Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

Answer: 630


Simão needs to find a number that is the code to the Treasure Chest, which is hidden in the table.

| 5 | 9 | 4 | 9 | 4 | 1 |
|---|---|---|---|---|---|
| 6 | 3 | 7 | 3 | 4 | 8 |
| 8 | 2 | 4 | 2 | 5 | 5 |
| 7 | 4 | 5 | 7 | 5 | 2 |
| 2 | 7 | 6 | 1 | 2 | 8 |
| 5 | 2 | 3 | 6 | 7 | 1 |

To discover the code, he needs to form groups of 3 digits that are in successive cells, either horizontally or vertically, and whose sum is 14. Once these groups are identified, the code is the sum of the numbers that do not appear in these groups. What is this code?

Answer: 29


Let \( \omega^3 = 1 \), where \( \omega \neq 1 \). Show that \( z_1 \), \( z_2 \), and \(-\omega z_1 - \omega^2 z_2\) are the vertices of an equilateral triangle.

Answer: z_1,z_2,-\omegaz_1-\omega^2z_2


Find the minimum point of the function $f(x)=x+2\cos x$ on the interval $[0, \pi]$.

Answer: \dfrac{5\pi}{6}


Consider the following sequence  $$ (a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots) $$  Find all pairs  $(\alpha, \beta)$  of positive real numbers such that  $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$ .

(Proposed by Tomas Barta, Charles University, Prague)

Answer: (\frac{3}{2},\frac{\sqrt{2}}{3})


There is a positive real number $x$ not equal to either $\tfrac{1}{20}$ or $\tfrac{1}{2}$ such that\[\log_{20x} (22x)=\log_{2x} (202x).\]The value $\log_{20x} (22x)$ can be written as $\log_{10} (\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 112


Let \( P \in \mathbf{C}[X] \) be such that for all \( z \in \mathbf{C} \) with \( |z| = 1 \), \( P(z) \) is real.

Show that \( P \) is constant.

Answer: P


Find all functions \( f \) from \(\mathbb{R}_{+}^{*}\) to \(\mathbb{R}_{+}^{*}\) that satisfy the following condition:

\[
\forall x, y \in \mathbb{R}_{+}^{*}, \quad f(x+f(y))=f(x+y)+f(y)
\]

Answer: f(x)=2x


Given point \( A \) has coordinates \( (0,3) \), and points \( B \) and \( C \) are on the circle \( O: x^{2}+y^{2}=25 \), satisfying \(\angle B A C=90^{\circ}\). Find the maximum area of \(\triangle A B C\).

Answer: \frac{25+3\sqrt{41}}{2}


Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing?

Answer: 126.5625


Find the biggest natural number  $m$  that has the following property: among any five 500-element subsets of  $\{ 1,2,\dots, 1000\}$  there exist two sets, whose intersection contains at least  $m$  numbers.

Answer: 200


Let the sequence  $(a_{n})$  be deﬁned by  $a_{1} = t$  and  $a_{n+1} = 4a_{n}(1 - a_{n})$  for  $n \geq 1$ . How many possible values of t are there, if  $a_{1998} = 0$ ?

Answer: 2^{1996}+1


On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one letter per magnet. Two vowels and three consonants fall off and are put away in a bag. If the Ts are indistinguishable, how many distinct possible collections of letters could be put in the bag?

Answer: 75


From the numbers $1, 2, \cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$.

Answer: 16


Assume real numbers  $a_i,b_i\,(i=0,1,\cdots,2n)$  satisfy the following conditions:
(1) for  $i=0,1,\cdots,2n-1$ , we have  $a_i+a_{i+1}\geq 0$ ;
(2) for  $j=0,1,\cdots,n-1$ , we have  $a_{2j+1}\leq 0$ ;
(2) for any integer  $p,q$ ,  $0\leq p\leq q\leq n$ , we have  $\sum_{k=2p}^{2q}b_k>0$ .
Prove that  $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$ , and determine when the equality holds.

Answer: \sum_{i=0}^{2n}(-1)^ia_ib_i\ge0


Given real numbers \( x \) and \( y \) satisfy \( x^2 + y^2 = 20 \). Find the maximum value of \( xy + 8x + y \).

Answer: 42


Let \( M \) and \( N \) be two points on the Thales' circle of segment \( AB \), distinct from \( A \) and \( B \). Let \( C \) be the midpoint of segment \( NA \), and \( D \) be the midpoint of segment \( NB \). The circle is intersected at the point \( E \) a second time by the line \( MC \), and at point \( F \) by the line \( MD \). What is the value of the expression

\[
MC \cdot CE + MD \cdot DF
\]

if \( AB = 2 \) units?

Answer: 1


Find all binary operations  $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$  (meaning  $\diamondsuit$  takes pairs of positive real numbers to positive real numbers) such that for any real numbers  $a, b, c > 0$ ,

- the equation  $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$  holds; and
- if  $a\ge 1$  then  $a\,\diamondsuit\, a\ge 1$ .

*Evan Chen*

Answer: \diamondsuit\frac{}{b}


How many positive, three-digit integers contain at least one $4$ as a digit but do not contain a $6$ as a digit?

Answer: 200


In an equilateral triangle $\triangle PRS$, if $QS=QT$ and $\angle QTS=40^\circ$, what is the value of $x$?

Answer: 80


A quadratic function $f(x) = x^2 - ax + b$ has one root in the interval $[-1, 1]$ and another root in the interval $[1, 2]$. Find the minimum value of $a - 2b$.
A. 0
B. $-1$
C. -2
D. 1

Answer: -1


Can you find two numbers such that the difference of their squares is a cube, and the difference of their cubes is a square? What are the two smallest numbers with this property?

Answer: 106


Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([4, 6]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{9 - \sqrt{19}}{2}\right)) \ldots) \). If necessary, round the answer to two decimal places.

Answer: 6.68


Let $ABC$ be an equilateral triangle. Suppose that the points on sides $AB$, $BC$, and $AC$, including vertices $A$, $B$, and $C$, are divided arbitrarily into two disjoint subsets. Then at least one of the subsets contains the vertex of a right triangle.

Answer: 1


Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each running at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \mathrm{~m}$ apart. They start at the same time, but when Luíza finishes the race, Ana still has $120 \mathrm{~m}$ to reach point $B$. Yesterday, Luíza gave Ana a chance: "We will start at the same time, but I will start some meters before point $A$ so that we arrive together." How many meters before point $A$ should Luíza start?

Answer: 125


Let $R$ be the set of all real numbers. Find all functions $f: R \rightarrow R$ such that for all $x, y \in R$, the following equation holds:
$$
f\left(x^{2}+f(y)\right)=y+(f(x))^{2}.
$$

Answer: f(x)=x


Find the millionth digit after the decimal point in the decimal representation of the fraction \( \frac{3}{41} \).

Answer: 7


Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area.

Answer: \frac{5}{2}


Complex numbers \(a, b, c\) form an equilateral triangle with side length 18 in the complex plane. If \(|a+b+c|=36\), find \(|bc + ca + ab|\).

Answer: 432


Given a set of paired data $(18,24)$, $(13,34)$, $(10,38)$, $(-1,m)$, the regression equation for these data is $y=-2x+59.5$. Find the correlation coefficient $r=$______(rounded to $0.001$).

Answer: -0.998


The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9.

Answer: 84


The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon.  What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
pair[] bigHexagon = new pair[6];
bigHexagon[0] = dir(0);
pair[] smallHexagon = new pair[6];
smallHexagon[0] = (dir(0) + dir(60))/2;
for(int i = 1; i <= 7; ++i){

bigHexagon[i] = dir(60*i);

draw(bigHexagon[i]--bigHexagon[i - 1]);

smallHexagon[i] = (bigHexagon[i] + bigHexagon[i - 1])/2;

draw(smallHexagon[i]--smallHexagon[i - 1]);
}
dot(Label("$A$",align=dir(0)),dir(0)); dot(Label("$B$",align=dir(60)),dir(60)); dot(Label("$C$",align=dir(120)),dir(120)); dot(Label("$D$",align=dir(180)),dir(180)); dot(Label("$E$",align=dir(240)),dir(240)); dot(Label("$F$",align=dir(300)),dir(300));
[/asy]

Answer: \frac{3}{4}


In the Cartesian coordinate system $xoy$, point $P(0, \sqrt{3})$ is given. The parametric equation of curve $C$ is $\begin{cases} x = \sqrt{2} \cos \varphi \\ y = 2 \sin \varphi \end{cases}$ (where $\varphi$ is the parameter). A polar coordinate system is established with the origin as the pole and the positive half-axis of $x$ as the polar axis. The polar equation of line $l$ is $\rho = \frac{\sqrt{3}}{2\cos(\theta - \frac{\pi}{6})}$.

(Ⅰ) Determine the positional relationship between point $P$ and line $l$, and explain the reason;
(Ⅱ) Suppose line $l$ intersects curve $C$ at two points $A$ and $B$, calculate the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$.

Answer: \sqrt{14}


Let $x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}$. What is the greatest integer that does not exceed $100x$?

Answer: 241


Consider the natural implementation of computing Fibonacci numbers:

\begin{tabular}{l}
1: \textbf{FUNCTION}  $\text{FIB}(n)$ :
2: $\qquad$  \textbf{IF}  $n = 0$  \textbf{OR}  $n = 1$  \textbf{RETURN} 1
3: $\qquad$  \textbf{RETURN}  $\text{FIB}(n-1) + \text{FIB}(n-2)$ \end{tabular}

When  $\text{FIB}(10)$  is evaluated, how many recursive calls to  $\text{FIB}$  occur?

Answer: 176


Find the number of ways in which the letters in "HMMTHMMT" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMTTHH" satisfies this property, but "HHTMMMTM" does not.

Answer: 12


Place 7 different-colored goldfish into 3 glass tanks numbered 1, 2, and 3, such that the number of fish in each tank is not less than the number on its label. Determine the number of different ways to do this.

Answer: 455


Given an ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) (where \(a > b > 0\)) with the left focal point \(F\), and a point \(P(x_{0}, y_{0})\) on the ellipse where \(x_{0} > 0\). A tangent to the circle \(x^{2} + y^{2} = b^{2}\) is drawn at point \(P\), intersecting the ellipse at a second point \(Q\). Let \(I\) be the incenter of triangle \(PFQ\), and \(\angle PFQ = 2\alpha\). Then \( |FI|\cos \alpha = \) ___.

Answer: a


Let $n$ be a natural number with the following property: If 50 different numbers are randomly chosen from the numbers $1, 2, \ldots, n$, there will necessarily be two numbers among them whose difference is 7. Find the maximum value of such $n$.

Answer: 98


Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function such that for all \( x, y \in \mathbb{R} \):

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

Let \(\alpha=f(0)\). Prove that:

(i) \(f(\alpha) f(-\alpha)=0\),

(ii) \(\alpha=0\),

(iii) \(f(x)=x\) for all \( x \in \mathbb{R} \).

Answer: f(x)=x


A certain lottery has tickets labeled with the numbers \(1, 2, 3, \ldots, 1000\). The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize.
You have ticket number 1000. What is the probability that you get a prize?

Answer: \frac{1}{501}


Six people stand in a row, with exactly two people between A and B. Calculate the number of different ways for them to stand.

Answer: 144


A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$

Answer: \textbf{(E)}\26


When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man?

Answer: 13


Find all positive integers $n > 3$ such that there exist $n$ points $A_{1}, A_{2}, \cdots, A_{n}$ in the plane and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ satisfying the following conditions:
(1) Any 3 points among $A_{1}, A_{2}, \cdots, A_{n}$ are not collinear;
(2) For each triplet of points $\left\{A_{i}, A_{j}, A_{k}\right\}$ (where $1 \leq i < j < k \leq n$), the area of triangle $\triangle A_{i} A_{j} A_{k}$, denoted by $S_{ijk}$, is equal to $r_{i} + r_{j} + r_{k}$.

(Note: This is a problem from the 36th IMO in 1995)

Answer: 4


Answer: 719


A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts.

Answer: \dfrac{3}{10}


Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.

Answer: \sqrt{2}


Let $ABCD$ be a parallelogram with $\angle ABC=135^\circ$, $AB=14$ and $BC=8$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=3$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of segment $FD$.
A) $\frac{6}{17}$
B) $\frac{18}{17}$
C) $\frac{24}{17}$
D) $\frac{30}{17}$

Answer: \frac{24}{17}


We place flower pots with masses $1 \mathrm{~kg}$, $2 \mathrm{~kg}$, and $3 \mathrm{~kg}$ at the corners of an equilateral weightless triangular board. Where should we suspend the board so that the flower pots do not fall off?

Answer: (\frac{7}{12},\frac{\sqrt{3}}{4})


On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the digits is equal to 13. What is the distance from Yolkino to Palkino?

Answer: 49


Evaluate \(\left(a^a - a(a-2)^a\right)^a\) when \( a = 4 \).

Answer: 1358954496


It is known that there are a total of $n$ students in the first grade of Shuren High School, with $550$ male students. They are divided into layers based on gender, and $\frac{n}{10}$ students are selected to participate in a wetland conservation knowledge competition. It is given that there are $10$ more male students than female students among the participants. Find the value of $n$.

Answer: 1000


Answer: 671


Let \( M, K \), and \( L \) be points on \( AB \), \( BC \), and \( CA \), respectively. Prove that the area of at least one of the three triangles \( \triangle MAL \), \( \triangle KBM \), and \( \triangle LCK \) is less than or equal to one-fourth the area of \( \triangle ABC \).

Answer: 1


Answer: 1


We rotate the square \(ABCD\) with a side length of 1 around its vertex \(C\) by \(90^\circ\). What area does side \(AB\) sweep out?

Answer: \frac{\pi}{4}


In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$.  Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively.  What is the minimum possible value of $BE+DE+CD$?

Answer: 14


A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
[asy] for (int i=0; i<12; ++i){  for (int j=0; j<i; ++j){    //dot((-j+i/2,-i));    draw((-j+i/2,-i)--(-j+i/2+1,-i)--(-j+i/2+1,-i+1)--(-j+i/2,-i+1)--cycle);  } } [/asy]

Answer: 640


Given a triangle ABC, let the lengths of the sides opposite to angles A, B, C be a, b, c, respectively. If a, b, c satisfy $a^2 + c^2 - b^2 = \sqrt{3}ac$,
(1) find angle B;
(2) if b = 2, c = $2\sqrt{3}$, find the area of triangle ABC.

Answer: 2\sqrt{3}


Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. The largest possible value of $d$ can be written in the form $r-s\sqrt{t}$, where $r, s$ and $t$ are positive integers and $t$ is not divisible by the square of any prime. Find $r+s+t$.

Answer: 14


Seven natural numbers are written in a circle. Prove that there are two adjacent numbers whose sum is even.

Answer: 2


The function \( g(x) \) satisfies
\[ g(x) - 2 g \left( \frac{1}{x} \right) = 3^x + x \]
for all \( x \neq 0 \). Find \( g(2) \).

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


After watching a movie, viewers rated the movie sequentially with an integer score from 0 to 10. At any given moment, the movie rating was calculated as the sum of all given scores divided by their number. At a certain moment $T$, the rating turned out to be an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers who could have voted after moment $T ?$

Answer: 5


In triangle \(ABC\), \(AB = 15\), \(BC = 12\), and \(AC = 18\). In what ratio does the center \(O\) of the incircle of the triangle divide the angle bisector \(CM\)?

Answer: 2:1


Let  $n$  be the product of the first  $10$  primes, and let   $$ S=\sum_{xy\mid n} \varphi(x) \cdot y, $$  where  $\varphi(x)$  denotes the number of positive integers less than or equal to  $x$  that are relatively prime to  $x$ , and the sum is taken over ordered pairs  $(x, y)$  of positive integers for which  $xy$  divides  $n$ . Compute  $\tfrac{S}{n}.$

Answer: 1024


If $r$ is the remainder when each of the numbers $1059,~1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals
$\textbf{(A) }1\qquad \textbf{(B) }15\qquad \textbf{(C) }179\qquad \textbf{(D) }d-15\qquad  \textbf{(E) }d-1$

Answer: \textbf{(B)}15


Given positive real numbers \( x \) and \( y \) satisfy:
\[
\left(2 x+\sqrt{4 x^{2}+1}\right)\left(\sqrt{y^{2}+4}-2\right) \geqslant y
\]
then the minimum value of \( x + y \) is ______.

Answer: 2


How many ways can one color the squares of a \(6 \times 6\) grid red and blue such that the number of red squares in each row and column is exactly 2?

Answer: 67950


A jar contains $5$ different colors of gumdrops. $30\%$ are blue, $20\%$ are brown, $15\%$ are red, $10\%$ are yellow, and other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
$\textbf{(A)}\ 35\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64$

Answer: \textbf{(C)}\42


From a point \( M \) on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), two tangent lines are drawn to the circle with the minor axis of the ellipse as its diameter. The points of tangency are \( A \) and \( B \). The line \( AB \) intersects the \(x\)-axis and \(y\)-axis at points \( P \) and \( Q \) respectively. Find the minimum value of \(|PQ|\).

Answer: 10/3


Find the largest positive integer  $n$  such that the number  $(2n)!$  ends with  $10$  more zeroes than the number  $n!$ .

*Proposed by Andy Xu*

Answer: 42


Answer: 111111_{(2)}


Let \( A \) and \( B \) be two points in a plane, and let \( (d) \) be a line that does not intersect the segment \([A B]\). Determine (geometrically) the point \( M \) on \( (d) \) for which the angle \(\widehat{A M B}\) is maximal.

Answer: M


Add together all natural numbers less than 1980 for which the sum of their digits is even!

Answer: 979605


Say a real number  $r$  is \emph{repetitive} if there exist two distinct complex numbers  $z_1,z_2$  with  $|z_1|=|z_2|=1$  and  $\{z_1,z_2\}\neq\{-i,i\}$  such that
\[
z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1).
\]
There exist real numbers  $a,b$  such that a real number  $r$  is \emph{repetitive} if and only if  $a < r\le b$ . If the value of  $|a|+|b|$  can be expressed in the form  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ , find  $100p+q$ .

*Proposed by James Lin*

Answer: 2504


Niko counted a total of 60 birds perching in three trees. Five minutes later, 6 birds had flown away from the first tree, 8 birds had flown away from the second tree, and 4 birds had flown away from the third tree. He noticed that there was now the same number of birds in each tree. How many birds were originally perched in the second tree?

A) 14
B) 18
C) 20
D) 21
E) 22

Answer: 22


For positive constant  $a$ , let  $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$ . Denote by  $l(t)$  the length of the part  $a\leq y\leq t$  for  $C$  and denote by  $S(t)$  the area of the part bounded by the line  $y=t\ (a<t)$  and  $C$ . Find  $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

Answer: a


Let \( f_{1} = 9 \) and
\[ f_{n} = \begin{cases}
f_{n-1} + 3 & \text{if } n \text{ is a multiple of } 3 \\
f_{n-1} - 1 & \text{if } n \text{ is not a multiple of } 3
\end{cases} \]
If \( B \) is the number of possible values of \( k \) such that \( f_{k} < 11 \), determine the value of \( B \).

Answer: 5


The numbers $a, b, c, d$ belong to the interval $[-6.5, 6.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.

Answer: 182


A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing?

Answer: \frac{3}{20}


From a sheet of squared paper measuring $29 \times 29$ cells, 99 squares have been cut out, each consisting of four cells. Prove that it is possible to cut out one more square.

Answer: 1


Let  $n$  points be given inside a rectangle  $R$  such that no two of them lie on a line parallel to one of the sides of  $R$ . The rectangle  $R$  is to be dissected into smaller rectangles with sides parallel to the sides of  $R$  in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect  $R$  into at least  $n + 1$  smaller rectangles.

*Proposed by Serbia*

Answer: n+1


Let  $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$

(1) Evaluate  $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$ .

(2) Find the area of the region bounded by two curves  $ y\equal{}f(x)$  and  $ y\equal{}g(x)\ (0\leq x\leq \pi).$

Answer: \frac{8\sqrt{2+\sqrt{2}}-4}{3}


Given the equations \( x^{2} + px + q = 0 \) and \( x^{2} + rx + s = 0 \) where one root of the first equation is \(\alpha\) and one root of the second equation is \(\beta\), and \(\alpha / \beta = k\). Prove the following equality:

\[ \left(q - k^{2} s\right)^{2} + k(p - k r)(k p s - q r) = 0 \]

Answer: (k^{2})^{2}+k(p-kr)(kp-)=0


Place four small spheres, each with a radius of 1, into a larger sphere. What is the minimum possible radius of the larger sphere?

Answer: 1+\frac{\sqrt{6}}{2}


**
How many non-similar regular 500-pointed stars are there?

**

Answer: 99


What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped?

Answer: 1/24


How many of the smallest 216 positive integers written in base 6 use the digit 3 or 4 (or both) as a digit?

Answer: 168


Find any solution to the rebus

$$
\overline{A B C A}=182 \cdot \overline{C D}
$$

where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).

As an answer, write the four-digit number \(\overline{A B C D}\).

Answer: 2916


In a triangular pyramid \( SABC \), the lateral edge \( SC \) is equal to the edge \( AB \) and is inclined to the plane of the base \( ABC \) at an angle of \( 60^\circ \). It is known that the vertices \( A, B, C \) and the midpoints of the lateral edges of the pyramid are located on a sphere with a radius of 1. Prove that the center of this sphere lies on the edge \( AB \) and find the height of the pyramid.

Answer: \sqrt{3}


All integers from 1 to \(2n\) are written in a row. Then, the position number is added to each of these integers.

Prove that among the obtained sums, there are at least two that give the same remainder when divided by \(2n\).

Answer: 2


Given that the sine and cosine values of angle $α$ are both negative, and $\cos(75^{\circ}+α)=\frac{1}{3}$, find the value of $\cos(105^{\circ}-α)+\sin(α-105^{\circ})$ = \_\_\_\_\_\_.

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


Given a sequence $\{x_n\}$ that satisfies $x_{n+2}=|x_{n+2}-x_n|$ (where $n \in \mathbb{N}^*$), if $x_1=1$, $x_2=a$ (where $a \leqslant 1$ and $a \neq 0$), and $x_{n+3}=x_n$ for any positive integer $n$, then the sum of the first 2017 terms of the sequence $\{x_n\}$ is ______.

Answer: 1345


Given a four-digit number $\overline{A B C D}$ that satisfies the following properties: $\overline{A B}$, $\overline{B C}$, and $\overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ and $81 = 9^2$). What is the sum of all four-digit numbers that satisfy this property?

Answer: 13462


There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points)
(Ⅰ) If the 3 girls must stand together, how many different arrangements are there?
(Ⅱ) If no two girls are next to each other, how many different arrangements are there?
(Ⅲ) If there are exactly three people between person A and person B, how many different arrangements are there?

Answer: 720


A snail is crawling forward at a non-uniform speed (without moving backward). Several people observe its crawling in a 6-minute span, each starting their observation before the previous person finishes. Each observer watches the snail for exactly 1 minute. It is known that every observer finds that the snail crawled 1 meter in the minute they observed. Prove that the distance the snail crawled in these 6 minutes does not exceed 10 meters.

Answer: 10


The sequence  ${a_{n}}$   $n\in \mathbb{N}$  is given in a recursive way with  $a_{1}=1$ , $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$ , for all  $n\geq 2$ .
Determine the least number  $M$ , such that  $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$  for all  $m\in \mathbb{N}$

Answer: 2


A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 10. Points $V$, $W$, and $X$ are the midpoints of edges $PR$, $RQ$, and $QT$, respectively. Determine the perimeter of triangle $VWX$.

Answer: 5 + 10\sqrt{5}


Let $A$ be a set with $n$ elements. The $m$ subsets $A_{1}, A_{2}, \cdots, A_{m}$ of $A$ are pairwise disjoint. Prove that:
(1) $\sum_{i=1}^{m} \frac{1}{C_{n}^{|A_{i}|}} \leq 1$;
(2) $\sum_{i=1}^{m} C_{n}^{|A_{i}|} \geq m^{2}$.

Answer: 1


The ternary sequence $\left(x_{n}, y_{n}, z_{n}\right), n \in \mathbb{N}$ is determined by the following relations: $x_{1}=2, y_{1}=4, z_{1}=\frac{6}{7}$, $x_{n-1}=\frac{2 x_{n}}{x_{n}^{2}-1}, y_{n+1}=\frac{2 y_{n}}{y_{n}^{2}-1}, z_{n+1}=\frac{2 z_{n}}{z_{n}^{2}-1}$.
1. Prove that the process of forming the ternary sequence can continue indefinitely.
2. Is it possible that at some step, the obtained ternary sequence $\left(x_{n}, y_{n}, z_{n}\right)$ satisfies the equation $x_{n}+y_{n}+z_{n}=0$?

Answer: No


Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combination is unique across the cards. A set of three cards is defined as complementary if:
i. Each card has a different symbol or all have the same symbol.
ii. Each card has a different color or all have the same color.
iii. Each card has a different intensity or all have the same intensity.
Determine the number of different complementary three-card sets available.

Answer: 117


Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order).

Answer: 2: 3: 4


In a right triangle \(ABC\) (with \(\angle C = 90^\circ\)), height \(CD\) is drawn. The radii of the circles inscribed in triangles \(ACD\) and \(BCD\) are 0.6 cm and 0.8 cm, respectively. Find the radius of the circle inscribed in triangle \(ABA\).

Answer: 1


Let \( \triangle ABC \) be an acute triangle, with \( M \) being the midpoint of \( \overline{BC} \), such that \( AM = BC \). Let \( D \) and \( E \) be the intersection of the internal angle bisectors of \( \angle AMB \) and \( \angle AMC \) with \( AB \) and \( AC \), respectively. Find the ratio of the area of \( \triangle DME \) to the area of \( \triangle ABC \).

Answer: \frac{2}{9}


Prove that if a pair of values for the variables \( x \) and \( y \) satisfies the equations
$$
x^{2}-3 x y+2 y^{2}+x-y=0
$$

and
$$
x^{2}-2 x y+y^{2}-5 x+7 y=0,
$$

then this pair also satisfies the equation
$$
x y-12 x+15 y=0.
$$

Answer: xy-12x+15y=0


Let  $\triangle ABC$  be a triangle with circumcenter  $O$  satisfying  $AB=13$ ,  $BC = 15$ , and  $AC = 14$ .  Suppose there is a point  $P$  such that  $PB \perp BC$  and  $PA \perp AB$ .  Let  $X$  be a point on  $AC$  such that  $BX \perp OP$ . What is the ratio  $AX/XC$ ?

*Proposed by Thomas Lam*

Answer: \frac{169}{225}


John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was  $2002$  ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number?

Answer: 1001


Answer: ismultipleofn


There is a rectangle  $ABCD$  such that  $AB=12$  and  $BC=7$ .  $E$  and  $F$  lie on sides  $AB$  and  $CD$  respectively such that  $\frac{AE}{EB} = 1$  and  $\frac{CF}{FD} = \frac{1}{2}$ . Call  $X$  the intersection of  $AF$  and  $DE$ . What is the area of pentagon  $BCFXE$ ?

Proposed by Minseok Eli Park (wolfpack)

Answer: 47


On the sides \(BC\) and \(AC\) of an isosceles triangle \(ABC\) (\(AB = AC\)), points \(D\) and \(E\) are found such that \(AE = AD\) and \(\angle EDC = 18^\circ\). Find the measure of the angle \(\angle BAD\).

Answer: 36


It is known that the graph of the function \(y = x^2 + px + q\) passes through points \(A(1, 1)\) and \(B(3, 1)\). Does this graph pass through point \(C(4, 5)\)?

Answer: 0


Given that \( G \) is a simple graph with 20 vertices and 100 edges, we can find 4050 ways to identify a pair of non-intersecting edges. Prove that \( G \) is regular.
(2004 Iran Mathematics Olympiad)

Answer: G


Non-negative real numbers \( x, y, z \) satisfy \( x^{2} + y^{2} + z^{2} = 10 \). Find the maximum and minimum values of \( \sqrt{6 - x^{2}} + \sqrt{6 - y^{2}} + \sqrt{6 - z^{2}} \).

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


Let \( f(n) \) be the number of real solutions of \( x^3 = \lfloor x^3 \rfloor + (x - \lfloor x \rfloor)^3 \) in the range \( 1 \le x < n \). Find \( f(n) \).

Answer: n^3-n


On the board, the natural number \( N \) was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers?

(I. Efremov)

Answer: 6


Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \).

Answer: 1890


Given \(\alpha\) and \(\beta\) satisfy the equations
\[
\begin{array}{c}
\alpha^{3}-3 \alpha^{2}+5 \alpha-4=0, \\
\beta^{3}-3 \beta^{2}+5 \beta-2=0 .
\end{array}
\]
find \(\alpha + \beta\).

Answer: 2


Find all the pairs of positive integers  $(m,n)$  such that the numbers  $A=n^2+2mn+3m^2+3n$ ,  $B=2n^2+3mn+m^2$ ,  $C=3n^2+mn+2m^2$  are consecutive in some order.

Answer: (,n)=(k,k+1)


Given \(\triangle ABC\) is an acute triangle, \(b = 2c\), and \(\sin B - \sin(A+B) = 2 \sin C \cos A\). Find the range of values for \((\cos B + \sin B)^{2} + \sin 2C\).

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


A rectangular pasture is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 15 meters along the fence including at the points where the fence meets the rock wall. Given the dimensions of the pasture are 36 m by 75 m, find the minimum number of posts required.

Answer: 14


Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number  $k$  such that there may be  $k$  equal numbers between them.

Answer: 4


Let \(\triangle ABC\) be a scalene triangle. Let \(h_{a}\) be the locus of points \(P\) such that \(|PB - PC| = |AB - AC|\). Let \(h_{b}\) be the locus of points \(P\) such that \(|PC - PA| = |BC - BA|\). Let \(h_{c}\) be the locus of points \(P\) such that \(|PA - PB| = |CA - CB|\). In how many points do all of \(h_{a}, h_{b}\), and \(h_{c}\) concur?

Answer: 2


Let  $H$  be a convex, equilateral heptagon whose angles measure (in degrees)  $168^\circ$ ,  $108^\circ$ ,  $108^\circ$ ,  $168^\circ$ ,  $x^\circ$ ,  $y^\circ$ , and  $z^\circ$  in clockwise order.  Compute the number  $y$ .

Answer: 132


Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist?

Answer: 12


How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there?

Answer: 27


Let  $S$  be a finite set of points in the plane. A linear partition of  $S$  is an unordered pair  $\{A,B\}$  of subsets  of  $S$  such that  $A\cup B=S,\ A\cap B=\emptyset,$  and  $A$  and  $B$  lie on opposite sides of some straight line disjoint from  $S$  ( $A$  or  $B$  may be empty). Let  $L_{S}$  be the number of linear partitions of  $S.$  For each positive integer  $n,$  find the maximum of  $L_{S}$  over all sets  $S$  of  $n$  points.

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


Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.

Answer: 23


Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shaded when triangles APQ and CSR are shaded.

[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,2)--(2,4)--(0,4)--(0,2)--cycle,white,linewidth(1));
filldraw((4,2)--(2,0)--(4,0)--(4,2)--cycle,white,linewidth(1));
label("P",(0,2),W);
label("Q",(2,4),N);
label("R",(4,2),E);
label("S",(2,0),S);
[/asy]

Answer: \frac{1}{4}


In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\circ$, what is the perimeter of the shaded region?

Answer: 16 + \frac{32}{3}\pi


Given an arithmetic sequence $\\{a_{n}\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{4}=-12$ and $a_{8}=-4$:

$(1)$ Find the general term formula for the sequence;
$(2)$ Find the minimum value of $S_{n}$ and the corresponding value of $n$.

Answer: -90


Three chiefs of Indian tribes are sitting by a fire with three identical pipes. They are holding a war council and smoking. The first chief can finish a whole pipe in ten minutes, the second in thirty minutes, and the third in an hour. How should the chiefs exchange the pipes among themselves in order to prolong their discussion for as long as possible?

Answer: 20


Answer: a_n=b_n=c_n


On a table, there are 10 number cards labeled $0$ to $9$. Three people, A, B, and C, each take three of these cards. They then compute the sum of all different three-digit numbers that can be formed with the three cards they took. The results for A, B, and C are $1554$, $1688$, and $4662$ respectively. What is the remaining card? (Note: $6$ and $9$ cannot be considered as each other.)

Answer: 9


A certain district's education department wants to send 5 staff members to 3 schools for earthquake safety education. Each school must receive at least 1 person and no more than 2 people. How many different arrangements are possible? (Answer with a number)

Answer: 90


Four princesses thought of two-digit numbers, and Ivan thought of a four-digit number. After they wrote their numbers in a row in some order, the result was 132040530321. Find Ivan's number.

Answer: 5303


On the island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 20 inhabitants of the island lined up in order of their height (from tallest to shortest, the tallest being the first) for a game. Each of them had to say one of the following phrases: "There is a liar below me" or "There is a knight above me". As a result, those standing in positions from the third to the seventh said the first phrase, while the others said the second phrase. How many knights were there among these 20 people, if all the inhabitants have different heights?

Answer: 17


The points $A=\left(4, \frac{1}{4}\right)$ and $B=\left(-5,-\frac{1}{5}\right)$ lie on the hyperbola $x y=1$. The circle with diameter $A B$ intersects this hyperbola again at points $X$ and $Y$. Compute $X Y$.

Answer: \sqrt{\frac{401}{5}}


Let  $M$  be the midpoint of the base  $AC$  of an acute-angled isosceles triangle  $ABC$ . Let  $N$  be the reflection of  $M$  in  $BC$ . The line parallel to  $AC$  and passing through  $N$  meets  $AB$  at point  $K$ . Determine the value of  $\angle AKC$ .

(A.Blinkov)

Answer: 90


There are 4 boys and 3 girls lining up for a photo. The number of ways they can line up such that no two boys are adjacent is \_\_\_\_ (answer with a number), and the number of ways they can line up such that no two girls are adjacent is \_\_\_\_ (answer with a number).

Answer: 1440


Given the set $\{2,3,5,7,11,13\}$, add one of the numbers twice to another number, and then multiply the result by the third number. What is the smallest possible result?

Answer: 22


A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?

$\textbf{(A)}\ 2.18\qquad\textbf{(B)}\ 2.24\qquad\textbf{(C)}\ 2.31\qquad\textbf{(D)}\ 2.37\qquad\textbf{(E)}\ 2.41$

Answer: \textbf{(D)}\2.37


Answer: -2


If \( x = \frac{1}{4} \), which of the following has the largest value?
(A) \( x \)
(B) \( x^2 \)
(C) \( \frac{1}{2} x \)
(D) \( \frac{1}{x} \)
(E) \( \sqrt{x} \)

Answer: 4


Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it by appropriately selecting stamps of different denominations and using no more than three stamps. Determine the maximum value of R and provide a corresponding design.

Answer: 14


Given the set \( X = \{1, 2, \cdots, 100\} \), determine whether there exist 1111 different subsets of \( X \) such that the intersection of any two (including possibly the same subset) contains a number of elements that is a perfect square. Note: Perfect squares include \( 0, 1, 4, 9, 16, \cdots \).

Answer: Yes


Consider a positive real number  $a$  and a positive integer  $m$ . The sequence  $(x_k)_{k\in \mathbb{Z}^{+}}$  is defined as: $x_1=1$ ,  $x_2=a$ ,  $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$ . $a)$  Prove that the sequence is converging. $b)$  Find  $\lim_{n\rightarrow \infty}{x_n}$ .

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


In triangle $ABC$, $AB=125$, $AC=117$ and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.

Answer: 56


Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.

Answer: 16


In an acute triangle \( ABC \), angle \( A \) is \( 35^\circ \). Segments \( BB_1 \) and \( CC_1 \) are altitudes, points \( B_2 \) and \( C_2 \) are the midpoints of sides \( AC \) and \( AB \) respectively. Lines \( B_1C_2 \) and \( C_1B_2 \) intersect at point \( K \). Find the measure (in degrees) of angle \( B_1KB_2 \).

Answer: 75


Construct a cross-section of a cube.

Given:
- Line KL, where K is the midpoint of \( D_1 C_1 \), and L is the midpoint of \( C_1 B_1 \), lies in the plane face \( A_1 B_1 C_1 D_1 \).
- This line intersects the extended edges \( A_1 B_1 \) and \( A_1 D_1 \) at points F and E respectively.
- It's easy to calculate that \( D_1 E = \frac{1}{2} A_1 D_1 = \frac{1}{3} A_1 E \) and similarly \( B_1 F = \frac{1}{2} A_1 B_1 = \frac{1}{3} A_1 F \).
- The points E and A lie on the same face. It can be shown that the line AE intersects the edge \( D D_1 \) at point N, dividing this edge in the ratio 2:1.
- Similarly, the line AF intersects the edge \( B_1 B \) at point M, also dividing this edge in the ratio 2:1. Therefore, the cross-section will be the pentagon \( AMLKN \).

The projection of this pentagon onto the lower base will be the pentagon \( ABL_1 K_1 D \), and its area is \(\frac{7}{8}\).

If \( T \) is the midpoint of \( KL \) and \( T_1 \) is the projection of point T onto the lower base, the angle \( TAT_1 \) will be the angle between the cross-section and the lower face of the cube. Then, \(\cos(TAT_1) = \frac{3}{\sqrt{17}} \). Using the formula for the area projection, the area of the cross-section will be \(\frac{7\sqrt{17}}{24}\).

Answer: \frac{7\sqrt{17}}{24}


Answer: 190


Of the following sets of data the only one that does not determine the shape of a triangle is:
$\textbf{(A)}\ \text{the ratio of two sides and the inc{}luded angle}\\ \qquad\textbf{(B)}\ \text{the ratios of the three altitudes}\\ \qquad\textbf{(C)}\ \text{the ratios of the three medians}\\ \qquad\textbf{(D)}\ \text{the ratio of the altitude to the corresponding base}\\ \qquad\textbf{(E)}\ \text{two angles}$

Answer: \textbf{(D)}


The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is completed exactly on time. How many days would it take for Kevin and Dave to complete the job if they work together?

Answer: 12


In the universe of Pi Zone, points are labeled with  $2 \times 2$  arrays of positive reals. One can teleport from point  $M$  to point  $M'$  if  $M$  can be obtained from  $M'$  by multiplying either a row or column by some positive real. For example, one can teleport from  $\left( \begin{array}{cc} 1 & 2  3 & 4 \end{array} \right)$  to  $\left( \begin{array}{cc} 1 & 20  3 & 40 \end{array} \right)$  and then to  $\left( \begin{array}{cc} 1 & 20  6 & 80 \end{array} \right)$ .

A *tourist attraction* is a point where each of the entries of the associated array is either  $1$ ,  $2$ ,  $4$ ,  $8$  or  $16$ .  A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?

*Proposed by Michael Kural*

Answer: 17


The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables.

Answer: 79


Given an odd function \( f(x) \) defined on \(\mathbf{R}\) that is symmetric about the line \( x=2 \), and when \( 0<x \leq 2 \), \( f(x)=x+1 \). Calculate \( f(-100) + f(-101) \).

Answer: 2


Answer: 12


Print 90,000 five-digit numbers
$$
10000, 10001, \cdots, 99999
$$

on cards, with each card displaying one five-digit number. Some numbers printed on the cards (e.g., 19806 when reversed reads 90861) can be read in two different ways and may cause confusion. How many cards will display numbers that do not cause confusion?

Answer: 89100


In a convex quadrilateral $ABCD$, the midpoint of side $AD$ is marked as point $M$. Line segments $BM$ and $AC$ intersect at point $O$. It is known that $\angle ABM = 55^\circ$, $\angle AMB = 70^\circ$, $\angle BOC = 80^\circ$, and $\angle ADC = 60^\circ$. What is the measure of angle $BCA$ in degrees?

Answer: 35


Determine the number of solutions to the equation
\[\tan (10 \pi \cos \theta) = \cot (10 \pi \sin \theta)\]
where $\theta \in (0, 2 \pi).$

Answer: 56


Rebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistance of \(a\) and \(b\) ohms respectively, and combines them into one by one of the following methods:
- Connect them in series, which produces a resistor with resistance of \(a+b\) ohms;
- Connect them in parallel, which produces a resistor with resistance of \(\frac{a b}{a+b}\) ohms;
- Short-circuit one of the two resistors, which produces a resistor with resistance of either \(a\) or \(b\) ohms.
Suppose that after three minutes, Rebecca has a single resistor with resistance \(R\) ohms. How many possible values are there for \(R\)?

Answer: 15


Given that the circumference of a sector is $20\,cm$ and its area is $9\,cm^2$, find the radian measure of the central angle of the sector.

Answer: \frac{2}{9}


Points \( A \) and \( B \) lie on a circle with center \( O \) and radius 6, and point \( C \) is equidistant from points \( A, B, \) and \( O \). Another circle with center \( Q \) and radius 8 is circumscribed around triangle \( A C O \). Find \( B Q \).

Answer: 10


Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?

Answer: 4


Compute the perimeter of the triangle that has area  $3-\sqrt{3}$  and angles  $45^\circ$ ,  $60^\circ$ , and  $75^\circ$ .

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


The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the smallest possible number of members of the committee?

Answer: 134


If $\left\{a_{n}\right\}$ is an increasing sequence and for any positive integer $n$, $a_{n} = n^{2} + \lambda n$ always holds, then the value range of the real number $\lambda$ is ______ .

Answer: (-3,+\infty)


A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
$\textbf{(A)}\ 585 \qquad \textbf{(B)}\ 594 \qquad \textbf{(C)}\ 672 \qquad \textbf{(D)}\ 679 \qquad \textbf{(E)}\ 694$

Answer: 585\\textbf{(A)}


In the triangular prism $S-ABC$, the lateral edge $SC$ is equal in length to the base edge $AB$, and forms a $60^{\circ}$ angle with the base plane $ABC$. The vertices $A$, $B$, and $C$ as well as the midpoints of the lateral edges of the prism all lie on a sphere with radius 1. Prove that the center of this sphere lies on the edge $AB$, and determine the height of the prism.

Answer: \sqrt{3}


An angle is inscribed with a circle of radius \( R \), and the length of the chord connecting the points of tangency is \( a \). Two tangents are drawn parallel to this chord, forming a trapezoid. Find the area of this trapezoid.

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


Let $a, b, c, d$ be real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a, b, c, d)$ such that the minimum value is achieved.

Answer: -\frac{1}{8}


The integer $y$ has 24 positive factors. The numbers 20 and 35 are factors of $y$. What is the smallest possible value of $y$?

Answer: 1120


Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots\]of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 36$

Answer: \textbf{(C)}19


Consider polynomials  $P$  of degree  $2015$ , all of whose coefficients are in the set  $\{0,1,\dots,2010\}$ . Call such a polynomial *good* if for every integer  $m$ , one of the numbers  $P(m)-20$ ,  $P(m)-15$ ,  $P(m)-1234$  is divisible by  $2011$ , and there exist integers  $m_{20}, m_{15}, m_{1234}$  such that  $P(m_{20})-20, P(m_{15})-15, P(m_{1234})-1234$  are all multiples of  $2011$ . Let  $N$  be the number of good polynomials. Find the remainder when  $N$  is divided by  $1000$ .

*Proposed by Yang Liu*

Answer: 460


Three segments, not lying in the same plane, intersect at one point and are divided into halves by this point. Prove that there are exactly two tetrahedrons in which these segments connect the midpoints of opposite edges.

Answer: 2


Find the fraction with the smallest denominator in the open interval \(\left(\frac{47}{245}, \frac{34}{177}\right)\).

Answer: \frac{19}{99}


How many diagonals within a regular nine-sided polygon span an odd number of vertices between their endpoints?

Answer: 18


In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households for a survey. It was found that a total of 120 households own 3 or more sets of housing, among which there are 40 ordinary households and 80 high-income households. Based on these data and combining your statistical knowledge, what do you think is a reasonable estimate of the proportion of households in the area that own 3 or more sets of housing?

Answer: 4.8\%


Let  $2000 < N < 2100$  be an integer. Suppose the last day of year  $N$  is a Tuesday while the first day of year  $N+2$  is a Friday. The fourth Sunday of year  $N+3$  is the  $m$ th day of January. What is  $m$ ?

*Based on a proposal by Neelabh Deka*

Answer: 23


Let the function \( f(x) = x^2 + ax + b \) where \( a \) and \( b \) are real constants. It is known that the inequality \( |f(x)| \leqslant |2x^2 + 4x - 30| \) holds for any real number \( x \). A sequence \( \{a_n\} \) and \( \{b_n\} \) is defined as: \( a_1 = \frac{1}{2} \), \( 2a_n = f(a_{n-1}) + 15 \) for \( n = 2, 3, 4, \ldots \), and \( b_n = \frac{1}{2+a_n} \) for \( n = 1, 2, 3, \ldots \). Let \( S_n \) be the sum of the first \( n \) terms of the sequence \( \{b_n\} \) and \( T_n \) be the product of the first \( n \) terms of the sequence \( \{b_n\} \).

(1) Prove that \( a = 2 \) and \( b = -15 \).
(2) Prove that for any positive integer \( n \), \( 2^{n+1} T_n + S_n \) is a constant value.

Answer: 2


Let $\triangle ABC$ be an acute isosceles triangle with circumcircle $\omega$. The tangents to $\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \cdot CZ = 478$. Find $BZ \cdot CZ$.

Answer: 144


Let  $p$  be a prime.  It is given that there exists a unique nonconstant function  $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$  such that  $\chi(1) = 1$  and  $\chi(mn) = \chi(m)\chi(n)$  for all  $m, n \not\equiv 0 \pmod p$  (here the product  $mn$  is taken mod  $p$ ). For how many positive primes  $p$  less than  $100$  is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual  $a^{-1}$  denotes multiplicative inverse.

*Proposed by David Altizio*

Answer: 24


All edges (including the sides of the base) of a triangular pyramid are equal. Find the ratio of the radius of the sphere inscribed in the pyramid to its height.

Answer: 1:4


Natural numbers $1, 2, 3, 4, \cdots$ are arranged in order according to the direction indicated by arrows, and they turn at positions of numbers like $2, 3, 5, 7, 10, \cdots$ and so on.
(1) If 2 is counted as the first turn, then the number at the 45th turn is $\qquad$ .
(2) From 1978 to 2010, the natural numbers that are exactly at turning positions are $\qquad$ .

Answer: 1981


Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with one problem each?

Answer: 33


Answer: -1


It is given isosceles triangle  $ABC$  ( $AB=AC$ ) such that  $\angle BAC=108^{\circ}$ . Angle bisector of angle  $\angle ABC$  intersects side  $AC$  in point  $D$ , and point  $E$  is on side  $BC$  such that  $BE=AE$ . If  $AE=m$ , find  $ED$

Answer: m


$(MON 1)$  Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

Answer: 1944


Chords \(AB\) and \(CD\) of a circle with center \(O\) both have a length of 5. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) intersect at point \(P\), where \(DP=13\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL:LC\).

Answer: 13/18


The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

Answer: 75


A man and his faithful dog simultaneously started moving along the perimeter of a block from point \( A \) at time \( t_{0} = 0 \) minutes. The man moved at a constant speed clockwise, while the dog ran at a constant speed counterclockwise. It is known that the first time they met was \( t_{1} = 2 \) minutes after starting, and this meeting occurred at point \( B \). Given that they continued moving after this, each in their original direction and at their original speed, determine the next time they will both be at point \( B \) simultaneously. Note that \( A B = C D = 100 \) meters and \( B C = A D = 200 \) meters.

Answer: 14


There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children?

Answer: 98


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


The product of the digits of a four-digit number is 810. If none of the digits is repeated, what is the sum of the digits?
(A) 18
(B) 19
(C) 23
(D) 25
(E) 22

Answer: 23


Compute the unique positive integer $m$ such that
\[1 \cdot 2^1 + 2 \cdot 2^2 + 3 \cdot 2^3 + \dots + m \cdot 2^m = 2^{m + 8}.\]

Answer: 129


There are \( n \) real numbers written on a blackboard. It is allowed to erase any two numbers, say \( a \) and \( b \), and write another number \(\frac{1}{4}(a+b)\) on the board. This operation is performed \( n-1 \) times, resulting in only one number left on the blackboard. Initially, all \( n \) numbers on the board are 1. Prove that the final number remaining on the blackboard is not less than \(\frac{1}{n}\).

Answer: \frac{1}{n}


Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]

*Proposed by Karthik Vedula*

Answer: 6


It is known that the ellipse $C_1$ and the parabola $C_2$ have a common focus $F(1,0)$. The center of $C_1$ and the vertex of $C_2$ are both at the origin. A line $l$ passes through point $M(4,0)$ and intersects the parabola $C_2$ at points $A$ and $B$ (with point $A$ in the fourth quadrant).
1. If $|MB| = 4|AM|$, find the equation of line $l$.
2. If the symmetric point $P$ of the origin $O$ with respect to line $l$ lies on the parabola $C_2$, and line $l$ intersects the ellipse $C_1$ at common points, find the minimum length of the major axis of the ellipse $C_1$.

Answer: \sqrt{34}


Define functions  $f$  and  $g$  as nonconstant, differentiable, real-valued functions on  $R$ . If  $f(x+y)=f(x)f(y)-g(x)g(y)$ ,  $g(x+y)=f(x)g(y)+g(x)f(y)$ , and  $f'(0)=0$ , prove that  $\left(f(x)\right)^2+\left(g(x)\right)^2=1$  for all  $x$ .

Answer: (f(x))^2+((x))^2=1


Let  $a$ ,  $b$  and  $c$  be real numbers such that  $abc(a+b)(b+c)(c+a)\neq0$  and  $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$ Prove that  $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$

Answer: 2017


The extension of the altitude $BH$ of triangle $ABC$ intersects its circumcircle at point $D$ (points $B$ and $D$ are on opposite sides of line $AC$). The degree measures of arcs $AD$ and $CD$ that do not contain point $B$ are $120^{\circ}$ and $90^{\circ}$, respectively. Determine the ratio in which segment $BD$ is divided by side $AC$.

Answer: 1:\sqrt{3}


On a board, there are $N \geq 9$ distinct non-negative numbers less than one. It turns out that for any eight distinct numbers from the board, there exists a ninth number on the board, different from them, such that the sum of these nine numbers is an integer. For which $N$ is this possible?

Answer: 9


The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?
$\text{(A)}\ 24\% \qquad \text{(B)}\ 27\% \qquad \text{(C)}\ 36\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 73\%$

Answer: (B)\27


For which values of the parameter \( a \) does the equation

$$
5^{x^{2}-6 a x+9 a^{2}}=a x^{2}-6 a^{2} x+9 a^{3}+a^{2}-6 a+6
$$

have exactly one solution?

Answer: 1


Given vectors $\overrightarrow{a}=(1,\sin x)$, $\overrightarrow{b}=(\sin x,-1)$, $\overrightarrow{c}=(1,\cos x)$, where $x\in(0,\pi)$.
(Ⅰ) If $(\overrightarrow{a}+ \overrightarrow{b})\nparallel \overrightarrow{c}$, find $x$;
(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $B$ is the $x$ found in (Ⅰ), $2\sin^2B+2\sin^2C-2\sin^2A=\sin B\sin C$, find the value of $\sin \left(C- \frac{\pi}{3}\right)$.

Answer: \frac{1-3\sqrt{5}}{8}


Given a natural number \( a \), let \( S(a) \) represent the sum of its digits (for example, \( S(123) = 1 + 2 + 3 = 6 \) ). If a natural number \( n \) has all distinct digits, and \( S(3n) = 3S(n) \), what is the maximum value of \( n \)?

Answer: 3210


Answer: \frac{224\sqrt{3}+240}{3}


Two numbers \( x \) and \( y \) satisfy the equation \( 26x^2 + 23xy - 3y^2 - 19 = 0 \) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.

Answer: -3


A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better-ranked team always wins, and the worse-ranked team is eliminated. Let \( p \) be the probability that the second-best ranked team is eliminated in the last round. Compute \( \lfloor 2021 p \rfloor \).

Answer: 674


For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?

Answer: 409


Four friends, One, Two, Five, and Ten, are located on one side of a dark tunnel and have only one flashlight. It takes one minute for person One to walk through the tunnel, two minutes for Two, five minutes for Five, and ten minutes for Ten. The tunnel is narrow, and at most two people can walk at the same time with the flashlight. Whenever two people walk together, they walk at the speed of the slower one. Show that all four friends can go from one side of the tunnel to the other in 17 minutes.

Answer: 17


Given $\log _{2} 3=a$ and $\log _{3} 7=b$, find the value of $\log _{42} 56$.

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


The positive reals \( x \) and \( y \) satisfy \( x^2 + y^3 \geq x^3 + y^4 \). Show that \( x^3 + y^3 \leq 2 \).

Answer: 2


A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest  $\ell > 0$  such that there exists a square of side  $\ell$  contained in a cube with edge  $1$ .

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


One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?

Answer: 9


Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAA$, while $BBAB$ is not such a sequence. How many such sequences have length 14?

Answer: 172


Person A and person B are playing a game with the following rules: In odd-numbered rounds, the probability of A winning is $\frac{3}{4}$; in even-numbered rounds, the probability of B winning is $\frac{3}{4}$. There are no ties in any round. The game ends when one person has won 2 more rounds than the other. Determine the expected number of rounds played when the game ends.

Answer: \frac{16}{3}


For the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On segment \(BC\), point \(E\) is marked such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)?

Answer: 17


An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point?

Answer: 221


The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ?

Answer: 84


A sequence of numbers is arranged in the following pattern: \(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \cdots\). Starting from the first number on the left, find the sum of the first 99 numbers.

Answer: 1782


Complex numbers $z_1,$ $z_2,$ and $z_3$ are zeros of a polynomial $Q(z) = z^3 + pz + s,$ where $|z_1|^2 + |z_2|^2 + |z_3|^2 = 300$. The points corresponding to $z_1,$ $z_2,$ and $z_3$ in the complex plane are the vertices of a right triangle with the right angle at $z_3$. Find the square of the hypotenuse of this triangle.

Answer: 450


Calculate the area enclosed by the Bernoulli lemniscate \( r^{2} = a^{2} \cos 2\varphi \).

Answer: ^2


In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.

[asy]
defaultpen(1);

pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);

label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);

[/asy]

Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$?

Answer: \frac{1}{4}


It is given regular  $n$ -sided polygon,  $n \geq 6$ . How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?

Answer: \frac{n(n-4)(n-5)}{6}


Let \( S = \{1, 2, 3, \ldots, 9, 10\} \). A non-empty subset of \( S \) is considered "Good" if the number of even integers in the subset is more than or equal to the number of odd integers in the same subset. For example, the subsets \( \{4,8\}, \{3,4,7,8\} \) and \( \{1,3,6,8,10\} \) are "Good". How many subsets of \( S \) are "Good"?

Answer: 637


Of the following
(1) $a(x-y)=ax-ay$
(2) $a^{x-y}=a^x-a^y$
(3) $\log (x-y)=\log x-\log y$
(4) $\frac{\log x}{\log y}=\log{x}-\log{y}$
(5) $a(xy)=ax \cdot ay$
$\textbf{(A)}\text{Only 1 and 4 are true}\qquad\\\textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\\textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\\textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\\textbf{(E)}\ \text{Only 1 is true}$

Answer: \textbf{(E)}


Fully factor the following expression: $2x^2-8$

Answer: (2) (x+2) (x-2)


Acute triangle  $ABP$ , where  $AB > BP$ , has altitudes  $BH$ ,  $PQ$ , and  $AS$ . Let  $C$  denote the intersection of lines  $QS$  and  $AP$ , and let  $L$  denote the intersection of lines  $HS$  and  $BC$ . If  $HS = SL$  and  $HL$  is perpendicular to  $BC$ , find the value of  $\frac{SL}{SC}$ .

Answer: \frac{1}{3}


To actively create a "demonstration school for the prevention and control of myopia in children and adolescents in the city" and cultivate students' good eye habits, a certain school conducted a competition on correct eye knowledge this semester. $20$ student answer sheets were randomly selected and their scores (denoted by $x$, unit: points) were recorded as follows:
$86\ \ 82\ \ 90\ \ 99\ \ 98\ \ 96\ \ 90\ \ 100\ \ 89\ \ 83$
$87\ \ 88\ \ 81\ \ 90\ \ 93\ \ 100\ \ 96\ \ 100\ \ 92\ \ 100$
Organize the data:

$80\leqslant x \lt 85$
$85\leqslant x \lt 90$
$90\leqslant x \lt 95$
$95\leqslant x\leqslant 100$

$3$
$4$
$a$
$8$

Analyze the data:

Mean
Median
Mode

$92$
$b$
$c$

Based on the above information, answer the following questions:
$(1)$ Write down the values of $a$, $b$, and $c$ directly from the table;
$(2)$ $2700$ students from the school participated in the knowledge competition. Estimate the number of people with scores not less than $90$ points;
$(3)$ Choose one quantity from the median and mode, and explain its significance in this question.

Answer: 1755


The crafty rabbit and the foolish fox made an agreement: every time the fox crosses the bridge in front of the rabbit's house, the rabbit would double the fox's money. However, each time the fox crosses the bridge, he has to pay the rabbit a toll of 40 cents. Hearing that his money would double each time he crossed the bridge, the fox was very happy. However, after crossing the bridge three times, he discovered that all his money went to the rabbit. How much money did the fox initially have?

Answer: 35


In a certain city, the rules for selecting license plate numbers online are as follows: The last five characters of the plate must include two English letters (with the letters "I" and "O" not allowed), and the last character must be a number. How many possible combinations meet these requirements?

Answer: 3456000


Given a positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \). Determine the maximum value of \( r \).

Answer: 4\sqrt{2}


Given that $α \in \left( \frac{π}{2}, π \right)$ and $3\cos 2α = \sin \left( \frac{π}{4} - α \right)$, find the value of $\sin 2α$.

Answer: -\frac{17}{18}


Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?

Answer: \frac{4}{7}


Find the smallest positive number $\alpha$ such that there exists a positive number $\beta$ satisfying the inequality
$$
\sqrt{1+x}+\sqrt{1-x} \leqslant 2-\frac{x^{\alpha}}{\beta}
$$
for $0 \leqslant x \leqslant 1$.

Answer: 2


Let  $ABC$  be a triangle with  $\angle BCA=90^{\circ}$ , and let  $D$  be the foot of the altitude from  $C$ . Let  $X$  be a point in the interior of the segment  $CD$ . Let  $K$  be the point on the segment  $AX$  such that  $BK=BC$ . Similarly, let  $L$  be the point on the segment  $BX$  such that  $AL=AC$ . Let  $M$  be the point of intersection of  $AL$  and  $BK$ .

Show that  $MK=ML$ .

*Proposed by Josef Tkadlec, Czech Republic*

Answer: MK=ML


In triangle \(ABC\), \(AB = BC\) and \(\angle B = 20^\circ\). Point \(M\) on the base \(AC\) is such that \(AM : MC = 1 : 2\), and point \(H\) is the projection of \(C\) onto \(BM\). Find the angle \(\angle AHB\).

Answer: 100


A 2 by 2003 rectangle consists of unit squares as shown below.  The middle unit square of each row is shaded.  If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square?  Express your answer as a common fraction. [asy]
size(7cm);
defaultpen(linewidth(0.7));
dotfactor=4;
int i,j;

fill((6,0)--(7,0)--(7,2)--(6,2)--cycle,gray);

for(i=0;i<=3;++i)

{

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

draw((i+5,0)--(i+5,2));

draw((i+10,0)--(i+10,2));

}
for(j=0;j<=2;++j)

{

draw((0,j)--(3.3,j));

draw((0,j)--(3.3,j));

draw((4.7,j)--(8.3,j));

draw((4.7,j)--(8.3,j));

draw((9.7,j)--(13,j));

draw((9.7,j)--(13,j));

}

real x;

for(x=3.7;x<=4.3;x=x+0.3)

{

dot((x,0));

dot((x,2));

dot((x+5,0));

dot((x+5,2));

}[/asy]

Answer: \dfrac{1001}{2003}


In the final phase of a professional bowling competition, the top five players compete as follows: first, the fifth and fourth place players compete, and the loser gets the 5th place prize; the winner then competes with the third place player, and the loser gets the 4th place prize; the winner then competes with the second place player, and the loser gets the 3rd place prize; the winner then competes with the first place player, and the loser gets the 2nd place prize, and the winner gets the 1st place prize. How many different award sequences are possible?

Answer: 16


Prove that for any three positive real numbers \(x, y, z\), the following inequality holds:
\[
\left(x^{2}+y^{2}\right)^{2} \geq (x+y+z)(x-y+z)(x+y-z)(y+z-x)
\]
Indicate all triples \(x, y, z\) for which equality is achieved.

Answer: x,x,x\sqrt{2}


A box contains three balls, each of a different color. Every minute, Randall randomly draws a ball from the box, notes its color, and then returns it to the box. Consider the following two conditions: (1) Some ball has been drawn at least three times (not necessarily consecutively). (2) Every ball has been drawn at least once. What is the probability that condition (1) is met before condition (2)?

Answer: \frac{13}{27}


Let  $n\geq 2$  be a given integer. Initially, we write  $n$  sets on the blackboard and do a sequence of moves as follows: choose two sets  $A$  and  $B$  on the blackboard such that none of them is a subset of the other, and replace  $A$  and  $B$  by  $A\cap B$  and  $A\cup B$ . This is called a  $\textit{move}$ .
Find the maximum number of moves in a sequence for all possible initial sets.

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


Let  $ABC$  be an isosceles obtuse-angled triangle, and  $D$  be a point on its base  $AB$  such that  $AD$  equals to the circumradius of triangle  $BCD$ . Find the value of  $\angle ACD$ .

Answer: 30


Consider the sets  $A_1=\{1\}$ ,  $A_2=\{2,3,4\}$ ,  $A_3=\{5,6,7,8,9\}$ , etc. Let  $b_n$  be the arithmetic mean of the smallest and the greatest element in  $A_n$ . Show that the number  $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$  is a prime integer.

Answer: 1999


Among the following 4 propositions, the correct one is:
(1) If a solid's three views are completely identical, then the solid is a cube;
(2) If a solid's front view and top view are both rectangles, then the solid is a cuboid;
(3) If a solid's three views are all rectangles, then the solid is a cuboid;
(4) If a solid's front view and left view are both isosceles trapezoids, then the solid is a frustum.

Answer: (3)


For how many values of $n$ with $3 \leq n \leq 12$ can a Fano table be created?

Answer: 3


Answer: -60


The cost of purchasing a car is 150,000 yuan, and the annual expenses for insurance, tolls, and gasoline are about 15,000 yuan. The maintenance cost for the first year is 3,000 yuan, which increases by 3,000 yuan each year thereafter. Determine the best scrap year limit for this car.

Answer: 10


What is the largest value of $x$ that satisfies the equation $\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form.

Answer: \frac{1}{\sqrt[3]{12}}


Given a triangle $C_{1} C_{2} O$. A bisector $C_{2} C_{3}$ is drawn, then in the triangle $C_{2} C_{3} O$ a bisector $C_{3} C_{4}$ is drawn, and so on.

Prove that the sequence of angles $\gamma_{n} = C_{n+1} C_{n} O$ converges, and find this limit if $\angle C_{1} O C_{2} = \alpha$.

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


There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?

Answer: 10


A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

Answer: 12\%


A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$.

Answer: \frac{1}{16}


In the addition sum shown, \(J\), \(K\), and \(L\) stand for different digits. What is the value of \(J + K + L\)?

A) 6
B) 8
C) 9
D) 10
E) 11

\[
\begin{array}{r}
J K L \\
J L L \\
+J K L \\
\hline 479
\end{array}
\]

Answer: 11


A numerical sequence is defined by the conditions: \( a_{1} = 1 \), \( a_{n+1} = a_{n} + \left\lfloor \sqrt{a_{n}} \right\rfloor \).

How many perfect squares are there among the first terms of this sequence that do not exceed \( 1{,}000{,}000 \)?

Answer: 10


A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square?

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


What is the sum of all the even integers between $200$ and $400$?

Answer: 30300


In the equation, $\overline{\mathrm{ABCD}}+\overline{\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\qquad$?

Answer: 31


Two adjacent faces of a tetrahedron, each of which is a regular triangle with a side length of 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points)

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


In the equation \(\overline{ABC} \times \overline{AB} + C \times D = 2017\), the same letters represent the same digits, and different letters represent different digits. If the equation holds true, what two-digit number does \(\overline{\mathrm{AB}}\) represent?

Answer: 14


For how many rational numbers  $p$  is the area of the triangle formed by the intercepts and vertex of  $f(x) = -x^2+4px-p+1$  an integer?

Answer: 4


In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and
obtains $8$ and $2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and
find $-9$ and $-1$ for the roots. The correct equation was:
$\textbf{(A)}\ x^2-10x+9=0 \qquad \textbf{(B)}\ x^2+10x+9=0 \qquad \textbf{(C)}\ x^2-10x+16=0\\  \textbf{(D)}\ x^2-8x-9=0\qquad \textbf{(E)}\ \text{none of these}$

Answer: \textbf{(A)}\x^2-10x+9=0


What amount can a worker, who is a tax resident, expect after paying personal income tax if they are credited with 45000 for their work?

Answer: 39150


A point $(x, y)$ is selected uniformly at random from the unit square $S=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\}$. If the probability that $(3x+2y, x+4y)$ is in $S$ is $\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$.

Answer: 820


In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $b=\sqrt{2}a$. Find the maximum value of the area of $\triangle ABC$.

Answer: 2\sqrt{2}


A square is divided into  $25$  unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?

Answer: 12


The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle ACB$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle CDE$?

Answer: \frac{50-25\sqrt{3}}{2}


After adding the coefficients \( p \) and \( q \) with the roots of the quadratic polynomial \( x^2 + px + q \), the result is 2, and after multiplying them, the result is 12. Find all such quadratic polynomials.

Answer: x^2+3x+2


In a diagram, $\triangle ABC$ and $\triangle BDC$ are right-angled, with $\angle ABC = \angle BDC = 45^\circ$, and $AB = 16$. Determine the length of $BC$.

Answer: 8\sqrt{2}


The average of a set of distinct primes is 27. What is the largest prime that can be in this set?

Answer: 139


For the set $\{1,2, \cdots, n\}$ and each of its non-empty subsets, we define the "alternating sum" as follows: arrange the numbers in the set in ascending order, then alternately add and subtract each number starting from the largest one (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$, and the alternating sum of $\{5\}$ is just 5). For $n=7$, find the total sum of all these alternating sums.

Answer: 448


A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list?

Answer: 6


Two regular tetrahedra are inscribed in a cube in such a way that four vertices of the cube serve as the vertices of one tetrahedron, and the remaining four vertices of the cube serve as the vertices of the other. What fraction of the volume of the cube is occupied by the volume of the intersection of these tetrahedra?

Answer: 1/6


A jar contains  $2$  yellow candies,  $4$  red candies, and  $6$  blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the  $2$  yellow candies will be eaten before any of the red candies are eaten is given by the fraction  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

Answer: 16


Prove that if an $n$-gon is divided arbitrarily into $k$ triangles, then $k \geq n-2$.

Answer: k\gen-2


Given a function \( f(x) \) defined on the set of all real numbers \(\mathbf{R}\) that satisfies the following conditions: (1) For any \( x \in \mathbf{R} \), \( 0 < f(x) \leq a \) (where \( a \) is a fixed constant); (2) For any \( x, y \in \mathbf{R} \), \( \sqrt{f(x) f(y)} \geq f\left(\frac{x+y}{2}\right) \). Prove that \( f(x) \) is a constant function.

Answer: f(x)


Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number?

Answer: 4423


Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence.  Compute the greatest possible value of the common ratio of this geometric sequence.

Answer: -2


Through a fixed point \( M(m, 0) \), draw an arbitrary line that intersects the parabola \( y^2 = 16x \) at points \( P \) and \( Q \). If \( \frac{1}{|PM|^2} + \frac{1}{|QM|^2} \) is a constant, find all possible values of the real number \( m \).

Answer: 8


Find the angle of inclination of the tangent line to the curve $y=\frac{1}{3}x^3-5$ at the point $(1,-\frac{3}{2})$.

Answer: \frac{\pi}{4}


Problems A, B, and C were posed in a mathematical contest. 25 competitors solved at least one of the three. Amongst those who did not solve A, twice as many solved B as solved C. The number solving only A was one more than the number solving A and at least one other. The number solving just A equalled the number solving just B plus the number solving just C. How many solved just B?

Answer: 6


In the $x-y$ plane, draw a circle of radius 2 centered at $(0,0)$. Color the circle red above the line $y=1$, color the circle blue below the line $y=-1$, and color the rest of the circle white. Now consider an arbitrary straight line at distance 1 from the circle. We color each point $P$ of the line with the color of the closest point to $P$ on the circle. If we pick such an arbitrary line, randomly oriented, what is the probability that it contains red, white, and blue points?

Answer: \frac{2}{3}


Answer: \frac{3}{4}


In a tetrahedron \(ABCD\) with a volume of 12, points \(E\), \(F\), and \(G\) are on edges \(AB\), \(BC\), and \(AD\) respectively. Given that \(AE = 2EB\), \(BF = FC\), and \(AG = 2GD\), a cross-section \(EFHG\) is made through points \(E\), \(F\), and \(G\). The perpendicular distance from point \(C\) to this cross-section is 1. Find the area of this cross-section.

Answer: 7


An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly $8$ moves that ant is at a vertex of the top face on the cube is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer: 49


In the Cartesian coordinate system, the coordinates of the two foci of an ellipse are $F_{1}(-2\sqrt{2},0)$ and $F_{2}(2\sqrt{2},0)$. The minimum distance from a point on ellipse $C$ to the right focus is $3-2\sqrt{2}$.

$(1)$ Find the equation of ellipse $C$;
$(2)$ Suppose a line with a slope of $-2$ intersects curve $C$ at points $E$ and $F$. Find the equation of the trajectory of the midpoint $N$ of segment $EF$;
$(3)$ Suppose a line passing through point $F_{1}(-2\sqrt{2},0)$ intersects curve $C$ and forms a chord segment $PQ$. Find the maximum area of $\triangle PQO$ ($O$ is the origin).

Answer: \dfrac{3}{2}


Let the three-digit number \( n = \overline{abc} \). If \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle, how many such three-digit numbers are there?

Answer: 165


From the 30 numbers $1, 2, 3, 4, \cdots, 30$, if we randomly select 10 consecutive numbers, how many situations are there where exactly 2 of the selected numbers are prime?

Answer: 4


Let \( K(x) \) be the number of irreducible fractions \( \frac{a}{b} \) such that \( a < x \) and \( b < x \) (where \( a \) and \( b \) are natural numbers).

For example, \( K\left(\frac{5}{2}\right) = 3 \) (the fractions are \( 1, 2, \frac{1}{2} \)).

Compute the sum \( K(100) + K\left(\frac{100}{2}\right) + K\left(\frac{100}{3}\right) + \ldots + K\left(\frac{100}{99}\right) + K\left(\frac{100}{100}\right) \).

Answer: 9801


Let  $S = \{1, 22, 333, \dots , 999999999\}$ . For how many pairs of integers  $(a, b)$  where  $a, b \in S$  and  $a < b$  is it the case that  $a$  divides  $b$ ?

Answer: 14


Let \( n \) be a positive integer and \(\left(x_{1}, x_{2}, \ldots, x_{2n}\right)\), where \( x_i = 0 \) or \( 1 \) for \( i = 1, 2, \ldots, 2n \), be a sequence of \( 2n \) integers. Let \( S_n \) be the sum
\[
S_{n}=x_{1} x_{2}+x_{3} x_{4}+\cdots+x_{2n-1} x_{2n} .
\]

If \(O_n\) is the number of sequences such that \(S_n\) is odd and \(E_n\) is the number of sequences such that \(S_n\) is even, prove that
\[
\frac{O_{n}}{E_{n}}=\frac{2^{n}-1}{2^{n}+1} .
\]

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


In how many ways can the letters of the word "ОЛИМПИАДА" be rearranged so that the subsequence "ЛАМПА" does not appear consecutively in that order (the identical rearrangement is also considered)? (12 points)

Answer: 90660


Given acute angles $\alpha$ and $\beta$ such that $\sin^2 \alpha + \sin^2 \beta = \sin(\alpha + \beta)$, prove that $\alpha + \beta = \frac{\pi}{2}$.

Answer: \alpha+\beta=\frac{\pi}{2}


Answer: 4


A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side?

Answer: \frac{1}{2}


Let \( P(0, a) \) be any point other than the origin on the \( y \)-axis. A line passing through the point \( P \) and parallel to the \( x \)-axis intersects the curve \( y = \frac{1}{a} \ln x \) at point \( Q \). The tangent to the curve \( y = \frac{1}{a} \ln x \) at point \( Q \) intersects the \( y \)-axis at point \( R \). Find the minimum value of the area of \( \triangle PQR \).

Answer: \frac{e^{\frac{1}{2}}}{\sqrt{2}}


Given the function $f(x)=x-\frac{1}{x}-\ln x$:
1. If $f(x)=x-\frac{1}{x}-\ln x$ has equal derivatives at $x=x_{1}$ and $x=x_{2}\left(x_{1} \neq x_{2}\right)$, prove that $f\left(x_{1}\right)+f\left(x_{2}\right)>3-2 \ln 2$.
2. If for any $k \in(-\infty, 1)$, the line $y=k x+b$ intersects the curve $y=f(x)$ at a unique point, find the range of the real number $b$.

Answer: [-\ln2,+\infty)


Divide an \( m \)-by-\( n \) rectangle into \( m n \) nonoverlapping 1-by-1 squares. A polyomino of this rectangle is a subset of these unit squares such that for any two unit squares \( S \) and \( T \) in the polyomino, either

(1) \( S \) and \( T \) share an edge, or

(2) there exists a positive integer \( n \) such that the polyomino contains unit squares \( S_1, S_2, S_3, \ldots, S_n \) such that \( S \) and \( S_1 \) share an edge, \( S_n \) and \( T \) share an edge, and for all positive integers \( k < n \), \( S_k \) and \( S_{k+1} \) share an edge.

We say a polyomino of a given rectangle spans the rectangle if for each of the four edges of the rectangle the polyomino contains a square whose edge lies on it.

What is the minimum number of unit squares a polyomino can have if it spans a 128-by-343 rectangle?

Answer: 470


If organisms do not die but only divide, then the population will certainly never die out.

The conditions are satisfied by the function whose graph is highlighted in the image.

$$
x(p)=\left\{\begin{array}{l}
1, \text { if } 0 \leq p \leq \frac{1}{2} \\
\frac{q}{p}, \text { if } \frac{1}{2}<p \leq 1
\end{array}\right.
$$

In our case $p=0.6>\frac{1}{2}$, therefore, $x=\frac{q}{p}=\frac{2}{3}$.

Answer: \frac{2}{3}


In acute triangle  $ABC$ , points  $D$  and  $E$  are the feet of the angle bisector and altitude from  $A$  respectively.  Suppose that  $AC - AB = 36$  and  $DC - DB = 24$ .  Compute  $EC - EB$ .

Answer: 54


Determine all positive integers \( n \) such that \( 3^n + 1 \) is divisible by \( n^2 \).

Answer: 1


Suppose that  $a$  and  $b$  are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of  $p$ ?

Answer: 5


Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\dots$, $C_6$. All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Find the product of the lengths of these twelve chords.

Answer: 28672


The sum of the following seven numbers is exactly 19: $a_1 = 2.56$, $a_2 = 2.61$, $a_3 = 2.65$, $a_4 = 2.71$, $a_5 = 2.79$, $a_6 = 2.82$, $a_7 = 2.86$. It is desired to replace each $a_i$ by an integer approximation $A_i$, $1\le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $\| A_i-a_i\|$, the maximum absolute value of the difference, is as small as possible. For this minimum $M$, what is $100M$?
Explanation of the Question
 Note: please read the explanation AFTER YOU HAVE TRIED reading the problem but couldn't understand.
 For the question. Let's say that you have determined 7-tuple $(A_1,A_2,A_3,A_4,A_5,A_6,A_7)$. Then you get the absolute values of the $7$ differences. Namely, \[|A_1-a_1|, |A_2-a_2|, |A_3-a_3|, |A_4-a_4|, |A_5-a_5|, |A_6-a_6|, |A_7-a_7|\] Then $M$ is the greatest of the $7$ absolute values. So basically you are asked to find the 7-tuple $(A_1,A_2,A_3,A_4,A_5,A_6,A_7)$ with the smallest $M$, and the rest would just be a piece of cake.

Answer: 61


A corner square of an $8 \times 8$ chessboard is removed. We want to cover the remaining board with congruent triangles without overlap. What is the minimum number of triangles needed?

Answer: 18


A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $ \frac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<{1000}$. Find $r+s+t.$

Answer: 330


Complex numbers $a, b, c$ form an equilateral triangle with side length 18 in the complex plane. If $|a+b+c|=36$, find $|b c+c a+a b|$.

Answer: 432


A triangle has integer side lengths \( a \), \( b \), and \( c \), and the length of one of its altitudes is equal to the sum of the lengths of the other two altitudes. Prove that the number \( a^{2} + b^{2} + c^{2} \) is a perfect square.

Answer: ^2+b^2+^2


In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a-4\cos C=c\cos B$.
$(1)$ Find the value of $b$;
$(2)$ If $a^{2}+b^{2}+c^{2}=2\sqrt{3}ab\sin C$, find the area of $\triangle ABC$.

Answer: 4\sqrt{3}


Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12  \qquad \textbf{(B)}\ 16 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 40$

Answer: \textbf{(C)}\28


Simplify the expression $\dfrac{45}{28} \cdot \dfrac{49}{75} \cdot \dfrac{100}{63}$.

Answer: \frac{5}{3}


The first term of a sequence is $3107$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $614^{\text{th}}$ term of the sequence?

Answer: 20


Given the equation \( abcd = aad \times e \), where different letters represent different digits, what is the maximum value of the four-digit number \( abcd \)?

Answer: 3015


Let \(I\) be the set of points \((x, y)\) in the Cartesian plane such that
\[ x > \left( \frac{y^4}{9} + 2015 \right)^{1/4} \]

Let \(f(r)\) denote the area of the intersection of \(I\) and the disk \(x^2 + y^2 \leq r^2\) of radius \(r > 0\) centered at the origin \((0,0)\). Determine the minimum possible real number \(L\) such that \(f(r) < L r^2\) for all \(r > 0\).

Answer: \frac{\pi}{3}


How many natural numbers \( n > 1 \) exist for which there are \( n \) consecutive natural numbers whose sum is equal to 2016?

Answer: 5


All the sequences consisting of five letters from the set $\{T, U, R, N, I, P\}$ (with repetitions allowed) are arranged in alphabetical order in a dictionary. Two sequences are called "anagrams" of each other if one can be obtained by rearranging the letters of the other. How many pairs of anagrams are there that have exactly 100 other sequences between them in the dictionary?

Answer: 0


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

Answer: 150


The numbers \(a, b, c, d\) belong to the interval \([-10.5, 10.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).

Answer: 462


(1) In an arithmetic sequence $\{ a_n \}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.

(2) Determine the interval of decrease for the function $y = \log_{\frac{1}{2}}(x^2 + 2x - 3)$.

(3) In triangle $ABC$, given $a = \sqrt{3}$, $b = 1$, and $\angle A = \frac{\pi}{3}$, find the value of $\cos B$.

(4) Point $A$ lies on circle $O$ with diameter $BC$ and is distinct from points $B$ and $C$. Point $P$ is outside the plane $ABC$ and plane $PBC$ is perpendicular to plane $ABC$. Given $BC = 3$, $PB = 2\sqrt{2}$, and $PC = \sqrt{5}$, find the surface area of the circumscribed sphere of the tetrahedron $P-ABC$.

Answer: 10\pi


Fix integers  $n\ge k\ge 2$ . We call a collection of integral valued coins  $n-diverse$  if no value occurs in it more than  $n$  times. Given such a collection, a number  $S$  is  $n-reachable$  if that collection contains  $n$  coins whose sum of values equals  $S$ . Find the least positive integer  $D$  such that for any  $n$ -diverse collection of  $D$  coins there are at least  $k$  numbers that are  $n$ -reachable.

*Proposed by Alexandar Ivanov, Bulgaria.*

Answer: n+k-1


The rabbits sawed several logs. They made 10 cuts and received 16 pieces. How many logs did they saw?

Answer: 6\,


Let  $N \ge 5$  be given. Consider all sequences  $(e_1,e_2,...,e_N)$  with each  $e_i$  equal to  $1$  or  $-1$ . Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length  $N$ .

Answer: 16


If we select numbers \(a_1, a_2, a_3\) from the set \(\{1, 2, \cdots, 14\}\) in increasing order such that \(a_2 - a_1 \geq 3\) and \(a_3 - a_2 \geq 3\), how many different ways are there to choose the numbers that satisfy these conditions?

Answer: 120


Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units.

Answer: \frac{11\pi}{24}


Let \( m, n, p, q \) be non-negative integers. For all \( x > 0 \):
$$
\frac{(x+1)^m}{x^n} - 1 = \frac{(x+1)^p}{x^q}
$$

Determine the value of \( \left(m^2 + 2n + p\right)^{2q} \).

Answer: 9


What is the greatest possible sum of the digits in the base-nine representation of a positive integer less than $3000$?

Answer: 24


Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a\cos C=c\sin A$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $a > 2$ and $b-c=1$, find the minimum perimeter of triangle $\triangle ABC$.

Answer: 9 + 6\sqrt{2}


Seven people stand in a row. (Write out the necessary process, and use numbers for the answers)
(1) How many ways can person A and person B stand next to each other?
(2) How many ways can person A and person B stand not next to each other?
(3) How many ways can person A, person B, and person C stand so that no two of them are next to each other?
(4) How many ways can person A, person B, and person C stand so that at most two of them are not next to each other?

Answer: 4320


Let $ABC$ be a triangle and $O$ the center of its circumcircle. Let $P$ and $Q$ be two points on the sides $[AC]$ and $[AB]$ respectively. Denote $K$, $L$, and $M$ as the midpoints of $[PB]$, $[QC]$, and $[PQ]$. Show that if the circle $KLM$ is tangent to $(PQ)$ then $OP = OQ$.

Answer: OP=OQ


In a \(3 \times 3\) table, the digits from 1 to 9 are placed in the cells. Then, the sums of the digits in each row are calculated. What is the maximum number of these sums that can be a perfect square?

Answer: 2


\(ABCD\) is a convex quadrilateral in which \(AC\) and \(BD\) meet at \(P\). Given \(PA = 1\), \(PB = 2\), \(PC = 6\), and \(PD = 3\). Let \(O\) be the circumcenter of \(\triangle PBC\). If \(OA\) is perpendicular to \(AD\), find the circumradius of \(\triangle PBC\).

Answer: 3


Find all prime numbers \( p \) and \( q \) such that \( p^{q+1} + q^{p+1} \) is a square of a natural number.

Answer: 2