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


A segment of a line connecting the center of the base of a regular triangular pyramid with the midpoint of a lateral edge is equal to the side of the base. Find the cosine of the angle between adjacent lateral faces.

Answer: \frac{7}{15}


Given four points  $O,\ A,\ B,\ C$  on a plane such that  $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of  $\triangle{ABC}$ .

Answer: 2\sqrt{7} + \frac{3\sqrt{3}}{2}


Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:
(1) Person A and Person B must stand at the two ends;
(2) All boys must stand together;
(3) No two boys stand next to each other;
(4) Exactly one person stands between Person A and Person B.

Answer: 1200


How many distinct triangles can be drawn using three of the dots below as vertices?
[asy]
dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));
[/asy]

Answer: 18


An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$.  The sum of the numbers is $ 10{,}000$.  Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term.  What is the value of $ G - L$?

Answer: \frac{8080}{199}


Given a complete bipartite graph on  $n,n$  vertices (call this  $K_{n,n}$ ), we colour all its edges with  $2$  colours , red and blue . What is the least value of  $n$  such that for any colouring of the edges of the graph , there will exist at least one monochromatic  $4$  cycle ?

Answer: 5


Find all values of \( a \) for which the system

$$
\left\{\begin{array}{l}
|y|+|y-x| \leqslant a-|x-1| \\
(y-4)(y+3) \geqslant(4-x)(3+x)
\end{array}\right.
$$

has exactly two solutions.

Answer: 7


In the plane rectangular coordinate system xOy, the origin is taken as the pole, the positive semi-axis of the x-axis is taken as the polar axis, and the polar coordinate system is established. The same unit length is adopted in both coordinate systems. The polar coordinate equation of the curve C is ρ = 4$\sqrt {2}$cos(θ - $\frac {π}{4}$), and the parameter equation of the line is $\begin{cases} x=2+ \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{cases}$ (t is the parameter).

1. Find the rectangular coordinate equation of curve C and the general equation of line l.
2. Suppose point P(2, 1), if line l intersects curve C at points A and B, find the value of $| \frac {1}{|PA|} - \frac {1}{|PB|} |$.

Answer: \frac { \sqrt {31}}{7}


Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?

Answer: 1400


In an isosceles triangle \(ABC\), it is known that \(AC = 4\) and \(AB = BC = 6\). The angle bisector of \(\angle C\) intersects side \(AB\) at point \(D\). A circle is drawn through point \(D\), touching side \(AC\) at its midpoint, and intersects the segment \(AD\) at point \(E\). Find the area of the triangle \(DEC\).

Answer: \frac{44\sqrt{2}}{45}


Let \( O \) be the origin in the complex plane, and let \( Z_{1} \) and \( Z_{2} \) be two points in the complex plane that satisfy the following conditions:
1. The arguments of the complex numbers corresponding to \( Z_{1} \) and \( Z_{2} \) are constants \(\theta\) and \(-\theta\), respectively, where \( 0 < \theta < \frac{\pi}{2} \).
2. The area of the triangle \( O Z_{1} Z_{2} \) is a constant \( S \).

Find the equation of the locus of the centroid \( Z \) of the triangle \( O Z_{1} Z_{2} \) and determine the minimum modulus of the complex number corresponding to the point \( Z \).

Answer: \frac{2}{3}\sqrt{S\cot\theta}


The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^{2}$, $8\text{ in}^{2}$, and $6\text{ in}^{2}$ respectively is:
$\textbf{(A)}\ 576\text{ in}^{3}\qquad\textbf{(B)}\ 24\text{ in}^{3}\qquad\textbf{(C)}\ 9\text{ in}^{3}\qquad\textbf{(D)}\ 104\text{ in}^{3}\qquad\textbf{(E)}\ \text{None of these}$

Answer: \mathrm{(B)}\24in^3


From point \( A \), which is 5 units away from the center of a circle with a radius of 3 units, two secants \( A K C \) and \( A L B \) are drawn. The angle between them is \( 30^{\circ} \) (where \( K, C, L, B \) are the points of intersection of the secants with the circle). Find the area of triangle \( A K L \) if the area of triangle \( A B C \) is 10.

Answer: \frac{8}{5}


How many ways are there to arrange 5 identical red balls and 5 identical blue balls in a line if there cannot be three or more consecutive blue balls in the arrangement?

Answer: 126


Find the range of the function \( f(x)=\frac{\sin x-1}{\sqrt{3-2 \cos x-2 \sin x}} \) for \( 0 \leq x \leq 2\pi \).

Answer: [-1,0]


When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is calculated.

Answer: 18185


In triangle \( \triangle ABC \), with \( \angle A \leq \angle B \leq \angle C \), suppose
$$
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \sqrt{3},
$$
find the value of \( \sin B + \sin 2B \).

Answer: \sqrt{3}


In the diagram, \( B, C \) and \( D \) lie on a straight line, with \(\angle ACD=100^{\circ}\), \(\angle ADB=x^{\circ}\), \(\angle ABD=2x^{\circ}\), and \(\angle DAC=\angle BAC=y^{\circ}\). The value of \( x \) is:

Answer: 20


In the coordinate plane, a point with both integer x-coordinate and y-coordinate is called a lattice point, and a triangle with all vertices as lattice points is called a lattice triangle. How many lattice right triangles \( \mathrm{Rt} \triangle OAB \) exist with the incenter at \( I(2015, 7 \times 2015) \) and the right-angle vertex at the origin \( O \)?

Answer: 54


Shuai Shuai memorized words for 7 days. Starting from the second day, he memorized 1 more word each day than the previous day. The sum of the number of words memorized in the first 4 days is equal to the sum of the number of words memorized in the last 3 days. How many words did Shuai Shuai memorize in total over these 7 days?

Answer: 84


Answer: 10


Given vectors $\overrightarrow {a} = (\sin\theta, \cos\theta - 2\sin\theta)$ and $\overrightarrow {b} = (1, 2)$.
(1) If $\overrightarrow {a} \parallel \overrightarrow {b}$, find the value of $\tan\theta$;
(2) If $|\overrightarrow {a}| = |\overrightarrow {b}|$ and $0 < \theta < \pi$, find the value of $\theta$.

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


Given that $\theta=\arctan \frac{5}{12}$, find the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$.

Answer: \frac{\pi}{4}


A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?

Answer: 8 \frac{3}{4}


In the Cartesian coordinate plane, a parabola given by \( y = kx^2 \) (where \( k > 0 \)) and a circle given by \( (x - a)^2 + (y - b)^2 = r^2 \) have at least 3 points in common. One of these points is the origin, and the other two points lie on the line \( y = kx + b \). What is the minimum value of the real number \( b \)?

Answer: 2


Find the angle at the vertex of the axial section of a cone if the central angle in the development of its lateral surface is $\alpha$ radians.

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


Given the number $500 \ldots 005$ (with 80 zeros in between). We need to replace any two zeros with non-zero digits so that the resulting number is divisible by 165. In how many ways can this be done?

Answer: 17280


Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\]

Answer: \frac{131}{11}


For any three-digit number \( n \), let \( \overline{n} \) represent the number obtained by removing all digits equal to 0 from \( n \). For example, if \( n = 102 \), then \( \overline{n} = 12 \). How many three-digit numbers \( n \) satisfy \( \overline{n} < n \) and \( \overline{n} \) is a divisor of \( n \)?

Answer: 93


If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?

Answer: 6410


Points \( A(0,0) \) and \( B(1000,0) \), as well as points \( C_{1}(1,1) \), \( C_{2}(2,1) \), \(\ldots\), \( C_{999}(999,1) \), are marked on the coordinate plane. Then all possible lines \( A C_{i} \) and \( B C_{i} \) (for \( 1 \leq i \leq 999 \)) are drawn. How many integer-coordinate points of intersection do all these lines have? (An integer-coordinate point is one where both coordinates are integers.)

Answer: 2326


Find all  $10$ -digit whole numbers  $N$ , such that first  $10$  digits of  $N^2$  coincide with the digits of  $N$  (in the same order).

Answer: 1000000000


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
  13x +  by +  cz &= 0 \\
   ax + 23y +  cz &= 0 \\
   ax +  by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$.  What is the value of
\[ \frac{a}{a - 13} + \frac{b}{b - 23} + \frac{c}{c - 42} \, ?\]

Answer: 1


Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/500$. Find $n$.

Answer: 13


Let  $k<<n$  denote that  $k<n$  and  $k\mid n$ . Let  $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$  be such that, if  $n\leq 2013$  and  $k<<n$ , then  $f(k)<<f(n)$ . What’s the least possible value of  $M$ ?

Answer: 1024


Find the number of integers between 1 and 200 inclusive whose distinct prime divisors sum to 16.

Answer: 6


Jimmy finds that 8 bananas weigh the same as 6 oranges. If Jimmy has 40 oranges, and the weight of 4 oranges equals the weight of 5 apples, how many apples and bananas would Jimmy need to equal the weight of his 40 oranges?

Answer: 50


Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ ($a>0, b>0$), with left and right foci $F_{1}$ and $F_{2}$ respectively, a line passing through the focus $F_{2}$ intersects the right branch of the hyperbola at points $P$ and $Q$, and $P Q \perp P F_{1}$. If $|P Q| = \frac{5}{12} |P F_{1}|$, determine the eccentricity of the hyperbola.

Answer: \frac{\sqrt{37}}{5}


When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.
What is the remainder when $x+2uy$ is divided by $y$?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2u \qquad \textbf{(C)}\ 3u \qquad \textbf{(D)}\ v \qquad \textbf{(E)}\ 2v$

Answer: \textbf{(D)}\v


How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0 ! and 1 ! to be distinct.

Answer: 39


In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a=2b\sin A$.
$(1)$ Find angle $B$;
$(2)$ If $b=\sqrt{7}$, $c=3$, and $D$ is the midpoint of side $AC$, find $BD$.

Answer: \frac{\sqrt{19}}{2}


Determine the smallest constant \( k > 0 \) such that
$$
\frac{a b}{a+b+2 c}+\frac{b c}{b+c+2 a}+\frac{c a}{c+a+2 b} \leq k(a+b+c),
$$
for all \( a, b, c > 0 \).

Answer: \frac{1}{4}


In the arithmetic sequence $\{a_{n}\}$, given that $a_{3}=-2, a_{n}=\frac{3}{2}, S_{n}=-\frac{15}{2}$, find the value of $a_{1}$.

Answer: -\frac{19}{6}


If point \( P \) is on the curve \( y=\frac{1}{2} e^{x} \) and point \( Q \) is on the curve \( y=\ln (2 x) \), then the minimum value of \( |PQ| \) is \( \qquad \).

Answer: \sqrt{2}(1 - \ln 2)


In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, participants are ranked according to their score (if several participants have the same score, their order is chosen randomly). Each participant has earned half of their points in their matches against the last ten participants (in the ranking). How many people participated in this tournament?

Answer: 25


In a triangle, the lengths of the three sides are integers $l, m, n$, with $l > m > n$. It is known that $\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\}$, where $\{x\}=x-[x]$, and $[x]$ denotes the greatest integer not exceeding $x$. Find the minimum perimeter of such a triangle.

Answer: 3003


A "progressive number" refers to a positive integer in which each digit is greater than the digit to its left (such as 1 458). If the four-digit "progressive numbers" are arranged in ascending order, then the 30th number is \_\_\_\_\_\_\_\_.

Answer: 1359


The last two digits of $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (where [x] denotes the greatest integer less than or equal to $x$).

Answer: 8


There are 19 cards, each with a number written on it from $15, 16, 17, \ldots, 33$ respectively (one number per card). Vasya, Petya, and Misha, members of a math club, want to divide all these cards among themselves such that each person gets at least one card, and no one has a pair of cards where the difference between the numbers is odd. How many ways are there to divide the cards in this manner?

Answer: 4596


Express the quotient $1021_3 \div 11_3$ in base $3$.

Answer: 22_3


The Valbonne middle school has 367 students. Demonstrate that, among these students, there are at least two who have the same birthday.

Answer: 2


How many of the first 1000 positive integers can be expressed as \([2x] + [4x] + [6x] + [8x]\)?

Answer: 600


In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 15$, $\tan Z = \frac{4}{3}$, and $\tan X = \frac{3}{2}$, what is the length of $XY$?

Answer: \frac{20\sqrt{13}}{3}


The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs.

Answer: 12\pi


Let  $n \ge 4$  be a natural number. Let  $A_1A_2 \cdots A_n$  be a regular polygon and  $X = \{ 1,2,3....,n \} $ . A subset  $\{ i_1, i_2,\cdots, i_k \} $  of  $X$ , with  $k \ge 3$  and  $i_1 < i_2 < \cdots < i_k$ , is called a good subset if the angles of the polygon  $A_{i_1}A_{i_2}\cdots A_{i_k}$  , when arranged in the increasing order, are in an arithmetic progression. If  $n$  is a prime, show that a proper good subset of  $X$  contains exactly four elements.

Answer: 4


Let $A B C$ be a triangle with a perimeter of 2. Denote $a=B C, b=A C$ and $c=A B$. Show that:

$$
a^{2}+b^{2}+c^{2}<2(1-a b c)
$$

Answer: ^2+b^2+^2<2(1-abc)


Given that the operation "※" satisfies the following equations:
\(4 ※ 2 = 14\),
\(5 ※ 3 = 22\),
\(3 ※ 5 = 4\),
\(7 ※ 18 = 31\).

According to these rules, what is \(6 ※ 8\)?

Answer: 28


Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$.

Answer: 70


Given a rectangle \(ABCD\), a circle intersects the side \(AB\) at points \(K\) and \(L\), and the side \(CD\) at points \(M\) and \(N\). Find the length of segment \(MN\) if \(AK = 10\), \(KL = 17\), and \(DN = 7\).

Answer: 23


Eighty bricks, each measuring \(3'' \times 8'' \times 15''\), are to be stacked to form a column. Each brick can be oriented to contribute \(3''\), \(8''\), or \(15''\) to the total column height. Determine how many different total column heights can be achieved using all eighty bricks.

Answer: 961


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


Let \( S \) be the set of points \((x, y)\) in the plane such that the sequence \( a_n \) defined by \( a_0 = x \), \( a_{n+1} = \frac{a_n^2 + y^2}{2} \) converges. What is the area of \( S \)?

Answer: 4+\pi


Given an arithmetic sequence $\{a_n\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\{b_n\}$.

Answer: \frac {175}{264}


Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, let $F_1$ and $F_2$ be the left and right foci, and let $P$ and $Q$ be two points on the right branch. If $\overrightarrow{PF_2} = 2\overrightarrow{F_2Q}$ and $\overrightarrow{F_1Q} \cdot \overrightarrow{PQ} = 0$, determine the eccentricity of this hyperbola.

Answer: \frac{\sqrt{17}}{3}


Let  $\lambda$  be a positive real number satisfying  $\lambda=\lambda^{2/3}+1$ . Show that there exists a positive integer  $M$  such that  $|M-\lambda^{300}|<4^{-100}$ .

*Proposed by Evan Chen*

Answer: M


The height of an isosceles triangle, dropped perpendicularly to its lateral side, divides it into segments equal to 2 and 1, counting from the vertex of the triangle. Find the base of the triangle.

Answer: \sqrt{6}


There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let  $ N$  be the total number of intersection points of these circles. Determine the smallest possible values of  $ N$ .

Answer: 3012


Solve the equation \(x^{3}-3 y^{3}-9 z^{3}=0\) in integers.

Answer: (0,0,0)


A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?

Answer: 9


You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open all the doors and disarm all the mines. In the room is a panel with 3 buttons, which contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially, 3 doors are closed, and 3 mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?

Answer: 9


Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included.

Answer: 345


Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there of lining them up from left to right?

Answer: 12


Given a convex quadrilateral \(ABCD\) where \(AB=AD=1\), \(\angle A=80^\circ\), and \(\angle C=140^\circ\), find the length of the diagonal \(AC\).

Answer: 1


In triangle KIA, let point \( O \) be the foot of the median from vertex \( K \); \( Y \) be the foot of the perpendicular drawn from point \( I \) to the angle bisector of angle \( IOK \); \( Z \) be the foot of the perpendicular drawn from point \( A \) to the angle bisector of angle \( AOK \). Let \( X \) be the intersection point of segments \( KO \) and \( YZ \). Prove that \( YX = ZX \).

Answer: YX=ZX


A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly  $ k $  rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly  $\ell$  rectangles. How much the number of rectangles can be?

*I. Bogdanov, D. Fon-Der-Flaass*

Answer: k\ell


If the program flowchart on the right is executed, determine the value of the output S.

Answer: 55


What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{1001}+1\right)^{3/2}$?

Answer: 743


A math extracurricular activity group at a certain school designed a tree planting plan on graph paper for a desert as follows: The $k$-th tree is planted at point $P_{k}(x_{k}, y_{k})$, where $x_{1}=1$ and $y_{1}=1$. For $k \geq 2$, the coordinates are determined by:

\[
\begin{cases}
x_{k} = x_{k-1} + 1 - 5 \left[\frac{k-1}{5}\right] + 5 \left[\frac{k-2}{5}\right], \\
y_{k} = y_{k-1} + \left[\frac{k-1}{5}\right] - \left[\frac{k-2}{5}\right],
\end{cases}
\]

where $[a]$ denotes the integer part of the real number $a$, for example, $[2.6] = 2$ and $[0.6] = 0$. According to this plan, the coordinates of the 2008th tree are _____

Answer: (3,402)


A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is:
$\text{(A) } \frac{2}{n-1}\quad \text{(B) } \frac{n-1}{2}\quad \text{(C) } \frac{2}{n}\quad \text{(D) } 2n\quad \text{(E) } 2(n-1)$

Answer: \textbf{(E)}2(n-1)


A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\triangle DEF$.

Answer: 225


Consider the matrix
\[\mathbf{N} = \begin{pmatrix} 2x & -y & z \\ y & x & -2z \\ y & -x & z \end{pmatrix}\]
and it is known that $\mathbf{N}^T \mathbf{N} = \mathbf{I}$. Find $x^2 + y^2 + z^2$.

Answer: \frac{2}{3}


In English class, you have discovered a mysterious phenomenon -- if you spend  $n$  hours on an essay, your score on the essay will be  $100\left( 1-4^{-n} \right)$  points if  $2n$  is an integer, and  $0$  otherwise. For example, if you spend  $30$  minutes on an essay you will get a score of  $50$ , but if you spend  $35$  minutes on the essay you somehow do not earn any points.

It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?

*Proposed by Evan Chen*

Answer: 75


Real numbers \( r, s, t \) satisfy \( 1 \leq r \leq s \leq t \leq 4 \). Find the minimum value of \( (r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2} \).

Answer: 12-8\sqrt{2}


Let $r$ and $s$ be positive integers such that\[\frac{5}{11} < \frac{r}{s} < \frac{4}{9}\]and $s$ is as small as possible. What is $s - r$?

Answer: 11


The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?

Answer: 36


A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.
$(1)$ Find the probability that exactly one red ball is drawn;
$(2)$ Let the random variable $X$ represent the number of red balls drawn. Find the distribution of the random variable $X$.

Answer: \frac{3}{10}


Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could turn out to be rational numbers?

Answer: 1250


Given  $w$  and  $z$  are complex numbers such that  $|w+z|=1$  and  $|w^2+z^2|=14$ , find the smallest possible value of  $|w^3+z^3|$ . 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}


A student plotted the graph of the function \( y = ax^2 + bx + c \) (\(a \neq 0\)) by choosing 7 different values for the variable \( x \): \( x_1 < x_2 < \cdots < x_7 \). They set \( x_2 - x_1 = x_3 - x_2 = \cdots = x_7 - x_6 \) and calculated the corresponding \( y \) values, creating the following table:
\[
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ \\
\hline
$y$ & 51 & 107 & 185 & 285 & 407 & 549 & 717 \\
\hline
\end{tabular}
\]

However, due to a calculation error, one of the \( y \) values is incorrect. Which \( y \) value is incorrect, what should the correct value be, and why?

Answer: 551


Given that \( P \) is the circumcenter of \( \triangle ABC \) and \(\overrightarrow{PA} + \overrightarrow{PB} = \lambda \overrightarrow{PC}\), where \(\tan C = \frac{12}{5}\), find the real number \(\lambda\).

Answer: -\frac{10}{13}


Square \(ABCD\) has a side length of 4. Point \(E\) is the midpoint of \(AB\), and \(F\) is a variable point on side \(BC\). Triangles \(\triangle ADE\) and \(\triangle DCF\) are folded along \(DE\) and \(DF\) respectively, so that points \(A\) and \(C\) coincide at point \(A'\). Find the maximum distance from point \(A'\) to the plane \(DEF\).

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


Find the maximum value of the parameter \(a\) for which the equation \((|x-2|+2a)^{2}-3(|x-2|+2a)+4a(3-4a)=0\) has three solutions. Specify the largest value in your answer.

Answer: 0.5


Let  $\mathbb{N}$  denote the set of positive integers, and let  $S$  be a set. There exists a function  $f :\mathbb{N} \rightarrow S$  such that if  $x$  and  $y$  are a pair of positive integers with their difference being a prime number, then  $f(x) \neq f(y)$ . Determine the minimum number of elements in  $S$ .

Answer: 4


In the Land of Fools, there are \(N^2\) cities arranged in a square grid, with the distance between neighboring cities being 10 km. The cities are connected by a road system consisting of straight sections that are parallel to the sides of the square. What is the minimum length of such a road system if it is known that it is possible to travel from any city to any other city?

Answer: 10(N^2-1)


Let $H$ be the orthocenter of triangle $ABC.$  For all points $P$ on the circumcircle of triangle $ABC,$
\[PA^2 + PB^2 + PC^2 - PH^2\]is a constant.  Express this constant in terms of the side lengths $a,$ $b,$ $c$ and circumradius $R$ of triangle $ABC.$

Answer: ^2+b^2+^2-4R^2


Let \( A_n \) denote the \((n-1) \times (n-1)\) matrix \( (a_{ij}) \) with \( a_{ij} = i + 2 \) for \( i = j \), and 1 otherwise. Is the sequence \( \frac{\det A_n}{n!} \) bounded?

Answer: No


Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible number of integer sums that could have resulted?

Answer: 511


Given that the height of a cylinder is $2$, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of $2\sqrt{6}$, calculate the surface area of the cylinder.

Answer: (10+4\sqrt{5})\pi


In the two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$.

Answer: 32


*Nocycleland* is a country with  $500$  cities and  $2013$  two-way roads, each one of them connecting two cities. A city  $A$  *neighbors*  $B$  if there is one road that connects them, and a city  $A$  *quasi-neighbors*  $B$  if there is a city  $C$  such that  $A$  neighbors  $C$  and  $C$  neighbors  $B$ .
It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities  $A$ ,  $B$ ,  $C$  and  $D$  such that  $A$  neighbors  $B$ ,  $B$  neighbors  $C$ ,  $C$   neighbors  $D$ , and  $D$  neighbors  $A$ .
Show that there is at least one city that quasi-neighbors at least  $57$  other cities.

Answer: 57


11 groups perform at a festival. Each day any groups not performing watch the others (but groups performing that day do not watch the others). What is the smallest number of days for which the festival can last if every group watches every other group at least once during the festival?

Answer: 6


Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.

Answer: \frac{54+9 \sqrt{3}}{8}


Given a cube \(ABCD-A_1B_1C_1D_1\) with edge length 1, a sphere is centered at vertex \(A\) with radius \(\frac{2 \sqrt{3}}{3}\). What is the length of the curve formed by the intersection of the sphere's surface with the surface of the cube?

Answer: \frac{5\sqrt{3}\pi}{6}


Find all positive integers  $n$  such that the set  $S=\{1,2,3, \dots 2n\}$  can be divided into  $2$  disjoint subsets  $S_1$  and  $S_2$ , i.e.  $S_1 \cap S_2 = \emptyset$  and  $S_1 \cup S_2 = S$ , such that each one of them has  $n$  elements, and the sum of the elements of  $S_1$  is divisible by the sum of the elements in  $S_2$ .

*Proposed by Viktor Simjanoski*

Answer: n\not\equiv5\pmod{6}


A cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}, \overline{P_1P_3}, \overline{P_1P_4}, \overline{P_1'P_2'}, \overline{P_1'P_3'},$ and $\overline{P_1'P_4'}$ with each vertex positioned $\frac{2}{3}$ of the distance from $P_1$ or $P_1'$.

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


If non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}| = |\overrightarrow{b}|$ and $(\sqrt{3}\overrightarrow{a} - 2\overrightarrow{b}) \cdot \overrightarrow{a} = 0$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________.

Answer: \frac{\pi}{6}


The angle at vertex $A$ of trapezoid $ABCD$ is $\alpha$. The leg $AB$ is twice the length of the shorter base $BC$. Find the angle $BAC$.

Answer: \arctan(\frac{\sin\alpha}{2+\cos\alpha})


Given positive integers \(a_{1}, a_{2}, \ldots, a_{2023}\) such that \(a_{k}=\sum_{i=1}^{2023}\left|a_{k}-a_{i}\right|\) for all \(1 \leq k \leq 2023\), find the minimum possible value of \(a_{1}+a_{2}+\cdots+a_{2023}\).

Answer: 2046264


Given four Chinese idioms: "一家之言", "言扬行举", "举世皆知", and "知行合一", each character in these idioms represents one of 11 consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. If the sum of the numbers represented by the four characters in each idiom is 21, what is the maximum number that "行" can represent?

Answer: 8


Given the set \( S = \{1, 2, \ldots, 10\} \), some five-element subsets of \( S \) are to be chosen such that any two elements of \( S \) appear together in at most two of the chosen five-element subsets. What is the maximum number of such five-element subsets that can be chosen?

Answer: 8


For an acute-angled triangle,

$$
\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma \leqslant \sin (\alpha+\β)+\sin (\beta+\gamma)+\sin (\γ+\α)
$$

Answer: \sin2\alpha+\sin2\beta+\sin2\gamma\le\sin(\alpha+\beta)+\sin(\beta+\gamma)+\sin(\gamma+\alpha)


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

Answer: 334000


There are 100 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers?

Answer: 34


The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression?

Answer: 32


From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers.

Answer: 525


Consider a circle of radius  $6$ . Let  $B,C,D$  and  $E$  be points on the circle such that  $BD$  and  $CE$ , when extended, intersect at  $A$ . If  $AD$  and  $AE$   have length  $5$  and  $4$  respectively, and  $DBC$  is a right angle, then show that the length of  $BC$  is  $\frac{12+9\sqrt{15}}{5}$ .

Answer: \frac{12+9\sqrt{15}}{5}


Does there exist an infinite increasing sequence of natural numbers \(a_{1}, a_{2}, a_{3}, \ldots\) such that the sum of any two distinct terms of the sequence is coprime with the sum of any three distinct terms of the sequence?

Answer: Yes


Among nine beauties, some are always honest and tell the truth (angels), while the rest always lie (devils). The first beauty says: "There is exactly 1 devil among us"; the second beauty says: "There are exactly 2 angels among us"; the third beauty says: "There are exactly 3 devils among us"; the fourth beauty says: "There are exactly 4 angels among us"; the fifth beauty says: "There are exactly 5 devils among us"; the sixth beauty says: "There are exactly 6 angels among us"; the seventh beauty says: "There are exactly 7 devils among us"; the eighth beauty says: "There are exactly 8 angels among us"; the ninth beauty says: "There are exactly 9 devils among us." How many angels are there among these beauties?

Answer: 2


Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers.

Answer: 1/4


Find all positive integers  $n$  that have 4 digits, all of them perfect squares, and such that  $n$  is divisible by 2, 3, 5 and 7.

Answer: 4410


Let \( n \) be the smallest positive integer that satisfies the following conditions: (1) \( n \) is a multiple of 75; (2) \( n \) has exactly 75 positive integer factors (including 1 and itself). Find \(\frac{n}{75}\).

Answer: 432


 $ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$   satisfy the given conditions

 $ a)$   $ f(0,0)\equal{}1$  ,  $ f(0,1)\equal{}1$  ,

 $ b)$   $ \forall k \notin \left\{0,1\right\}$    $ f(0,k)\equal{}0$  and

 $ c)$   $ \forall n \geq 1$  and  $ k$  ,  $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$

find the sum  $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$

Answer: 2^{2008}


Given two lines $l_1: ax+2y+6=0$, and $l_2: x+(a-1)y+a^2-1=0$.
(1) If $l_1 \perp l_2$, find the value of $a$;
(2) If $l_1 \parallel l_2$, find the value of $a$.

Answer: -1


In the rectangle PQRS, side PQ is of length 2 and side QR is of length 4. Points T and U lie inside the rectangle such that ∆PQT and ∆RSU are equilateral triangles. Calculate the area of quadrilateral QRUT.

Answer: 4 - \sqrt{3}


Given that $O$ is the circumcenter of $\triangle ABC$, and $D$ is the midpoint of $BC$, if $\overrightarrow{AO} \cdot \overrightarrow{AD} = 4$ and $BC = 2\sqrt{6}$, find the length of $AD$.

Answer: \sqrt{2}


In an archery competition held in a certain city, after ranking the scores, the average score of the top seven participants is 3 points lower than the average score of the top four participants. The average score of the top ten participants is 4 points lower than the average score of the top seven participants. How many points more is the sum of the scores of the fifth, sixth, and seventh participants compared to the sum of the scores of the eighth, ninth, and tenth participants?

Answer: 28


Find the solutions to $z^3 = -8.$  Enter the solutions, separated by commas.

Answer: -2, 1 + i \sqrt{3}, 1 - i \sqrt{3}


How many ordered triples of nonzero integers  $(a, b, c)$  satisfy  $2abc = a + b + c + 4$ ?

Answer: 6


Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$
[asy] import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(1,1.63),W); dot((1,3),ds); label("$A$",(1,3),N); dot((0,0),ds); label("$B$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]

Answer: 616


The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out of the tournament. If 3 of the 16 teams are Spanish, what is the probability that there are 2 Spanish teams in the final round?

Answer: $\frac{4}{5}$


Find the number of pairs  $(n,C)$  of positive integers such that  $C\leq 100$  and  $n^2+n+C$  is a perfect square.

Answer: 180


Given $\left(a + \frac{1}{a}\right)^2 = 5$, find the value of $a^3 + \frac{1}{a^3}$.

Answer: 2\sqrt{5}


Find the smallest integer starting with the digit 7 that becomes three times smaller when this digit is moved to the end. Find all such numbers.


Sequences  $a_n$  and  $b_n$  are defined for all positive integers  $n$  such that  $a_1 = 5,$   $b_1 = 7,$    $$ a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2}, $$  and   $$ b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}. $$   $ $
How many integers  $n$  from 1 to 1000 satisfy the property that  $a_n, b_n$  form the legs of a right triangle with a hypotenuse that has integer length?

Answer: 24


The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).

Find the angle \( \angle AMB \).

Answer: 72


Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standard is 2 yuan per hour (with fractions of an hour calculated as a full hour). Suppose two individuals, A and B, each rent a bicycle once. The probability that A and B return their bicycles within two hours is $\frac{1}{4}$ and $\frac{1}{2}$, respectively. The probability that they return their bicycles between two and three hours is $\frac{1}{2}$ and $\frac{1}{4}$, respectively. Neither A nor B will rent a bicycle for more than four hours.

(I) Calculate the probability that the bicycle rental fees paid by A and B are the same.
(II) Let $\xi$ be the random variable representing the sum of the bicycle rental fees paid by A and B. Find the distribution of $\xi$ and its mathematical expectation $E_{\xi}$.

Answer: \frac{7}{2}


Given tetrahedron $P-ABC$, if one line is randomly selected from the lines connecting the midpoints of each edge, calculate the probability that this line intersects plane $ABC$.

Answer: \frac{3}{5}