Given a positive number $k(k>2)$ and a positive integer $n(n \geqslant 3)$. Find the maximum positive number $\lambda$, such that if positive numbers $a_{1}$, $a_{2}, \cdots, a_{n}$ satisfy
$$
\left(a_{1}+a_{2}+\cdots+a_{n}\right)\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)<\lambda,
$$

then it must be true that $a_{1}+a_{2}<k a_{3}$.

Answer: \left( \sqrt{k + 5 + \dfrac{4}{k}} + n - 3 \right)^2


3. The equation $a \cos (x+1)+b \cos (x+2)+c \cos (x+3)$ $=0$ has at least two roots in the open interval $(0, \pi)$. Then all the roots of this equation are $\qquad$ .

Answer: \text{All real numbers}


Problem 6.5. Cat Matroskin, Uncle Fyodor, Postman Pechkin, and Sharik sat down at a round table. In front of each of them was a plate with 15 sandwiches. Every minute, three of them ate a sandwich from their own plate, while the fourth ate a sandwich from their neighbor's plate. After 5 minutes of the meal, there were 8 sandwiches left in Uncle Fyodor's plate. What is the minimum number of sandwiches that could have remained on Cat Matroskin's plate?

Answer: 7


12. Arrange all positive integers $m$ whose digits are no greater than 3 in ascending order to form a sequence $\left\{a_{n}\right\}$. Then $a_{2007}=$ $\qquad$ .

Answer: 133113


Answer: \dfrac{3^{1024} - 2049}{2^{47}}


B1. Find all pairs of natural numbers $a$ and $b$ for which

$$
v=a b-2 a-4 b
$$

where $v$ is the least common multiple of $a$ and $b$.

Answer: (6, 12)


14. Find all positive integers $n$ such that $\min _{k \in \mathbb{N}_{+}}\left(k^{2}+\left[\frac{n}{k^{2}}\right]\right)=2006$.

Answer: [1005568, 1006591]


5. On a long stick, there are three types of graduation lines. The first type divides the stick into 10 equal parts;

The second type divides the stick into 12 equal parts; The third type divides the stick into 15 equal parts. If each graduation line cuts the stick, how many pieces will the stick be cut into?

Answer: 28


## Task A-1.5.

Martin assigns each of the 12 edges of a cube a number, either 1 or -1. Then he assigns to each of the six faces of the cube the product of the 4 numbers on the edges of that face. Finally, Martin sums all 18 numbers assigned to the edges and faces of the cube.

What is the smallest sum Martin can achieve?

Answer: -12


Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.

Determine the smallest possible degree of $f$.

(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)

[i]Ankan Bhattacharya[/i]

Answer: 2n


The base of the quadrilateral pyramid \( P A B C D \) is the parallelogram \( A B C D \). \( M \) is the foot of the perpendicular dropped from point \( A \) to \( B D \). It is known that \( B P = D P \). Prove that the distance from point \( M \) to the midpoint of edge \( A P \) is equal to half the length of edge \( C P \).

Answer: \dfrac{CP}{2}


A 20-row and several-column matrix filled with 0s and 1s satisfies the following conditions: all columns are unique, and in any two columns, the rows that contain 1s in the same positions do not occur more than twice. Find the minimum number of 1s in the matrix when the number of columns is maximized.

Answer: 3820


There are 5 paths leading to the top of a mountain. Two people ascend and then descend. How many different ways can this happen if the two people are not distinguished and

a) each path can be used by at most one person and at most once,

b) each path can be used twice but only in different directions,

c) there are no restrictions,

d), e), and f) are the same as a), b), and c), but the two people are distinguished.

Answer: 625


Let \( f(n) \) be the number of ways to write \( n \) as a sum of powers of 2, where the order of the summation is considered. For example, \( f(4) = 6 \) because 4 can be written as \( 4 \), \( 2 + 2 \), \( 2 + 1 + 1 \), \( 1 + 2 + 1 \), \( 1 + 1 + 2 \), and \( 1 + 1 + 1 + 1 \). Find the smallest \( n \) greater than 2013 for which \( f(n) \) is odd.

Answer: 2047


Every edge of a square-based pyramid is 2 units. Another pyramid with edges of equal length is attached to one of the pyramid's faces as a base. The total length of the edges of the resulting solid is 18 units. What is the volume of this solid?

Answer: 2\sqrt{2}


Write a system of equations for a line that passes through the origin and forms equal angles with the three coordinate axes. Determine the measure of these angles. How many solutions does the problem have?

Answer: 4


There is a calculator with digits from 0 to 9 and two operation symbols. Initially, the display shows the number 0. You can press any buttons. The calculator performs actions in the sequence of button presses. If an operation symbol is pressed multiple times in a row, the calculator only remembers the last press. A scattered scientist pressed a lot of buttons in a random sequence. Estimate the probability that the result of the resulting chain of operations is an odd number.

Answer: \frac{1}{3}


A pedestrian walks along a highway at a speed of 5 km/h. Buses travel along this highway in both directions at the same speed, meeting every 5 minutes. At 12 o'clock, the pedestrian noticed that the buses met near him and, continuing to walk, began counting the passing and overtaking buses. At 2 o'clock, the buses met again near him. It turned out that during this time, the pedestrian encountered four more buses than those overtaking him. Find the speed of the bus.

Answer: 30


A football invitational tournament has 29 cities participating, with each city sending two teams, Team A and Team B. According to the competition rules, each pair of teams competes in at most one match, and teams from the same city do not compete against each other. After several match days, statistics show that, except for Team A from City $A$, the number of matches played by each of the other teams is different. How many matches has Team B from City $A$ played? Provide a proof for your conclusion.

Answer: 28


Chris and Paul each rent a different room of a hotel from rooms 1 to 60. The hotel manager mistakes them for one person and assigns them a room number formed by concatenating Chris's and Paul's room numbers. For example, if Chris has room 15 and Paul has room 9, their concatenated room number would be 159. If there are 360 rooms in the hotel, what is the probability that the concatenated room number is a valid room number within the range 1 to 360?

Answer: \frac{153}{1180}


Using the diagram below, a seven-digit integer can be created as follows: trace a path that uses each line segment exactly once and use the labels on the line segments as digits. For example, the path that goes from \( C \) to \( A \) to \( B \) to \( C \) to \( E \) to \( B \) to \( D \) and finally to \( E \) gives the positive integer 3264715. What is the largest possible positive integer that can be created in this way?

![diagram](https://cdn.mathpix.com/cropped/2024_04_17_d13f9eaf1d614231018ag-1.jpg?height=608&width=612&top_left_y=1905&top_left_x=819)

Answer: 7645123


On the coordinate plane, draw the following lines:
$$
y = k, \quad y = \sqrt{3} x + 2k, \quad y = -\sqrt{3} x + 2k
$$
where \( k = 0, \pm 1, \pm 2, \ldots, \pm 10 \). These 63 lines divide the plane into several equilateral triangles. Find the number of equilateral triangles with a side length of \(\frac{2}{\sqrt{3}}\).

Answer: 660


Tanya sequentially wrote out numbers of the form $\{\lfloor n \wedge 7 - 1 \rfloor\}$ for natural numbers $n = 2, 3,$ and so on, and noticed that for $n = 8$, the number obtained is divisible by 337. What is the smallest $n > 1$ for which she will obtain a number divisible by 2022?

Answer: 79


The polygon enclosed by solid lines consists of 5 congruent squares joined edge-to-edge in a cross shape. One more congruent square is attached to an edge at one of the twelve positions indicated. Determine the number of the resulting polygons that can be folded to form a cube.

Answer: 4


In a regional athletics competition, $275$ sprinters participate in a $100-$meter dash. The track can accommodate $8$ runners at a time, and in each race, the top 2 runners qualify for the next round while the others are eliminated. Determine the number of races required to declare the overall winner.

Answer: 49


A square with integer side length is cut into 12 squares, 9 of which have an area of 1 and 3 of which have an area of 4 each. Calculate the smallest possible side length of the original square.

Answer: 5


Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $3$, $2$, and $4$, respectively, and constant terms $2$, $3$, and $6$, respectively. If $P(z)$ and $Q(z)$ each have $z=-1$ as a root, determine the minimum possible value of the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$.

Answer: 1


In trapezoid EFGH, the bases EF and GH are not equal, where EF = 10 units and GH = 14 units. The lengths of the non-parallel sides are EG = 7 units and FH = 7 units. Calculate the perimeter of trapezoid EFGH.

Answer: 38


A square with area $16$ is inscribed in a square with area $25$, where each vertex of the smaller square touches a side of the larger square. One vertex of the smaller square divides a side of the larger square into two segments, and one length is three times the other length. Find the product of the lengths of the two segments.

Answer: \frac{75}{16}


Three congruent circles are centered at points $A$, $B$, and $C$ such that each circle passes through the center of the other two circles. The line containing both $A$ and $B$ is extended to intersect the circles at points $D$ and $E$ respectively. The circles centered at $A$ and $C$ intersect at a point $F$ distinct from where they intersect with the circle centered at $B$. Calculate the degree measure of $\angle DFE$.

Answer: 120^{\circ}


Given a triangle with vertices $A=(2,2)$, $B=(6,2)$, and $C=(5,5)$ plotted on an $8\times6$ grid, and a smaller rectangle within the grid defined by its diagonal with endpoints $(1,1)$ and $(6,5)$. Determine the fraction of the grid area covered by the portion of the triangle that lies within the smaller rectangle.

Answer: \frac{1}{8}


Given Peter's family ordered a 16-slice pizza, Peter ate 2 slices alone, shared 1 slice equally with his brother Paul, and shared 1 slice equally with their sister Sarah, and brother Paul. What fraction of the pizza did Peter eat in total?

Answer: \frac{17}{96}


Ralph walked down a street and passed five houses in a row, each painted a different color: green, blue, orange, red, and yellow. He passed the orange house before the red house and after the green house. He also passed the blue house before the yellow house, which was not next to the blue house. Additionally, the blue house was not next to the orange house. Determine the total number of orderings of the colored houses.

Answer: 3


Julia leaves Green Bay at 7:00 AM heading for Appleton on her scooter, traveling at a uniform rate of 15 miles per hour, and Mark leaves Appleton at 7:45 AM heading for Green Bay on his scooter, traveling at a uniform rate of 20 miles per hour. They travel on the same 85-mile route between Green Bay and Appleton. Determine the time at which they meet.

Answer: 9:51


Four red beads, two white beads, and two blue beads are placed in line in random order. Calculate the probability that no two beads of the same color are neighbors.

Answer: \frac{1}{14}


In $\bigtriangleup ABC$, $E$ is a point on side $\overline{AB}$, and $D$ is a point on side $\overline{BC}$ such that $BD=DE=EC$. Let $\angle BDE$ be $90^{\circ}$. Determine the degree measure of $\angle AED$.

Answer: 45^\circ


Maria buys computer disks at a price of 5 for $7 and sells them at a price of 4 for $7. Find the number of computer disks Maria must sell in order to make a profit of $125.

Answer: 358


53. A store sells yogurt, each bottle of yogurt costs 4 yuan, and to recycle empty bottles, every three yogurt bottles can be exchanged for one bottle of yogurt. Xiaoliang spent a total of 96 yuan on yogurt, so the maximum number of bottles of yogurt he could have drunk is $\qquad$.

(You can borrow empty bottles, but you must return them)

Answer: 36


364. Once I decided to take a ride on a chairlift. At some point, I noticed that the chair coming towards me had the number 95, and the next one had the number 0, followed by 1, 2, and so on. I looked at the number on my chair; it turned out to be 66. Have I passed the halfway point? At which chair will I pass the halfway point?

Answer: 18


On the blackboard, there is a number: 321321321321. By erasing some digits, we can get a number that is divisible by 9. The largest possible number is $\qquad$.

Answer: 3213212121


A chess piece can move 8 or 9 squares horizontally or vertically. It is forbidden to move onto the same square twice.

What is the maximum number of squares this piece can cover on a $15 \times 15$ board? (The tour can start from any square.)

Answer: 196


Five points lie on a straight line. Alex finds the distances between every pair of points. He obtains, in increasing order, 2, 5, 6, 8, 9, \( k \), 15, 17, 20, and 22. What is the value of \( k \)?

Answer: 14


Using the digits 1, 3, and 5, Monica forms three-digit numbers that are greater than 150. How many numbers can Monica form?

Answer: 21


In the figure, \(PQRS\) is a square, \(\overline{QE} = 17\), and \(\overline{PH} = 12\).

Calculate \(\overline{SE}\).

Answer: 7


As shown in the figure, \( \triangle ABC \) is an isosceles right triangle. Point \( E \) is on side \( BC \) such that \( BE = 3CE \). Points \( D \), \( A \), and \( F \) are collinear. Let the length of the altitude from \( B \) to \( BE \) in \( \triangle DBE \) be \( h_1 \), and the length of the altitude from \( E \) to \( EC \) in \( \triangle FEC \) be \( h_2 \). We have \( 3h_1 + h_2 = 3 \) cm. The combined area of \( \triangle DBE \) and \( \triangle FEC \) is 6 square centimeters. What is the area of \( \triangle ABC \) in square centimeters?

Answer: 64


Triangle \( ABC \) has side lengths \( AB = 19 \), \( BC = 20 \), and \( CA = 21 \). Points \( X \) and \( Y \) are selected on sides \( AB \) and \( AC \), respectively, such that \( AY = XY \) and \( XY \) is tangent to the incircle of \(\triangle ABC\). If the length of segment \( AX \) can be written as \(\frac{a}{b}\), where \( a \) and \( b \) are relatively prime positive integers, compute \( 100a + b \).

Answer: 6710


Two numbers, 2 and 4, are written first, then their sum, which is 6. Then the sum of the last two single-digit numbers ($4+6=10$), followed by $1+0=1$, and so on. As a result, the following sequence is obtained:

$$
2461101112358133
$$

Which digits cannot appear in this sequence if the sequence is continued indefinitely? What digit will be in the hundredth position?

Answer: 5


Three cones with vertex \(A\) and slant height 6 are externally tangent to each other. For two of the cones, the angle between the slant height and the axis of symmetry is \(\frac{\pi}{8}\), while for the third cone, it is \(\frac{\pi}{4}\). Find the volume of the pyramid \(O_{1} O_{2} O_{3} A\), where \(O_{1}\), \(O_{2}\), and \(O_{3}\) are the centers of the bases of the cones.

Answer: 9\sqrt{\sqrt{2}+1}


The watermelon seller weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) one by one while waiting for customers, balancing a watermelon on one pan of the scale with one or two weights on the other pan (possibly identical). At the same time, the seller noted down which weights he used. What is the minimum number of different numbers that could have appeared in his records if the weight of each weight is an integer number of kilograms?

Answer: 6


| $\left[\begin{array}{l}\text { [ Right triangle with an angle of \$30^{\circ}\$ } \\ \text { [ Parallel lines, properties and criteria. Transversals ] }\end{array}\right.$ |
| :---: | :---: |
|  | $\left[\begin{array}{ll}\text { Chords and secants (other) }\end{array}\right]$ |

In a circle, a diameter $A B$ and a chord $C D$ parallel to it are drawn such that the distance between them is half the radius of the circle (see figure). Find the angle $C A B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-09.jpg?height=434&width=483&top_left_y=1301&top_left_x=793)

Answer: 75^\circ


3. From the 6 face diagonals of the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, two tetrahedra $A-B_{1} C D_{1}$ and $A_{1}-B C_{1} D$ can be formed. If the combined volume of the two tetrahedra is 1 (overlapping parts are counted only once), then the volume of the rectangular prism is $\qquad$ .

Answer: 2


8. (40 points) On a strip, numbers from 1 to 1598 were written in order and the strip was cut into several parts. It turned out that the arithmetic mean of all numbers in the first part is some natural number $n$, the second part - the number $2 n$, the third part - the number $3 n$, and so on. Explain for which $n$ this is possible.

Answer: 1


8. Given the set of lattice points $S=\{(x, y) \mid x, y \in\{0,1,2,3,4,5\}\}$, find the number of triangles with vertices in $S$.

Answer: 6768


2. Let $M$ be a moving point on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$. Given points $F(1,0)$ and $P(3,1)$. Then the maximum value of $2|M F|-|M P|$ is $\qquad$.

Answer: 1


Let \( x_{1}, \ldots, x_{n} \) be real numbers strictly greater than 1. Consider the number of real numbers of the form \( \varepsilon_{1} x_{1} + \cdots + \varepsilon_{n} x_{n} \) where \( \varepsilon_{i} \in \{0, 1\} \) that lie within a given interval of length 1. What is the maximum number that can be obtained this way?

Answer: \dbinom{n}{\lfloor n/2 \rfloor}


7. As shown in Figure 3, in $\triangle A B C$, it is known that $\angle B=$ $40^{\circ}, \angle B A D=30^{\circ}$. If $A B$ $=C D$, then the size of $\angle A C D$ is $\qquad$ (degrees).

Answer: 40


## Problem 2

Let $f:(0, \pi) \rightarrow \mathbf{R}, f(x)=\frac{\cos ^{2} x}{\sin ^{3} x} \cdot e^{-x}$. Determine the primitive $F$ of $f$ for which $F\left(\frac{\pi}{4}\right)=0$.

Matlab

Answer: \frac{e^{-x} (\sin x - \cos x)}{2 \sin^{2} x}


Problem 9.3. Find all odd prime numbers $p$ which divide the number $1^{p-1}+$ $2^{p-1}+\cdots+2004^{p-1}$.

Kerope Tchakerian

Answer: 17


10. (3 points) Cars A and B both depart from point A, pass through point B, and head to point C. The distance between A and B is equal to the distance between B and C. Car B's speed is 80% of Car A's speed. It is known that Car B departs 11 minutes earlier than Car A, but stops for 7 minutes at point B, while Car A continues non-stop to point C. In the end, Car B arrives at point C 4 minutes later than Car A. Then, Car A surpasses Car B $\qquad$ minutes after Car B departs.

Answer: 27


2. As shown in Figure 3, in $\triangle A B C$, $B D$ bisects $\angle A B C$, $C D \perp B D$ at point $D$, $\angle B A C=\angle A C D, B C=$ $5, B A=11$. Then the area of $\triangle A B C$ is

Answer: \dfrac{132}{5}


Problem 3. Let $n$ and $m$ be natural numbers such that $m+i=$ $a_{i} b_{i}^{2}$ for $i=1,2, \ldots, n$, where $a_{i}$ and $b_{i}$ are natural numbers and $a_{i}$ is not divisible by a square of a prime number. Find all $n$ for which there exists an $m$ such that $a_{1}+a_{2}+\cdots+a_{n}=12$.

Answer: 3


1. Calculate the value of the expression

$$
S=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}
$$

where $a_{1}>0, a_{2}>0, \ldots, a_{n}>0$ and $a_{1}, a_{2}, \ldots, a_{n}$ form an arithmetic progression.

Answer: \dfrac{n-1}{\sqrt{a_1} + \sqrt{a_n}}


12.20*. (SFRY, 75). On a plane, $n+4$ points are marked, four of which are located at the vertices of a square, and the remaining $n$ lie inside this square. Any of the marked points are allowed to be connected by segments, provided that no constructed segment contains marked points other than its endpoints, and no two of the constructed segments have common points other than, possibly, their endpoints. Find the maximum number of segments that can be constructed in this way.

## § 13. Geometric Inequalities

(see Appendix G: definitions $1,35,38$; theorems $1,6,64,69,70,74,75,77,80$ )

Answer: 3n + 5


Three, (25 points) Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+2 b^{2}+3 c^{2}=2008$.

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

Answer: (22, 6, 22)


Inside a certain square, the distances from a point $P$ to the vertices $A, B, C$ of the square are $a, b, c$. What is the area of the square? What is it in the case where $a=b=c$?

Answer: \dfrac{a^2 + c^2 + \sqrt{4b^2(a^2 + c^2 - b^2) - (a^2 - c^2)^2}}{2}


B3. Jure put 2012 black frnikols in a row. First, he replaced every third frnikol in the row with a red frnikol. Then he replaced every fifth frnikol in the row with an orange frnikol. Finally, he replaced every seventh black frnikol in the row with a blue frnikol. How many black frnikols did he have left in the row at the end?

## Problems for 3rd Grade

Time for solving: 120 minutes. In section A, a correct answer will be worth two points, while a wrong answer will result in one point being deducted. Enter your answers for section A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_66bdf7a61d9bfcb32ca4g-06.jpg?height=164&width=802&top_left_y=546&top_left_x=640)

Answer: 921


6. The ball invited 2018 couples, who were arranged in areas numbered $1,2, \cdots, 2018$. The ball organizers stipulated: when the ball reaches the $i$-th minute, the couple standing in area $s_{i}$ (if there is someone in this area) will move to area $r_{i}$, and the couple originally in area $r_{i}$ (if there is someone in this area) will exit the dance, where,
$$
s_{i} \equiv i(\bmod 2018), r_{i} \equiv 2 i(\bmod 2018),
$$

and $1 \leqslant s_{i} 、 r_{i} \leqslant 2018$. According to this rule, when the ball has been going on for $2018^{2}$ minutes, how many couples are still dancing (Note: if $s_{i}=r_{i}$, then the couple in area $s_{i}$ remains in the original area and does not exit the dance)?

Answer: 1009


II. (25 points) Given 15 quadratic equations $x^{2}-p_{i} x+q_{i}=0(i=1,2, \cdots, 15)$ with coefficients $p_{i} 、 q_{i}$ taking values from $1,2, \cdots, 30$, and these coefficients are all distinct. If an equation has a root greater than 20, it is called a "good equation." Find the maximum number of good equations.

Answer: 10


Five soccer teams compete, with each pair of teams playing one match: the winner gets 3 points, the loser gets 0 points, and in case of a draw each team gets 1 point. After the matches, the total points of each team do not exceed 9 points, and there are exactly two teams that have the same points. Let the scores of the five teams from highest to lowest be $A, B, C, D, E$ (where there are exactly two equal scores). If $\overline{\mathrm{ABCDE}}$ (the concatenation of the scores) is a multiple of 15, how many draws were there in total?

Answer: 3


Russian 2001 Problem 5 2001 coins, each value 1, 2 or 3 are arranged in a row. Between any two coins of value 1 there is at least one coin, between any two of value 2 there are at least two coins, and between any two of value 3 there are at least three coins. What is the largest number of value 3 coins that could be in the row?

Answer: 501


In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$.

Answer: 40


The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off each of the mirrors once, and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\frac{AB}{AX} = 64$, and $\frac{CA}{AY} = \frac{p}{q}$ for coprime positive integers $p$ and $q$, find $p + q$.

Answer: 131


The figure on the right shows the ellipse \( \frac{(x-19)^2}{19} + \frac{(y-98)^2}{98} = 1998 \).

Let \( R_1, R_2, R_3, \) and \( R_4 \) denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of \( R_1 - R_2 + R_3 - R_4 \).

Answer: 7448


Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2} = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?

Answer: 462


Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. Note that numbers like $012$ and $007$ are not considered $3$-digit numbers.

Answer: 400185


Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

Answer: 13


The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x) \cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$.

Answer: 100


Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose
\[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

Answer: 702


\( \triangle ABC \) has side lengths \( 13 \), \( 14 \), and \( 15 \). Let the feet of the altitudes from \( A \), \( B \), and \( C \) be \( D \), \( E \), and \( F \), respectively. The circumcircle of \( \triangle DEF \) intersects \( AD \), \( BE \), and \( CF \) at \( I \), \( J \), and \( K \) respectively. What is the area of \( \triangle IJK \)?

Answer: 21


8. (5 points) As shown in the figure, a square wall with a side length of 50 meters. Person A and Person B start from points $A$ and $C$ respectively at the same time, moving along the wall in a clockwise direction. It is known that A walks 5 meters per second, and B walks 3 meters per second. Then, they will walk to the same side of the square after at least $\qquad$ seconds.

Answer: 30


Find all positive real numbers $\lambda$ such that every sequence $a_{1}, a_{2}, \ldots$ of positive real numbers satisfying

$$
a_{n+1}=\lambda \cdot \frac{a_{1}+a_{2}+\ldots+a_{n}}{n}
$$

for all $n \geq 2024^{2024}$ is bounded.
Remark: A sequence $a_{1}, a_{2}, \ldots$ of positive real numbers is bounded if there exists a real number $M$ such that $a_{i}<M$ for all $i=1,2, \ldots$
Answer: All positive real numbers $\lambda \leq 1$.

Answer: \lambda \leq 1


# Problem 4. (3 points)

Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.

At some point, the first runner meets the second. Twenty minutes later, the second runner meets the third for the first time. Another thirty minutes after that, the third runner meets the first for the first time.

How many minutes does it take for one runner to complete the entire track?

Answer: 100


Joe knows that to convert from pounds to kilograms, you need to divide the mass in pounds by 2 and then decrease the resulting number by 10%. From this, Joe concluded that to convert from kilograms to pounds, you need to multiply the mass in kilograms by 2 and then increase the resulting number by 10%. By what percentage does he err from the correct mass in pounds?

Answer: 1


In all the figures of this question, we see a triangle \(ABC\) divided into four parts; in these triangles, \(D\) is the midpoint of \(AB\), \(E\) is the midpoint of \(AC\), and \(FG\) measures \(\frac{1}{2} BC\).

a) The quadrilaterals \(DJMA\) and \(ELNA\) are obtained by rotating the quadrilaterals \(DHFB\) and \(EIGC\) by \(180^{\circ}\) around \(D\) and \(E\), respectively. Explain why the points \(M\), \(A\), and \(N\) are collinear, that is, why the measure of angle \(MAN\) is equal to \(180^{\circ}\).

b) In the figure below, point \(K\) is the intersection of lines \(JM\) and \(LN\). Explain why the triangles \(FGI\) and \(MNK\) are congruent.

The above items show that \(HJKL\) is a rectangle formed with the four parts into which triangle \(ABC\) was divided.

c) Show that \(LH = EF\).

d) In the figure below, triangle \(ABC\) has area 9 and \(HJKL\) is a square. Calculate the length of \(EF\).

Answer: 3


How many ways can the word MATHEMATICS be partitioned so that each part contains at least one vowel? For example, MA-THE-MATICS, MATHE-MATICS, MATHEM-A-TICS, and MATHEMATICS are such partitions.

Answer: 36


2 hours after departure, the train stopped for 30 minutes. On the remaining part of the journey to the station, repair work was being carried out, and the train was allowed to travel at a speed that was $1 / 3$ of the initial speed, causing the train to arrive at the station 1 hour and 10 minutes late. The next day, the stop occurred 14 km closer to the final station, and under the same conditions, the delay was reduced to 50 minutes. Determine the distance between the stations and the speed of the train.

Answer: 84


There is an oak tree growing by the road from Yolkino to Palkino, and it is twice as close to Yolkino as it is to Palkino. Fedya, traveling at a constant (and positive) speed from Yolkino to Palkino, was twice as close to the oak tree as he was to Yolkino at 12:00. At 12:40, it turned out again that Fedya was twice as close to the oak tree as he was to Yolkino. When will Fedya arrive in Palkino?

Answer: 13:10


Let's define the distance between two numbers as the absolute value of their difference. It is known that the sum of the distances from thirty-three consecutive natural numbers to a certain number \(a\) is 3168, and the sum of the distances from these same thirty-three numbers to a certain number \(b\) is 924. Find all possible values of \(a\), given that \(a + b = 120\).

Answer: 122


A right triangle has acute angles of $60^{\circ}$ and $30^{\circ}$. Two congruent circles are inscribed in the triangle, touching the hypotenuse, each other, and one of the legs. How many times is the smaller leg of the triangle the radius of the circles?

Answer: 2+\sqrt{3}


In triangle \(ABC\), \(AC = 2\sqrt{3}\), \(AB = \sqrt{7}\), and \(BC = 1\). A point \(K\) is taken outside the triangle such that the segment \(KC\) intersects segment \(AB\) at a point different from \(B\), and the triangle with vertices \(K\), \(A\), and \(C\) is similar to the original triangle. Find the angle \(AKC\), given that the angle \(KAC\) is obtuse.

Answer: 30


Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10000\lfloor x \rfloor = 10000x$?

Answer: 199


The set of real numbers $x$ for which
$$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge 1$$is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals?

Answer: 3


Deanna has $44$ pieces of candy. What is the largest number of kids whom she can give an equal number of candies such that every kid gets at least $3$ pieces?

Answer: 14


The foci of the ellipse $\frac{x^2}{49} + \frac{y^2}{33} = 1$ are $F_1$ and $F_2,$ as shown below.  Let $P$ be a point on the circle $x^2 + (y - 3)^2 = 4.$  Line $F_2 P$ intersects the ellipse again at $Q,$ where the $y$-coordinate of $Q$ is positive.  Find the maximum value of $PQ + F_1 Q.$

[asy]
unitsize(0.4 cm);

pair P, Q;
pair[] F;

path ell = yscale(sqrt(33))*xscale(7)*Circle((0,0),1);
F[1] = (4,0);
F[2] = (-4,0);
P = (0,3) + 2*dir(240);
Q = intersectionpoint(P--interp(F[2],P,5),ell);

draw(ell);
draw((-8,0)--(8,0));
draw((0,-7)--(0,7));
draw(Circle((0,3),2));
draw(F[1]--Q--F[2]);

dot("$F_1$", F[1], S);
dot("$F_2$", F[2], S);
dot("$P$", P, S);
label("$Q$", Q, NE);
[/asy]

Answer: 11


Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of triangle $ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$?

Answer: 6


A parallelogram has three of its vertices at $(-1,0)$, $(2,4)$ and $(2,-4)$.  What is the positive difference between the greatest possible perimeter and the least possible perimeter of the parallelogram?

Answer: 6


Let \( n \) be a set of integers. \( S(n) \) is defined as the sum of the elements of \( n \). Consider the set \( T = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Let \( A \) and \( B \) be subsets of \( T \) such that \( A \cup B = T \) and \( A \cap B = \varnothing \). The probability that \( S(A) \geq 4S(B) \) can be expressed as \( \frac{p}{q} \). Compute \( p + q \).

Answer: 545


Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$.

Answer: 468


To each pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:

1. $a*(b*c) = (a*b)*c$ for all $a, b, c$.
2. $a*a = 1$ for all $a$.

Solve the equation $x*36 = 216$.

Answer: 7776


Let $\Gamma$ be a circle with center $A$, radius $1$, and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$, and diameter $DY$, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.

Answer: 16


In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$.

Answer: 25


39 students participated in a math competition. The exam consisted of 6 problems, and each problem was worth 1 point for a correct solution and 0 points for an incorrect solution. For any 3 students, there is at most 1 problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the 39 students. Find the smallest possible value of $B$.

Answer: 165


Let $S$ be a subset of the natural numbers such that $0 \in S$, and for all $n \in \mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?

Answer: 47


Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9.

For example, if \( n = 2024 \) and we choose \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 3 \), then \( m = 4177 \) is an alero of \( n \), but if we choose the digits \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 6 \), then we don't obtain an alero of \( n \), because \( 4 + 6 \) exceeds 9.

Find the smallest \( n \) which is a multiple of \( 2024 \) that has an alero which is also a multiple of \( 2024 \).

Answer: 10120


7. (3 points) Five children $A, B, C, D, E$ are sitting in a circle (as shown in the right figure). The teacher gives $A, B, C, D, E$ 2, 4, 6, 8, 10 balls respectively. Then, starting from $A$, they play a game in a clockwise direction: if the number of balls the left neighbor has is less than their own, they give 2 balls to the left neighbor; if the left neighbor has more or the same number of balls, they do not give any. This continues until the fourth round. The number of balls each of them has at the end is

Answer: 6


5. A knight is placed in each cell of a chessboard. What is the minimum number of knights that can be removed from the board so that no knight remains that attacks exactly four other knights? (A knight attacks the squares that are one square away horizontally and two squares away vertically, or vice versa.)

Answer: 8


23. Calculate the sum $1 \cdot x+2 x^{2}+3 x^{3}+\ldots+n x^{n}$.

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


9.1. Distribute 24 pencils of four colors (6 pencils of each color) to 6 students, with each student getting 4 pencils. It is known that no matter how the pencils are distributed, there will always be $n$ students such that the $4 n$ pencils they have are of four colors. Find the minimum value of $n$.

Answer: 3


A natural number is called a square if it can be written as the product of two equal numbers. For example, 9 is a square, since $9=3 \times 3$. The first squares are 1, 4, 9, 16, 25, ... A natural number is called a cube if it can be written as the product of three equal numbers. For example, 8 is a cube, since $8=2 \times 2 \times 2$. The first cubes are $1,8,27,64,125 \ldots$

One day, the square and cube numbers decided to go on strike. This is how the other natural numbers had to take new positions:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-21.jpg?height=113&width=1082&top_left_y=763&top_left_x=544)

a) What is the number that ended up in the $12^{\text {th }}$ position?

b) What are the numbers less than or equal to 2013 that are both squares and cubes?

c) What is the new position occupied by the number 2013?

d) Find out which number ended up in the $2013^{\text{th}}$ position.

Answer: 17


2. Given is the set $A=\{1,2,3, \ldots, 62,63\}$. A subset $X$ of the set $A$ will be called good if for every element $x \in X$ it holds that $x$ does not divide the sum of the remaining elements of the set $X$. Determine the maximum number of elements that a good subset can have.

Answer: 61


A board  $ n \times n$  ( $n \ge 3$ ) is divided into  $n^2$  unit squares. Integers from  $O$  to  $n$  included, are written down: one integer in each unit square, in such a way that the sums of integers in each  $2\times 2$  square of the board are different. Find all  $n$  for which such boards exist.

Answer: n = 3, 4, 5, 6


Task 1 - 120711 Klaus had set his wristwatch to the time signal from Radio DDR at 12:00 on a Saturday. He noticed at 12:00 on the following Sunday when listening to the time signal that his watch was exactly 6 minutes slow, but he forgot to set it correctly.

He wanted to leave early on the following Monday at exactly 8:00 AM.

What time did his watch show at this time, assuming that his watch ran at a constant rate throughout?

Answer: 7:49


11. From the three-digit numbers $100, 101, 102, \ldots, 699, 700$, if $n$ different numbers are taken, such that there are always three numbers among them with the same sum of digits. Then the minimum value of $n$ is $\qquad$

Answer: 47


8. [6] $A B C D$ is a convex quadrilateral such that $A B<\underline{A D}$. The diagonal $\overline{A C}$ bisects $\angle B A D$, and $m \angle A B D=130^{\circ}$. Let $E$ be a point on the interior of $\overline{A D}$, and $m \angle B A D=40^{\circ}$. Given that $B C=$ $C D=D E$, determine $m \angle A C E$ in degrees.

Answer: 55


13. Let $M$ be a finite set of numbers. If it is known that among any three elements of $M$, there are always two numbers whose sum belongs to $M$, how many elements can $M$ have at most?

Answer: 7


Two circles touch each other externally at point $D$. We draw one of the external tangents $AB$ to these circles. We rotate the entire figure around the central line of the two circles. What is the ratio of the volumes of the solids formed by the rotation of the region defined by the lines $AB$, $AD$, and $BD$ (forming triangle $ABD$) and the region defined by the arcs $AD$ and $BD$ and the line $AB$?

Answer: 2


When the polynomial \( x^{1951}-1 \) is divided by \( x^{4}+x^{3}+2x^{2}+x+1 \), a quotient and a remainder are obtained. Find the coefficient of \( x^{14} \) in the quotient.

Answer: -1


3 [ Constructions on a Projection Drawing ]

In a regular quadrilateral pyramid with a lateral edge of 20, the angle between the lateral edges lying in the same face is $\frac{\pi}{3}$. A line perpendicular to one of the lateral edges and passing through a point on this edge intersects the height of the pyramid. Find the length of the segment of this line that lies inside the pyramid, if the point of intersection of this line with the height divides the height in the ratio 3:7, counting from the vertex.

Answer: 17


In circle \( \odot O \), let \( M \) be the midpoint of chord \( GH \). Draw chords \( AB \) and \( CD \) through point \( M \). Given that \( \angle AMG = \alpha \) and \( \angle CMG = \beta \), prove that \( \frac{\sin \alpha}{\sin \beta} = \frac{MB - MA}{MC - MD} \).

Answer: \frac{\sin \alpha}{\sin \beta} = \frac{MB - MA}{MC - MD}


There are nonzero integers $a$ , $b$ , $r$ , and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$ . For each possible combination of $a$ and $b$ , let ${p}_{a,b}$ be the sum of the zeros of $P(x)$ . Find the sum of the ${p}_{a,b}$ 's for all possible combinations of $a$ and $b$ .
Please give the answer directly without any intermediate steps.

Answer: 80


Given \( a, b, c \) are the sides opposite to \(\angle A, \angle B, \angle C\) respectively in \(\triangle ABC\), and \( S \) is the area of \(\triangle ABC\). If \( a = 4 \), \( b = 5 \), and \( S = 5 \sqrt{3} \), find the length of \( c \).

Answer: \sqrt{21}


8. Given that the perimeter of the regular pentagon square $A B C D E$ is $2000 \mathrm{~m}$, two people, A and B, start from points $A$ and $C$ respectively at the same time, walking around the square in the direction of $A \rightarrow B \rightarrow C \rightarrow D \rightarrow E \rightarrow A \rightarrow \cdots$. Person A's speed is $50 \mathrm{~m} / \mathrm{min}$, and person B's speed is $46 \mathrm{~m} / \mathrm{min}$. Then, after $\qquad$ $\min$, A and B will first start walking on the same side.

Answer: 104


Given 6 different colours, how many ways can we colour a cube so that each face has a different colour? Show that given 8 different colours, we can colour a regular octahedron in 1680 ways so that each face has a different colour.

Answer: 30


If we divide a square into parts with three lines, then among the resulting polygons, there is always one whose diameter is not less than $\sqrt{13}$. What is the minimum length of the side of the square? (The diameter of a polygon is the distance between its two farthest vertices.)

Translating the text as requested, while preserving the original formatting and line breaks.

Answer: 6


Margot wants to prepare a jogging timetable. She wants to jog exactly twice a week, and on the same days every week. She does not want to jog on two consecutive days. How many different timetables could Margot prepare?

Answer: 14


A real number $x$ satisfies the equation $\frac{1}{x}+\frac{x}{80}=\frac{7}{30}$. What are the possible values of $x$ ?

Answer: 12


There are 100 weights of different masses on a table. A weight is called "lucky" if its mass is equal to the sum of the masses of some two other weights on the table. What is the minimum number of lucky weights that guarantees that the masses of some two weights differ by at least three times?

Answer: 87


On the card, a four-digit natural number is written, from which we can strike out any two digits and always get a two-digit natural number that is divisible by 5 without a remainder. How many such four-digit natural numbers exist? (Note, for example, the number 06 is not a two-digit number.)

(L. Šimünek)

Answer: 18


Let $\xi_{1}, \ldots, \xi_{n}$ be independent random variables uniformly distributed on $[0,1]$. Using the inclusion-exclusion formula from Problem II.8.60, prove the Laplace formula

$$
\mathrm{P}\left(\xi_{1}+\ldots+\xi_{n} \leqslant x\right)=\frac{1}{n!} \sum_{k=0}^{[x]}(-1)^{k} C_{n}^{k}(x-k)^{n}, \quad 0 \leqslant x \leqslant n
$$

where $[x]$ is the integer part of $x$.

Answer: \mathrm{P}\left(\xi_{1}+\ldots+\xi_{n} \leqslant x\right)=\frac{1}{n!} \sum_{k=0}^{[x]}(-1)^{k} C_{n}^{k}(x-k)^{n}


12. As shown in the figure, in quadrilateral $\mathrm{ABCD}$, $\mathrm{AC}$ and $\mathrm{BD}$ intersect at point $\mathrm{O}$. Draw the altitude $\mathrm{DE}$ of triangle $\mathrm{DBC}$, and connect $\mathrm{AE}$. If the area of triangle $\mathrm{ABO}$ is equal to the area of triangle $\mathrm{DCO}$, and $\mathrm{DC}=17$ cm, $\mathrm{DE}=15$ cm, then what is the area of the shaded part in square centimeters?

Answer: 60


5. In a $3 \times 3$ table, the following numbers are written:

| 10 | 20 | 40 |
| :---: | :---: | :---: |
| 32 | 61 | 91 |
| 100 | 1000 | 2000 |

It is allowed to swap any two numbers in one move. What is the minimum number of moves required to achieve that the sum of the numbers in each column is divisible by $3 ?$

Answer: 2


Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.

Answer: 1


[ Concurrency of heights. Angles between heights. [Trigonometric ratios in a right-angled triangle ]

In triangle $A B C$, angle $\angle B$ is obtuse, the extensions of heights $A M$ and $C N$ intersect at point $O, \angle B A C=\alpha, \angle B C A=\gamma, A C=b$. Find the distance from point $O$ to the line $A C$.

Answer: \dfrac{b}{\tan \alpha + \tan \gamma}


2.2. Find the length of the arc $A B$ of the curve $e^{2 y}\left(e^{2 x}-1\right)=e^{2 x}+1$, if $x_{A}=1, x_{B}=2$.

Answer: \frac{1}{2} \ln\left(e^{4} + 1\right) - 1


A set of points $M$ in the plane satisfies the following conditions:
(a) No three points are collinear;
(b) Each point in $M$ can be colored either red or blue such that every triangle with vertices of the same color has at least one point of the other color inside it.

Determine the maximum possible number of elements in $M$.

Answer: 8


A one-hundred-digit natural number \( N \) is composed of ones and twos, and between any two twos there is an even number of digits. It is known that \( N \) is divisible by 3. How many ones and how many twos are there in the number \( N \)?

Answer: 98 \text{ ones and } 2 \text{ twos}


The upper side of a paper square is white, and the lower side is red. A point F is randomly chosen within the square. Then the square is folded so that one randomly chosen vertex coincides with point F. Find the expected number of sides of the resulting red polygon.

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-14.jpg?height=515&width=512&top_left_y=1655&top_left_x=773)

Answer: 5 - \dfrac{\pi}{2}


Example 5 In space, there are four spheres with radii of $2$, $2$, $3$, and $3$. Each sphere is externally tangent to the other three spheres. There is another smaller sphere that is externally tangent to these four spheres. Find the radius of the smaller sphere.
(1995, China Mathematical Olympiad)

Answer: \dfrac{6}{11}


In Italy, they produce clocks in which the hour hand completes one full rotation per day, and the minute hand completes 24 rotations per day. As usual, the minute hand is longer than the hour hand. Consider all positions of the two hands and the zero marking on the Italian clock that coincide with those on a regular clock. How many such positions exist on the Italian clock within a day? (The zero marking on the Italian clock represents 24 hours, while on a regular clock it represents 12 hours.)

Answer: 12


Given a permutation \( \left(a_{0}, a_{1}, \ldots, a_{n}\right) \) of the sequence \(0, 1, \ldots, n\). A transposition of \(a_{i}\) with \(a_{j}\) is called legal if \(a_{i}=0\) for \(i>0\), and \(a_{i-1}+1=a_{j}\). The permutation \( \left(a_{0}, a_{1}, \ldots, a_{n}\right) \) is called regular if after a number of legal transpositions it becomes \((1, 2, \ldots, n, 0)\). For which numbers \(n\) is the permutation \((1, n, n-1, \ldots, 3, 2, 0)\) regular?

Answer: \(n=2\) and \(n=2^{k}-1, k\) positive integer.

Answer: n=2


The smallest three positive proper divisors of an integer $n$ are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$.

Answer: 42


Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isosceles triangle with $a$, $b$, and $c$ as the lengths of sides. Determine how many possible three-digit numbers $N$ there are.

($N=\overline{abc}$ means that $a$, $b$, and $c$ are digits of $N$, and not $N=a \times b \times c$.)

Answer: 165


Suppose that a polynomial of the form $p(x) = x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?

Answer: 1005


David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?

Answer: 5


We will say that a rearrangement of the letters of a word has no *fixed letters* if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance, $HBRATA$ is a rearrangement with no fixed letter of $BHARAT$. How many distinguishable rearrangements with no fixed letters does $BHARAT$ have? (The two $A$s are considered identical.)

Answer: 84


You have a $1 \times 2024$ grid of squares in a column, vertices labelled with coordinates $(0,0)$ to $(1,2024)$. Place a weed at $(0,0)$. When a weed is attempting to be placed at coordinates $(x,y)$, it will be placed with a $50\%$ probability if and only if exactly one of the vertices $(x-1, y)$ or $(x, y-1)$ has a weed on it, otherwise the attempt will fail with probability $1$. The placement attempts are made in the following order: For each vertex with $x$ coordinate $0$, attempt a placement for each vertex starting from $y$ coordinate $0$, incrementing by $1$ until $2024$. Then, attempts will be made on the vertices with $x$ coordinate $1$ in the same fashion. Each placement attempt is made exactly once.

The probability that a weed appears on $(1,2024)$ after placing the weed at $(0,0)$ and attempting to place weeds on every vertex is $p$. Estimate $9p \cdot 2^{2025}$ to the nearest integer.

Answer: 18


A five-digit positive integer $abcde_{10}$ ($a \neq 0$) is said to be a *range* if its digits satisfy the inequalities $a < b > c < d > e$. For example, $37452$ is a range. How many ranges are there?

Answer: 8622


Let $\sigma_k(n)$ be the sum of the $k^{th}$ powers of the divisors of $n$. For all $k \ge 2$ and all $n \ge 3$, we have that \[\frac{\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.\] Find the largest possible value of $m$.

Answer: 0


A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

Answer: 60


The first $9$ positive integers are placed into the squares of a $3 \times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest numbers. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest numbers. How many ways can we distribute the numbers into the chessboard such that $a = b = 4$?

Answer: 25920


The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

Answer: 1342


[Inscribed quadrilateral with perpendicular diagonals]

$A B C D$ is an inscribed quadrilateral with perpendicular diagonals. $O$ is the center of the circumscribed circle of quadrilateral $A B C D$. $P$ is the point of intersection of the diagonals.

Find the sum of the squares of the diagonals, given the length of the segment $O P$ and the radius of the circle $R$.

Answer: 8R^2 - 4OP^2


3. Encipher the message SURRENDER IMMEDIATELY using the affine transformation $C \equiv 11 P+18(\bmod 26)$.

Answer: IEXXKFZKXCUUKZCSTKJW


Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an
integer. Find the smallest $k$ for which the set A can be partitioned into two subsets
having the same number of elements, the same sum of elements, the same sum
of the squares of elements, and the same sum of the cubes of elements.

Answer: 4


For each positive integer \( n \), define the function
\[ f(n) = \begin{cases}
0 & \text{ if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor & \text{ if } n \text{ is not a perfect square.}
\end{cases} \]
(where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \)). Find the value of \( \sum_{k=1}^{240} f(k) \).

Answer: 768


Question 199: Two people, A and B, play a game with the following rules: In the odd-numbered rounds, A's probability of winning is $\frac{3}{4}$; in the even-numbered rounds, B's probability of winning is $\frac{3}{4}$. There are no ties in any round. The rule is: The game ends when one person wins 2 more rounds than the other. Try to find: When the game ends, the expected number of rounds played by A and B.

Answer: \dfrac{16}{3}


Let \( p \) be a prime number. Let \( \mathbb{F}_{p} \) denote the integers modulo \( p \), and let \( \mathbb{F}_{p}[x] \) be the set of polynomials with coefficients in \( \mathbb{F}_{p} \). Define \( \Psi: \mathbb{F}_{p}[x] \rightarrow \mathbb{F}_{p}[x] \) by
$$
\Psi\left(\sum_{i=0}^{n} a_{i} x^{i}\right)=\sum_{i=0}^{n} a_{i} x^{p^{i}} .
$$

Prove that for nonzero polynomials \( F, G \in \mathbb{F}_{p}[x] \),
$$
\Psi(\operatorname{gcd}(F, G))=\operatorname{gcd}(\Psi(F), \Psi(G))
$$

Answer: \Psi(\operatorname{gcd}(F, G)) = \operatorname{gcd}(\Psi(F), \Psi(G))


10.027. Each of the three equal circles of radius $r$ touches the other two. Find the area of the triangle formed by the common external tangents to these circles.

Answer: r^2 (6 + 4\sqrt{3})


In the image on the right, a regular icosahedron is depicted. Five points adjacent to the top vertex lie in a horizontal plane. Another five points, adjacent to the bottom vertex, also lie in a horizontal plane. How many ways are there to travel from the top vertex to the bottom vertex, always moving downwards or within the same horizontal plane, and not visiting any vertex more than once?

Answer: 810


1. (USS) Three problems $A, B$, and $C$ were given on a mathematics olympiad. All 25 students solved at least one of these problems. The number of students who solved $B$ and not $A$ is twice the number of students who solved $C$ and not $A$. The number of students who solved only $A$ is greater by 1 than the number of students who along with $A$ solved at least one other problem. Among the students who solved only one problem, half solved $A$. How many students solved only $B$?

Answer: 6


Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.

Answer: 37


Problem 11.5. In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 people, are competing against each other. During the tournament, each schoolchild must play against each student exactly once, and each person must play no more than one game per day. The number of games played on different days may vary.

At some point in the tournament, the organizer noticed that the schedule for the next day can be arranged in exactly 1 way with 15 games, and in $N$ ways with 1 game (the order of the games in the schedule does not matter, only who plays against whom). Find the maximum possible value of $N$.

Answer: 120


Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$, and $(0, 3, 0)$, and whose apex is the point $(1, 1, 3)$. Let $Q$ be a square pyramid whose base is the same as the base of $P$, and whose apex is the point $(2, 2, 3)$. Find the volume of the intersection of the interiors of $P$ and $Q$.

Answer: \dfrac{27}{4}


13. Task $\Gamma$. Dudeney. If one and a half chickens lay one and a half eggs in one and a half days, then how many chickens plus another half chicken, laying eggs one and a half times faster, will lay ten eggs and a half in one and a half weeks?

Answer: \dfrac{1}{2}


2. The digits 1 to 8 are placed into the cells of the grid on the right, making four three-digit numbers when read clockwise. For which values of $k$ from 2 to 6 is it possible to create an arrangement such that all four of the three-digit numbers are multiples of $k$ ?

Answer: 2


16. Given real numbers $x, y$ satisfy $2^{x+1}+2^{y+1}=4^{x}+4^{y}$, find the range of $M=8^{x}+8^{y}$.

Answer: (8,16]


## Task 2.

Let $N$ be a natural number. We call a staircase a part of a square plate of dimensions $N \times N$ that consists of the first $K$ fields in the $K$-th row for $K=1,2, \ldots, N$. In how many ways can the staircase be cut into rectangles of different areas that consist of the fields of the given plate?

Answer: 2^{N-1}


A triangle is cut into several (no less than two) triangles. One of them is isosceles (not equilateral), and the others are equilateral. Find the angles of the original triangle.

#

Answer: 30^\circ


Example 1. Place four equal small balls on a plane so that the line connecting their centers forms a square, with each side having two balls touching, and place one more equal ball on top of these four balls so that it touches all of them. Given the highest point of the top ball is a. Find the radius $x$ of the small balls.

Answer: \dfrac{a(2 - \sqrt{2})}{2}


26. In the diagram below, $A B C D$ is a cyclic quadrilateral with $A B=A C$. The line $F G$ is tangent to the circle at the point $C$, and is parallel to $B D$. If $A B=6$ and $B C=4$, find the value of $3 A E$.

Answer: 10


4. Given that the three interior angles $A, B, C$ of $\triangle A B C$ form an arithmetic sequence, and the corresponding sides are $a, b, c$, and $a, c, \frac{4}{\sqrt{3}} b$ form a geometric sequence, then $S_{\triangle A B C}: a^{2}=$ $\qquad$ .

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


11.45*. What is the maximum number of points that can be placed on a segment of length 1 so that on any segment of length $d$ contained in this segment, there are no more than $1+1000 d^{2}$ points?

See also problems $15.1,17.20$.

## §7. Extremal Properties of Regular Polygons

Answer: 32


When I divide the number $X$ by the number $Y$, I get the number $Z$ and a remainder of 27. When I divide the number $X$ by the number $Z$, I get the number $1.1 \cdot Y$ and a remainder of 3.

Which numbers $X, Y, Z$ satisfy the given conditions? Determine all possibilities.

(L. Hozová)

Answer: 267


The rectangular spiral shown in the diagram is constructed as follows. Starting at $(0,0)$, line segments of lengths $1,1,2,2,3,3,4,4, \ldots$ are drawn in a clockwise manner, as shown. The integers from 1 to 1000 are placed, in increasing order, wherever the spiral passes through a point with integer coordinates (that is, 1 at $(0,0), 2$ at $(1,0), 3$ at $(1,-1)$, and so on). What is the sum of all of the positive integers from 1 to 1000 which are written at points on the line $y=-x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_ff0a25f37b4b6ded9838g-3.jpg?height=567&width=561&top_left_y=215&top_left_x=1183)

Answer: 10944


A man had seven married sons. Before his death, he distributed his oxen among his family members. The eldest son received a certain number of oxen, and his wife received one-ninth of the remainder. The second son received one more ox than his elder brother, and his wife also received one-ninth of the remainder. The third son similarly received one more ox than the second son, and his wife also received one-ninth of the remainder, and so on for the rest of his sons and their wives. In this way, each family received the same number of oxen. How many oxen did the man have?

Answer: 56


8. (5 points) In the cells of a $5 \times 7$ table, the numbers 1, 2, and 3 are arranged such that in any $2 \times 2$ square, there are all three different numbers. What is the smallest value that the sum of the numbers in the entire table can take?

Answer: 55


8. (10 points) As shown in the figure, a square piece of paper $A B C D$ is cut along the diagonal $B D$ into two triangles. In the first operation, triangle $A B D$ is moved vertically downward by 3 cm to triangle $E F G$; in the second operation, triangle $E F G$ is moved vertically downward by another 5 cm to triangle $H I J$. The overlapping area of the two pieces of paper after the first operation is equal to the overlapping area of the two pieces of paper after the second operation. What is the area of the square piece of paper $A B C D$ in square centimeters? $\qquad$

Answer: 121


2. Given the equalities $\log _{a} b+\log _{b} c+\log _{c} a=\log _{b} a+\log _{c} b+\log _{a} c=-1.5$. Find $\log _{a} b$.

If there are multiple answers, list them in ascending order separated by a semicolon.

Answer: -2;-0.5;1


7. Let the dihedral angle between a lateral face and the base of a regular quadrilateral pyramid $P-ABCD$ be $\alpha$, and the dihedral angle between two adjacent lateral faces be $\beta$. Then $2 \cos \beta + \cos 2 \alpha =$ $\qquad$ .

Answer: -1


12. A little ant lives in a rectangular cave (as shown in the figure, the lines represent paths). It crawls from point $A$ to point $B$, then to point $C$, and finally to point $D$, covering a total of 22 cm; It crawls from point $A$ to point $D$, then to point $C$, and finally to point $B$, covering a total of 29 cm; It crawls from point $A$ to point $B$, then to point $C$, and finally back to point $A$, covering a total of 30 cm. If the little ant starts from point $A$, crawls along every path in the cave, and finally arrives at point $C$, then it has crawled at least $\qquad$ cm.

Answer: 47


Example 2 As shown in Figure 2 of the hopscotch game: a person can only enter the first
square from outside; in the squares, each time they can
jump forward 1 square or 2 squares.
Then, the number of ways a person can jump
from outside to the sixth square is $\qquad$.

Answer: 8


In a single-round-robin tournament, 10 chess players are participating. After what minimum number of rounds can it happen that a sole winner has already been determined prematurely? (In each round, the participants are divided into pairs. Win - 1 point, draw - 0.5 points, loss - 0).

Answer: 7


14. 3 piles of peaches have the numbers $93, 70, 63$, respectively. A monkey moves peaches between the 3 piles. It is known that the monkey can move a maximum of 5 peaches each time, and will eat 1 peach during the move from one pile to another. When the number of peaches in the 3 piles is equal, the monkey has eaten at least $\qquad$ peaches.

Answer: 4


[ Arithmetic. Mental calculation, etc.]

From the ten-digit number 2946835107, five digits were erased. What is the largest number that could result from this?

#

Answer: 98517


7. A sequence of numbers is arranged according to the following rule:
$$
\frac{1}{3}, \frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11} \text {, }
$$
$\qquad$
Starting from the $\qquad$ term, the difference between each term and the previous term is less than $\frac{1}{2021}$.

Answer: 45


8. (10 points) King Arthur summoned 6 knights to the palace. Each knight has exactly 2 friends. They sat around a round table (the names and seats of the knights are shown in the figure), and it turned out that in this arrangement, any two adjacent knights are friends. King Arthur wants to rearrange the seats, so how many different ways can King Arthur arrange the seats such that no knight sits next to his friends (rotations that result in the same arrangement are considered the same method)?

Answer: 6


4. In a right-angled triangle where the legs are in the ratio 5:12, the circle with diameter $MN$ where $M$ and $N$ are the midpoints of the legs, intersects the hypotenuse at points $P$ and $Q$. Determine the ratio in which points $P$ and $Q$ divide the hypotenuse.

Answer: 50:119:169


## Task A-4.7.

Višnja decided to write all natural numbers from 1 to 2014 in some order on the board. Višnja's brother Marijan will write the absolute value of the difference between each pair of adjacent numbers, and then erase all the original numbers. Marijan will repeat this process until only one number remains on the board.

Determine the largest possible number that can remain on the board at the end.

Answer: 2012


1. On an island, there are two tribes: the tribe of knights, who always tell the truth, and the tribe of liars, who always lie. On the main holiday, 2017 islanders sat around a large round table. Each islander said the phrase: "my neighbors are from the same tribe." It turned out that two liars made a mistake and accidentally told the truth. How many liars can sit at this table?

Answer: 1344


10. (12 points) The natural numbers $1, 2, 3, \cdots, 2014, 2015, 2016$ are arranged in a circle clockwise. Starting from the number 1, the following operations are performed clockwise:
First step: Cross out 1, keep 2;
Second step: Cross out $3, 4$, keep 5;
Third step: Cross out $6, 7, 8$, keep 9;
Fourth step: Cross out $10, 11, 12, 13$, keep $14; \cdots$;
That is, in the nth step, first cross out n numbers, then keep 1 number, and continue this operation until all numbers are crossed out.
What is the last number to be crossed out? $\qquad$

Answer: 2015


5 In a rectangle $A B C D$ with an area of 1 (including the boundary) there are 5 points, no three of which are collinear. Find the minimum number of triangles, formed by these 5 points as vertices, that have an area not greater than $\frac{1}{4}$.

Answer: 2


152. Math Puzzler $1 / 78$

Due to the ongoing crisis in capitalist states, the French Franc loses about 10 percent of its value annually.

How much would a good that costs 1 Franc today, on average, cost after three years, assuming a constant inflation rate?

Answer: 1.37


5. Bear Big and Bear Small jog on a circular track that is 1500 meters long every day.
On the first day, both start from the starting point at the same time and run the entire way. When Bear Big completes 4 laps and returns to the starting point, Bear Small has completed 3 laps and an additional 300 meters.
On the second day, Bear Big runs the entire way, while Bear Small alternates between running and walking, with the running speed being twice the walking speed. Both start from the starting point at the same time, and when Bear Big completes 3 laps and returns to the starting point, Bear Small also returns to the starting point.
So, the distance Bear Small walked on the second day is $\qquad$ meters.

Answer: 600


*5. In a football championship, each team is required to play a match against all other teams. Each match, the winning team gets 2 points, a draw results in 1 point for each team, and the losing team gets 0 points. It is known that one team has the highest score, but it has won fewer matches than any other team. If there are at least $n$ teams participating, then $n=$ $\qquad$ .

Answer: 6


11. (10 points) The emblem of XiXi's army is the regular octagon pattern shown in the figure. Given that the side length of the regular octagon is 18, what is the area of the shaded part?

Answer: 324


B5. For a regular decagon $A B C D E F G H I J$ with sides of length 12, let $M$ be the center. Let $S$ be the intersection of $M C$ and $A D$.

Calculate the difference between the perimeter of quadrilateral $A B C D$ and the perimeter of triangle $D M S$.

## C-Problems

For the C-Problems, the answer is not the only thing that matters; there should also be a reasoning that shows your answer is correct. Each correctly worked-out C-Problem is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper as well.

NOTE: Solve each C-Problem on a separate sheet of paper and also submit the corresponding scratch paper per problem.

Answer: 24


3.48 A young man was returning home from vacation on a bicycle. At first, after traveling several kilometers, he spent one day more than half the number of days remaining after this until the end of his vacation. Now the young man has two options to travel the remaining distance to arrive home on time: to travel $h$ km more daily than originally planned, or to maintain the original daily travel distance, exceeding it only once - on the last day of the journey by $2 h$ km. How many days before the end of his vacation did the young man set out for home?

Answer: 4


7. Arrange the ten digits from 0 to 9 into a ten-digit number without repetition and with the first digit not being zero. The number of such ten-digit numbers that can be divided by 99 is $\qquad$.

Answer: 285120


Consider the number obtained by writing the numbers $1, 2, \ldots, 1990$ one after another. In this number, every digit at an even position is omitted. In the resulting number, every digit at an odd position is omitted. Then, in the new number, every digit at an even position is omitted, and so on. What will be the last remaining digit?

Answer: 9


Peter wants to create a unique die where each face displays a different positive integer. For any two adjacent faces, the numbers must differ by at least two. Determine the minimum possible sum of the six numbers on the die.

Answer: 36


Starting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365$. Find $y + d$.

Answer: 317


$P(x)$ is a polynomial of degree at most $6$ such that $P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$, and $P(7)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P(8)$?

Answer: 312


Let $f$ be a function satisfying $f(x+y+z) = f(x) + f(y) + f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1) = 1$, $f(2) = 2$. Then $\lim \limits_{n \to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals:

1. 4
2. 6
3. 12
4. 24

Answer: 4


Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x)$, $f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that:

1. When $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and  $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$;
2. When $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$.

Find the maximal value of $M$.

Answer: 8


Andrew and Blair are bored in class and decide to play a game. They pick a pair \((a, b)\) with \(1 \le a, b \le 100\). Andrew says the next number in the geometric series that begins with \(a, b\) and Blair says the next number in the arithmetic series that begins with \(a, b\). For how many pairs \((a, b)\) is Andrew's number minus Blair's number a positive perfect square?

Answer: 281


Sam has 1 Among Us task left. He and his task are located at two randomly chosen distinct vertices of a 2021-dimensional unit hypercube. Let $E$ denote the expected distance he has to walk to get to his task, given that he is only allowed to walk along edges of the hypercube. Compute $\lceil 10E \rceil$.

Answer: 10105


Elfs and trolls are seated at a round table, with a total of 60 creatures. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everyone claims to sit between an elf and a troll, but exactly two elfs made a mistake. How many trolls are there at this table?

Answer: 20


Two distinct squares of the $8 \times 8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$, there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.

Answer: 9


Richard has four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars. (One dollar is 100 cents.) What is the expected value of this final sum of money, in cents?

Answer: 1025


Ted loves creating haikus, which are Japanese three-line poems with syllable patterns of 5, 7, and 5 respectively. However, his vocabulary is limited. Specifically, for integers $1 \le n \le 7$, he knows $2n$ words that each contain $n$ syllables. Words cannot cross between lines but may be repeated.

If Ted can create $N$ distinct haikus, find the remainder when $N$ is divided by $100$.

Answer: 28


Given the function $f(x)=k x^{2}+k x+k+3$, and for any $x \in[-2,3]$, $f(x) \geqslant 0$ always holds, find the range of real number $k$.

To find the range of $k$ for which $f(x) \geqslant 0$ for all $x \in[-2,3]$, we need to analyze the function $f(x)=k x^{2}+k x+k+3$.

1. **Determine the vertex of the parabola:**
   The function $f(x)$ is a quadratic function, and its graph is a parabola. The vertex of the parabola can be found using the formula $x = -\frac{b}{2a}$, where $a = k$ and $b = k$. Thus, the vertex is at:
   \[
   x = -\frac{k}{2k} = -\frac{1}{2}
   \]

2. **Evaluate $f(x)$ at the vertex:**
   Substitute $x = -\frac{1}{2}$ into $f(x)$:
   \[
   f\left(-\frac{1}{2}\right) = k \left(-\frac{1}{2}\right)^2 + k \left(-\frac{1}{2}\right) + k + 3 = k \cdot \frac{1}{4} - k \cdot \frac{1}{2} + k + 3 = \frac{k}{4} - \frac{k}{2} + k + 3 = \frac{k}{4} + \frac{2k}{4} + 3 = \frac{3k}{4} + 3
   \]
   For $f(x) \geqslant 0$ at the vertex, we need:
   \[
   \frac{3k}{4} + 3 \geqslant 0 \implies \frac{3k}{4} \geqslant -3 \implies 3k \geqslant -12 \implies k \geqslant -4
   \]

3. **Evaluate $f(x)$ at the endpoints of the interval $[-2, 3]$:**
   - At $x = -2$:
     \[
     f(-2) = k(-2)^2 + k(-2) + k + 3 = 4k - 2k + k + 3 = 3k + 3
     \]
     For $f(-2) \geqslant 0$, we need:
     \[
     3k + 3 \geqslant 0 \implies 3k \geqslant -3 \implies k \geqslant -1
     \]

   - At $x = 3$:
     \[
     f(3) = k(3)^2 + k(3) + k + 3 = 9k + 3k + k + 3 = 13k + 3
     \]
     For $f(3) \geqslant 0$, we need:
     \[
     13k + 3 \geqslant 0 \implies 13k \geqslant -3 \implies k \geqslant -\frac{3}{13}
     \]

4. **Combine the conditions:**
   The conditions from the vertex and the endpoints are:
   \[
   k \geqslant -4, \quad k \geqslant -1, \quad k \geqslant -\frac{3}{13}
   \]
   The most restrictive condition is $k \geqslant -\frac{3}{13}$.

Therefore, the range of $k$ is:
\[
\boxed{k \geqslant -\frac{3}{13}}
\]

Answer: [-\frac{3}{13}, \infty)


5. Seven balls of different colors are placed into three boxes numbered 1, 2, and 3. It is known that the number of balls in each box is not less than its number. The number of different ways to place the balls is $\qquad$

Answer: 455


Natural numbers $1, 2, 3, \cdots, 2014, 2015, 2016$ are arranged in a circle in clockwise order.
Starting from number 1, we perform the following operations in clockwise order.

Step 1: Cross out 1 and keep 2.

Step 2: Cross out 3 and 4, and keep 5.

Step 3: Cross out 6, 7, and 8, and keep 9.

Step 4: Cross out 10, 11, 12, and 13, and keep 14;

and so on: in the nth step, first cross out n numbers, then keep 1 number. Repeat these operations until all numbers are crossed out.

What is the last number to be crossed out?

Answer: 2015


10.1. Find the sum of all such integers $a \in[0 ; 400]$, for each of which the equation $x^{4}-6 x^{2}+4=\sin \frac{\pi a}{200}-2\left[x^{2}\right]$ has exactly six roots. Here the standard notation is used: $[t]$ - the integer part of the number $t$ (the greatest integer not exceeding $t$).

Answer: 60100


A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + b}{c \pi}$ for some positive integers $a, b, c$ with $a$ and $c$ relatively prime. Find $a+b+c$.

Answer: 34


Three. (20 points) In the pyramid $S-ABC$, $SA=4, SB \geqslant 7, SC \geqslant 9, AB=5, BC \leqslant 6, AC \leqslant 8$. Try to find the maximum volume of the pyramid $S-ABC$.

Answer: 8\sqrt{6}


4A. Find the largest natural number that is less than the sum of the squares of its digits.

Answer: 99


11. A farmer has 100 pigs and 100 chickens. He has four adjacent square yards, forming a $2 \times 2$ grid. The farmer wants to distribute the livestock among the yards according to the following requirements: the first row has 120 heads, the second row has 300 feet; the first column has 100 heads, the second column has 320 feet. Then there are $\qquad$ different ways to distribute them.

Answer: 341


A square is inscribed in an equilateral triangle, with each vertex of the square on the perimeter of the triangle. One side of the square cuts off a smaller equilateral triangle; into this, we again inscribe a square in the same manner, and so on. What fraction of the triangle is covered by the infinite sequence of squares?

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


3. The area of $\triangle A B C$ is $S, \angle A=45^{\circ}$, line $M N$ divides the area of $\triangle A B C$ into two equal parts, and $M$ is on $A B$, $N$ is on $A C$. Then the shortest length of $M N$ is $\qquad$

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


10. As shown in the figure, two squares $\mathrm{ABEG}, \mathrm{GECD}$, point $\mathrm{H}$ is the midpoint of $\mathrm{GE}$, $\frac{D F}{D C}=\frac{1}{3}$. Connect $\mathrm{DH}$, $\mathrm{CH}$, $\mathrm{AF}$, $\mathrm{BF}$. The area of square $\mathrm{ABEG}$ is $\mathrm{m}$ square centimeters, and the area of the shaded part is $\mathrm{n}$ square centimeters. Given that $\mathrm{m}$ and $\mathrm{n}$ are positive integers, and $\mathrm{m}$ has 9 divisors, then the side length of square $\mathrm{ABEG}$ is $\qquad$ centimeters.

Answer: 10


The third question: An exam consists of $m$ questions, with $n$ students participating, where $m, n \geqslant 2$ are given integers. The scoring rule for each question is: if exactly $x$ students fail to answer the question correctly, then each student who answers the question correctly gets $x$ points, and those who fail to answer correctly get zero points. Each student's total score is the sum of their scores on the $m$ questions. The total scores of all students are arranged in descending order as $p_{1} \geqslant p_{2} \geqslant \cdots \geqslant p_{n}$. Find the maximum possible value of $p_{1}+p_{\mathrm{n}}$.

Answer: m(n - 1)


449. Ten Barrels. A merchant had 10 barrels of sugar, which he arranged into a pyramid as shown in the figure. Each barrel, except one, was marked with its own number. It turned out that the merchant had accidentally placed the barrels in such a way that the sum of the numbers along each row was 16.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-157.jpg?height=500&width=580&top_left_y=327&top_left_x=744)

Could you rearrange the barrels so that the sum of the numbers along each row equals the smallest possible number? Of course, the central barrel (which happened to be barrel number 7 in the figure) does not count in the sum.

Answer: 13


30. 20 identical balls are placed into 4 identical boxes, with no box left empty. How many ways are there to do this? If empty boxes are allowed, how many ways are there?

Answer: 64


## Task A-3.2.

Determine all pairs of natural numbers $x$ and $y$ for which $\frac{x y^{2}}{x+y}$ is a prime number.

Answer: (2, 2)


4. In acute triangle $A B C, M$ and $N$ are the midpoints of sides $A B$ and $B C$, respectively. The tangents to the circumcircle of triangle $B M N$ at $M$ and $N$ meet at $P$. Suppose that $A P$ is parallel to $B C$, $A P=9$ and $P N=15$. Find $A C$.

Answer: 20\sqrt{2}


For every \( n \in \mathbb{N} \) let \( d(n) \) denote the number of (positive) divisors of \( n \). Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) with the following properties:
(i) \( d(f(x))=x \) for all \( x \in \mathbb{N} \);
(ii) \( f(x y) \) divides \( (x-1) y^{x y-1} f(x) \) for all \( x, y \in \mathbb{N} \).

Answer: f(n) = \prod_{p \mid n} p^{p^{k_p} - 1}


Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.

Answer: n^2


## Task B-1.1.

How many ordered pairs of natural numbers $(a, b)$ are there such that $a+b=1000$ and neither of the numbers $a$ and $b$ contains the digit 0 in their decimal representation?

Answer: 738


12. (20 points) As shown in Figure 2, given $k>0$ and $k \neq 1$, the line $l: y=k x+1$ is symmetric to the line $l_{1}: y=k_{1} x+1$ with respect to the line $y=x+1$. The lines $l$ and $l_{1}$ intersect the ellipse $E$: $\frac{x^{2}}{4}+y^{2}=1$ at points $A, M$ and $A, N$ respectively.
(1) Find the value of $k k_{1}$;
(2) Prove that for any $k$, the line $M N$ always passes through a fixed point.

Answer: 1


II. A sequence of numbers where all terms are 0 or 1 is called a 0,1 sequence. Let $A$ be a finite 0,1 sequence, and let $f(A)$ denote the 0,1 sequence obtained by changing each 1 in $A$ to 0,1 and each 0 to 1,0. For example, $f((1,0,0,1))=(0,1,1,0,1,0,0,1)$, and let $f^{n}(A)$ denote $f(f(\cdots f(A) \cdots))(n$ times $f)$.
Try to find: In $f^{n}((1))$, how many pairs of consecutive terms are 0?

---

The translation preserves the original text's formatting and line breaks.

Answer: \dfrac{2^{n-1} + (-1)^n}{3}


3. As shown in Figure 3, in isosceles $\triangle ABC$, $AB = AC$, and $P$, $Q$ are points on sides $AC$, $AB$ respectively, such that $AP = PQ = QB = BC$. Then $\angle PCQ =$ $\qquad$

Answer: 30^\circ


9.5. What is the maximum number of integers that can be written in a row so that the sum of any five consecutive ones is greater than zero, and the sum of any seven consecutive ones is less than zero?

Answer: 10


## Task 3 - 330833

For every triangle $ABC$, the following notations are introduced:

The line through $A$ and $B$ is $u$, the line through $B$ and $C$ is $v$, the line through $C$ and $A$ is $w$. When $u$ is reflected across $v$, it becomes the line $p$. When $u$ is reflected across $w$, it becomes the line $q$.

As usual, the measures of the interior angles $\angle BAC$ and $\angle ABC$ are denoted by $\alpha$ and $\beta$, respectively. For the following tasks, it is always assumed that $\alpha = 55^\circ$.

a) At what angle do the lines $p$ and $q$ intersect if $\beta = 75^\circ$?

b) How large must $\beta$ be for $p$ and $q$ to be perpendicular to each other?

c) Give a value of $\beta$ such that $p$ and $q$ can be shown to be parallel to each other!

d), e) Create a drawing in which $p$ and $q$ intersect at a point on the same side of the line $u$ as $C$; choose the triangle $ABC$

d) with an acute interior angle at $B$.

e) with an obtuse interior angle at $B$.

(For d) and e), no justification is required.)

Answer: 80^\circ


6.2. There are 7 safes and 7 codes for them, but it is unknown which code belongs to which safe. What is the minimum number of attempts required to guarantee matching the codes to the safes?

Answer: 21


Dona Lígia has a square plot of land. She decides to divide it into five regions, consisting of four rectangles and one square as illustrated in the figure below:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-20.jpg?height=416&width=417&top_left_y=503&top_left_x=725)

$\mathrm{In}$ the figure above, we have that:

- The central square has an area of $64 \mathrm{~m}^{2}$;
- The longer sides of the four rectangles have the same length;
- The five regions have the same perimeter.

Determine the area of Dona Lígia's plot of land.

Answer: 256


The neighbor's bubbling spring fills its clear water

We three Eroses fill - see - this basin.

I with my wings. This one with a jug. The other, through the tip of the arrow.

"The sun needs no more than a sixth,

To fill me - spoke - with a gentle murmur the winged one."

"I need four hours and I will fill this jug"

"I can fill this basin from my arrow in half a day"

Then spoke the Jug, - and the Archer as well

"If we all fill it at once - said the friend -

How long will it take for the basin to be full?"

Answer: \dfrac{12}{7}


Yatsenko I.V.

In the elections to the 100-seat parliament, 12 parties participated. Parties that received strictly more than $5\%$ of the voters' votes enter the parliament. Among the parties that entered the parliament, seats are distributed proportionally to the number of votes they received. After the elections, it turned out that each voter voted for exactly one party (there were no invalid ballots, votes "against all" and the like) and each party received an integer number of seats. At the same time, the Party of Mathematics Enthusiasts received 25% of the votes. What is the maximum number of seats in the parliament that it could have received?

Answer: 50


8. Given that a quadratic curve is tangent to the line $x-2 y+3=0$ at point $P(-1,1)$, tangent to the line $5 x+2 y-6=0$ at point $Q(2,-2)$, and passes through point $R(1,0)$. Find the equation of this quadratic curve.

Answer: x^2 + x + 2y - 2 = 0


Mr. Kutil wanted to paint star ornaments on 20 tiles in the bathroom. On the can of paint, it was written that the paint would cover $750 \mathrm{~cm}^{2}$. How many cans of paint did Mr. Kutil have to buy at least, if one square of the grid has an area of $1 \mathrm{~cm}^{2}$? The ornament on one tile is shown in the picture.

(L. Hozová)

![](https://cdn.mathpix.com/cropped/2024_04_17_151e4384192b9f5c88c8g-1.jpg?height=454&width=448&top_left_y=752&top_left_x=838)

Answer: 1


A regular tetrahedron with 2 unit edges has two of its vertices on the circumference of the base and the top of a right circular cylinder. What is the volume of the cylinder?

Answer: \sqrt{2} \pi


In an acute triangle \(ABC\), heights \(CH\) and \(AH_1\) are drawn. It is known that \(AC = 2\), and the area of the circle circumscribed around triangle \(HBH_1\) is \(\pi / 3\). Find the angle between height \(CH\) and side \(BC\).

Answer: 30^\circ


6.33 8 people participate in a chess round-robin tournament, where the winner gets 1 point, a draw gets 0.5 points, and the loser gets 0 points. It is known that the 8 people's scores are all different, and the second-place player's score is the sum of the scores of the last 4 players. What was the result of the match between the third-place player and the seventh-place player?

Answer: 1


There is an unlimited supply of test tubes of three types - A, B, and C. Each test tube contains one gram of solution of the same substance. Test tubes of type A contain a 10% solution of the substance, type B contain a 20% solution, and type C contain a 90% solution. Sequentially, one by one, the contents of the test tubes are poured into a container. At the same time, test tubes of the same type cannot be used consecutively. It is known that a 20.17% solution was obtained in the container by performing the minimum number of pourings. What is the maximum number of type C test tubes that can be used in this process?

Answer: 73


A rectangular prism has one of its diagonals $A A^{\prime}$. From vertex $A$, we draw a plane through the endpoints of the edges $(B, C, D)$. Into what parts does this plane divide $A A^{\prime}$?

Answer: 1:2


(The 2021 ICO P4)

The path index of a graph $G$ is the minimum number of paths needed to pass through each vertex of $G$ exactly once. Given a connected graph $G$, what is the maximum possible value for its path index, knowing that the largest set of vertices in $G$ that are pairwise non-adjacent is $n>1$ (independence number)?

Answer: n-1


Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a)  n=2019$ ;  $b)  n=2020$.

Answer: 1009


$\underline{\text { by Banov I.S }}$

The height and the radius of the base of a cylinder are equal to 1. What is the smallest number of spheres with a radius of 1 that can completely cover this cylinder?

Answer: 3


12.320. The lateral faces of a regular triangular prism are squares. Find the angle between the diagonal of a lateral face and a non-intersecting side of the base of the prism.

Answer: \arctan{\sqrt{7}}


11. 6 teams participate in a round-robin tournament (each team plays one match against each of the other teams), with 3 points awarded for a win, 1 point for a draw, and 0 points for a loss. In the end, the 6 teams have different points, the first and second place teams differ by 4 points, the fourth and fifth place teams also differ by 4 points, and the third place team won 2 matches, while the sixth place team drew 2 matches. Therefore, the points of the third place team are $\qquad$ points.

Answer: 8


6. Let $a>1, m>p>0$. If the solution to the equation $x+\log _{a} x=m$ is $p$. Then the solution to the equation $x+a^{x}=m$ is

Answer: m - p


9.2. Can the products of all non-zero digits of two consecutive natural numbers differ by exactly 54 times?

Answer: Yes


8. Given $x \leqslant-\frac{1}{2}$. Then the minimum value of the bivariate function
$$
\begin{array}{l}
f(x, y) \\
=\sqrt{x^{2}+y^{2}}+\sqrt{x^{2}-2 x+2}+\sqrt{y^{2}-2 y+5}
\end{array}
$$

is . $\qquad$

Answer: \dfrac{\sqrt{13} + \sqrt{29}}{2}


3. As shown in Figure 4, in the "dart-shaped" quadrilateral $ABCD$, $AB=4\sqrt{3}$, $BC=8$, $\angle A=\angle B=\angle C=30^{\circ}$. Then the distance from point $D$ to $AB$ is $\qquad$

Answer: 1


# Task № 3.3

## Condition:

Alina makes phone cases on order for a tech store. Each case has a pattern and a charm.

The case can be silicone, leather, or plastic. Alina has charms in the shapes of a bear, a dinosaur, a raccoon, and a fairy, and she can draw the moon, the sun, and clouds on the case.

Alina is only satisfied when three cases are laid out in a row from left to right on the display according to the following rules:

- There must be a silicone case with a bear charm;
- Next to the case with the bear charm, there must be a leather case on the left and a plastic case on the right;
- The three cases in the row must have different materials, charms, and patterns.

How many ways are there to make Alina happy?

Answer: 36


25. Arrange the numbers $0,0,1,1,2,2,4,4$ in a row (0 can be placed first), so that no four consecutive numbers form 2014. There are $\qquad$ ways.

Answer: 2401


A square steel plate has had a rectangular piece of width 3 cm and another rectangular piece of width 5 cm cut off (as shown in the figure). The total area removed is 81 square centimeters. What was the area of the original square? $\qquad$ square centimeters.

Answer: 144


From letter cards, the word "КАРАКАТИЦА" can be formed. How many different words (not necessarily meaningful) can be formed from these cards where the letters "Р" and "Ц" are adjacent?

Answer: 15120


In this problem assume \( s_{1} = 3 \) and \( s_{2} = 2 \). Determine, with proof, the nonnegative integer \( k \) with the following property:
1. For every board configuration with strictly fewer than \( k \) blank squares, the first player wins with probability strictly greater than \( \frac{1}{2} \); but
2. There exists a board configuration with exactly \( k \) blank squares for which the second player wins with probability strictly greater than \( \frac{1}{2} \).

Answer: 3


In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numbers possible?

Answer: 1008


Two straight roads intersect perpendicularly. On one road, a car starts from a distance of $l_{1}=80 \mathrm{~km}$ from the intersection and heads towards the intersection at a constant speed of $v_{1}=80 \mathrm{~km/h}$. On the other road, another car starts at the same time from a distance of $l_{2}=60 \mathrm{~km}$ from the intersection, traveling towards the intersection at a speed of $v_{2}=60 \mathrm{~km/h}$. How fast is the distance between the cars changing? What is the situation if the initial distances are $l_{1}=l_{2}^{\prime}=80 \mathrm{~km}$? How much time after departure will the two cars be closest to each other, and what is this distance in both cases?

Answer: 16


Three similar triangles \(PAB\), \(AQB\), and \(ABR\) are constructed on the same side of line segment \(AB\). Similarly, three similar triangles \(P'AB\), \(A Q'B\), and \(ABR'\) are constructed symmetrically about the perpendicular bisector of \(AB\). Prove that the six points \(P, Q, R, P', Q'\) and \(R'\) lie on the same circle.

Answer: 6


Rectangle \( W X Y Z \) has \( W X = 4 \), \( W Z = 3 \), and \( Z V = 3 \). The rectangle is curled without overlapping into a cylinder so that sides \( W Z \) and \( X Y \) touch each other. In other words, \( W \) touches \( X \) and \( Z \) touches \( Y \). The shortest distance from \( W \) to \( V \) through the inside of the cylinder can be written in the form \( \sqrt{\frac{a+b \pi^{2}}{c \pi^{2}}} \) where \( a, b \) and \( c \) are positive integers. The smallest possible value of \( a+b+c \) is
(A) 12
(B) 26
(C) 18
(D) 19
(E) 36

Answer: 18


All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.

Answer: 111112


Let triangle \( A B C \) have incircle \(\omega\), which touches \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\), respectively. Then, let \(\omega_{1}\) and \(\omega_{2}\) be circles tangent to \(AD\) and internally tangent to \(\omega\) at \(E\) and \(F\), respectively. Let \(P\) be the intersection of line \(EF\) and the line passing through the centers of \(\omega_{1}\) and \(\omega_{2}\). If \(\omega_{1}\) and \(\omega_{2}\) have radii 5 and 6, respectively, compute \(PE \cdot PF\).

Answer: 3600


In the diagram below, points $E$ and $F$ are located on sides $AB$ and $BD$ of triangle $\triangle ABD$ such that $AE = AC$ and $CD = FD$. If $\angle ABD = 60^\circ$, determine the measure of angle $\angle ECF$.

Answer: 60


It is known that the number $2^{333}$ has 101 digits and starts with the digit 1. How many numbers in the series $2, 4, 8, 16, \ldots, 2^{333}$ start with the digit 4?

Answer: 33


An engineer named Paul received for examination an object with a volume of approximately 100 oil samples (a container designed for 100 samples, which was almost full). Each sample is categorized based on sulfur content—either low-sulfur or high-sulfur, and density—either light or heavy. The relative frequency (statistical probability) that a randomly chosen sample is heavy oil is $\frac{1}{7}$. Additionally, the relative frequency that a randomly chosen sample is light low-sulfur oil is $\frac{9}{14}$. How many total samples of high-sulfur oil are there in the object if none of the heavy oil samples are low-sulfur?

Answer: 35


For an integer \( x \geq 1 \), let \( p(x) \) be the least prime that does not divide \( x \), and define \( q(x) \) to be the product of all primes less than \( p(x) \). In particular, \( p(1)=2 \). For \( x \) such that \( p(x)=2 \), define \( q(x)=1 \). Consider the sequence \( x_{0}, x_{1}, x_{2}, \ldots \) defined by \( x_{0}=1 \) and

\[ x_{n+1}=\frac{x_{n} p(x_{n})}{q(x_{n})} \]

for \( n \geq 0 \). Find all \( n \) such that \( x_{n}=1995 \).

Answer: 142


Four statements were made about triangle \(ABC\), and we know that out of these statements, two are true and two are false. What could be the perimeter of triangle \(ABC\) if \(BC = 1\)?

I. Triangle \(ABC\) is a right triangle.
II. The angle at \(A\) is \(30^\circ\).
III. \(AB = 2 \times BC\).
IV. \(AC = 2 \times BC\).

Answer: 3+\sqrt{5}


As illustrated, to set up a "gesture password" on a mobile phone in a $2 \times 2$ grid containing 9 points, start at one point and consecutively connect multiple points with line segments. Each segment cannot pass through any unused points, and the endpoints of each segment cannot both be previously used points. If a person's gesture password starts at the center point and uses only three points, how many connection methods are there?

Answer: 48


The numbers from 1 to 36 can be written in a counterclockwise spiral. In this spiral, all the terms on the diagonal beginning from the upper left corner sum up to 76. Suppose this spiral is continued all the way until 2015. Find the sum of all the terms on the diagonal beginning from the upper left corner of the resulting (incomplete) square.

Answer: 30405