9. In $\triangle A B C$, $\angle A<\angle B<\angle C$, $\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\sqrt{3}$.
Then $\angle B=$

Answer: \dfrac{\pi}{3}


5. In a $7 \times 7$ unit square grid, there are 64 grid points, and there are many squares with these grid points as vertices. How many different values can the areas of these squares have?
(21st Jiangsu Province Junior High School Mathematics Competition)

Answer: 18


2、The following is a calendar for May of a certain year. Using a $2 \times 2$ box that can frame four numbers (excluding Chinese characters), the number of different ways to frame four numbers is $\qquad$.

---

Note: The original text includes a placeholder for the answer (indicated by $\qquad$). This has been preserved in the translation.

Answer: 20


5. Call two vertices of a simple polygon "visible" to each other if and only if they are adjacent or the line segment connecting them lies entirely inside the polygon (except for the endpoints which lie on the boundary). If there exists a simple polygon with $n$ vertices, where each vertex is visible to exactly four other vertices, find all possible values of the positive integer $n$.

Note: A simple polygon is one that has no holes and does not intersect itself.

Answer: 5


2. Initially, the numbers 2, 3, and 4 are written on the board. Every minute, Anton erases the numbers written on the board and writes down their pairwise sums instead. After an hour, three enormous numbers are written on the board. What are their last digits? List them in any order, separated by semicolons.

Example of answer format:

$1 ; 2 ; 3$

Answer: 7 ; 8 ; 9


12. (18 points) The inverse function of $f(x)$ is $y=\frac{x}{1+x}$, $g_{n}(x)+\frac{1}{f_{n}(x)}=0$, let $f_{1}(x)=f(x)$, and for $n>1\left(n \in \mathbf{N}_{+}\right)$, $f_{n}(x)=f_{n-1}\left(f_{n-1}(x)\right)$. Find the analytical expression for $g_{n}(x)\left(n \in \mathbf{N}_{+}\right)$.

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


Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies
$$2\mid x_1,x_2,\ldots,x_k$$
$$x_1+x_2+\ldots+x_k=n$$
then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.

Answer: 2


Example 6 In the complete quadrilateral CFBEGA, the line containing diagonal $CE$ intersects the circumcircle of $\triangle ABC$ at point $D$. The circle passing through point $D$ and tangent to $FG$ at point $E$ intersects $AB$ at point $M$. Given $\frac{AM}{AB}=t$. Find $\frac{GE}{EF}$ (expressed in terms of $t$).

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


17. In $\triangle A B C, A C>A B$, the internal angle bisector of $\angle A$ meets $B C$ at $D$, and $E$ is the foot of the perpendicular from $B$ onto $A D$. Suppose $A B=5, B E=4$ and $A E=3$. Find the value of the expression $\left(\frac{A C+A B}{A C-A B}\right) E D$.

Answer: 3


Consider the polynomial \(x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\) with integer coefficients. Determine all possible values for \(m\), which represents the exact number of integer roots of the polynomial, counting multiplicity.

Answer: 0, 1, 2, 3, 4, 5


A square and a right triangle are adjacent to each other, with each having one side on the $x$-axis. The lower right vertex of the square and the lower left vertex of the triangle are at $(12, 0)$. They share one vertex, which is the top right corner of the square and the hypotenuse-end of the triangle. The side of the square and the base of the triangle on the x-axis each equal $12$ units. A segment is drawn from the top left vertex of the square to the farthest vertex of the triangle, creating a shaded area. Calculate the area of this shaded region.

Answer: 36 \text{ square units}


Consider a truncated pyramid formed by slicing the top off a regular pyramid. The larger base now has 6 vertices, and the smaller top base is a regular hexagon. The ant starts at the top vertex of the truncated pyramid, walks randomly to one of the six adjacent vertices on the larger base, and then from there, it walks to one of the six adjacent vertices on the smaller top base. What is the probability that this final vertex is the initially started top vertex? Express your answer as a common fraction.

Answer: \frac{1}{6}


The planet Zephyr moves in an elliptical orbit with its star at one of the foci. At its nearest point (perihelion), it is 3 astronomical units (AU) from the star, while at its furthest point (aphelion) it is 15 AU away. Determine the distance of Zephyr from its star when it is exactly halfway through its orbit.

Answer: 9 \text{ AU}


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 2cf(x)y\]
for all real numbers $x$ and $y$, where $c$ is a constant. Determine all possible values of $f(2)$, find the sum of these values, and calculate the product of the number of possible values and their sum.

Answer: 8


Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote $\sin x$, $\tan x$, and $\sec x$ on three different cards. He then distributed these cards to three students: Isabelle, Edgar, and Sam, one card to each. After sharing the values on their cards with each other, only Isabelle could accurately identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Isabelle's card.

Answer: 1


The planet Zytron orbits in an elliptical path around its star with the star located at one of the foci of the ellipse. Zytron’s closest approach to its star (perihelion) is 3 astronomical units (AU), and its furthest point (aphelion) is 8 AU away. How far is Zytron from its star when it is exactly halfway in its orbit along the major axis?

Answer: 5.5 \text{ AU}


A quadrilateral with one pair of parallel sides (trapezoid) has its consecutive angles in an arithmetic sequence. The sum of the smallest and largest angles equals $200^{\circ}$. If the second-smallest angle is $70^{\circ}$, find the measure of the third-largest angle.

Answer: 130^\circ


Utilizing the twelve-letter alphabet of the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, S, T, U, V), imagine the scenario for generating six-letter license plates. How many valid six-letter license plates can be formed that start with either A or E, end with R, cannot contain the letter V, and have no repeating letters?

Answer: 6048


Alice throws a fair eight-sided die each morning. If Alice throws a perfect square number, she chooses porridge for breakfast. For any prime numbers, she chooses toast. If she throws an 8, she will throw the die again. How many times can Alice be expected to throw her die in a leap year?

Answer: \frac{2928}{7}


At a sandwich shop, there are five types of bread, seven kinds of meat, and six kinds of cheese. A sandwich is made up of one type of bread, one type of meat, and one type of cheese. Turkey, salami, Swiss cheese, and rye bread are available at the shop. If a customer does not order a sandwich with a turkey/Swiss cheese combination nor a sandwich with rye bread/salami, how many different sandwiches can the customer order?

Answer: 199


Rosa received a perfume bottle in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After two weeks of using the perfume, $0.45$ liters remained in the bottle. What fraction represents the volume of perfume that Rosa has already used?

Answer: \frac{49\pi-45}{49\pi}


Several circles are depicted in the picture, connected by line segments. Tanya selects a natural number \( n \) and arranges different natural numbers in the circles so that the following property holds for all these numbers:

If the numbers \( a \) and \( b \) are not connected by a line segment, then the sum \( a^{2}+b^{2} \) must be coprime with \( n \). If they are connected, the numbers \( a^{2}+b^{2} \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?

Answer: 65


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: \arccos\frac{\sqrt{2}}{4}


Let \(\omega_{1}, \omega_{2}, \ldots, \omega_{100}\) be the roots of \(\frac{x^{101}-1}{x-1}\) (in some order). Consider the set
\[
S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}.
\]
Let \(M\) be the maximum possible number of unique values in \(S\), and let \(N\) be the minimum possible number of unique values in \(S\). Find \(M-N\).

Answer: 98


Every week, based on listeners' SMS votes, the ten most popular songs are selected. It is known that: 1) the same set of songs in the same order is never selected two weeks in a row; 2) a song that has once dropped in the ranking does not climb up again. What is the maximum number of weeks that the same 10 songs can remain in the ranking?

Answer: 46


In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. You are playing a version of this game with only two kinds of cards: planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card held. Starting with no planet cards and no hegemon cards, on each turn (starting at turn one), you take either a planet card or a hegemon card, whichever is worth more points given your current hand. Define a sequence \(\{a_{n}\}\) for all positive integers \(n\) by setting \(a_{n}\) to be 0 if on turn \(n\) you take a planet card and 1 if you take a hegemon card. What is the smallest value of \(N\) such that the sequence \(a_{N}, a_{N+1}, \ldots\) is necessarily periodic (meaning that there is a positive integer \(k\) such that \(a_{n+k}=a_{n}\) for all \(n \geq N\))?

Answer: 503


Let \( P(n) = \left(n-1^3\right)\left(n-2^3\right) \ldots\left(n-40^3\right) \) for positive integers \( n \). Suppose that \( d \) is the largest positive integer that divides \( P(n) \) for every integer \( n > 2023 \). If \( d \) is a product of \( m \) (not necessarily distinct) prime numbers, compute \( m \).

Answer: 48


I5.1 In Figure 2, find \( a \).

I5.2 If \( b = \log_{2} \left( \frac{a}{5} \right) \), find \( b \).

I5.3 A piece of string, \( 20 \text{ m} \) long, is divided into 3 parts in the ratio of \( b-2 : b : b+2 \). If \( N \text{ m} \) is the length of the longest portion, find \( N \).

I5.4 Each interior angle of an \( N \)-sided regular polygon is \( x^{\circ} \). Find \( x \).

Answer: 144


A square and four smaller squares are contained within it, each smaller square circumscribing a circle with a radius of 3 cm. What is the area of the larger square and the total area occupied by the four smaller squares?

Answer: 144 \text{ cm}^2


Consider the polynomial $g(x) = x^{2009} + 19x^{2008} + 1$ with distinct roots $s_1, s_2, ..., s_{2009}$. A polynomial $Q$ of degree $2009$ has the property that
\[ Q\left(s_j + \dfrac{1}{s_j}\right) = 0 \] for $j = 1, \ldots, 2009$. Determine the value of $\frac{Q(1)}{Q(-1)}$.

Answer: 1


In a basketball league, the number of games won by five teams is displayed in a graph, but the team names are missing. The clues below provide information about the teams:

1. The Sharks won more games than the Falcons.
2. The Knights won more games than the Wolves, but fewer than the Dragons.
3. The Wolves won more than 22 games.

Given the bar graph data: 24 wins, 27 wins, 33 wins, 36 wins, 38 wins. How many games did the Knights win?

Answer: 33


In the Zorgian language, there are 4 words: "zor", "glib", "mek", and "troz". In a sentence, "zor" cannot come directly before "glib", and "mek" cannot come directly before "troz"; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Zorgian?

Answer: 48


Consider the polynomial equation \[ x^4 + ax^3 + bx^2 + cx + d = 0, \] where \( a, b, c, \) and \( d \) are rational numbers. This polynomial has roots \( 2-\sqrt{5} \) and \( 1 \). Find the value of \( c \) if the polynomial also has a root that is three times an integer.

Answer: -8


In the right-angled triangle $LMN$, suppose $\sin N = \frac{3}{5}$. If the length of $LM$ is 15, calculate the length of $LN$.

Answer: 25


The score on a certain 150-point test varies directly with the square of the time a student spends preparing for the test. If a student receives 90 points on a test for which she spent 2 hours preparing, what score would she receive on the next test if she spent 3 hours preparing?

Answer: 202.5


In triangle $PQR$, $PQ = PR$, and $\overline{PS}$ is an altitude. Point $T$ is on the extension of $\overline{QR}$ such that $PT = 12$. The values of $\tan \angle RPT$, $\tan \angle SPT$, and $\tan \angle QPT$ form a geometric progression, and the values of $\cot \angle SPT$, $\cot \angle RPT$, $\cot \angle SPQ$ form an arithmetic progression. Determine the area of triangle $PQR$.

Answer: 24


Humanity discovers 15 habitable planets, of which 7 are Earth-like and 8 are Mars-like. Earth-like planets require 3 units of colonization effort, whereas Mars-like planets need only 1 unit. If a total of 18 units of colonization effort is available, how many different combinations of planets can be colonized, given that all planets are unique?

Answer: 2163


Suppose in a right triangle $PQR$, with $\angle PQR = 90^\circ$, we have $\cos Q = \frac{3}{5}$. If $PR = 5$, find the length of $PQ$.

Answer: 3


14. As shown in the figure, in acute $\triangle A B C$, $A C=B C=10$, and $D$ is a point on side $A B$. The incircle of $\triangle A C D$ and the excircle of $\triangle B C D$ that is tangent to side $B D$ both have a radius of 2. Find the length of $A B$.

 untranslated text remains the same as the source text.

Answer: 4\sqrt{5}


Through the altitude of an equilateral triangle with side length \(a\), a plane is drawn perpendicular to the plane of the triangle, and within this plane a line \(l\) is taken parallel to the altitude of the triangle. Find the volume of the solid obtained by rotating the triangle around the line \(l\).

Answer: \dfrac{\sqrt{3}}{24} \pi a^3


Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.

Answer: 2


![](https://cdn.mathpix.com/cropped/2024_05_06_7adbd1f2e5e559c8684fg-19.jpg?height=186&width=2015&top_left_y=1598&top_left_x=17)

$ABCD$ - a convex quadrilateral. It is known that $\angle CAD = \angle DBA = 40^{\circ}, \angle CAB = 60^{\circ}, \angle CBD = 20^{\circ}$. Find the angle $\angle CDB$.

Answer: 30^{\circ}


23. The birth date of Albert Einstein is 14 March 1879. If we denote Monday by 1, Tuesday by 2, Wednesday by 3, Thursday by 4, Friday by 5, Saturday by 6 and Sunday by 7, which day of the week was Albert Einstein born? Give your answer as an integer from 1 to 7 .

Answer: 5


Let $\frac{3}{2} \leqslant x \leqslant 5$. Prove the inequality $2 \sqrt{x+1}+\sqrt{2x-3}+\sqrt{15-3x}<2 \sqrt{19}$.

Answer: 2 \sqrt{x+1}+\sqrt{2x-3}+\sqrt{15-3x} < 2 \sqrt{19}


101. The number $N=2^{4 n+1}-4^{n}-1$ is divisible by 9 for an integer $n \geqslant 0$. The number $8 N+9$ is a perfect square.

10

Answer: (4^{n+1} - 1)^2


## Task 2 - 191242

Let $M$ be the set of all square areas $Q$ that lie in a given plane $\epsilon$, have a given point $Z$ of the plane $\epsilon$ as their center, and have a given segment length $a$ as their side length.

For arbitrary square areas $Q, Q^{\prime}$ from this set $M$, let $u\left(Q \cup Q^{\prime}\right)$ denote the perimeter of the polygonal area that results from the intersection of the square areas $Q$ and $Q^{\prime}$.

Investigate whether there are square areas $Q, Q^{\prime}$ in $M$ with the smallest possible $u\left(Q \cap Q^{\prime}\right)$.

If this is the case, determine (in dependence on $a$) this smallest possible value of $u\left(Q \cap Q^{\prime}\right)$.

Answer: 8a(\sqrt{2} - 1)


9.6. What is the smallest natural number $a$ for which there exist integers $b$ and $c$ such that the quadratic trinomial $a x^{2}+b x+c$ has two distinct positive roots, each not exceeding $\frac{1}{1000} ?$

(A. Khryabrov)

Answer: 1001000


11.4. In a row, $n$ integers are written such that the sum of any seven consecutive numbers is positive, and the sum of any eleven consecutive numbers is negative. For what largest $n$ is this possible?

Answer: 16


13 The line $l$ is the left directrix of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, $F_{1}$ is the left focus, and the moving point $P$ is on $l$. The third vertex of the equilateral triangle with side $P F_{1}$ is $M$, and the points $P$, $F_{1}$, and $M$ are arranged counterclockwise. Find the equation of the trajectory of point $M$.

Answer: x - \sqrt{3} y + 7 = 0


3. Find the smallest natural number $k$ such that every $k$-element set of pairwise coprime three-digit numbers contains at least one prime number.

Answer: 12


1. If we consider a pair of skew lines as one pair, then among the 12 lines formed by the edges of a regular hexagonal pyramid, the number of pairs of skew lines is $\qquad$ pairs.

Answer: 24


Given a square with side length \(a\), on each side of the square, a trapezoid is constructed outside such that the top bases and the lateral sides of these trapezoids form a regular dodecagon. Compute its area.

Answer: \dfrac{3}{2} a^2


A natural number $n$ was alternately divided by $29$, $41$ and $59$. The result was three nonzero remainders, the sum of which equals $n$. Find all such $n$

Answer: 79


Task B-3.4. A sphere is circumscribed around a right circular cone. A plane containing the center of this sphere and parallel to the base of the cone divides the cone into two parts of equal volumes. Determine the cosine of the angle at the vertex of the axial section of the cone.

Answer: \sqrt[3]{2} - 1


2.4. Find all values of $x$ for which the smallest of the numbers $8-x^{2}$ and $\operatorname{ctg} x$ is not less than -1. In the answer, record the total length of the found intervals on the number line, rounding it to hundredths if necessary.

Answer: 4.57


Four. (50 points) $n$ points $P_{1}, P_{2}, \cdots, P_{n}$ are sequentially on a straight line, each point is colored one of white, red, green, blue, or purple. If for any two adjacent points $P_{i}, P_{i+1}$ $(i=1,2, \cdots, n-1)$, either these two points are the same color, or at least one of them is white, it is called a good coloring. Find the total number of good colorings.

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


Bakayev E.V.

Cheburashka has a set of 36 stones with masses of 1 g, 2 g, ..., 36 g, and Shapoklyak has super glue, one drop of which can glue two stones together (thus, three stones can be glued with two drops, and so on). Shapoklyak wants to glue the stones in such a way that Cheburashka cannot select one or several stones from the resulting set with a total mass of 37 g. What is the minimum number of drops of glue she needs to achieve her goal?

Answer: 9


Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD=60^\circ$. What is the area of rhombus $BFDE$?

[asy]
pair A,B,C,D,I,F;
A=(0,0);
B=(10,0);
C=(15,8.7);
D=(5,8.7);
I=(5,2.88);
F=(10,5.82);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(D--I--B--F--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",I,W);
label("$F$",F,E);
[/asy]

Answer: 8


A square has sides of length 2.  Set $\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square.  The midpoints of the line segments in set $\cal S$ enclose a region whose area to the nearest hundredth is $k$.  Find $100k$.

Answer: 86


The same eight people sit in a certain church pew every week, but not always in the same order. Every week, each person hugs the people immediately to his or her left and right. How many weeks does it take (at a minimum) for every pair of people to hug at least once?

Answer: 4


In the graph below, each grid line counts as one unit. The line shown below passes through the point $(1001,n)$ (not shown on graph).  Find $n$.
[asy]size(250,0);
add(shift(-10,-10)*grid(20,20));
draw((-10,0)--(10,0),linewidth(2));
draw((0,-10)--(0,10),linewidth(2));
label("x",(10,0),E);
label("y",(0,10),N);
draw((-10,-2.71) -- (10,8.71),blue,Arrows);[/asy]

Answer: 575


Find the last two digits of $\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g., if the last two digits are $05$, just write $5$).

Answer: 20


Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of
\[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\]
where the sum is over all $12$ symmetric terms.

Answer: 112


Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.

Answer: 141


There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$.

In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values?

For example, we can choose $\{\Delta A1, \Delta B2, \odot C3\}$, but we cannot choose $\{\Delta A1, \square B2, \Delta C1\}$.

Answer: 1278


On the blackboard, there are $25$ points arranged in a grid as shown in the figure below. Gastón needs to choose $4$ points that will serve as the vertices of a square. How many different ways can he make this choice?

$$
\begin{matrix}
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet
\end{matrix}
$$

Answer: 50


Let $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the number $\{x + x^2 + \cdots + x^{20}\}$, where $\{a\}$ denotes the fractional part of $a$.

Answer: 41


A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_d x^d + a_{d-1} x^{d-1} + \cdots + a_1 x + a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d > 0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for which $P(\sqrt{2} + \sqrt{3}) = c$.

Answer: 9


Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test consists of $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat answers the $n$th problem correctly with a probability of $\frac{1}{n^2}$, while Q answers the $n$th problem correctly with a probability of $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly.

Just before turning in their test, the ducks and ducklings decide to share answers. For any question on which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores?

Answer: 1020


On Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on January 15th. On how many days in the period from Thursday, January 1, 2015, until (and including) Saturday, December 31, 2016, is it possible for Anna to arrange all her books on all her shelves such that there is an equal number of books on each shelf?

Answer: 89


For every integer $a > 1$, an infinite list of integers is constructed $L(a)$, as follows:

1. $a$ is the first number in the list $L(a)$.
2. Given a number $b$ in $L(a)$, the next number in the list is $b + c$, where $c$ is the largest integer that divides $b$ and is smaller than $b$.

Find all the integers $a > 1$ such that $2002$ is in the list $L(a)$.

Answer: 2002


Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:

1. The quadratic sends integers to integers.
2. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
3. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.

How many quadratics are there with these properties?

Answer: 478


Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$, rounded to the nearest integer?

Answer: 167


For an integer $k$, let $T_k$ denote the number of $k$-tuples of integers $(x_1, x_2, \ldots, x_k)$ with $0 \le x_i < 73$ for each $i$, such that $73 \mid x_1^2 + x_2^2 + \ldots + x_k^2 - 1$. Compute the remainder when $T_1 + T_2 + \ldots + T_{2017}$ is divided by $2017$.

Answer: 2


There are 6 positive integers \( a, b, c, d, e, f \) arranged in order, forming a sequence, where \( a=1 \). If a positive integer is greater than 1, then the number that is one less than it must appear to its left. For example, if \( d > 1 \), then one of \( a, b, \) or \( c \) must have the value \( d-1 \). For instance, the sequence \( 1,1,2,1,3,2 \) satisfies the requirement; \( 1,2,3,1,4,1 \) satisfies the requirement; \( 1,2,2,4,3,2 \) does not satisfy the requirement. How many different permutations satisfy the requirement?

Answer: 203


Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is \( \frac{1}{3} \). When Ryan swings, the probability that he will hit a home run is \( \frac{1}{2} \). In one round, what is the probability that Doug will hit more home runs than Ryan?

Answer: \frac{1}{5}


Let \(a \neq b\) be positive real numbers and \(m, n\) be positive integers. An \(m+n\)-gon \(P\) has the property that \(m\) sides have length \(a\) and \(n\) sides have length \(b\). Further suppose that \(P\) can be inscribed in a circle of radius \(a+b\). Compute the number of ordered pairs \((m, n)\), with \(m, n \leq 100\), for which such a polygon \(P\) exists for some distinct values of \(a\) and \(b\).

Answer: 940


Some \(1 \times 2\) dominoes, each covering two adjacent unit squares, are placed on a board of size \(n \times n\) so that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is 2008, find the least possible value of \(n\).

Answer: 77


One hundred points labeled 1 to 100 are arranged in a \(10 \times 10\) grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels 1 to 10, the second row has labels 11 to 20, and so on).
Convex polygon \(\mathcal{P}\) has the property that every point with a label divisible by 7 is either on the boundary or in the interior of \(\mathcal{P}\). Compute the smallest possible area of \(\mathcal{P}\).

Answer: 63


If \( a \) is the sum of all the positive factors of 2002, find the value of \( a \).

It is given that \( x>0, y>0 \) and \( \sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y}) \). If \( b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y} \), find the value of \( b \).

Given that the equation \(||x-2|-1|=c\) has only 3 integral solutions, find the value of \( c \).

If \( d \) is the positive real root of the equation \(\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2\), find the value of \( d \).

Answer: 2


Let $n$ be a strictly positive integer. Denote by $\mathfrak{S}_{n}$ the set of bijections from the set $\{1,2, \ldots, n\}$ to itself. Let $\sigma \in \mathfrak{S}_{n}$; we say that $(i, j)$ (for $1 \leq i, j \leq n$) is an inversion of $\sigma$ if $i\sigma(j)$. We also say that $i$ (for $1 \leq i \leq n$) is a record of $\sigma$ if for all $j$ with $1 \leq j \leq i-1$ we have $\sigma(j)<\sigma(i)$ (thus, 1 is always a record). Finally, denote $\operatorname{inv}(\sigma)$ the number of inversions of $\sigma$ and $\operatorname{rec}(\sigma)$ its number of records.

Calculate the following sum:

$$
\sum_{\sigma \in \mathfrak{S}_{n}} 2^{\operatorname{inv}(\sigma)+\operatorname{rec}(\sigma)}
$$

Answer: 2^{n(n+1)/2}


Consider two intersecting circles of radius $r$, whose centers are a distance $O_{1} O_{2}=1$ apart. The common chord is bisected at point $O$. A ray starting from $O$ and making an angle $\alpha$ with the common chord intersects the given circles at points $A$ and $B$. What is the length of $A B$?

Answer: \sin \alpha


Pat and Mat dug a well. On the first day, Pat dug a hole $40 \mathrm{~cm}$ deep. The second day, Mat continued and dug to three times the depth. On the third day, Pat dug as much as Mat did the previous day and hit water. At that moment, the ground was $50 \mathrm{~cm}$ above the top of his head.

Determine how tall Pat was.

(M. Dillingerová)

Answer: 150


Let $a>1$ be a positive integer.  The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$.  Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct.

Answer: 46


5. In a row, all natural numbers from 1 to 100 inclusive are written in ascending order. Under each number in this row, the product of its digits is written. The same procedure is applied to the resulting row, and so on. How many odd numbers will be in the fifth row?

Answer: 19


3. If on the interval $[1,2]$, the function $f(x)=x^{2}+p x+q$ and $g(x)=x+\frac{1}{x^{2}}$ take the same minimum value at the same point, then the maximum value of $f(x)$ on this interval is $(\quad)$.

Answer: 4 + \sqrt[3]{4} - \dfrac{5}{2}\sqrt[3]{2}


Consider the natural numbers from 0 to $2^n - 1$. These numbers are divided into two categories: $A$ contains all numbers whose binary representation has an odd number of 1s, and $B$ contains all the other numbers. Find the value of $\sum_{x \in A} x^n - \sum_{y \in B} y^n$.

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


Problem 10.1. Find all real numbers $x$ such that $\tan \left(\frac{\pi}{12}-x\right)$, $\tan \frac{\pi}{12}$ and $\tan \left(\frac{\pi}{12}+x\right)$ form (in some order) a geometric progression.

Answer: x = k\pi


On the side $AB$ of the convex quadrilateral $ABCD$, a point $M$ is chosen such that $\angle AMD = \angle ADB$ and $\angle ACM = \angle ABC$. The tripled square of the ratio of the distance from point $A$ to the line $CD$ to the distance from point $C$ to the line $AD$ is $2$, and $CD = 20$. Find the radius of the inscribed circle of triangle $ACD$.

Answer: 4\sqrt{10} - 2\sqrt{15}


1. Given that $a$, $b$, $c$, and $d$ are prime numbers, and $a b c d$ is the sum of 77 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$

Answer: 32


26. A sphere passes through the vertices of one face of a cube and touches the sides of the opposite face of the cube. Find the ratio of the volumes of the sphere and the cube.

Answer: \dfrac{41\sqrt{41}}{384} \pi


44. As shown in the figure, on the graph of the inverse proportion function $y=\frac{2}{x}(x>0)$, there are points $P_{1}, P_{2}, P_{3}, P_{4}$, whose x-coordinates are $1,2,3,4$ respectively. Perpendicular lines to the x-axis and y-axis are drawn through these points, and the areas of the shaded parts from left to right in the figure are $S_{1}, S_{2}, S_{3}$, respectively. Then $S_{1}+S_{2}+S_{3}=$ $\qquad$ .

Answer: \dfrac{3}{2}


Lynne is tiling her long and narrow rectangular front hall. The hall is 2 tiles wide and 13 tiles long. She is going to use exactly 11 black tiles and exactly 15 white tiles. Determine the number of distinct ways of tiling the hall so that no two black tiles are adjacent (that is, share an edge).

## PART B

For each question in Part B, your solution must be well organized and contain words of explanation or justification when appropriate. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

Answer: 486


For a triangle with angles $\alpha, \beta, \gamma$,

$$
\sin \alpha+\sin \beta=(\cos \alpha+\cos \beta) \sin \gamma
$$

is satisfied. What is the measure of angle $\gamma$?

Answer: 90^\circ


In an ancient cipher known as "Scytale," a strip of papyrus was wound around a cylindrical rod, coil to coil, without gaps or overlaps. Then, with the rod in a horizontal position, the text of the message was written line by line on the papyrus. After that, the strip of papyrus with the written text was sent to the recipient, who had an identical rod, allowing them to read the message. We received a message encrypted using the "Scytale" cipher. However, its author, concerned with ensuring even lines, drew horizontal lines while writing, which remained on the strip in the form of dashes between the letters. The angle of inclination of these dashes to the edge of the strip is $\alpha$, the width of the strip is $d$, and the width of each line is $h$. Indicate how, using the given data, to read the text.

Answer: \frac{d}{h}


1. Two trains, each containing 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after the first cars of the trains met, passenger Sasha, sitting in the third car, passed passenger Valera from the oncoming train, and another 32 seconds later, the last cars of these trains had completely passed each other. In which car, by count, was Valera traveling?

Answer: 12.

Answer: 12


A rope of length 2018 is subjected to the following operation: it is first divided into two ropes of positive integer lengths, and the lengths of the two ropes are recorded. One of these ropes is then selected and the procedure is repeated until there are 2018 ropes each of length 1. If the lengths of the two ropes obtained in a particular operation are not equal, then that operation is called "good".

(1) Find the maximum and minimum number of good operations;
(2) Prove that in all processes in which the number of good operations reaches its minimum, the number of different lengths of the ropes recorded is the same.

Answer: 2016


In the quadrilateral $A B C D$, $A B=C D, A B C \angle=90^{\circ}$ and $B C D \angle=100^{\circ}$. The intersection point of the perpendicular bisectors of sides $A D$ and $B C$ is $M$. Determine the angle $B M C$.

---

The translation is provided as requested, maintaining the original formatting and structure.

Answer: 10^\circ


Two lines intersect at an angle $\gamma$. A grasshopper jumps from one line to the other; the length of each jump is 1 meter, and the grasshopper does not jump back unless it is possible. Prove that the sequence of jumps is periodic if and only if $\gamma / \pi$ is a rational number.

Answer: \gamma / \pi \text{ is a rational number}


6. The base of the pyramid $TABC$ is the triangle $ABC$, all sides of which are equal to $\sqrt{3}$, and the height of the pyramid coincides with the lateral edge $TA$. Find the area of the section of the pyramid by a plane passing through the center of the sphere circumscribed around the pyramid, forming an angle of $60^{\circ}$ with the base plane, intersecting the edge $AB$ at point $M$ such that $MB=2AM$, and intersecting the edge $BC$. It is known that the distance from point $A$ to the section plane is 0.25.

Answer: \dfrac{11\sqrt{3}}{30}


Given a quadratic function \( f(x) = ax^2 + bx + 2 \) (where \( a \neq 0 \)), and for any real number \( x \), it holds that \( |f(x)| \geq 2 \). Find the coordinates of the focus of this parabolic curve.

Answer: \left(0, 2 + \dfrac{1}{4a}\right)


Let $K$ and $N>K$ be fixed positive integers. Let $n$ be a positive integer and let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct integers. Suppose that whenever $m_{1}, m_{2}, \ldots, m_{n}$ are integers, not all equal to 0, such that $\left|m_{i}\right| \leqslant K$ for each $i$, then the sum

$$
\sum_{i=1}^{n} m_{i} a_{i}
$$

is not divisible by $N$. What is the largest possible value of $n$?

## Proposed by North Macedonia

Answer: \left\lfloor \log_{K+1} N \right\rfloor


10. In a square lawn with side length $a$, there are sprinkler devices installed at each of the four corners. The sprinkler devices can rotate to spray water at a $90^{\circ}$ angle, and each sprinkler can rotate from one side of its corner to the other side, with an effective range of $a$. What is the proportion of the lawn that can be covered simultaneously by all four sprinklers to the entire lawn? $\qquad$ .

Answer: 1 + \dfrac{\pi}{3} - \sqrt{3}


Problem 2. (2 points)

The sum of the sines of five angles from the interval $\left[0 ; \frac{\pi}{2}\right]$ is 3. What are the greatest and least integer values that the sum of their cosines can take?

Answer: 4


## Task A-3.3.

Given is a triangle $A B C$. Let point $D$ be the foot of the altitude from vertex $A$, and point $E$ be the intersection of the angle bisector of $\varangle C B A$ with the opposite side. If $\varangle B E A=45^{\circ}$, determine $\varangle E D C$.

Answer: 45^\circ


## 31. Child Psychology

The brightest hours of a little child are spent drawing. He depicts either Indians or Eskimos and draws a dwelling for each of them. Thus, next to the Eskimos, he usually draws an igloo, and next to the Indians - wigwams. But sometimes the child makes mistakes, and not infrequently, for example, an igloo can be found next to the Indians. One female psychologist took these drawings very seriously; she discovered that there were twice as many Indians as Eskimos, that there were as many Eskimos with wigwams as there were Indians with igloos, and that, speaking of Eskimos, one wigwam corresponds to three igloos. Then she decided to determine what part of the inhabitants of the wigwams are Indians (and derive from this a very complex IQ $^{1}$ formula for the child). But she has not yet succeeded in doing so. Help her.

Answer: \dfrac{7}{8}


Example 3 In $\triangle A B C$, $A B=A C, \angle A=$ $20^{\circ}, D$ is a point on $A C$, $\angle D B C=60^{\circ}, E$ is a point on $A B$, $\angle E C B=50^{\circ}$. Find the degree measure of $\angle B D E$.

Answer: 30


Question As shown in Figure 1, two rectangles $R$ and $S$ are placed inside an acute triangle $T$. Let $A(X)$ denote the area of polygon $X$. Find the maximum value of $\frac{A(R)+A(S)}{A(T)}$ or prove that the maximum value does not exist.

Answer: \dfrac{2}{3}


4. Place 27 balls numbered $1 \sim 27$ into three bowls, Jia, Yi, and Bing, such that the average values of the ball numbers in bowls Jia, Yi, and Bing are $15$, $3$, and $18$, respectively, and each bowl must contain no fewer than 4 balls. Then the maximum value of the smallest ball number in bowl Jia is $\qquad$

Answer: 10


## Task A-2.5.

Let $n$ be a natural number. If a regular $n$-gon is divided into $n-2$ triangles by drawing $n-3$ diagonals that do not have common interior points, we say that we have obtained a triangulation. A triangulation of an $n$-gon, some of whose vertices are red, is good if each of those $n-2$ triangles has at least two red vertices.

Determine the smallest natural number $k$, depending on $n$, such that we can color $k$ vertices of a regular $n$-gon red so that there exists at least one good triangulation.

Answer: \left\lceil \frac{n}{2} \right\rceil


3.95. The base of the pyramid is an isosceles triangle with an angle $\alpha$ between the lateral sides. The pyramid is placed in a certain cylinder such that its base is inscribed in the base of this cylinder, and the vertex coincides with the midpoint of one of the cylinder's generators. The volume of the cylinder is $V$. Find the volume of the pyramid.

Answer: \dfrac{V \sin \alpha \cos^2 \dfrac{\alpha}{2}}{3\pi}


4. Quadrilateral $ABCD$ is inscribed in a circle. The tangents to this circle, drawn at points $A$ and $C$, intersect on the line $BD$. Find the side $AD$, if $AB=2$ and $BC: CD=4: 5$.

Answer: $\frac{5}{2}$.

Answer: \dfrac{5}{2}


$\underline{115625}$ topics: [  Tangent circles [Mean proportionals in a right triangle]

Two circles touch each other externally at point $C$. A line is tangent to the first circle at point $A$ and to the second circle at point $B$. The line $A C$ intersects the second circle at point $D$, different from $C$. Find $B C$, if $A C=9$, $C D=4$.

#

Answer: 6


7.2 On the Island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 15 residents of the island lined up by height (from tallest to shortest, the tallest being the first) for a game. Each had to say one of the following phrases: "There is a liar below me" or "There is a knight above me." In the end, those standing from the fourth to the eighth position said the first phrase, while the rest said the second. How many knights were among these 15 people, given that all residents have different heights?

Answer: 11


Let  $A$  be a set of natural numbers, for which for  $\forall n\in \mathbb{N}$  exactly one of the numbers  $n$ ,  $2n$ , and  $3n$  is an element of  $A$ . If  $2\in A$ , show whether  $13824\in A$ .

Answer: 13824 \notin A


Let n be the natural number ( $n\ge 2$ ). All natural numbers from  $1$   up to  $n$  ,inclusive, are written on the board   in some order:  $a_1$ ,  $a_2$  ,  $...$  ,  $a_n$ . Determine all natural numbers  $n$  ( $n\ge 2$ ), for which the product  $$ P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n) $$   is an even number, whatever the arrangement of the numbers written on the board.

Answer: n


Let  $n$  be a positive integer. There are  $n$  purple and  $n$  white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?

Answer: n


A set of  $k$  integers is said to be a *complete residue system modulo*  $k$  if no two of its elements are congruent modulo  $k$ . Find all positive integers  $m$  so that there are infinitely many positive integers  $n$  wherein  $\{ 1^n,2^n, \dots , m^n \}$  is a complete residue system modulo  $m$ .

Answer: m


Let  $N \geqslant 3$  be an integer. In the country of Sibyl, there are  $N^2$  towns arranged as the vertices of an  $N \times N$  grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.

*Proposed by Sutanay Bhattacharya and Anant Mudgal*

Answer: N


In a square  $ABCD$  of diagonals  $AC$  and  $BD$ , we call  $O$  at the center of the square. A square  $PQRS$  is constructed with sides parallel to those of  $ABCD$  with  $P$  in segment  $AO, Q$  in segment  $BO, R$  in segment  $CO, S$  in segment  $DO$ .  If area of  $ABCD$  equals two times the  area of  $PQRS$ ,  and  $M$  is the midpoint of the  $AB$  side, calculate the measure of the angle  $\angle AMP$ .

Answer: \frac{45^\circ}{2}


Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An *operation* consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.

Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.

*Proposed by Anthony Wang*

Answer: (n, \alpha)


Let  $n$  be a positive integer. Find the number of permutations  $a_1$ ,  $a_2$ ,  $\dots a_n$  of the
sequence  $1$ ,  $2$ ,  $\dots$  ,  $n$  satisfying  $$ a_1 \le 2a_2\le 3a_3 \le \dots \le na_n $$  .

Proposed by United Kingdom

Answer: F_{n+1}


Example 7 When is $x$ a rational number such that the algebraic expression $9 x^{2}$ $+23 x-2$ is exactly the product of two consecutive positive even numbers?

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

Answer: x = -17


A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB|  = 4$. Let $E $ be the intersection  point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.

Answer: 2


Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$.

Answer: 60^\circ


5. Two circles touch internally at point $A$. From the center of the larger circle $O$, a radius $O B$ of the larger circle is drawn, which is also tangent to the smaller circle at point $C$. Determine $\measuredangle B A C$.

Answer: 45^\circ


10. Let two non-zero and non-collinear vectors $a, b$ in the same plane satisfy $b \perp (a-b)$. Then for any $x \in \mathbf{R}$, the range of $|a-b x|$ is $\qquad$ (expressed in terms of vectors $a, b$).

Answer: [|\mathbf{a} - \mathbf{b}|, +\infty)


Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)

Answer: n \geq 3


1. N1 (TWN) ${ }^{1 \mathrm{MO} 04}$ Find all pairs of positive integers $(x, p)$ such that $p$ is a prime, $x \leq 2 p$, and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$.

Answer: (1, p)


## [ equations in integers ] Decompositions and partitions $\quad]$ [ GCD and LCM. Mutual simplicity ]

Ostap Bender organized a giveaway of elephants to the population in the city of Fux. 28 union members and 37 non-members showed up for the giveaway, and Ostap distributed the elephants equally among all union members and equally among non-members.

It turned out that there was only one way to distribute the elephants (so that all elephants were distributed). What is the maximum number of elephants that Ostap Bender could have had? (It is assumed that each person who came received at least one elephant.)

Answer: 2072


53. As shown in the figure, in the rectangular paper piece $\mathrm{ABCD}$, $\mathrm{AB}=4, \mathrm{BC}=6$. The paper is folded so that point $\mathrm{B}$ coincides with point $\mathrm{D}$, and the fold line is $\mathrm{EF}$. The area of the part of the rectangle that does not overlap is $\qquad$ -

Answer: \dfrac{20}{3}


Given that $\triangle ABC$ is an acute-angled triangle, and $L$ is any straight line in the coordinate plane containing the triangle. Let $u$, $v$, and $w$ be the perpendicular distances from points $A$, $B$, and $C$ to the line $L$, respectively. Prove that \( u^{2} \tan A + v^{2} \tan B + w^{2} \tan C \geq 2S \), where $S$ is the area of $\triangle ABC$.

Answer: u^{2} \tan A + v^{2} \tan B + w^{2} \tan C \geq 2S


4. On the board, you can either write two ones, or erase two already written identical numbers $n$ and write the numbers $n+k$ and $n-k$, provided that $n-k \geqslant 0$. What is the minimum number of such operations required to obtain the number $2048?$

Answer: 3071


[ Arithmetic progression ]

[ Arithmetic. Mental calculation, etc. ]

What is the sum of the digits of all numbers from one to a billion?

Answer: 40500000001


7. On the side $B C$ of triangle $A B C$, a point $K$ is marked.

It is known that $\angle B+\angle C=\angle A K B, \quad A K=4, \quad B K=9, \quad K C=3$.

Find the area of the circle inscribed in triangle $A B C$.

Answer: \dfrac{35}{13}\pi


Problem 5. On the table, there are 10 stacks of playing cards (the number of cards in the stacks can be different, there should be no empty stacks). The total number of cards on the table is 2015. If a stack has an even number of cards, remove half of the cards. If the number of remaining cards in the stack is still even, remove half again, and so on, until the number of cards in the stack becomes odd. Do this for each stack. Explain:

a) What is the maximum possible number of cards remaining on the table?

b) What is the minimum possible number of cards remaining on the table?

In each case, provide an example of how the playing cards can be distributed among the stacks.

Answer: 2014


At a gathering, $n$ invitees arrived, including Mr. Balogh. Besides them, there is a journalist who wants to talk to Mr. Balogh. The journalist knows that no one knows Mr. Balogh, but Mr. Balogh knows everyone. The journalist can approach any of the invitees and ask if they know any other person.

a) Can the journalist definitely find Mr. Balogh with fewer than $n$ questions?

b) How many questions does the journalist need to find Mr. Balogh if they are lucky?

Answer: Yes


A family of sets $\mathscr{F}$ is called "perfect" if for any three sets $X_{1}, X_{2}, X_{3} \in \mathscr{F}$,
$$
\left(X_{1} \backslash X_{2}\right) \cap X_{3} \text{ and } \left(X_{2} \backslash X_{1}\right) \cap X_{3}
$$
have at least one empty set. Prove that if $\mathscr{F}$ is a perfect subfamily of a given finite set $U$, then
$$
|\mathscr{F}| \leqslant |U|+1,
$$
where $|X|$ represents the number of elements in the finite set $X$.

Answer: |\mathscr{F}| \leqslant |U| + 1


In a triangle $ABC$ ($\angle{BCA} = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference with diameter $BC$. Let $F$ be the intersection of $AC$ with a line tangent to the circumference. If $\angle{CAB} = 46^{\circ}$, find the measure of $\angle{CFD}$.

Answer: 92


3. (6 points) As shown in the figure, it is a geometric body composed of 26 identical small cubes, with its bottom layer consisting of $5 \times 4$ small cubes. If the entire outer surface (including the bottom) is painted red, then when this geometric body is disassembled, the number of small cubes that have 3 red faces is $\qquad$ pieces.

Answer: 14


2. As shown in the figure, $\angle A O B=\angle B O C=\angle C O D=\angle D O E=\angle E O F$, there are 15 acute angles in the figure, and the sum of these 15 acute angles is $525^{\circ}$, then $\angle A O D=$ $\qquad$ ${ }^{\circ}$.

Answer: 45


A string of 33 pearls has its largest and most valuable pearl in the middle. The other pearls are arranged so that their value decreases by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the entire string of pearls is 25 times the value of the fourth pearl from the middle, on the more expensive side.

Answer: 90000


3. Let $N$ be a regular nonagon, $O$ the center of its circumscribed circle, $P Q$ and $Q R$ two consecutive sides of $N$, $A$ the midpoint of $P Q$, and $B$ the midpoint of the radius perpendicular to $Q R$. Find the angle between $A O$ and $A B$.

Answer: 30^\circ


Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$.

Answer: 210


In the interior of the square $ABCD$, the point $P$ is located such that $\angle DCP = \angle CAP = 25^{\circ}$. Find all possible values of $\angle PBA$.

Answer: 40


A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be \( \frac{m}{n} \) where \( m, n \) are numbers such that \( m, n \) are positive integers and \( \gcd(m, n) = 1 \).

Take this answer and add the numerator and denominator.

Answer: 3004


For a positive integer $k$, let $f_{1}(k)$ be the square of the sum of the digits of $k$. For example, $f_{1}(123) = (1+2+3)^{2} = 36$. Define $f_{n+1}(k) = f_{1}(f_{n}(k))$. Determine the value of $f_{2007}(2^{2006})$. Justify your claim.

Answer: 169


Georg is putting his $250$ stamps in a new album. On the first page, he places one stamp, and then on every subsequent page, he places either the same number of stamps or twice as many as on the preceding page. In this way, he ends up using precisely all $250$ stamps. How few pages are sufficient for him?

Answer: 8


The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A \cup B$.

Answer: 5


A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be $\frac{m}{n}$ where $m,n$ are numbers such that $m,n$ are positive integers and $\gcd(m,n) = 1$. Take this answer and add the numerator and denominator.

Answer: 3004


Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 17


There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in pairs to each of the $2010$ players sitting around a round table. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules:
- If a player holds at least one red card, they pass one red card to the player sitting to their left.
- Otherwise, they transfer one white card to the left.

The game ends after the round when each player has one red card and one white card. Determine the maximum number of rounds possible.

Answer: 2009


A pentagon has vertices labeled $A, B, C, D, E$ in that order counterclockwise, such that $AB$ and $ED$ are parallel, and \( \angle EAB = \angle ABD = \angle ACD = \angle CDA \). Furthermore, suppose that \( AB = 8 \), \( AC = 12 \), \( AE = 10 \). If the area of triangle $CDE$ can be expressed as \( \frac{a \sqrt{b}}{c} \), where $a$, $b$, $c$ are integers such that $b$ is square-free, and $a$, $c$ are relatively prime, find $a + b + c$.

Answer: 56


In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved.

Answer: 3


Let $S = \{1, 2, \ldots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \ldots, S_n$ for some $n$, i.e., $S_i$ are nonempty, pairwise disjoint sets, and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of the elements of the set $S_i$. Define the score of this partition by:
\[
\frac{a_1 + a_2 + \ldots + a_n}{n}.
\]

Among all $n$ and partitions of $S$, determine the minimum possible score.

Answer: 10


Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots , P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?

Answer: 108


At a certain pizzeria, there are five different toppings available, and a pizza can be ordered with any (possibly empty) subset of them on it. Determine the number of ways to order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither.

Answer: 271


10. Transporting utility poles from a construction site by the roadside along a straight road in the same direction to plant them 500 m away on the roadside, plant one at the 500 m mark, and then plant one every 50 m along the roadside. Knowing that the transport vehicle can carry a maximum of 3 poles at a time, to complete the task of transporting and planting 20 poles, and returning to the construction site, the minimum total distance the transport vehicle must travel is $\qquad$ m.

Answer: 14000


19. Given an arbitrary triangle. On each side of the triangle, 10 points are marked. Each vertex of the triangle is connected by segments to all the marked points on the opposite side. Into what maximum number of parts could the segments divide the triangle?

Answer: 331


Let \( ABCDE \) be a convex pentagon such that \( BC = DE \). Assume there is a point \( T \) inside \( ABCDE \) with \( TB = TD \), \( TC = TE \), and \( \angle TBA = \angle AET \). Let lines \( CD \) and \( CT \) intersect line \( AB \) at points \( P \) and \( Q \), respectively, and let lines \( CD \) and \( DT \) intersect line \( AE \) at points \( R \) and \( S \), respectively. Assume that points \( P, B, A, Q \) and \( R, E, A, S \), respectively, are collinear and occur on their lines in this order. Prove that the points \( P, S, Q, R \) are concyclic.

Answer: P, S, Q, R \text{ are concyclic}


2. Find all integer solutions $x$ and $y$ that satisfy the equation $x^{2} + xy + y^{2} = 1$.

Answer: (1, 0)


Problem 4. We say a 2023-tuple of nonnegative integers $\left(a_{1}, a_{2}, \ldots a_{2023}\right)$ is sweet if the following conditions hold:

- $a_{1}+a_{2}+\ldots+a_{2023}=2023$,
- $\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leqslant 1$.

Determine the greatest positive integer $L$ such that

$$
a_{1}+2 a_{2}+\ldots+2023 a_{2023} \geqslant L
$$

holds for every sweet 2023-tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$.

(Ivan Novak)

Answer: 22228


3. A certain number of boys and girls went camping during the summer break. They planned an ecological action that they would finish in 29 days if each child worked evenly - working the same part of the job in any given days. The boys worked a bit faster; in the same time, 2 boys do as much work as 3 girls. Fortunately, after three days of starting the work, a larger group of children joined them: boys - 8 times more than the initial number of girls, and girls - 18 times more than the initial number of boys. The newly arrived children worked with the same dynamics as the initially arrived ones. How many days in total were working days?

Answer: 5


Variant 134. a) $36+8 \pi$ cm or $23+3 \sqrt{41}+8 \pi$ cm; b) not always.

Evaluation criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; $\mathbf{1 5}$ points - mostly correct solution and correct answer, but there are defects (for example: comparison of numbers in the last part of the solution is not performed or is performed incorrectly; it is not proven that the sum of the lengths of the arcs is equal to the length of the circumference); 10 points - the solution is mostly correct, but contains computational errors; 5 points - only one of the two cases is considered, and the solution and answer (for this case) are completely correct.

## Problem 4 (Variant 131).

When studying the operation of a new type of heat engine working cyclically, it was found that for part of the period it receives heat, with the absolute value of the heat supply power expressed by the law: $P_{1}(t)=P_{0} \frac{\sin \omega t}{100+\sin t^{2}}, 0<t<\frac{\pi}{\omega}$. The gas performs work, developing mechanical power $P_{2}(t)=3 P_{0} \frac{\sin (2 \omega t)}{100+\sin (2 t)^{2}}, 0<t<\frac{\pi}{2 \omega}$. The work done on the gas by external bodies is $2 / 3$ of the magnitude of the work done by the gas. Determine the efficiency of the engine.

Answer: \dfrac{1}{2}


4. Each diagonal of the convex pentagon $A B C D E$ cuts off a triangle of unit area from it. Calculate the area of the pentagon $A B C D E$.

Lemma. Let in a convex quadrilateral $A B C D$ the diagonals $A C$ and $B D$ intersect at point $O$. If triangles $A B O$ and $C O D$ are equal in area, then $A B C D$ is a trapezoid.

Proof. Since $S_{A O B}=S_{C O D}$, then $S_{A B D}=S_{C A D}$, which means the heights from points $B$ and $C$ to line $A D$ in triangles $A B D$ and $C A D$ are equal. This implies that all points on line $B C$ are equidistant from line $A D$, hence $B C \| A D$.

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


5. On the edge $S B$ of a regular quadrilateral pyramid $S A B C D$ with vertex $S$, a point $L$ is marked such that $B L: L S=2: 5$. The point $L$ is the vertex of a right circular cone, on the circumference of the base of which lie three vertices of the pyramid $S A B C D$.

a) Find the ratio $A S: C D$.

b) Suppose it is additionally known that the height of the pyramid $S A B C D$ is 7. Find the volume of the cone.

Answer: \dfrac{\sqrt{15}}{3}


* 6. Among $n(\geqslant 3)$ lines, exactly $p(\geqslant 2)$ are parallel, and no three lines among the $n$ lines intersect at the same point. Then the number of regions into which these $n$ lines divide the plane is

Answer: \dfrac{n^2 + n - p^2 + p + 2}{2}


The diagonals of a convex quadrilateral are 12 and 18 and intersect at point \( O \).

Find the sides of the quadrilateral formed by the points of intersection of the medians of triangles \( AOB, BOC, COD \), and \( AOD \).

Answer: 4


49th Putnam 1988 Problem B3 α n is the smallest element of the set { |a - b√3| : a, b non-negative integers with sum n}. Find sup α n . Solution

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


$\left[\begin{array}{l}\text { Symmetry helps solve the problem } \\ \text { [The inscribed angle is half the central angle]}\end{array}\right]$

$A B$ is the diameter of the circle; $C, D, E$ are points on the same semicircle $A C D E B$. On the diameter $A B$, points $F$ and $G$ are taken such that $\angle C F A = \angle D F B$ and $\angle D G A = \angle E G B$. Find $\angle F D G$, if the arc $A C$ is $60^{\circ}$ and the arc $B E$ is $20^{\circ}$.

Answer: 50^\circ


Task 2. Find the number of ways to color all natural numbers from 1 to 20 in blue and red such that both colors are used and the product of all red numbers is coprime with the product of all blue numbers.

Answer: 62


Given an isosceles triangle \(ABC\) with \(AB = BC\). In the circumcircle \(\Omega\) of triangle \(ABC\), a diameter \(CC'\) is drawn. A line passing through point \(C'\) parallel to \(BC\) intersects segments \(AB\) and \(AC\) at points \(M\) and \(P\) respectively. Prove that \(M\) is the midpoint of segment \(C'P\).

Answer: M \text{ is the midpoint of } C'P


17. (5 points)
As shown in the figure, there is a circular path and two straight paths on the Green Green Grassland. The two straight paths both pass through the center of the circle, with a radius of 50 meters. Village Chief Slow Sheep starts from point A and walks along the path. He needs to walk through all the paths and return to point A. He needs to walk at least $\qquad$ meters. (Take $\pi$ as 3.14)

Answer: 671


[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external segment. Auxiliary similar triangles [Mean proportional in a right triangle

From point $A$, two tangents (with points of tangency $M$ and $N$) and a secant intersecting the circle at points $B$ and $C$, and the chord $MN$ at point $P$, are drawn, with $AB: BC=2: 3$. Find $AP: PC$.

#

Answer: \dfrac{4}{3}


4. Given an isosceles triangle $A B C$, where $\angle A=30^{\circ}, A B=A C$. Point $D$ is the midpoint of $B C$. On segment $A D$, point $P$ is chosen, and on side $A B$, point $Q$ is chosen such that $P B=P Q$. What is the measure of angle $P Q C ?$ (S. S. Korechkova)

Answer: 15^\circ


15. There are 10 students standing in a row, and their birthdays are in different months. There are $n$ teachers who will select these students to join $n$ interest groups. Each student is selected by exactly one teacher, and the order of the students is maintained. Each teacher must select students whose birthdays are in months that are either strictly increasing or strictly decreasing (selecting one or two students is also considered strictly increasing or decreasing). Each teacher should select as many students as possible. For all possible orderings of the students, find the minimum value of $n$.

Answer: 4


If for all \( n \in \mathbf{Z}_{+} \), \( a^{n+2} + 3a^n + 1 \) is not a prime number, then the positive integer \( a \) is called "beautiful". Prove that in the set \( \{1, 2, \ldots, 2018\} \) there are at least 500 beautiful positive integers.

Answer: 500


Let \( P \) be the intersection point of the directrix \( l \) of an ellipse and its axis of symmetry, and \( F \) be the corresponding focus. \( AB \) is a chord passing through \( F \). Find the maximum value of \( \angle APB \) which equals \( 2 \arctan e \), where \( e \) is the eccentricity.

Answer: 2\arctane


A space probe, moving in a straight line at a constant speed, flies past Mars and measures the distance to the planet exactly at 12:00 noon every day. It is known that on February 1st, the distance was 5 million km, on February 10th, it was 2 million km, and on February 13th, it was 3 million km. Determine when the probe will be at the minimum distance from Mars. In this problem, Mars can be considered as a point.

Answer: 9


Place the numbers $1,2,\cdots,n$ on a circle such that the absolute difference between any two adjacent numbers is either 3, 4, or 5. Find the smallest $n$ that satisfies these conditions.

Answer: 7


Positive integers \( a, b \), and \( c \) have the property that \( a^b \), \( b^c \), and \( c^a \) end in 4, 2, and 9, respectively. Compute the minimum possible value of \( a + b + c \).

Answer: 17


In the Magic Land, one of the magical laws of nature states: "A flying carpet will only fly if it has a rectangular shape." Prince Ivan had a flying carpet of dimensions $9 \times 12$. One day, an evil dragon cut off a small piece of this carpet with dimensions $1 \times 8$. Prince Ivan was very upset and wanted to cut off another piece $1 \times 4$ to make a rectangle of $8 \times 12$, but the wise Vasilisa suggested a different approach. She cut the carpet into three parts and used magic threads to sew them into a square flying carpet of dimensions $10 \times 10$. Can you figure out how Vasilisa the Wise transformed the damaged carpet?

Answer: 10\times10


There are 29 students in a class. It is known that for any pair of students, there is at least one other student (not from the considered pair) who is friends with exactly one person from that pair. What is the minimum number of pairs of friends that can exist in the class?

Answer: 21


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Alice`, `Peter`
 - Each person has a favorite color: `blue`, `green`, `white`, `yellow`, `red`
 - People use unique phone models: `huawei p50`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`, `google pixel 6`
 - Each person has an occupation: `artist`, `teacher`, `doctor`, `engineer`, `lawyer`

## Clues:
1. The person who is an engineer is somewhere to the right of the person who is a lawyer.
2. Bob is in the second house.
3. The person who uses a Samsung Galaxy S21 is the person who is a doctor.
4. The person who is a doctor is the person who loves blue.
5. The person whose favorite color is green is not in the fifth house.
6. The person who is a lawyer is the person who uses a OnePlus 9.
7. The person who loves blue is directly left of the person whose favorite color is red.
8. The person who is a lawyer is somewhere to the right of the person who uses a Samsung Galaxy S21.
9. There is one house between the person who uses a Google Pixel 6 and the person who uses a Huawei P50.
10. Arnold is the person who is an engineer.
11. Alice is the person who loves yellow.
12. The person who uses a Google Pixel 6 is Eric.
13. The person who uses a Google Pixel 6 is the person who is a teacher.
14. The person whose favorite color is red is somewhere to the right of the person who is a teacher.
 What is the value of attribute Color for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{}

Answer: green


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Eric`, `Arnold`, `Peter`
 - Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `pizza`
 - People have unique favorite sports: `basketball`, `tennis`, `soccer`
 - People have unique hair colors: `black`, `blonde`, `brown`
 - People have unique favorite music genres: `classical`, `rock`, `pop`
 - People have unique favorite book genres: `science fiction`, `mystery`, `romance`

## Clues:
1. The person who loves basketball is somewhere to the left of the person who loves science fiction books.
2. The person who has brown hair is Eric.
3. The person who has brown hair is the person who is a pizza lover.
4. The person who loves romance books is the person who is a pizza lover.
5. Peter is the person who has blonde hair.
6. The person who loves science fiction books is the person who loves soccer.
7. The person who loves eating grilled cheese is the person who loves basketball.
8. The person who loves pop music is the person who has blonde hair.
9. The person who has brown hair is somewhere to the left of the person who loves rock music.
10. The person who loves classical music is not in the first house.
 What is the value of attribute FavoriteSport for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{}

Answer: tennis


There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
 - Each person has an occupation: `engineer`, `artist`, `teacher`, `doctor`

## Clues:
1. The person who is an engineer is not in the third house.
2. The person who is a teacher is directly left of Peter.
3. The person who is an engineer is Arnold.
4. The person who is a doctor is somewhere to the right of Alice.
5. Arnold is somewhere to the left of Eric.
6. There is one house between Arnold and Alice.
 What is the value of attribute Occupation for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}

Answer: teacher


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Peter`, `Eric`, `Arnold`, `Alice`
 - The people keep unique animals: `dog`, `fish`, `bird`, `horse`, `cat`
 - The people are of nationalities: `dane`, `swede`, `brit`, `german`, `norwegian`
 - Each mother is accompanied by their child: `Meredith`, `Bella`, `Timothy`, `Samantha`, `Fred`
 - Each person has a unique type of pet: `bird`, `fish`, `hamster`, `cat`, `dog`

## Clues:
1. The dog owner is not in the third house.
2. The person who keeps a pet bird is in the fourth house.
3. The German is Eric.
4. Bob is somewhere to the left of the person's child is named Fred.
5. The person with a pet hamster is Arnold.
6. The British person is the dog owner.
7. The person's child is named Meredith is not in the third house.
8. The fish enthusiast is the person's child is named Samantha.
9. The person who is the mother of Timothy is the German.
10. The person who has a cat is in the fifth house.
11. Peter is not in the fourth house.
12. The Swedish person and the person's child is named Samantha are next to each other.
13. The Swedish person is Peter.
14. The Dane is in the second house.
15. Eric and the cat lover are next to each other.
16. The German is in the first house.
17. The person with an aquarium of fish is somewhere to the right of the bird keeper.
 What is the value of attribute House for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: 3


There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`
 - People have unique favorite book genres: `mystery`, `science fiction`, `biography`, `fantasy`, `romance`
 - Each mother is accompanied by their child: `Samantha`, `Timothy`, `Fred`, `Meredith`, `Bella`

## Clues:
1. The person's child is named Samantha is somewhere to the left of the person who loves biography books.
2. Arnold is the person who loves biography books.
3. The person who is the mother of Timothy is the person who loves science fiction books.
4. The person's child is named Bella is Peter.
5. There is one house between the person who loves biography books and Bob.
6. Alice is somewhere to the right of Eric.
7. The person who loves mystery books is in the fourth house.
8. The person who loves biography books is directly left of the person who is the mother of Timothy.
9. The person who loves fantasy books is directly left of the person's child is named Meredith.
 What is the value of attribute House for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{}

Answer: 5


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Eric`, `Peter`
 - Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`
 - The people keep unique animals: `horse`, `cat`, `bird`
 - Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
 - People have unique favorite book genres: `science fiction`, `romance`, `mystery`
 - People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`

## Clues:
1. The person who loves mystery books is the person's child is named Fred.
2. The cat lover is Eric.
3. The person partial to Pall Mall is in the second house.
4. The person who keeps horses is the person's child is named Meredith.
5. The person's child is named Bella is the Prince smoker.
6. The person who uses an iPhone 13 is directly left of the person who uses a Samsung Galaxy S21.
7. The person's child is named Fred is directly left of Arnold.
8. Peter is somewhere to the left of Eric.
9. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
10. The person who loves science fiction books is in the third house.
11. The person who loves mystery books is not in the second house.
 What is the value of attribute House for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
 - Each person has a unique name: `Arnold`, `Peter`, `Eric`
 - People have unique favorite book genres: `romance`, `mystery`, `science fiction`
 - Each person has a unique birthday month: `sept`, `april`, `jan`
 - The people are of nationalities: `brit`, `dane`, `swede`

## Clues:
1. Arnold is the person who loves mystery books.
2. Peter is directly left of the person who loves romance books.
3. Peter is somewhere to the left of the person who loves mystery books.
4. The person whose birthday is in April is somewhere to the right of the person whose birthday is in January.
5. The person whose birthday is in September is the Swedish person.
6. The Dane is in the third house.
7. The person who loves science fiction books is the person whose birthday is in September.
 What is the value of attribute House for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}

Answer: 1


Let $ABCD$ be a convex cyclic quadrilateral. Suppose $P$ is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $\{PA, PB, PC, PD\} = \{3, 4, 6, 8\}$, what is the maximum possible area of $ABCD$?

Answer: 54


Given in the figure:
- The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$.
- The area of the triangle $\triangle FEC$ is $40$ cm$^2$.
- The sum of the areas of the triangles $\triangle ADF$ and $\triangle DBE$ is $150$ cm$^2$.

Determine whether the area of the triangle $\triangle DEF$ follows clearly from this information and, if necessary, calculate this area.

Answer: 60


The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$. The sides of $\triangle ABC$ have lengths $AB = 73$, $BC = 123$, and $AC = 120$. Find the length $EG$.

Answer: 119


We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.

How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$?

Answer: 694


There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ unit apart. Find the maximum possible value of $I$.

Answer: 3954


Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, \( \frac{4}{3} \) seconds, \( \frac{5}{3} \) seconds, and 2 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 1 minute? (Include first and last.)

Answer: 85


Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. Points $A$, $B$, and $C$ are on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle tangent to $AB$ and $BC$ at $Q$ and $R$, and also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ intersects $AC$ at $X$ and $Y$. The area $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$. Find $a+b+c$.

Answer: 19


Consider a string of $n$ 1's. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times, and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.

Answer: 9


Say an odd positive integer $n > 1$ is \textit{twinning} if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.

Answer: 12


A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m + n$.

Answer: 407


A circular disc is divided into $12$ equal sectors, each of which is colored using one of $6$ different colors. No two adjacent sectors can have the same color. Determine the number of distinct ways to color the disc under these conditions.

Answer: 20346485


Six different small books and three different large books are placed on a shelf. Three children can each choose either two small books or one large book. Determine the number of ways the three children can select their books.

Answer: 1176


Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define
\[f(A) = \sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.\]
If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$.

Answer: 441


In square $ABCD$ with $AB = 10$, point $P, Q$ are chosen on side $CD$ and $AD$ respectively such that $BQ \perp AP,$ and $R$ lies on $CD$ such that $RQ \parallel PA.$ $BC$ and $AP$ intersect at $X,$ and $XQ$ intersects the circumcircle of $PQD$ at $Y$. Given that $\angle PYR = 105^{\circ},$ $AQ$ can be expressed in simplest radical form as $b\sqrt{c}-a$ where $a, b, c$ are positive integers. Find $a+b+c.$

Answer: 23


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

[i]Proposed by David Altizio[/i]

Answer: 24


11 Given that $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ are on the ellipse $C$. If $x_{1}+x_{2}=\frac{1}{2}$, and $\overrightarrow{A F_{2}}=$ $\lambda \overrightarrow{F_{2} B}$, find the value of $\lambda$.

Answer: \dfrac{3 - \sqrt{5}}{2}


Square $AIME$ has sides of length $10$ units.  Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units.  Find the length of the altitude to $EM$ in $\triangle GEM$ .
Please give the answer directly without any intermediate steps.

Answer: 25


How many quadratic residues are there modulo $pq$? And modulo $p^n$? You can use the fact that $\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)^{*}$ is cyclic.

Answer: \frac{(p-1)(q-1)}{4}


B3. One of the four gnomes, Anne, Bert, Chris, and Dirk, has stolen gold from the king. Each of the gnomes, who know each other inside and out, makes two statements. If a gnome is a liar, at least one of those two statements is a lie. If a gnome is not a liar, both statements are true.

Anne says: "Bert is a liar." and "Chris or Dirk did it."

Bert says: "Chris is a liar." and "Dirk or Anne did it."

Chris says: "Dirk is a liar." and "Anne or Bert did it."

Dirk says: "Anne is a liar." and "Bert or Chris did it."

How many of these eight statements are true?

Answer: 5


## Problema 2.

Sea $\mathrm{P}$ un punto del lado $\mathrm{BC}$ de un triángulo $\mathrm{ABC}$. La paralela por $\mathrm{P}$ a $\mathrm{AB}$ corta al lado $\mathrm{AC}$ en el punto $\mathrm{Q}$ y la paralela por $\mathrm{P}$ a $\mathrm{AC}$ corta al lado $\mathrm{AB}$ en el punto $\mathrm{R}$. La razón entre las áreas de los triángulos RBP y QPC es $\mathrm{k}^{2}$.

Determínese la razón entre las áreas de los triángulos ARQ y ABC.

Answer: \dfrac{k}{(k + 1)^2}


31. Consider the identity $1+2+\cdots+n=\frac{1}{2} n(n+1)$. If we set $P_{1}(x)=\frac{1}{2} x(x+1)$, then it is the unique polynomial such that for all positive integer $n, P_{1}(n)=1+2+\cdots+n$. In general, for each positive integer $k$, there is a unique polynomial $P_{k}(x)$ such that
$$
P_{k}(n)=1^{k}+2^{k}+3^{k}+\cdots+n^{k} \quad \text { for each } n=1,2, \ldots .
$$

Find the value of $P_{2010}\left(-\frac{1}{2}\right)$.

Answer: 0


Problem 4. Inside a rectangular grid with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell boundaries (the hole does not contain any boundary cells). If the figure is cut along all horizontal grid lines, 20 strips 1 cell wide will be obtained. How many strips will be obtained if, instead, it is cut along all vertical grid lines? (A $1 \times 1$ square is also a strip!)

[6 points]

(A. V. Shapovalov)

Answer: 21


13. Teacher Li and three students, Xiao Ma, Xiao Lu, and Xiao Zhou, set off from the school one after another and walk along the same road to the cinema. The three students have the same walking speed, and Teacher Li's walking speed is 1.5 times that of the students. Now, Teacher Li is 235 meters away from the school, Xiao Ma is 87 meters away from the school, Xiao Lu is 59 meters away from the school, and Xiao Zhou is 26 meters away from the school. When they walk another $\qquad$ meters, the distance Teacher Li is from the school will be exactly the sum of the distances the three students are from the school.

Answer: 42


11. Given positive integers $x, y, z$ satisfying $x y z=(22-x)(22-y)(22-z)$, and $x+y+z<44, x^{2}+y^{2}+z^{2}$, the maximum and minimum values are denoted as $M$ and $N$, respectively, then $M+N=$ $\qquad$.

Answer: 926


5. Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with its left focus at $F$, the line $y=k x-1$ intersects the ellipse at points $A$ and $B$. When the perimeter of $\triangle F A B$ is maximized, the area of $\triangle F A B$ is $\qquad$

Answer: \dfrac{12\sqrt{2}}{7}


4. Extreme set (from 6th grade, $\mathbf{1}$ point). From the digits 1 to 9, three single-digit and three two-digit numbers are formed, with no digits repeating. Find the smallest possible arithmetic mean of the resulting set of numbers.

Answer: 16.5


6.2. On the plate, there were 15 doughnuts. Karlson took three times more doughnuts than Little Man, and Little Man's dog Bibbo took three times fewer than Little Man. How many doughnuts are left on the plate? Explain your answer.

Answer: 2


4. For a regular tetrahedron $V-ABC$ with base edge length $a$ and side edge length $b$, let $M$ be a point on the height $VO$ such that $\frac{VM}{MO}=\frac{b}{a}$. A plane passing through $M$ is parallel to the side edge $VA$ and the base edge $BC$. The area of the section formed by this plane cutting the regular tetrahedron is

Answer: \dfrac{2 a b^2 (3a + b)}{9 (a + b)^2}


10.284. A square and an equilateral triangle are described around a circle of radius $R$, with one side of the square lying on a side of the triangle. Calculate the area of the common part of the triangle and the square.

Answer: \dfrac{(18 - 4\sqrt{3})R^2}{3}


10. All positive integer solution pairs $(x, y)$ of the equation $x^{2}+y^{2}=208(x-y)$ are $\qquad$ .

Answer: (48, 32)


8. As shown in Figure $11, \angle A O B=$ $30^{\circ}, \angle A O B$ contains a fixed point $P$, and $O P=10, O A$ has a point $Q, O B$ has a fixed point $R$. If the perimeter of $\triangle P Q R$ is minimized, find its minimum value.

Answer: 10


9.1. Find the smallest natural number $n$ for which the decimal representation of $n$ together with $n^{2}$ uses all the digits from 1 to 9 exactly once.

Answer: 567


A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$.

Answer: 1


Problem 4. The number 1 is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a-1$ and $a+1$; if the number 0 occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show 1 after 0 steps; 2 after 1 step; 1,3 after 2 steps; 2, 2, 4 after 3 steps, and so on. How many numbers will there be on the blackboard after $n$ steps?

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


10. Let $n$ be a positive integer. If in the $2 n+1$ consecutive natural numbers including 2009, the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers, then the value of $n$ is $\qquad$.

Answer: 31


## Task 2

Subtract the number 5 from 9! Write down the equation for this!

Answer: 4


In a school, $2n$ subjects are studied. All students have grades of "four" and "five," and no two students have the same grades. It is not possible to say that one of any two students is studying better than the other. Prove that the number of students in the school does not exceed $\frac{(2n)}{n!n!}$!

Answer: \dfrac{(2n)!}{n! \cdot n!}


Petya is playing a shooting game. If he scores less than 1000 points, the computer will add $20 \%$ of his score. If he scores from 1000 to 2000 points, the computer will add $20 \%$ of the first thousand points and $30 \%$ of the remaining points. If Petya scores more than 2000 points, the computer will add $20 \%$ of the first thousand points, $30 \%$ of the second thousand, and $50 \%$ of the remaining points. How many bonus points did Petya receive if he had 2370 points at the end of the game?

#

Answer: 470


2. In $\triangle A B C$, $\angle B=2 \angle C, A D$ is the angle bisector of $\angle A$, $m A B=n B D(n>m>0)$. Then $\cos C=$ $\qquad$ .

Answer: \dfrac{m + n}{2n}


## Task 3.

On the side $\overline{A B}$ of the cyclic quadrilateral $A B C D$, there is a point $X$ with the property that the diagonal $\overline{B D}$ bisects the length of $\overline{C X}$, and the diagonal $\overline{A C}$ bisects the length of $\overline{D X}$.

What is the smallest possible ratio $|A B|:|C D|$ in such a quadrilateral?

Answer: 2


We folded a rectangular piece of paper along one of its diagonals. After the fold, the four vertices ended up at the four vertices of a trapezoid, three sides of which are of equal length. What is the length of the shorter side of the original rectangle if the longer side is 12 cm?

Answer: 4\sqrt{3}


In a right triangle \(ABC\) with right angle at \(A\), the altitude \(AH\) is drawn. On the extension of the hypotenuse \(BC\) beyond point \(C\), there is a point \(X\) such that

\[ HX = \frac{BH + CX}{3}. \]

Prove that \( 2 \angle ABC = \angle AXC \).

Answer: 2 \angle ABC = \angle AXC


Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$.
Find length of $XY$.

Answer: \dfrac{1}{2}


3. There are 12 students in the class. Each of them was asked how many friends they have in the class. Each number \( n \) was either mentioned exactly \( n \) times or not at all. What is the maximum value that the sum of all the numbers mentioned can take?

Answer: 90


10. Given that $\lambda$ is a positive real number, the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{3}, a_{n+1}=\frac{2 a_{n}^{2}}{4 a_{n}^{2}-2 a_{n}+1}, n=1,2,3, \cdots$. If for any $m \in \mathbf{N}^{*}$, we have $\sum_{k=1}^{m} a_{k}<\lambda$, find the minimum value of $\lambda$.

Answer: 1


Given a positive integer \( N \), let \( a_1, a_2, \ldots, a_n \) be \( n \) distinct positive integers less than \( N \). Denote the least common multiple of these \( n \) numbers as \( M \) and the greatest common divisor as \( m \). Given that \( M \mid m^2 \), prove that \( n^3 < 4 N \).

Answer: n^3 < 4N


Test: Given that $a$ is a natural number, a quadratic trinomial with integer coefficients and $a$ as the leading coefficient has two distinct positive roots less than 1. Find the minimum value of $a$.

---

The above text translated into English, preserving the original text's line breaks and format.

Answer: 5


The lines of the first given table are all arithmetic progressions of the same ratio, and the columns of this table are all arithmetic progressions of the same ratio. In the second given table, the same formation rule was used, but someone erased some numbers, leaving only three. What is the number that was in the position indicated with

| 5 | 8 | 11 | 14 | 17 |
| :---: | :---: | :---: | :---: | :---: |
| 12 | 15 | 18 | 21 | 24 |
| 19 | 22 | 25 | 28 | 31 |
| 26 | 29 | 32 | 35 | 38 |
| 33 | 36 | 39 | 42 | 45 |

|  |  | 39 |  |  |
| :--- | :--- | :--- | :--- | :--- |
|  |  |  |  |  |
|  |  |  |  | 87 |
| 56 |  |  |  |  |
|  |  |  | $\star$ |  |

Answer: 102


There are three printing machines. The first machine, with a card containing the numbers $a$ and $b$, outputs a card with the numbers $a+1$ and $b+1$; the second machine, with a card containing even numbers $a$ and $b$, outputs a card with the numbers $a / 2$ and $b / 2$; the third machine, with a pair of cards containing the numbers $a, b$ and $b, c$ respectively, outputs a card with the numbers $a$ and $c$. All machines return the cards placed in them. Is it possible, using these machines, to transform the card $(5, 19)$ into the card $(1, 1988)$?

Answer: \text{No}


2. Given a positive integer $n$, define $f(0, j)=f(i, 0)=0, f(1,1)=n$ and

$$
f(i, j)=\left\lfloor\frac{f(i-1, j)}{2}\right\rfloor+\left\lfloor\frac{f(i, j-1)}{2}\right\rfloor
$$

for all positive integers $i$ i $j,(i, j) \neq(1,1)$. How many ordered pairs of positive integers $(i, j)$ are there for which $f(i, j)$ is an odd number?

(Dušan Djukić)

Answer: n


2. Given $x \in \mathbf{R}, f(x)=x^{2}+12 x+30$. Then the solution set of the equation $f(f(f(f(f(x)))))=0$ is $\qquad$ .

Answer: \{-6 + 6^{1/32}, -6 - 6^{1/32}\}


6. Given $[x]$ represents the greatest integer not exceeding the real number $x$. If $P=\sum_{i=1}^{2010}\left[\frac{6^{i}}{7}\right]$, then the remainder when $P$ is divided by 35 is

Answer: 20


5. There is a method of setting a six-digit password: the first 2 digits are the last 2 digits of the birth year, the middle 2 digits are the birth month, and the last 2 digits are the birth date. For example, if born on April 23, 1998, the six-digit password is set as 980423. The number of possible passwords set in this way is $\qquad$ kinds.

Answer: 36525


2. Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+(p-2) x+1$ $=0$ and. $[1+\alpha(p+\alpha)-\beta][1+\beta(p+\beta)$ $-\alpha]=-\frac{7}{2}$, find the value of $p$.

Answer: \dfrac{9}{2}


# Problem 7.5 (7 points)

On a plane, 6 lines are drawn and several points are marked. It turned out that on each line exactly 3 points are marked. What is the minimum number of points that could have been marked?

Answer: 7


Answer: \dfrac{1}{2}


In a given group of numbers, one number is equal to the average of all, the largest number is 7 greater than the average, the smallest is 7 less than the average, and most of the numbers in the group have below-average values.

What is the smallest number of numbers that can be in the group?

(K. Pazourek)

Hint. What is the average of the three more closely described numbers from the group?

Answer: 7


5.9 The International Mathematical Olympiad Problem Selection Committee has 34 countries participating, with each country being represented by a team leader and a deputy leader. Before the meeting, the participants shake hands with each other, but the team leader and deputy leader from the same country do not shake hands. After the meeting, the team leader of the host country asks the participants how many times they shook hands, and the answers received are all different. How many people did the deputy leader of the host country shake hands with?

Answer: 33


3. Petrov lists the odd numbers: $1,3,5, \ldots, 2013$, while Vasechkin lists the even numbers: $2,4, \ldots, 2012$. Each of them calculated the sum of all digits of all their numbers and told the excellent student Masha. Masha subtracted Vasechkin's result from Petrov's result. What did she get?

Answer: 1007


Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$

Answer: 30^\circ


Viruses are capable of reproducing inside bacteria, essentially the bacteria's organism reproduces them. When a virus enters a bacterium, after a certain time, the bacterium dies and the resulting viruses are released.

Suppose a virus enters a colony of $n$ bacteria and immediately enters one of the bacteria; after one minute, 2 viruses are released from the infected bacterium, and during this time, the other bacteria each divide exactly once. (Each bacterium produces two new bacteria.) Thereafter, every newly released virus immediately infects one of the bacteria (each bacterium is infected by only one virus), and in these, 2-2 viruses are produced within one minute, while the other bacteria continue to divide.

Will the bacterial colony be wiped out - and if so, how long will it take?

Answer: n


4. As shown in the upper right figure, the radii of semicircles $A$ and $B$ are equal, they are tangent to each other, and both are internally tangent to the semicircle $O$ with radius 1. $\odot O_{1}$ is tangent to both of them, and $\odot O_{2}$ is tangent to $\odot O_{1}$, semicircle $O$, and semicircle $B$. Then the radius $r$ of $\odot O_{2}$ is $\qquad$

Answer: \dfrac{1}{6}