Given three different points \( L, M \), and \( N \) on a straight line (with \( M \) between \( L \) and \( N \), and \( LN \neq MN \)), semicircles are constructed with diameters \( LM \), \( MN \), and \( LN \) such that the centers of these semicircles are points \( A, B \), and \( C \) respectively. Point \( C \) lies on one side of the line \( LN \), while points \( A \) and \( B \) lie on the other side of the line \( LN \). Find the ratio of the area of the figure enclosed by these three semicircles to the area of the triangle \( ABC \).

Answer: \pi


67. Find three numbers such that the largest exceeds the middle by a given part $\left(\frac{1}{3}\right)$ of the smallest, the middle exceeds the smallest by a given part $\left(\frac{1}{3}\right)$ of the largest, and the smallest exceeds the number 10 by a given part $\left(\frac{1}{3}\right)$ of the middle number.

Answer: \frac{45}{2}


Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.

Answer: \dfrac{4n^2}{9}


The line of sight of a circular cone is 1, and the angle it forms with the base is $\theta$. The edge length of the inscribed cube in this cone is $\qquad$
$\qquad$ (4 vertices of the cube are on the base of the cone, and the other 4 vertices are on the side surface of the cone).

Answer: \dfrac{\sqrt{2} \sin\theta \cos\theta}{\sin\theta + \sqrt{2} \cos\theta}


3. Let's call a natural number special if one of its digits can be replaced by another digit so that all digits in the resulting number are distinct. Numbers in which all digits are already distinct are also considered special. How many special ten-digit numbers exist? (20 points)

Answer: 150232320


41. In $\triangle A B C$, it is known that $\angle A=60^{\circ}$, the $n$-section lines of $\angle A B C$ and the $n$-section lines of $\angle A C B$ intersect at $G_{1}$, $G_{2}$, $G_{3}$, $\cdots$, $G_{n-1}$. When $n=10$, $\angle B G_{9} C=$ $\qquad$ degrees. (where the integer $n \geqslant 2$)

Answer: 72


3. Two trees of heights 5 and 10 meters are at equal distances from a streetlight pole. These two trees cast shadows of lengths 5 and 15 meters. How tall is the pole?

Answer: 20


Answer: 1


One, (20 points) A research study group from Class 1, Grade 3 of Yuhong Middle School conducted a survey on students' lunch time at the school canteen. It was found that within a unit of time, the number of people buying lunch at each window and the number of people choosing to eat outside due to unwillingness to wait for a long time are both fixed numbers. Furthermore, it was found that if only 1 window is opened, it takes $45 \mathrm{~min}$ to ensure that all waiting students can buy lunch; if 2 windows are opened simultaneously, it takes $30 \mathrm{~min}$. It was also discovered that if all waiting students can buy lunch within $25 \mathrm{~min}$, the number of people eating outside can be reduced by $80 \%$ within a unit of time. Assuming the total number of students in the school remains constant and everyone needs to have lunch, to facilitate student dining, the research group suggests that the school canteen should sell out lunch within $20 \mathrm{~min}$. How many windows should be opened simultaneously at least?

Answer: 5


## Task 1 - 130621

A rectangular glass pane is $24 \mathrm{~cm}$ long and $22 \mathrm{~cm}$ wide. From this, rectangular panes of $8 \mathrm{~cm}$ length and $6 \mathrm{~cm}$ width are to be cut.

What is the maximum number of such panes that can be obtained?

Illustrate a way to achieve this maximum number in a drawing at a scale of 1 : 2!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0202.jpg?height=323&width=282&top_left_y=581&top_left_x=293)

Answer: 11


We are given a map divided into $13\times 13$ fields. It is also known that at one of the fields a tank of the enemy is stationed, which we must destroy. To achieve this we need to hit it twice with shots aimed at the centre of some field. When the tank gets hit it gets moved to a neighbouring field out of precaution. At least how many shots must we fire, so that the tank gets destroyed certainly?
[i]We can neither see the tank, nor get any other feedback regarding its position.[/i]

Answer: 254


Problem 12.2. A circle through the vertex $A$ of $\triangle A B C, A B \neq A C$, meets the sides $A B$ and $A C$ at points $M$ and $N$, respectively, and the side $B C$ at points $P$ and $Q$, where $Q$ lies between $B$ and $P$. Find $\Varangle B A C$, if $M P \| A C$, $N Q \| A B$ and $\frac{B P}{C Q}=\frac{A B}{A C}$

Oleg Mushkarov, Nikolai Nikolov

Answer: 120^\circ


Three thieves $A, B$ and $C$ divided the proceeds of a robbery equally. On the first night, while $C$ was sleeping, $A$ and $B$ took half of what he had and divided it equally between themselves. On the second night, while $A$ was sleeping, $B$ and $C$ took half of what he had and divided it equally between themselves. On the third night, while $B$ was sleeping, $A$ and $C$ took half of what he had and divided it equally between themselves. The next morning, they parted ways forever. When $B$ counted his money, he realized he had $\mathrm{R} \$ 10,000.00$. Determine the initial amount that was stolen by the thieves.

Answer: 38400


12. The positive integer solutions $(x, y)$ of the equation $2 x^{2}-x y-3 x+y+2006=0$ are $\qquad$ pairs.

Answer: 4


In $\triangle A B C, A B=2, B C=\sqrt{3}$ and $\angle A B C=150^{\circ}$. $P$ is a point on the plane such that $\angle A P B=45^{\circ}$ and $\angle B P C=120^{\circ}$. Find $B P$.
(2 marks)
In $\triangle A B C, A B=2, B C=\sqrt{3}$ and $\angle A B C=150^{\circ}$. $P$ is a point on the plane such that $\angle A P B=45^{\circ}$ and $\angle B P C=120^{\circ}$. Find $B P$.

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


Test $3 P$ is a point inside $\triangle A B C$, line $A C$ and $B P$ intersect at $Q$, line $A B$ and $C P$ intersect at $R$, it is known that $A R=R B=C P, C Q=P Q$. Find $\angle B R C$.

Answer: 120^\circ


1. Consider a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, where $A B C D$ is the bottom face with the letters assigned in a clockwise direction, and $A, B, C$, and $D$ are directly below $A^{\prime}, B^{\prime}, C^{\prime}$, and $D^{\prime}$ respectively. The parallelepiped is divided into eight pieces by three planes that are orthogonal to each other and parallel to the faces of the parallelepiped. For each vertex $P$ of the parallelepiped, let $V_{P}$ denote the volume of the piece of the parallelepiped that contains $P$. Given that $V_{A}=40, V_{C}=300, V_{B^{\prime}}=360$ and $V_{C^{\prime}}=90$, what is the volume of the parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$?

Answer: 2015


1. We will call a prime number nice if it can be written as the difference of two cubes of natural numbers. Determine the last digit of all nice prime numbers.

Answer: 1


3. The tennis player and the badminton player got tired of sports tournaments and decided to have a mathematical duel. They wrote down the numbers from 25 to 57 inclusive in a row and agreed to place either “+” or “-” (in any order, but only these signs) between them. If the result of the expression turns out to be even when all the signs between the numbers are placed, the tennis player wins, and if it is odd - the badminton player wins. Who turned out to be the winner and why?

Olympiad for schoolchildren "Hope of Energy". Final stage.

When all the signs between the numbers are placed, the tennis player wins, and if it is odd - the badminton player wins. Who turned out to be the winner and why?

Answer: \text{Badminton player}


Evdokimov M.A.

In triangle $A B C A B=B C, \angle B=20^{\circ}$. Point $M$ on the base $A C$ is such that $A M: M C=1: 2$, point $H$ is the projection of $C$ on $B M$. Find the angle $A H B$.

Answer: 100^\circ


7. As shown in Figure 2, a right-angle ruler $A B C$ is tangent to $\odot O$ at point $D$, and $A$ touches $\odot O$ at point $A$. It is measured that $A B=a, B D=b$. Then the radius of $\odot O$ is $\qquad$.

Answer: \dfrac{a^2 + b^2}{2a}


11.7. Given a polynomial

$$
P(x)=a_{2 n} x^{2 n}+a_{2 n-1} x^{2 n-1}+\ldots+a_{1} x+a_{0}
$$

where each coefficient $a_{i}$ belongs to the interval $[100,101]$. For what minimal $n$ can such a polynomial have a real root? (I. Bogdanov, K. Sukhov)

Answer: 100


On each non-boundary unit segment of an  $8\times 8$  chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?

Answer: 0


4. Given the set $S=\{1,2, \cdots, 2005\}, A \subseteq S, A$ such that the sum of any two numbers in $A$ is not divisible by 117, find the maximum value of $|A|$.

 untranslated text remains the same as requested. However, if you need any further assistance or a different translation, feel free to let me know!

Answer: 1003


Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realized that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?

Answer: 26


Is it possible to cut a cube with an edge length of 1 into parts using four planes in such a way that, for each part, the distance between any two points within the part is:

a) less than \( \frac{4}{5} \);

b) less than \( \frac{4}{7} \)?

It is assumed that all planes are cut simultaneously, and the cube and its parts do not move.

Answer: \text{Yes}


2) As shown in Figure 1, in a $4 \times 4$ grid composed of small squares with a side length of 1, there are 25 grid points. Among the right-angled triangles with grid points as vertices, the number of right-angled triangles with the lengths of the two legs being 1 and 3 is $\qquad$ .

Answer: 64


15. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{0}=1, a_{n}=\left[\sqrt{S_{n-1}}\right]$ $(n=1,2, \cdots,[x]$ represents the greatest integer not greater than $x$, $\left.S_{k}=\sum_{i=0}^{k} a_{i}(k=0,1, \cdots)\right)$. Find the value of $a_{2006}$.

Answer: 998


A right pyramid has a rectangular base with dimensions 8 cm by 15 cm. Its peak is 10 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges. Round your answer to the nearest whole number.

Answer: 99


In square $XYZW$, points $P$ and $S$ lie on $\overline{XZ}$ and $\overline{XW}$, respectively, such that $XP=XS=\sqrt{3}$. Points $Q$ and $R$ lie on $\overline{YZ}$ and $\overline{YW}$, respectively, and points $T$ and $U$ lie on $\overline{PS}$ so that $\overline{QT} \perp \overline{PS}$ and $\overline{RU} \perp \overline{PS}$. If triangle $XPS$, quadrilateral $YQTP$, quadrilateral $WSUR$, and pentagon $YRUTQ$ each has an area of $1.5$, find $QT^2$.
[asy]
real x = 3;
real y = 3*sqrt(3)/2;
real z = 3/sqrt(3);
pair X, Y, Z, W, P, Q, R, S, T, U;
X = (0,0);
Y = (4,0);
Z = (4,4);
W = (0,4);
P = (x,0);
Q = (4,y);
R = (y,4);
S = (0,x);
T = Q + z * dir(225);
U = R + z * dir(225);
draw(X--Y--Z--W--X);
draw(P--S);
draw(T--Q^^U--R);
draw(rightanglemark(Q, T, P), linewidth(.5));
draw(rightanglemark(R, U, S), linewidth(.5));
dot("$X$", X, SW);
dot("$Y$", Y, S);
dot("$Z$", Z, N);
dot("$W$", W, NW);
dot("$P$", P, S);
dot("$Q$", Q, E);
dot("$R$", R, N);
dot("$S$", S, W);
dot("$T$", T, SW);
dot("$U$", U, SW);
[/asy]

Answer: 3


In the diagram, there are two circles. The larger circle has center $O$ and an area of $100\pi$. The smaller circle has its center on the circumference of the larger circle and touches the larger circle internally at one point. Each circle is divided into three equal areas, and we want to find the total area of the shaded regions which cover two of the three parts for each circle.
```asy
size(120);
import graph;
fill(Arc((0,0),2,240,480)--cycle,mediumgray);
fill(Arc((1,0),1,60,300)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((1,0),1));
dot((0,0)); label("$O$", (0,0), W);
draw((-2,0)--(2,0));
draw((1,-1)--(1,1));
```

Answer: \frac{250\pi}{3}


On a 6 by 6 square grid where each dot is 1 cm apart from its nearest horizontal and vertical neighbors, consider a right triangle with vertices labeled $P$, $Q$, and $R$. The vertices $P$, $Q$, and $R$ are located at coordinates $(1, 5)$, $(5, 5)$, and $(1, 1)$, respectively. What is the product of the area of triangle $PQR$ and the perimeter of triangle $PQR$? Express your answer in simplest form.

Answer: 64 + 32\sqrt{2} \text{ cm}^3


Mrs. Lee has 12 grandchildren. Assuming that each grandchild is male with a probability of $\frac{2}{3}$ and female with a probability of $\frac{1}{3}$, independently of the others, what is the probability that Mrs. Lee has more grandsons than granddaughters or vice versa?

Answer: \frac{472305}{531441}


Six identical rectangles are arranged to form a larger rectangle $PQRS$, where the width of each smaller rectangle is $\frac{2}{5}$ of the length of the larger rectangle. If the area of $PQRS$ is $6000$, what is the length $x$, rounded off to the nearest integer?

Answer: 50


I have 6 shirts, 5 pairs of pants, and 6 hats. Each piece of clothing comes in six possible colors: tan, black, blue, gray, white, and yellow. I refuse to wear an outfit in which all 3 items are the same color, and I also refuse to wear pants and hats of the same color. How many outfit selections, consisting of one shirt, one pair of pants, and one hat, do I have?

Answer: 150


Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[
\frac{a^2}{kb} + \frac{b^2}{kc} + \frac{c^2}{ka}.
\]
for some positive constant \( k \).

Answer: \frac{3}{k}


An amoeba is placed in a controlled environment on day 1. Starting from the initial day, each amoeba splits into two amoebas every day, except that every third day, only half of the amoebas are capable of splitting due to environmental stress. After 10 days, how many amoebas will there be, assuming the controlled environment starts with one amoeba?

Answer: 432


IX OM - II - Task 2

Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to its initial position?

Answer: 4


9. Given a sphere with radius 4, there are two points $A, B$ on the sphere, $A B=4 \sqrt{2}$, and the center of the sphere is $O$. If a moving point $C$ on the sphere satisfies that the dihedral angle $C-A B-O$ is $60^{\circ}$, then the radius of the circumscribed sphere of the tetrahedron $O A B C$ is $\qquad$.

Answer: \dfrac{4\sqrt{6}}{3}


Answer: \dfrac{1 + \sqrt{5}}{2}


.

Let $D$ be the footpoint of the altitude from $B$ in the triangle $A B C$, where $A B=1$. The incentre of triangle $B C D$ coincides with the centroid of triangle $A B C$. Find the lengths of $A C$ and $B C$.

Answer: \dfrac{\sqrt{10}}{2}


4. We will write down all five-digit numbers in which each of the digits $4,5,6,7,8$ appears exactly once. Then we will strike out one (any of them) and sum all the remaining ones. What are the possible values of the digital sum of such a result?

The regional round of category B takes place

## on Tuesday, April 5, 2011

so that it starts in the morning and the contestants have 4 hours of clean time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphical display. These details will be communicated to the students before the start of the competition.

Answer: 33


11. There are 6 numbers: 1, 3, 3, 27, 81, 243. Each time, you can either take one of them, or take several different numbers to find their sum (each number can only be taken once, which can result in a new number). This way, a total of 63 new numbers can be obtained. If they are arranged in ascending order, they are 1, 3, 4, 9, ... What is the 58th number? $\qquad$ -

Answer: 354


Problem 3. Points $A$ and $B$, located on a circular avenue, are connected by a straight highway segment 4 km long, which is the diameter of the circular avenue. A pedestrian left his house and started walking along the avenue from point $A$. After 1 hour, he realized he had forgotten his keys and asked a neighbor-cyclist to bring them as quickly as possible. What is the minimum time it will take for the pedestrian to receive the keys if the cyclist's speed on the highway is 15 km/h, on the avenue - 20 km/h, and the pedestrian's speed is 7 km/h? The pedestrian can walk towards the cyclist.

Answer: \dfrac{4\pi -7}{27}


4. If we divide the number 13 by the three numbers 5,7 , and 9 , then these divisions leave remainders: when dividing by 5 the remainder is 3 , when dividing by 7 the remainder is 6 , and when dividing by 9 the remainder is 4 . If we add these remainders, we obtain $3+6+4=13$, the original number.

(a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$. Prove: if you divide $n$ by $a$ and $b$, then the sum of the two remainders never equals $n$.

(b) Determine all integers $n>229$ having the property that if you divide $n$ by 99,132 , and 229 , the sum of the three remainders is $n$.

Answer: 395


In a mathematics competition, 5 problems were set. Among the contestants, there were no two who solved exactly the same problems. However, if we ignore any one of the problems, then for any contestant, there is another contestant who solved the same problems from the remaining 4. How many people participated in the competition?

Answer: 32


Example 7 Given a real number $c \in\left(\frac{1}{2}, 1\right)$. Find the smallest constant $M$, such that for any integer $n \geqslant 2$ and real numbers $0<a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, as long as they satisfy
$$
\frac{1}{n} \sum_{k=1}^{n} k a_{k}=c \sum_{k=1}^{n} a_{k},
$$

we always have $\sum_{k=1}^{n} a_{k} \leqslant M \sum_{k=1}^{m} a_{k}$, where $m=[c n]$ is the greatest integer not exceeding the real number $c n$.
(2002, China Mathematical Olympiad)
Analysis: To find $M$, we need to establish a relationship between $\sum_{k=1}^{n} a_{k}$ and $\sum_{k=1}^{m} a_{k}$. But how can we derive $\sum_{k=1}^{m} a_{k}$ from the given conditions? Given $0<a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$ and $m=[c n]$, we think of using Chebyshev's inequality.

Answer: \dfrac{1}{1 - c}


N1. Determine all positive integers $M$ for which the sequence $a_{0}, a_{1}, a_{2}, \ldots$, defined by $a_{0}=\frac{2 M+1}{2}$ and $a_{k+1}-a_{k}\left\lfloor a_{k}\right\rfloor$ for $k-0,1,2, \ldots$, contains at least one integer term.

Answer: M


In the drawing to the side, triangle $ABC$ is equilateral and $BD = CE = AF = \frac{AB}{3}$. The ratio $\frac{EG}{GD}$ can be written in the form $\frac{m}{n}, \operatorname{gcd}(m, n)=1$. What is the value of $m+n$?

![](https://cdn.mathpix.com/cropped/2024_05_01_12f29789e10e6749ebb9g-37.jpg?height=460&width=516&top_left_y=1150&top_left_x=884)

#

Answer: 5


45th Putnam 1984 Problem A5 V is the pyramidal region x, y, z >= 0, x + y + z <= 1. Evaluate ∫ V x y 9 z 8 (1 - x - y - z) 4 dx dy dz. Solution

Answer: \dfrac{9! \cdot 8! \cdot 4!}{25!}


Theorem of Sines [ Ratios of linear elements of similar triangles ]

In an acute triangle $A B C$, heights $A P$ and $C Q$ are drawn from vertices $A$ and $C$. Find the side $A C$, given that the perimeter of triangle $A B C$ is 15, the perimeter of triangle $B P Q$ is 9, and the radius of the circumscribed circle of triangle $B P Q$ is $9 / 5$.

Answer: \dfrac{24}{5}


5.1. (14 points) In an acute-angled triangle $A B C$, angle $A$ is equal to $35^{\circ}$, segments $B B_{1}$ and $C C_{1}$ are altitudes, points $B_{2}$ and $C_{2}$ are the midpoints of sides $A C$ and $A B$ respectively. Lines $B_{1} C_{2}$ and $C_{1} B_{2}$ intersect at point $K$. Find the measure (in degrees) of angle $B_{1} K B_{2}$.

Answer: 75


3. The mathematical committee consists of $2 n$ members, $n \geqslant 3$. It is known that each member of the committee is in a dispute with exactly one other member of the committee (this relationship is symmetric). In how many ways can the committee be divided into three subcommittees: one for creating problems, one for evaluating problems, and one for organizing the competition, such that each subcommittee has at least two members and no two members of the committee who are in a dispute are in the same subcommittee?

Answer: 6^n - 3 \cdot 2^n (1 + 2n)


1. As shown in Figure 3, $O$ is a point inside $\triangle A B C$, extending $A O$ intersects $B C$ at point $D$, extending $B O$ intersects $A C$ at point $E$, the areas of $\triangle B O D$, $\triangle A O E$, and $\triangle A O B$ are 7, $14$, and $21$ respectively. Then $\frac{B D}{C D}=$ $\qquad$ .

Answer: \dfrac{7}{8}


Let $n \leq 100$ be an integer. Hare puts real numbers in the cells of a $100 \times 100$ table. By asking Hare one question, Wolf can find out the sum of all numbers of a square $n \times n$, or the sum of all numbers of a rectangle $1 \times (n - 1)$ (or $(n - 1) \times 1$). Find the greatest $n{}$ such that, after several questions, Wolf can find the numbers in all cells, with guarantee.

Answer: 51


10.425 A chord $AB$ is drawn through a point $M$ located on the diameter of a circle with a radius of 4 cm, forming an angle of $30^{\circ}$ with the diameter. A chord $BC$ is drawn through point $B$, perpendicular to the given diameter. Find the area of triangle $ABC$, if $AM: MB=2: 3$.

Answer: \dfrac{180\sqrt{3}}{19}


There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.

Answer: 11


4. Given the function $f(x)=\frac{2 x^{3}-6 x^{2}+13 x+10}{2 x^{2}-9 x}$

Determine all positive integers $x$ for which $f(x)$ is an integer.

Answer: 5


2. A triangle with a perimeter of $55 \mathrm{~mm}$ is divided by a certain line segment into two triangles. Calculate the length of the line segment by which the original triangle is divided if the perimeters of the resulting triangles are $41 \mathrm{~mm}$ and $25 \mathrm{~mm}$.

Answer: \dfrac{11}{2}


【Question 15】
Zhao, Qian, Sun, and Li, four children, all like playing chess. Among them, someone likes playing Chinese chess, someone likes playing Go, and someone likes both types of chess. Therefore, the different ways these four people can like these two types of chess are $\qquad$ kinds.

Answer: 36


Find all primes $p$ and $q$ such that $3 p^{q-1}+1$ divides $11^{p}+17^{p}$.

Time allowed: 4 hours and 30 minutes.
Each problem is worth 10 points.

Answer: p=3


## 10. Math Puzzle 3/66

Two hikers have planned a distance of $40 \mathrm{~km}$. Since one of them has a bicycle, he suggests to the other: "We will take turns using the bicycle, and you will ride for half an hour, then leave the bicycle by the road and continue on foot. In the meantime, I will walk to the bicycle, ride after you until I catch up, then you will ride for another half an hour, etc."

How much time will the two need if they walk at $5 \frac{\mathrm{km}}{\mathrm{h}}$ and cycle at $20 \frac{\mathrm{km}}{\mathrm{h}}$?

Answer: 5


Let \(ABC\) be a triangle, and let \(\ell_{1}\) and \(\ell_{2}\) be two parallel lines. For \(i=1, 2\), let \(\ell_{i}\) meet the lines \(BC, CA\), and \(AB\) at \(X_{i}, Y_{i}\), and \(Z_{i}\), respectively. Suppose that the line through \(X_{i}\) perpendicular to \(BC\), the line through \(Y_{i}\) perpendicular to \(CA\), and the line through \(Z_{i}\) perpendicular to \(AB\) determine a non-degenerate triangle \(\Delta_{i}\). Show that the circumcircles of \(\Delta_{1}\) and \(\Delta_{2}\) are tangent to each other.

Answer: \text{The circumcircles of } \Delta_1 \text{ and } \Delta_2 \text{ are tangent to each other.}


Answer: 26


3. Four congruent right triangles can be assembled to form the square string figure shown in Figure 2(a), or the rhombus shown in Figure 2(b). If the area of the large square in Figure 2(a) is 100, and the area of the small square is 4, then the cosine value of one acute angle in the rhombus in Figure 2(b) is

Answer: \dfrac{7}{25}


Segment $A B$ is the diameter of a circle. A second circle with center at point $B$ has a radius of 2 and intersects the first circle at points $C$ and $D$. Chord $C E$ of the second circle is part of a tangent to the first circle and is equal to 3. Find the radius of the first circle.

Answer: \dfrac{4\sqrt{7}}{7}


2. [4] You start with a number. Every second, you can add or subtract any number of the form $n$ ! to your current number to get a new number. In how many ways can you get from 0 to 100 in 4 seconds? ( $n!$ is defined as $n \times(n-1) \times(n-2) \times \cdots \times 2 \times 1$, so $1!=1,2!=2,3!=6,4!=24$, etc.)

Answer: 36


5. Determine all pairs $(x, y)$ of integers with $x \leq y$ for which $\frac{1}{x}+\frac{1}{y}=\frac{1}{4}$.

Answer: (-12, 3)


3. There are 16 members on the Height-Measurement Matching Team. Each member was asked, "How many other people on the team - not counting yourself - are exactly the same height as you?" The answers included six 1's, six 2's, and three 3's. What was the sixteenth answer? (Assume that everyone answered truthfully.)

Answer: 3


10. Let $a>1$ be a positive real number, and $n \geqslant 2$ be a natural number, and the equation $[a x]=x$ has exactly $n$ distinct solutions, then the range of values for $a$ is $\qquad$ .

Answer: \left[1 + \frac{1}{n}, 1 + \frac{1}{n-1}\right)


4. Let $\sigma=\left(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\right)$ be a permutation of $(1,2,3, \ldots, n)$. A pair $\left(a_{i}, a_{j}\right)$ is said to correspond to an inversion of $\sigma$, if $ia_{j}$. (Example: In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $(2,1)$, $(4,3),(4,1),(5,3),(5,1),(3,1)$.) How many permutations of $(1,2,3, \ldots n)$, $(n \geq 3)$, have exactly two inversions?

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


Problem 5.8. A boastful fisherman says the same phrase every day: "Today I caught more perch than I did the day before yesterday (2 days ago), but less than I did 9 days ago." What is the maximum number of days in a row that he can tell the truth?

Answer: 8


3. In the 100th year of Besmiki's tenure as the President of the Currency Authority, he decided to issue new gold coins. In this year, he put into circulation an unlimited number of gold coins with a face value of $2^{100}-1$ yuan. In the following year, he put into circulation an unlimited number of gold coins with a face value of $2^{101}-1$ yuan. This continued every year until the day when the face value of a newly issued gold coin equaled the sum of the face values of some gold coins issued in previous years, at which point he was dismissed. When did this situation occur in Besmiki's tenure as President?

Answer: 200


Given a function \( y = f(x) \) whose graph is both centrally symmetric about the point \((1,1)\) and axisymmetric about the line \( x + y = 0 \). If \( f(x) = \log_2(x + 1) \) for \( x \in (0,1) \), what is the value of \( f(\log_2 10) \)?

Answer: \frac{17}{5}


In the figure below, \( \triangle ABC \) is a triangle with side lengths \( AB = 4 \, \text{cm}, AC = 6 \, \text{cm} \). Additionally, \( \angle DAH = \angle HAB \), \( \angle AHB = 90^\circ \), and \( M \) is the midpoint of \( BC \). Find the length of the segment \( MH \).

![Figure](https://cdn.mathpix.com/cropped/2024_05_01_12f29789e10e6749ebb9g-25.jpg?height=549&width=465&top_left_y=590&top_left_x=867)

Answer: 1


For each positive integer \( n \) let \( S_{n} \) denote the set \( \{1,2,3, \ldots, n\} \). Compute the number of triples of subsets \( A, B, C \) of \( S_{2006} \) (not necessarily nonempty or proper) such that \( A \) is a subset of \( B \) and \( S_{2006}-A \) is a subset of \( C \).

Answer: 2^{4012}


In the drawing, a karting track layout is shown. The start and finish are at point \( A \), with the kart driver being able to return to point \( A \) and reenter the loop as many times as desired.

The time taken for the journey from \( A \) to \( B \) or back is one minute. The time taken to travel around the loop is also one minute. The loop can only be driven counterclockwise (the arrows show possible directions). The driver does not turn back midway and does not stop. The total duration of the race is 10 minutes. Determine the number of possible different routes (sequences of segments).

Answer: 34


The sum of two numbers is 50, and their product is 25. If the sum of their reciprocals is $a$, find $a$.

If the lines $ax + 2y + 1 = 0$ and $3x + by + 5 = 0$ are perpendicular, find $b$.

The area of an equilateral triangle is $100\sqrt{3} \mathrm{~cm}^2$. If its perimeter is $p \mathrm{~cm}$, find $p$.

If $x^3 - 2x^2 + px + q$ is divisible by $x + 2$, find $q$.

Answer: 136


A square is cut into four equal rectangles, which are then arranged to form the capital letter "P" as shown in the figure. The perimeter of the letter is 56. What is the perimeter of the original square?

Answer: 32


The time on a digital clock reads 3:33. What is the shortest length of time, in minutes, until all of the digits are again equal to each other?

Answer: 71


In the regular hexagon \( ABCDEF \) shown in the figure, point \( P \) is a point on \( AB \). It is known that the area of \( \triangle AFP = 8 \) and the area of \( \triangle CDP = 42 \). What is the area of \( \triangle EFP \)?

Answer: 33


4. Let the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ satisfy:
(1) If $x \leqslant y$, then $f(x) \leqslant f(y)$;
(2) When $x>0$,
$$
\begin{array}{l}
f\left(x^{4}\right)+f\left(x^{2}\right)+f(x)+f(1) \\
=x^{4}+x^{2}+x+1 .
\end{array}
$$

Find the function $f$.

Answer: f(x) = x


56. Person A and Person B start from points $A$ and $B$ respectively at the same time, heading towards each other. They meet for the first time 100 meters away from point $B$. After the meeting, Person A's speed doubles; Person A reaches $B$ and immediately turns back, catching up with Person B when Person B is still 50 meters away from $A$. Therefore, the distance between $A$ and $B$ is $\qquad$ meters.

Answer: 250


A set $A$ of positive integers is called [i]uniform[/i] if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements.

Answer: 7


[ Properties and characteristics of an isosceles triangle.] [Sum of the angles of a triangle. Theorem of the exterior angle.]

Triangle $A B C$ is isosceles $(A B=B C)$. Segment $A M$ divides it into two isosceles triangles with bases $A B$ and $M C$. Find angle $B$.

Answer: 36^\circ


For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \tfrac pq$ for coprime positive integers $p$, $q$, find $p + q$.

[i]Proposed by Justin Lee[/i]

Answer: 91


Anna and Zsófi take turns rolling a die, and they always add the rolled number to the sum of the numbers rolled so far. The one who, after their roll, makes the sum first divisible by 4 wins. If Anna starts, what is the probability that she wins?

Answer: \dfrac{52}{99}


8. Given that the volume of tetrahedron $P-ABC$ is $1$, $G$ and $K$ are the centroids of $\triangle ABC$ and $\triangle PBC$ respectively. A line through $G$ intersects $AB$ and $AC$ at points $M$ and $N$. Then the maximum volume of the quadrilateral pyramid $K-MNCB$ is

Answer: \dfrac{5}{27}


## Task B-1.4.

Three friends, Ante, Bojan, and Vinko, are guessing an unknown six-digit number composed of the digits $1,2,3,4,5,6$, with no repeated digits. Ante said the number is 123456, Bojan said 245163, and Vinko said 463215. None of them guessed the exact number, but Ante correctly guessed the positions of 3 digits, Bojan also correctly guessed the positions of 3 digits, and Vinko correctly guessed the position of 1 digit. Determine the unknown number.

Answer: 243156


10. A bag contains 10 balls each of red, white, and black colors. If 12 balls are drawn from it, but it is required that all three colors are included, then there are $\qquad$ different ways to do so.

Answer: 55


Two weightless glasses with different quantities of water are balanced on the beam of unequal-arm scales with length $L$. An amount of water with mass $m$ is poured from one glass into the other, causing the scales to lose balance. To restore balance, the support point of the beam needs to be shifted by a distance $l$ relative to its original position. Find the total mass of water in the glasses, ignoring the mass of the beam itself.

Answer: \dfrac{m L}{l}


Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.

Answer: n-1


Digital clocks show hours and minutes, such as 14:37.

How many minutes per day does at least one five light up on these clocks?

Answer: 450


Four, in an election, there are 12 candidates, and each member of the electoral committee casts 6 votes. It is known that any two members' votes have at most 2 candidates in common. Find the maximum number of members in the committee.
(Proposed by the Problem Committee)

Answer: 4


## Zadatak B-4.3.

Prvi, peti i jedanaesti član rastućeg aritmetičkog niza istovremeno su tri uzastopna člana geometrijskog niza. Odredite sto dvanaesti član aritmetičkog niza, ako je njegov prvi član jednak 136 .

Answer: 2023


Answer: -3


Problem 3. A square with side length 5 is divided into unit squares by parallel to its sides lines. Let $A$ be the set of the vertexes of the unit squares which are not on the sides of the given square. How many points from $A$ can be chosen at most in a way that no three of them are vertexes of isosceles rectangle triangle?

Answer: 6


3.1. How many numbers from 1 to 1000 can be represented as $[2 x]+[4 x]+[6 x]+[8 x]$ with positive $x$?

Answer: 600


Ding 0. .

Detective Nero Wolfe is investigating a crime. There are 80 people involved in the case, one of whom is the criminal, and another is a witness to the crime (but it is unknown who they are). Each day, the detective can invite one or several of these 80 people to his office, and if the witness is among the invited but the criminal is not, the witness will reveal who the criminal is. Can the detective definitely solve the case in 12 days?

Answer: Yes


## 15. The Great Mass

The main stained glass window of a modern cathedral is a circle with a diameter of 2 m, intersected by a cross formed by two perpendicular lines that intersect at a point 50 cm away from the center of the window. During the great mass, one slightly absent-minded Parisian decided to calculate the sum of the squares of the lengths of the two segments of this cross.

"Look at this! - he exclaimed. - The number of square meters is equal to the number of mortal sins ${ }^{2}$".

Why did this happen?

Answer: 7


As a seasoned editor of a comic magazine, you are planning a special issue featuring a collection of comic strips. You have reviewed 150 comic strips this month. For the special issue, you want to include only those that have been rated 4 stars or higher. Out of the 150 comic strips, 60% received a rating of 4 stars or higher.

1. How many comic strips received a rating of 4 stars or higher?
2. If you can only include 75% of those 4-star or higher-rated comic strips in the special issue, how many comic strips will be featured in the special issue?

Answer: 68


Given positive integer  $ m \geq 17$ ,  $ 2m$  contestants participate in a circular competition. In each round, we devide the  $ 2m$  contestants into  $ m$  groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for  $ 2m\minus{}1$  rounds so that each contestant has played a game with all the  $ 2m\minus{}1$  players. Find the least possible positive integer  $ n$ , so that there exists a valid competition and after  $ n$  rounds, for any  $ 4$  contestants, non of them has played with the others or there have been at least  $ 2$  games played within those  $ 4$ .

Answer: m-1


There are 51 senators in a senate. The senate needs to be divided into  $n$  committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does *not* necessarily hate senator A.) Find the smallest  $n$  such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

Answer: 7


Let  $p(x)$  be the monic cubic polynomial with roots  $\sin^2(1^{\circ})$ ,  $\sin^2(3^{\circ})$ , and  $\sin^2(9^{\circ})$ . Suppose that  $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$ , where  $0 <a,b \le 90$  and  $a,b,n$  are positive integers.  What is  $a+b+n$ ?

*Proposed by Andrew Yuan*

Answer: 216


Let  $n > 1$  be an integer. In a circular arrangement of  $n$  lamps  $L_0, \ldots, L_{n-1},$  each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps,  $Step_0, Step_1, \ldots .$   If  $L_{j-1}$  ( $j$  is taken mod  $n$ ) is ON then  $Step_j$  changes the state of  $L_j$  (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If  $L_{j-1}$  is OFF then  $Step_j$  does not change anything at all. Show that:

(i) There is a positive integer  $M(n)$  such that after  $M(n)$  steps all lamps are ON again,

(ii) If  $n$  has the form  $2^k$  then all the lamps are ON after  $n^2-1$  steps,

(iii) If  $n$  has the form  $2^k + 1$  then all lamps are ON after  $n^2 - n + 1$  steps.

Answer: n^2 - n + 1


We only know that the password of a safe consists of  $7$  different digits. The safe will open if we enter  $7$  different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than  $7$  attempts?

*(5 points for Juniors and 4 points for Seniors)*

Answer: 6


Let  $F$  be the family of all sets of positive integers with  $2010$  elements that satisfy the following condition:
The difference between any two of its elements is never the same as the difference of any other two of its elements. Let  $f$  be a function defined from  $F$  to the positive integers such that  $f(K)$  is the biggest element of  $K \in F$ . Determine the least value of  $f(K)$ .

Answer: 4040100


There are  $n+1$  containers arranged in a circle. One container has  $n$  stones, the others are empty. A move is to choose two containers  $A$  and  $B$ , take a stone from  $A$  and put it in one of the containers adjacent to  $B$ , and to take a stone from  $B$  and put it in one of the containers adjacent to  $A$ . We can take  $A = B$ . For which  $n$  is it possible by series of moves to end up with one stone in each container except that which originally held  $n$  stones.

Answer: n


Let  $n$  be a positive integer. We are given a  $3n \times 3n$  board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a  $2 \times 2$  square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all  $n$  for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.

Proposed by *Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina*

Answer: n


$n$  is a fixed natural number. Find the least  $k$  such that for every set  $A$  of  $k$  natural numbers, there exists a subset of  $A$  with an even number of elements which the sum of it's members is divisible by  $n$ .

Answer: k = 2n


A *smooth number* is a positive integer of the form  $2^m 3^n$ , where  $m$  and  $n$  are nonnegative integers.  Let  $S$  be the set of all triples  $(a, b, c)$  where  $a$ ,  $b$ , and  $c$  are smooth numbers such that  $\gcd(a, b)$ ,  $\gcd(b, c)$ , and  $\gcd(c, a)$  are all distinct.  Evaluate the infinite sum  $\sum_{(a,b,c) \in S} \frac{1}{abc}$ .  Recall that  $\gcd(x, y)$  is the greatest common divisor of  $x$  and  $y$ .

Answer: \frac{162}{91}


The adjoining figure shows two intersecting chords in a circle, with  $B$  on minor arc  $AD$ .  Suppose that the radius of the circle is 5, that  $BC = 6$ , and that  $AD$  is bisected by  $BC$ .  Suppose further that  $AD$  is the only chord starting at  $A$  which is bisected by  $BC$ .  It follows that the sine of the minor arc  $AB$  is a rational number.  If this fraction is expressed as a fraction  $m/n$  in lowest terms, what is the product  $mn$ ?
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1));
draw(Circle(origin, 1)^^A--D^^B--C);
real r=0.05;
pair M1=midpoint(M--D), M2=midpoint(M--A);
draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D)));
draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D)));
pair point=origin;
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $D$ ", D, dir(point--D));[/asy]

Answer: 175


A positive integer  $n$  is said to be  $m$ -free if  $n \leq m!$  and  $\gcd(i,n)=1$  for each  $i=1,2,...,m$ . Let  $\mathcal{S}_k$  denote the sum of the squares of all the  $k$ -free integers. Find the remainder when  $\mathcal{S}_7-\mathcal{S}_6$  is divided by  $1000$ .

*Proposed by **FedeX333X***

Answer: 80


Example 8 (2003 Bulgaria National Mathematical Olympiad) Find all real numbers $a$ that satisfy the condition $4[a n]=n+[a[a n]]$, where $n$ is any positive integer.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

(Note: The provided text is already in English, so no translation is needed. If you intended to translate the problem statement into English, it has been done as requested.)

Example 8 (2003 Bulgaria National Mathematical Olympiad) Find all real numbers $a$ that satisfy the condition $4[a n]=n+[a[a n]]$, where $n$ is any positive integer.

Answer: 2 + \sqrt{3}


4. Dana is a triangular grid of size $9 \times 9$ (see figure). At the top node of the grid, there is an ant colony, and at the rightmost node, there is their anthill. In all other nodes of the grid, there is one grain of rice. Each ant from the colony travels along the edges of the grid, but only in the directions right, down, or diagonally down-right. On its way, each ant collects all the grains of rice it encounters, and when it reaches the anthill, it stays there. What is the minimum number of ants that must be in the colony so that all the grains of rice can be collected?

![](https://cdn.mathpix.com/cropped/2024_06_07_77e56b77a012e45bd48fg-03.jpg?height=771&width=782&top_left_y=1219&top_left_x=1134)

## 59th Mathematical Competition for High School Students in Slovenia  Kranj, April 18, 2015

## Problems for 3rd Year

Solve the problems independently. You have 210 minutes for solving.

![](https://cdn.mathpix.com/cropped/2024_06_07_77e56b77a012e45bd48fg-04.jpg?height=163&width=417&top_left_y=364&top_left_x=1499)
The use of notes, literature, or a pocket calculator is not allowed.

Answer: 5


9.1. Several numbers are written on the board. It is known that the square of any written number is greater than the product of any two other written numbers. What is the maximum number of numbers that can be on the board

#

Answer: 3


35. The magic school has 164 students, 28 students choose the spell class, the number of students choosing the transformation class is twice that of the spell class, and the number of students choosing the flying class is twice that of the transformation class. If among the above three classes, each person must choose at least one and at most two, then the number of students choosing two classes is at least $\qquad$ people.

Answer: 32


# Task 2.

While a lion cub, who was 6 minutes away from the water hole, set off to drink, a second one, having already quenched his thirst, started heading back along the same path at 1.5 times the speed of the first. At the same time, a turtle, who was 32 minutes away from the water hole, set off along the same path to drink. After some time, the first lion cub stepped on the turtle, and 2 minutes and 24 seconds later, the second lion cub stepped on her. How many minutes after the second incident did the turtle reach the water hole, given that all three moved at constant speeds?

Answer: 28.8


## Task 2 - 060612

At the beginning of the school year, Heinz bought two different types of notebooks, one costing 8 Pf, the other 15 Pf per piece. He paid 1.31 MDN for 12 notebooks in total.

How many notebooks did he buy of each type?

Answer: 7


A [i]strip[/i] is the region between two parallel lines.  Let $A$ and $B$ be two strips in a plane.  The intersection of strips $A$ and $B$ is a parallelogram $P$.  Let $A'$ be a rotation of $A$ in the plane by $60^\circ$.  The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$.  Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$.  What is the greatest possible value of the number $x$?

Answer: 150


4. All natural numbers, the sum of the digits in the representation of which is divisible by 5, are listed in ascending order: $5,14,19,23,28,32, \ldots$ What is the smallest positive difference between consecutive numbers in this sequence? Provide an example and explain why it cannot be smaller.

---

The smallest positive difference between consecutive numbers in this sequence is 1. An example is the pair of numbers 14 and 15. The sum of the digits of 14 is $1 + 4 = 5$, which is divisible by 5. The sum of the digits of 15 is $1 + 5 = 6$, which is not divisible by 5. However, the next number in the sequence is 19, and the sum of its digits is $1 + 9 = 10$, which is divisible by 5. Therefore, the smallest difference is 1, as seen between 14 and 15.

To explain why it cannot be smaller: The smallest possible difference between two numbers is 1. If the difference were 0, the numbers would be the same, which is not possible for consecutive numbers in the sequence. If the difference were greater than 1, there would be a gap between the numbers, which means there would be a number in between that could potentially have a sum of digits divisible by 5, contradicting the requirement that the sequence lists all such numbers in ascending order. Therefore, the smallest possible difference is 1.

Answer: 1


3. Given $n$ as a fixed positive integer. For $-1 \leqslant x_{i} \leqslant 1(i=1,2, \cdots, 2 n)$, find
$$
\sum_{1 \leqslant r<s \leqslant 2 n}(s-r-n) x_{r} x_{s}
$$

the maximum value.

Answer: n(n -1)


The first two elements of the sequence $x_{n}$ are $x_{1}=1001, x_{2}=1003$ and if $n \geq 1$, then $x_{n+2}=\frac{x_{n+1}-2004}{x_{n}}$. What is the sum of the first 2004 elements of the sequence?

Answer: 1338004


Problem 4. A circle with radius 4 is inscribed in trapezoid $ABCD$, touching the base $AB$ at point $M$. Find the area of the trapezoid if $BM=16$ and $CD=3$.

Answer: 108


Problem 9.1. In a store, there are 20 items, the costs of which are different natural numbers from 1 to 20 rubles. The store has decided to have a promotion: when buying any 5 items, one of them is given as a gift, and the customer chooses which item to receive for free. Vlad wants to buy all 20 items in this store, paying as little as possible. How many rubles will he need? (Each of the 20 items is sold in 1 copy.)

Answer: 136


2. Find the integer solutions of $\frac{1}{x^{2}}+\frac{1}{x y}+\frac{1}{y^{2}}=1$.

Answer: (1, -1)


Given the height 455 as shown in Figure 3, the lines $AB$ and $AC$ are tangent to the circle $\odot O$ at points $B$ and $C$ respectively. $P$ is a moving point on the arc $\overparen{BC}$, $PD \perp AB$ at point $D$, $PE \perp AC$ at point $E$, and the lines $DE$ and $BC$ are connected. If the radius of $\odot O$ is 5, and the chord $BC$ is 8 units long, find the maximum area of $\triangle PDE$.

Answer: \dfrac{48}{25}


4. $F$ is the right focus of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{12}=1$, and $P$ is a moving point on the ellipse. For the fixed point $A(-2, \sqrt{3}),|P A|+$ $2|P F|$ the minimum value is $\qquad$ .

Answer: 10


In a certain convex \( n \)-gon (\( n>3 \)), all distances between the vertices are different.

a) Let's call a vertex uninteresting if the closest vertex to it is one of its neighboring vertices. What is the minimum possible number of uninteresting vertices (for the given \( n \))?

b) Let's call a vertex unusual if the farthest vertex from it is one of its neighboring vertices. What is the maximum possible number of unusual vertices (for the given \( n \))?

Answer: 2


8. The URL of the Hua Cup Competition is www.huabeisai.cn, and the letters form the following equation:
www+hua+bei+sai+cn=2008.

If each letter represents a digit from $0 \sim 9$, the same letter represents the same digit, different letters represent different digits, and $\mathrm{w}=8, \mathrm{~h}=6, \mathrm{a}=9, \mathrm{c}=7$, then the minimum value of the three-digit number $\overline{b e} i$ is _. $\qquad$

Answer: 103


Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall.  In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall.  One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room three feet above the floor.  Over the next few mornings, Tony moves the spider up three feet from the point where he finds it.  If the wall in the living room is $18$ feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?

Answer: 8


8. In Rt $\triangle A B C$, the incircle $\odot O$ touches the hypotenuse $A B$ at point $D$, and touches $B C, C A$ at points $E, F$ respectively. Draw $D K \perp A C$ at point $K$, and $D P \perp B C$ at point $P$. Given $A D$ $=m, B D=n$. Express the area $S$ of rectangle $C K D P$ in terms of $m, n$ as $\qquad$

Answer: \dfrac{2m^2n^2}{(m+n)^2}


In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.

Answer: 11


6. In a regular $(n+2)$-gon, two vertices are randomly selected, and a line passing through them is drawn. Let $p_{n}$ denote the probability that the difference in the number of vertices on either side of this line does not exceed 1. Then the value of $\sum_{i=1}^{2018} p_{i} p_{i+1}$ is $\qquad$

Answer: \dfrac{1009}{1010}


In an acute-angled triangle \(PQR\) with \(PQ > QR\), the altitudes \(PT\) and \(RS\) are drawn. \(QN\) is the diameter of the circumcircle of triangle \(PQR\). It is known that the acute angle between the altitudes \(PT\) and \(RS\) is \(\alpha\), and \(PR = a\). Find the area of the quadrilateral \(NSQT\).

Answer: \dfrac{a^2}{2} \cot \alpha


4. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which

$$
x+f(x f(y))=f(y)+y f(x)
$$

holds for all real numbers $x$ and $y$.

Answer: f(x) = x - 1


A rock travelled through an n x n board, stepping at each turn to the cell neighbouring the previous one by a side, so that each cell was visited once. Bob has put the integer numbers from 1 to n^2 into the cells, corresponding to the order in which the rook has passed them. Let M be the greatest difference of the numbers in neighbouring by side cells. What is the minimal possible value of M?

Answer: 2n - 1


[Similar auxiliary triangles $]$ [ Right triangle with an angle at ]

On the cathetus $A C$ of the right triangle $A B C$, a circle is constructed with $A C$ as its diameter. It intersects side $A B$ at point $E$. A point $G$ is taken on side $B C$ such that segment $A G$ intersects the circle at point $F$, and segments $E F$ and $A C$ are parallel, $B G=2 C G$ and $A C=2 \sqrt{3}$. Find $G F$.

Answer: 1


9. (20 points) In $\triangle A B C$, the measures of the three interior angles satisfy $\frac{\angle A}{\angle B}=\frac{\angle B}{\angle C}=\frac{1}{3}$.
Find the value of $T=\cos A+\cos B+\cos C$.

Answer: \dfrac{1 + \sqrt{13}}{4}


3. A five-digit number consists of only the three different digits $1, 3, 5$. Then the number of such five-digit numbers is $\qquad$.

Answer: 150


10. Let $f(x)=\frac{1}{x^{3}-x}$, find the smallest positive integer $n$ that satisfies the inequality $f(2)+f(3)+\cdots+f(n)>\frac{499}{2020}$.

Answer: 13


9. Parallelogram $A E C F$ is inscribed in square $A B C D$. It is reflected across diagonal $A C$ to form another parallelogram $A E^{\prime} C F^{\prime}$. The region common to both parallelograms has area $m$ and perimeter $n$. Compute the value of $\frac{m}{n^{2}}$ if $A F: A D=1: 4$.

Answer: \dfrac{7}{400}


We draw a circle around the midpoint of each side of a triangle, where the radius of the circle is $1/4$ of the respective side. We know that any two of these circles have a common point. Prove that the triangle is equilateral.

Answer: \text{The triangle is equilateral.}


5. (3 points) Petya and Vasya are playing a game. There are a total of three moves in the game. On the first move, Petya breaks a stick that is 10 cm long into two parts. Then Vasya breaks one of the resulting sticks into two parts. On the last move, Petya breaks one of the three resulting sticks into two parts. Vasya wins if a triangle can be formed from any three of the resulting pieces, and Petya wins otherwise. Who will win with correct play?

Answer: Petya


Given a 29-digit number \(X=\overline{a_{1}} \overline{-} \overline{-} \overline{29}\) where \(0 \leq a_{k} \leq 9\) and \(a_{1} \neq 0\). It is known that for every \(k\), the digit \(a_{k}\) occurs \(a_{30-k}\) times in the number. For example, if \(a_{10}=7\), then the digit \(a_{20}\) occurs seven times. Find the sum of the digits of the number \(X\).

Answer: 201


Problem 3. On a board, Zoran and Petar wrote two numbers. Zoran wrote a three-digit number which has the digit in the units place 1, the digit in the tens place which is 3 times larger than the digit in the units place, and the digit in the hundreds place which is 3 larger than the digit in the units place. Petar wrote a number which is the predecessor of the largest odd number from the first ten of the second hundred. Sanja added the written numbers, and then reduced the sum by their difference. What number did Sanja get?

Answer: 216


Given any five distinct points on a plane, let the ratio of the maximum distance to the minimum distance between them be $\lambda$. Prove that:
$$
\lambda \geqslant 2 \sin 54^{\circ}.
$$

Also, discuss the necessary and sufficient conditions for equality to hold.

Answer: \lambda \geqslant 2 \sin 54^{\circ}


Given a positive integer $n$, consider a triangular array with entries $a_{ij}$ where $i$ ranges from $1$ to $n$ and $j$ ranges from $1$ to $n-i+1$. The entries of the array are all either $0$ or $1$, and, for all $i > 1$ and any associated $j$ , $a_{ij}$ is $0$ if $a_{i-1,j} = a_{i-1,j+1}$, and $a_{ij}$ is $1$ otherwise. Let $S$ denote the set of binary sequences of length $n$, and define a map $f \colon S \to S$ via $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$. Determine the number of fixed points of $f$.

Answer: 2^{\lceil n/2 \rceil}


Five. (20 points) Given $a_{1}=1, a_{n+1}=2 a_{n}+(-1)^{n} n+1(n=1,2,3, \cdots)$. Find the general term formula $a_{n}$ (require it to be written as a finite sum, and the number of terms should be independent of $n$).

Answer: \dfrac{8 \cdot 2^{n} - (-1)^{n}(3n - 1) - 9}{9}


[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 1

Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.

Answer: -2\sqrt{2}


3. Let the integer $N>1, 1=d_{1}<d_{2}<\cdots<d_{s}=N$ be all the positive divisors of $N$. It is known that
$$
\left(d_{1}, d_{2}\right)+\left(d_{2}, d_{3}\right)+\cdots+\left(d_{s-1}, d_{s}\right)=N-2 \text {. }
$$

Find all possible values of $N$.

Answer: 3


Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.

Answer: n^2 - n + 1


Five, try to find two different natural numbers, whose arithmetic mean $A$ and geometric mean $G$ are both two-digit numbers. Among $A, G$, one can be obtained by swapping the units and tens digits of the other.

Answer: 32


11. As shown in Figure 3(a), the eight vertices of a cube are assigned values $a, b, c, d, e, f, g, h$, respectively. Then, the arithmetic mean of the values of the three adjacent vertices of each vertex, denoted as $\bar{a}, \bar{b}, \bar{c}, \bar{d}, \bar{e}, \bar{f}, \bar{g}, \bar{h}$, is placed at the corresponding vertex of another cube, as shown in Figure 3(b).

If $\bar{a}=9, \bar{b}=8, \bar{c}=11, \bar{d}=10, \bar{e}=13, \bar{f}=$ $12, \bar{g}=15, \bar{h}=14$, then the value of $a+g$ is

Answer: 20


There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that $A$ gives $B$ a coin and $B$ gives $A$ a coin at the same time.) Suppose that this process continues indefinitely. That is, for any positive integer $n$, there exists a person who will give away coins during the $n$th minute. What is the smallest number of coins that could be at the party?

[i]Proposed by Ray Li[/i]

Answer: 190


Example 6.20. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$.

Answer: 0.034


69. A reconnaissance aircraft is flying in a circle centered at point $A$. The radius of the circle is $10 \mathrm{km}$, and the speed of the aircraft is 1000 km/h. At some moment, a missile is launched from point $A$ with the same speed as the aircraft, and it is controlled such that it always remains on the line connecting the aircraft to point $A$. How long after the launch will the missile catch up to the aircraft?

Answer: \dfrac{\pi}{200}


Anička got a cake in the shape of a rectangle for her birthday. She cut it using two straight cuts. One cut went through the longer sides of the rectangle, intersecting them at one-third of their length, and the other cut went through the shorter sides, intersecting them at one-fifth of their length. Neither cut was parallel to the sides of the rectangle, and at each vertex of the rectangle, either two shorter segments or two longer segments of the divided sides always met.

![](https://cdn.mathpix.com/cropped/2024_04_17_24585cf941083de07254g-3.jpg?height=402&width=552&top_left_y=553&top_left_x=752)

Anička ate the gray-marked piece of the cake. Determine what fraction of the cake this was.

(A. Bohiniková)

Answer: \dfrac{2}{15}


Example 8. In $\triangle ABC$, $D$ is the midpoint of $AC$.
(1) If $BC$ is trisected, with division points from $C$ being $C_{1}, C_{2}$, and $C_{1}A, C_{2}A$ intersect $BD$ at $D_{1}$, $D_{2}$. Find $S \wedge N_{1} D_{2}: S \triangle A C_{1} C_{2}$;
(2) If $BC$ is divided into four equal parts, with division points from $C$ being $C_{1}, C_{2}, C_{3}$, and $C_{1}A, C_{2}A$ intersect $BD$ at $D_{1}, D_{2}$. Find $S \triangle A D_{1} D_{2}: S \triangle A C_{1} C_{2}$;
(3) If $BC$ is divided into $n$ equal parts, with division points from $C$ being $C_{1}, C_{2}, \cdots, C_{n-1}$, and $C_{1}A, C_{2}A$ intersect $BD$ at $D_{1}, D_{2}$, then $S \triangle A D_{1} D_{2}: S \triangle A C_{1} C_{2}=$ ?

Answer: \dfrac{n^2}{2(n - 1)(2n - 1)}


32. The Golden Merry ship sails from Port A to Port B via Port C. The journey from Port A to Port C is against the current, and from Port C to Port B is with the current. The distance from Port A to Port C is $\frac{2}{3}$ of the distance from Port C to Port B. The ship's speed against the current is half of its speed with the current. The ship takes a total of 7 hours to sail from Port A to Port B via Port C. The ship will take $\qquad$ hours to sail from Port B to Port A via Port C.

Answer: 8


In triangle \( \triangle ABC \), the coordinates of vertex \( A \) are \( A(1,4) \). The equation of the angle bisector of \( \angle ABC \) is \( x - 2y = 0 \) and the equation of the angle bisector of \( \angle ACB \) is \( x + y - 1 = 0 \). Find the equation of the line \( BC \).

Answer: 4x + 17y + 12 = 0


2. Given the sum of $2 n+1\left(n \in \mathbf{N}_{+}\right)$ consecutive positive integers is $a$, and the difference between the sum of the squares of the last $n$ numbers and the sum of the squares of the first $n$ numbers is $b$. If $\frac{a}{b}=\frac{11}{60}$, then the value of $n$ is

Answer: 5


2.10. Determine the volume of a regular truncated square pyramid if its diagonal is $18 \mathrm{~cm}$, and the side lengths of the bases are 14 and 10 cm.

Answer: 872


8. Let real numbers $a, b, c, d$ satisfy $ad + bc = 1$. Then
$$
u=a^{2}+b^{2}+c^{2}+d^{2}+(a+c)^{2}+(b-d)^{2}
$$

the minimum value of $u$ is $\qquad$.

Answer: 2\sqrt{3}


3. Given a square $A B C D$. Inside the square lies a point $U$ such that $B U$ and $A B$ are of equal length. The intersection of $B U$ and the diagonal $A C$ is $V$. Angle $D A U$ is equal to 28 degrees.

What is the angle at $V$ in triangle $B V C$?

![](https://cdn.mathpix.com/cropped/2024_04_17_227837a5972cf599de5fg-2.jpg?height=331&width=371&top_left_y=1699&top_left_x=1476)

Answer: 101^\circ


The triangle \(ABC\) is rotated around a line that lies in the plane of the triangle and passes outside of it through vertex \(A\), making equal angles with sides \(AB\) and \(AC\). Find the volume of the solid of revolution if \(AB = a\), \(AC = b\), and \(\angle BAC = \alpha\).

Answer: \dfrac{\pi a b (a + b) \sin \alpha \cos \dfrac{\alpha}{2}}{3}


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

Answer: 9


All arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?

Answer: 738826


A woman has $\$1.58$ in pennies, nickels, dimes, quarters, half-dollars and silver dollars. If she has a different number of coins of each denomination, how many coins does she have?

Answer: 18


$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily consecutive). What is the smallest number of $1$'s $a + b$ could possibly have?

Answer: 1


Right triangle $XY Z$ has right angle at $Y$ and $XY = 228$, $Y Z = 2004$. Angle $Y$ is trisected, and the angle trisectors intersect $XZ$ at $P$ and $Q$ so that $X$, $P$, $Q$,$Z$ lie on $XZ$ in that order. Find the value of $(PY + Y Z)(QY + XY )$.

Answer: 1370736


Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games - that is, into unordered pairs of unordered pairs of people?

Answer: 315


A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex.

Answer: 8


Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?

Answer: 33


A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.

Answer: 13


Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$.  Let $D$ be the point such that $AB\parallel CD$ and $CA\parallel BD$.  Let $R$ be a point within triangle $BCD$.  Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively.  Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively.  If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.

Answer: 180


In triangle $ABC$, $\angle ABC$ is obtuse.  Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$.  Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.

Answer: 10


Let $ C_1$ and $ C_2$ be externally tangent circles with radius 2 and 3, respectively. Let $ C_3$ be a circle internally tangent to both $ C_1$ and $ C_2$ at points $ A$ and $ B$, respectively. The tangents to $ C_3$ at $ A$ and $ B$ meet at $ T$, and $ TA \equal{} 4$. Determine the radius of $ C_3$.

Answer: 8


A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.

Answer: 60


Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $ n < 60$ (both Kelly and Jason know that $ n < 60$, but that they don't know what the value of $ n$ is). The computer tells Kelly the unit digit of $ n$, and it tells Jason the number of divisors of $ n$. Then, Kelly and Jason have the following dialogue:

Kelly: I don't know what $ n$ is, and I'm sure that you don't know either. However, I know that $ n$ is divisible by at least two different primes.

Jason: Oh, then I know what the value of $ n$ is.

Kelly: Now I also know what $ n$ is.

Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $ n$?

Answer: 10


Trodgor the dragon is burning down a village consisting of $ 90$ cottages.  At time $ t \equal{} 0$ an angry peasant arises from each cottage, and every $ 8$ minutes ($ 480$ seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage.  It takes Trodgor $ 5$ seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many [b]seconds[/b] does it take Trodgor to burn down the entire village?

Answer: 1920


Kermit the frog enjoys hopping around the inﬁnite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?

Answer: 10201


Let \(S_1\) be the area of the regular pentagon \(ABCDE\). And let \(S_2\) be the area of the regular pentagon whose sides lie on the lines \(AC, CE, EB, BD, DA\). The original answer is in the form \(\frac{k+m\sqrt{5}}{n}\). Please find the value of \(k+m+n\).

Answer: 12


We draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of the circle is placed in the middle of one of the squares. Find the number of squares through which the circle passes. Please provide your answer as an integer.

Answer: 40


If $a$, $b$, and $d$ are the lengths of a side, the shortest diagonal, and the longest diagonal, respectively, of a regular nonagon (see adjoining figure), determine the relationship between these variables in the form of an equation. The original answer is in the form $d = m \cdot a + n \cdot b + p$. Please give the value of $m + n + p$.

Answer: 2


A right circular cone and a right cylinder with the same height $20$ do not have the same circular base, but the circles are coplanar, and their centers are the same. If the cone and the cylinder are on the same side of the plane and their base radii are $20$ and $10$, respectively, find the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder.

Answer: 1


The integer $k$ is a \textit{good number}, if we can divide a square into $k$ squares. Find how many good numbers not greater than $2006$ there are.

Answer: 2003


Side \(AB\) of triangle \(ABC\) has length \(8\) inches. Line \(DEF\) is drawn parallel to \(AB\) so that \(D\) is on segment \(AC\), and \(E\) is on segment \(BC\). Line \(AE\) extended bisects angle \(FEC\). If \(DE\) has length \(5\) inches, then find the length of \(CE\) in inches. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 43


Acute-angled triangle \(ABC\) is inscribed in a circle with center at \(O\). The arcs are given as follows: \(\overarc{AB} = 120^\circ\) and \(\overarc{BC} = 72^\circ\). A point \(E\) is located on the minor arc \(AC\) such that \(OE\) is perpendicular to \(AC\). Determine the ratio of the magnitudes of angles \(\angle OBE\) and \(\angle BAC\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 4


A quadrilateral has consecutive sides of lengths $70$, $90$, $130$, and $110$. It is both inscribed in a circle and has a circle inscribed within it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find the value of $|x-y|$.

Answer: 13


Let \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find the value of \(m + n + p\).

Answer: 16


3. When $a=$ $\qquad$ , the equation $|x+2021|-2022|=a$ has exactly three roots.

Answer: 2022


$[\quad$ Evaluation + example $\quad]$

The hostess baked a pie for her guests. She may have either 10 or 11 guests. Into what smallest number of slices should she cut the pie in advance so that it can be evenly divided among either 10 or 11 guests?

Answer: 20


A machine produces parts of two types. On its conveyor belt, 75 parts are laid out in a single line. While the conveyor is moving, the machine prepares a part of the type that is less represented on the belt. Each minute, the next part falls off the belt and the prepared part is placed at the end of it. After some number of minutes following the conveyor's start, it might happen that the arrangement of parts on the belt repeats the initial arrangement for the first time. Find a) the minimum such number, b) all such numbers.

Answer: 2


Each of the ten radio operators at point A tries to establish communication with each of the twenty radio operators at point B. How many different possible communication scenarios are there?

Answer: 2^{200}


Suppose that  $A, B, C$ , and  $X$  are any four distinct points in the plane with   $$ \max \,(BX,CX) \le AX \le BC. $$  Prove that  $\angle BAC \le 150^o$ .

Answer: \angle BAC \leq 150^\circ


Problem 6.

It is known that three numbers $a_{1}, a_{2}, a_{3}$ were obtained as follows. First, a natural number $A$ was chosen and the numbers $A_{1}=[A]_{16}, A_{2}=$ $[A / 2]_{16}, A_{3}=[A / 4]_{16}$ were found, where $[X]_{16}$ is the remainder of the integer part of the number $X$ divided by 16 (for example, $[53 / 2]_{16}=10$). Then an integer $B$ was chosen such that 0 $\leq \mathrm{B} \leq 15$. The numbers $A_{1}, A_{2}, A_{3}$ and $B$ are written in binary, i.e., each is represented as a string of 0s and 1s of length 4, adding the necessary number of zeros on the left. We agree to add such strings symbol by symbol "in a column" without carrying to the next digit according to the rule: $1+1=0+0=0$ and $0+1=1+0=0$, and denote the operation of symbol-by-symbol addition by the symbol $\diamond$. For example, $3 \diamond 14=(0011) \diamond$ $(1110)=(1101)=13$. Let $a_{1}=A_{1} \diamond B, a_{2}=A_{2} \diamond B, a_{3}=A_{3} \diamond B$. Find the sum of all possible values of the number $a_{3}$, given that $a_{1}=2, a_{2}=$ 9. Choose the correct answer:

#

Answer: 16


Problem 10.6. Find the largest possible number $A$ having the following property: if the numbers $1,2, \ldots, 1000$ are ordered in arbitrary way then there exist 50 consecutive numbers with sum not less than $A$.

Ivan Landjev

Answer: 25025


5. Draw the tangent line to the curve $y=3x-x^{3}$ through the point $A(2,-2)$. Then the equation of the tangent line is $\qquad$ .

Answer: y = -2


8.1. A four-digit number is called "beautiful" if it is impossible to append a digit to the right so that the resulting five-digit number is divisible by 11. How many beautiful numbers are there that are greater than 3100 and less than 3600?

Answer: 46


5. For $n$ consecutive natural numbers, if each number is written in its standard prime factorization form, and each prime factor appears an odd number of times, such $n$ consecutive natural numbers are called a “consecutive $n$ strange group” (for example, when $n=3$, $22=2^{1} \times 11^{1}$, $23=23^{1}$, $24=2^{3} \times 3^{1}$, then $22, 23, 24$ form a “consecutive 3 strange group”). The maximum possible value of $n$ in a consecutive $n$ strange group is . $\qquad$

Answer: 7


2. Let $x_{1}=1, x_{n}=\frac{x_{n-1}-1}{2 x_{n-1}+4}(n \geqslant 2)$. The general term formula of the sequence $\left\{x_{n}\right\}$ is $x_{n}=$ $\qquad$

Answer: \dfrac{2^{n-2} - 3^{n-2}}{2 \cdot 3^{n-2} - 2^{n-2}}


## Task 2 - V00502

Take a three-digit number; however, with the restriction that the first and last digit do not match; write the digits in reverse order below and determine the difference! Write the reversal of this number again below! Then add the reversal and the difference! Work through the task with two examples! What do you notice?

Answer: 1089


A $5 \times 5$ chessboard consists of unit squares, 7 of which are red and 18 are blue. Two of the red squares are located on the edge of the board. The segments that separate two adjacent red squares are also colored red. The segments that separate two adjacent blue squares are colored blue. All other segments, including the edges of the board, are black. In this way, a total of 35 black segments are formed. How many red segments are there?

Answer: 5


A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\tfrac{p}{2}$ such that $\tfrac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.

Answer: 2


Prove that the typical algorithms for addition and subtraction of two integers \(m\) and \(n\), as studied in elementary school, have time complexities of \(\mathcal{O}(\log (m)+\log (n))\). Then, show that the algorithms for multiplication and Euclidean division have time complexities of \(\mathcal{O}(\log (m) \log (n))\).

Algorithm 3: Sieve of Eratosthenes
Arguments: \( n \) (we want to test if \( n \) is prime)
- \( m \leftarrow n-1 \quad \text{(Naive version)} \)
- \( m \leftarrow \lfloor \sqrt{n} \rfloor \quad \text{(More reasonable version)} \)
- For \( d \) ranging from 2 to \( m \):
  - If \( n \equiv 0 \pmod{d} \) then return "\( n \) is composite!" (i.e., \( d \) divides \( n \))
- Return "\( n \) is prime!" (i.e., \( n \) has no proper divisor \( d \leqslant m \))

Answer: \mathcal{O}(\log m + \log n)


We divide a segment into four parts with three randomly chosen points. What is the probability that these four sub-segments can be the four sides of some quadrilateral?

Answer: \dfrac{1}{2}


4. The base edge of a right regular quadrilateral pyramid $A B C D T$ ($T$ is the apex of the pyramid) is $\sqrt{2}$, and the lateral edge is 2. A plane passes through vertex $A$ and is perpendicular to edge $T C$ ($A$ and $C$ are diagonally opposite on the base) and intersects the edges $T B, T C$, and $T D$ at points $P, Q$, and $R$ respectively.

Determine the volume of the intersected pyramid $A B C D P Q R$.

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


4. Let $n$ be a positive integer. A regular hexagon with side $n$ is divided into equilateral triangles with side 1 by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of equilateral triangles.

(United Kingdom)

Time allowed: 270 minutes.

Each problem is worth 10 points.

Answer: \left( \dfrac{n(n+1)}{2} \right)^2


There are 2012 students lined up in a row and numbered sequentially from 1 to 2012 from left to right. In the first round, the students count off "1, 2", and the students who count off 2 remain. Starting from the second round, the remaining students count off "1, 2, 3", and the students who count off 3 remain. This process continues until only one student is left. What is the number of this last remaining student?

Answer: 1458


3. Given the sets $M=\{1,2,3\}, N=\{1,2,3,4,5\}$, define the function $f: M \rightarrow N$. Let points $A(1, f(1)), B(2, f(2)), C(3, f(3)), \triangle A B C$ have the circumcenter $D$, and $\underset{D A}{\operatorname{mu}}+D C=\lambda D B(\lambda \in R)$. Then the number of functions $f(x)$ that satisfy the condition is $\qquad$.

Answer: 20


4. Let $n \in \mathbf{Z}_{+}$. A volleyball team has $n$ male players and $n$ female players. Initially, each player is assigned to one of the positions numbered $1,2, \cdots, 2 n$, with only positions 1 and $n+1$ being outside the court. During the game, position swaps occur, and each swap involves moving the player in position $i(i=1,2, \cdots, 2 n)$ to position $i+1$, where position $2 n+1$ is the same as position 1. How many different initial allocation methods are there such that, regardless of how many rounds of swapping occur, at least $n-1$ female players are on the court?

Answer: 2^n \cdot n! \cdot n!


Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the intersection point of the sides $AB$ and $CD$,  $O$ be the intersection point of  $BM$ and $AC, N$ be the intersection point of  $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

Answer: \dfrac{1}{2}


Given the three sides of a triangle: $a, b$, and $c$. Determine the area of the triangle formed by the intersections of the internal angle bisectors and the corresponding sides.

Answer: \dfrac{2abc}{(a + b)(b + c)(c + a)} \sqrt{s(s - a)(s - b)(s - c)}


18. (17 points) Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of the integers 1, 2, $\cdots, n$, and satisfy the following conditions:
(1) $a_{1}=1$;
(2) $\left|a_{i}-a_{i+1}\right| \leqslant 2(i=1,2, \cdots, n-1)$.
Let the number of such permutations be $f(n)$. Find the remainder when $f(2010)$ is divided by 3.

Answer: 1


## Task 1 - 030831

What is the smallest eight-digit number that consists of all different digits and is divisible by 36? Justify that it is the smallest such number!

Answer: 10237896


3. (10 points) Add the tens, hundreds, and thousands digits of 2015, and write the sum after the units digit of 2015, resulting in the natural number 20153; add the tens, hundreds, and thousands digits of the new number, and write the sum after the units digit of 20153, resulting in 201536; perform this operation 2 more times, resulting in 201536914. Continue this process for a total of 2015 times, resulting in a very large natural number. The sum of all the digits of this natural number is $\qquad$

Answer: 8479


3. In a convex quadrilateral $A B C D: A B=A C=A D=B D$ and $\angle B A C=\angle C B D$. Find $\angle A C D$.

Answer: 70^\circ


In a Cartesian coordinate plane, there are two fixed points \( A \) and \( B \) on the \( x \)-axis, different from the origin \( O(0,0) \). A moving point \( C \) varies along the \( y \)-axis, and line \( l \) passes through \( O(0,0) \) and is perpendicular to line \( AC \). Find the trajectory of the intersection of line \( l \) with line \( BC \).

Answer: \frac{(x - \frac{b}{2})^2}{(\frac{b}{2})^2} + \frac{y^2}{(\frac{\sqrt{ab}}{2})^2} = 1


In the figure below ,$4$ of the $6$ disks are to be colored black and $2$ are to be colored white. Two colorings that can be obtained from one another by rotation or reflection of the entire figure are considered the same. [img]https://i.imgur.com/57nQwBI.jpg[/img]

There are only four such colorings for the given  two colors, as shown.

In how many ways can we color the $6$ disks such that $2$ are colored black , $2$ are colored white, $2$ are colored blue with the given identification condition?

Answer: 18


6. (10 points) There are some playing cards on the table. Jia took a prime number of cards, leaving a number that is a multiple of 5; Yi then took a prime number of cards, leaving a number that is a multiple of 3; Bing took a prime number of cards, leaving a number that is a multiple of 2; Ding took a prime number of cards, leaving a prime number of cards for Wu. It is known that the number of cards taken by Jia, Yi, Bing, Ding, and Wu is decreasing. How many cards were there on the table at least? $\qquad$

Answer: 63


\section*{Problem 4 - 121014}

In an old textbook, a problem reports on the following trade:

A farmer wanted to buy several animals from a livestock dealer. The dealer demanded the same price for each animal. The farmer managed to negotiate this price down by exactly the same percentage of the demanded price as it was supposed to be (in groschen).

He now paid 21 groschen per animal. At the original price, his money would have been enough for exactly 3 animals. Now he could buy more animals, spending all his money.

How many animals could the farmer buy in total?

Answer: 10


$$
\begin{array}{l}
\text { Three. (20 points) Let } x_{1}=3, \\
x_{n+1}=\left(\frac{2}{n^{2}}+\frac{3}{n}+1\right) x_{n}+n+1
\end{array}
$$
$(n=1,2, \cdots)$. Find the general formula for $x_{n}$.

Answer: 2n^3 + n^2


Problem 1. (5 points) In a country, there are 100 cities, which we consider as points in a plane. In a directory, the distance between each pair of cities is listed (4950 distances are listed).

a) One of the distances is erased. Can it always be uniquely restored using the remaining ones?

b) Suppose $k$ distances are erased and it is known that no three cities lie on the same line. What is the largest value of $k$ for which the erased distances can always be uniquely restored?

Answer: 96


## 194. Math Puzzle $7 / 81$

From a textbook by Adam Ries, who lived from 1492 to 1559, this problem was taken:

A son asks his father how old he is. The father answers: "If you were as old as I am, and half as old, and a quarter as old, and one year more, you would be 134 years old."

How old is the father?

Answer: 76


11.3. All cells of a square table $n \times n$ are numbered in some order with numbers from 1 to $n^{2}$. Petya makes moves according to the following rules. On the first move, he places a rook in any cell. On each subsequent move, Petya can either place a new rook in some cell or move a rook from a cell numbered $a$ horizontally or vertically to a cell with a number greater than $a$. Each time a rook lands in a cell, that cell is immediately colored; placing a rook on a colored cell is prohibited. What is the minimum number of rooks Petya will need to, regardless of the initial numbering, be able to color all the cells of the table in several moves?

(D. Khramov)

Answer: n


Consider natural numbers \(a\), \(b\), and \(c\) such that the fraction

\[
k = \frac{ab + c^2}{a + b}
\]

is a natural number less than \(a\) and \(b\). What is the smallest number of natural divisors that \(a + b\) can have?

Answer: 3


The side of the base of a regular quadrilateral pyramid $S A B C D$ ($S$ - vertex) is 10. Points $E$ and $F$ are located on the edges $D C$ and $B C$ respectively, with $C E=6, C F=9$. It is known that for this pyramid, there exists a unique cone whose vertex coincides with point $E$, the center of the base lies on the line $S A$, and the segment $E F$ is one of the generators. Find the volume of this cone.

Answer: 63\sqrt{6}\pi


Let \(\left\{a_{n}\right\}\) be a sequence of non-negative real numbers. Define:
\[ X_{k}=\sum_{i=1}^{2^{k}} a_{i}, \quad Y_{k}=\sum_{i=1}^{2^{k}}\left[\frac{2^{k}}{i}\right] a_{i}, \]
where \( k=0,1, \ldots \). Prove that for any positive integer \( n \), the following inequality holds:
\[ X_{n} \leq Y_{n}-\sum_{i=0}^{n-1} Y_{i} \leq \sum_{i=0}^{n} X_{i}. \]

Answer: X_{n} \leq Y_{n} - \sum_{i=0}^{n-1} Y_{i} \leq \sum_{i=0}^{n} X_{i}


28. Let $n$ be a positive integer. A sequence of $n$ positive integers (which may be equal) $a_{1}, a_{2}, \cdots, a_{n}$ is called a "full sequence" if it satisfies the following condition: for any positive integer $k (k \geqslant 2)$, if $k$ is one of the terms in the sequence, then $k-1$ is also a term in the sequence, and the first occurrence of $k-1$ in the sequence is before the last occurrence of $k$. How many "full sequences" are there for each $n$?
（43rd IMO Shortlist Problem）

Answer: n!


Example 2 Let $m, n$ be positive integers, $m<2001, n<$ 2002, and there are $2001 \times 2002$ different real numbers. Fill these numbers into the squares of a $2001 \times 2002$ chessboard so that each square contains exactly one number. If a number in a square is less than at least $m$ numbers in its column and also less than at least $n$ numbers in its row, then this square is called a "bad square". For each way of filling the numbers, find the minimum value of the number of bad squares $S$.
(2002, Vietnam Mathematical Olympiad)

Answer: (2001 - m)(2002 - n)


Jamie is a diversity and inclusion officer who works with various film production companies to increase representation in the film industry. This year, Jamie is working on a project with three different film studios. Studio A plans to produce 10 films, Studio B plans to produce 15 films, and Studio C plans to produce 5 films. Jamie's goal is to ensure that at least 40% of all films produced by these studios feature diverse casts and crews.

If Studio A agrees to have 5 of its films meet this goal, Studio B agrees to have 6 films, and Studio C agrees to have 3 films, will the combined effort of these studios meet Jamie's overall goal of 40% representation? How many of the films in total need to feature diverse casts and crews to meet the goal?

Answer: 14


Mr. Thompson, a stay-at-home dad, has a cleaning schedule to keep his house spotless. He spends 15 minutes cleaning the kitchen, 20 minutes vacuuming the living room, and 25 minutes dusting the bedrooms each day. On Saturdays, he spends an additional 30 minutes tidying up the garage. How many minutes in total does Mr. Thompson spend cleaning the house in one week, including his extra Saturday garage cleaning?

Answer: 450


An executive is inspired by a professor's innovative solutions to solve complex business challenges. In a recent project, the executive decided to apply a creative approach to increase the company's revenue. Initially, the company made $50,000 per month. By implementing the professor's strategies, the revenue increased by 20% in the first month. The following month, the revenue increased by an additional 10%. What is the total revenue the company made after these two months of implementing the professor's strategies?

Answer: 126000


A Tibetan Buddhist lama is organizing a series of meditation sessions focused on cultivating compassion and wisdom. Each session includes a discussion and a meditation period. For a special week-long retreat, the lama plans 3 sessions per day. Each session lasts for 45 minutes, and there is a 15-minute break between sessions. If the retreat lasts for 7 days, what is the total number of hours spent on discussions and meditation during the entire retreat?

Answer: 15.75


A venture capitalist named Alex invests in various startups to support innovation and economic growth. In one year, Alex invested in three startups: a green energy company, a healthcare technology firm, and an educational platform. The green energy company received $2 million, which is twice the amount given to the healthcare technology firm. The educational platform received $500,000 more than the healthcare technology firm. If the total investment in all three startups is $6 million, how much did the healthcare technology firm receive?

Answer: 1375000


Alex is a computer programmer who develops algorithms for dynamic pricing models. One day, Alex decides to experiment with a simple pricing model for a new online service. The base price for the service is $10 per user per month. However, Alex decides to increase the price by $2 for every additional 10 users, up to a maximum of 50 users.

To test the ethical implications of the model, Alex runs a simulation with 30 users. Calculate the total monthly revenue generated by the service under this pricing model for the 30 users.

Answer: 420


Alex, a politically conservative book editor, is organizing a book drive in support of her community's local library. She has 120 books to donate, which she believes reflect her values and can encourage thoughtful discussion. During a bipartisan event, she decides to divide the books equally among 4 local libraries that align with her values. After the donation, each library decides to host a reading session and invites 5 schools, ensuring that every school receives an equal share of the books from each library. How many books does each school receive from the libraries?

Answer: 6


An art collector named Mr. Smith, who is seeking legal advice to establish an art trust, has a collection of 150 paintings. He decides to divide his collection into three sections to be displayed in different galleries. The first gallery will receive 40% of the paintings, the second gallery will receive 35% of the paintings, and the remainder will be kept in his private collection.

How many paintings will Mr. Smith keep in his private collection?

Answer: 38


Emily, a neighborhood kid who grew up playing in the streets of Chelmsford, loves organizing games with her friends. One sunny afternoon, she decided to organize a game of street soccer. She invited 18 kids from the neighborhood to join. She wants to make sure that each team has an equal number of players, so she plans to divide the kids into four teams. After forming the teams, Emily realized that she had forgotten to include herself in the count. How many players will each team have if she adds herself to one of the teams and then redistributes the players equally again?

Answer: 5


## Task B-4.5.

An equilateral triangle of area $P$ is rotated in its plane by $30^{\circ}$ about its centroid. What is the area of the intersection of the original triangle and the triangle obtained by the rotation?

Answer: (\sqrt{3} - 1) P


Example 3 As shown in Figure 4, in a $4 \times 4$ grid square,
construct a grid point $\triangle A B C$
$(A B=\sqrt{5}, B C=$
$\sqrt{13}, C A=\sqrt{10}$ ). How many
grid point triangles congruent to
$\triangle A B C$ (including
$\triangle A B C)$ can be
constructed in Figure 4?

Answer: 32


8,9

In a sphere of radius 9, three equal chords $A A 1, B B 1$, and $C C 1$ are drawn through a point $S$ such that $A S=4, A 1 S=8, B S < B 1 S, C S < C 1 S$. Find the radius of the sphere circumscribed about the pyramid $S A B C$.

Answer: 7


15. A bouncy ball falls from point $A$ to the ground, bounces up to point $B$, then falls to a platform $20 \mathrm{~cm}$ high, bounces up to point $C$, and finally falls to the ground. Each time it bounces, the height it reaches is $80 \%$ of the height from which it fell. It is known that point $A$ is $68 \mathrm{~cm}$ higher above the ground than point $C$. Find the height of point $C$ above the ground.

Answer: 132


Let $n>2$ be an integer. In a country there are $n$ cities and every two of them are connected by a direct road. Each road is assigned an integer from the set $\{1, 2,\ldots ,m\}$ (different roads may be assigned the same number). The [i]priority[/i] of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest $m$ for which it is possible that all cities have a different priority.

Answer: 3


$2 \cdot 111$ two-digit number set $\{00,01, \cdots, 98,99\}$ has a subset $X$ with the following property: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. How many elements does $X$ have at minimum?

Answer: 55


4.077 The sum of four numbers forming a geometric progression is -40, and the sum of their squares is 3280. Find this progression.

Answer: -54


4. Given $\frac{1}{\cos \theta}-\frac{1}{\sin \theta}=\frac{4}{3}, 0<\theta<\frac{\pi}{2}$. Then $\theta=$

Answer: \arctan\left( \dfrac{4 + \sqrt{7}}{3} \right)


4. Alina, Bogdan, Claudia, Denisa, and Eugen are playing doubles in table tennis. In each game, one of the children stands on the side while the other four play two against the other two. Each pair plays against each of the pairs formed from the remaining players exactly once.

a) How many games were played in total?

b) If Alina won 12 games, and Bogdan won 6 games, how many games did Claudia win? Find all possibilities and justify your answer.

Answer: 15


Five. (15 points) If for any $n$ consecutive positive integers, there always exists a number whose sum of digits is a multiple of 8. Determine the minimum value of $n$. And explain the reason.

---

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

Answer: 15


The altitudes $A D$ and $B E$ of an acute-angled triangle $A B C$ intersect at point $H$. The circumcircle of triangle $A B H$ intersects sides $A C$ and $B C$ at points $F$ and $G$ respectively. Find $F G$, if $D E=5$ cm.

Answer: 10


In the diagram, $A B C D$ is a square. Points $A(1,4)$ and $B$ are on a parabola that is tangent to the $x$-axis. If $C$ has coordinates $\left(\frac{39}{4}, \frac{37}{4}\right)$, determine the equation of the parabola.

![](https://cdn.mathpix.com/cropped/2024_04_30_22929074a9efc78b4168g-2.jpg?height=658&width=636&top_left_y=877&top_left_x=796)

Answer: y = \dfrac{1}{4}(x - 5)^2


4. As shown in Figure 1, the circumcenter of $\triangle A B C$ is $O$, the incenter is $I, O I \perp A I, A B=10$, $A C=18$. Then the length of $B C$ is $\qquad$ .

Answer: 14


Problem 4. In a box, there are 50 cards on which the first 100 non-zero natural numbers are written, as follows: on the first card, the numbers 1 (on one side) and 2 (on the other side) are written, on the second card, the numbers 3 (on one side) and 4 (on the other side) are written, and so on, up to the 50th card, on which the numbers 99 (on one side) and 100 (on the other side) are written.

Eliza takes out four cards from the box and calculates the sum of the eight numbers written on them. How many distinct sums can Eliza obtain?

Answer: 185


In triangle \(ABC\), the angle bisector \(BE\) and the perpendicular bisector of side \(AB\) are drawn. It turns out that \(BE = EC\), and the perpendicular bisector intersects side \(BC\). Prove that angle \(C\) is less than 36 degrees.

Answer: \angle C < 36^\circ


10.4. Let point $P$ be the intersection of the diagonals of a convex quadrilateral $K L M N$. The areas of triangles $K L M, L M N$, and $N K P$ are $8 \mathrm{~m}^{2}, 9 \mathrm{~m}^{2}$, and $10 \mathrm{~m}^{2}$, respectively. Find the area of quadrilateral $K L M N$.

Answer: 24


4. As shown in the figure, triangles $A B C$ and $A D E$ are isosceles right triangles with $\angle A=90$. Point $M$ is the midpoint of $B C$. It is known that $A B=A C=D F=F M=E G=G M, \angle F D E=\angle G E D=9^{\circ}$, and points $F$ and $G$ are outside triangle ADE. What is the degree measure of $\angle \mathrm{FMG}$?

Answer: 54


18th Iberoamerican 2003 Problem B1 A ⊆ {1, 2, 3, ... , 49} does not contain six consecutive integers. Find the largest possible value of |A|. How many such subsets are there (of the maximum size)?

Answer: 41


636*. Find all five-digit numbers that are perfect squares and remain perfect squares when the first, the first two, or the first three digits are erased.

Answer: 15625