10. Given that the largest angle of an isosceles triangle is 4 times the smallest angle, then the difference between the largest and smallest angles is $(\quad)$ degrees. Exam point: Isosceles triangle

Answer: 90


10. (20 points) Given the parabola $C: y=\frac{1}{2} x^{2}$ and the circle $D: x^{2}+\left(y-\frac{1}{2}\right)^{2}=r^{2}(r>0)$ have no common points, a tangent line is drawn from a point $A$ on the parabola $C$ to the circle $D$, with the points of tangency being $E$ and $F$. When point $A$ moves along the parabola $C$, all points not on the line $E F$ form a planar region $\Omega$. Find the range of the area of region $\Omega$.

Answer: \left(0, \dfrac{\pi}{16}\right)


A cube with side length \(a\), labeled \(ABCD-A_1B_1C_1D_1\), is cut by a plane passing through its diagonal \(AC_1\). What is the minimum area of the cross-section?

Answer: \dfrac{\sqrt{6}}{2} a^2


Let \(ABCD\) be a convex quadrilateral where only \(\angle D\) is obtuse. The quadrilateral is divided into \(n\) obtuse triangles using some line segments, with no vertices of these obtuse triangles lying on the perimeter of the quadrilateral, except for \(A\), \(B\), \(C\), and \(D\). Prove that the necessary and sufficient condition for \(n\) is \(n \geq 4\).

Answer: n \geq 4


# Problem No. 7 (15 points)

For a body completely submerged in a liquid to be in equilibrium, a force $F=5 \mathrm{H}$ is applied to it. Determine the density of the body if its volume $V=1$ l, and the density of the liquid $\rho_{\text {ж }}=1000$ kg $/ \mathrm{m}^{3}$.

Answer: $1500 \mathrm{\kappa z} / \boldsymbol{m}^{3}$

#

Answer: 1500


Franklin 5.

A convex $n$-gon $P$, where $n>3$, is cut into equal triangles by diagonals that do not intersect inside it.

What are the possible values of $n$, if the $n$-gon is cyclic?

Answer: 4


Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

Answer: 3


129. In a parallelogram, there are two circles of radius 1, each touching the other and three sides of the parallelogram. It is also known that one of the segments of a side of the parallelogram from a vertex to the point of tangency is $\sqrt{3}$. Find the area of the parallelogram.

Answer: \dfrac{12 + 8\sqrt{3}}{3}


8. (10 points) Place seven consecutive natural numbers at the intersection points $A, B, C, D, E, F, G$ of five circles, such that the sum of the numbers on each circle is equal. If all the numbers filled in are greater than 0 and no more than 10, then the number filled at point $G$ is.

Answer: 6


Human explorers discover a solar system with 15 planets, 7 of which are Earth-like, and 8 are Mars-like. Each Earth-like planet requires 2 units of colonization effort, while each Mars-like planet requires only 1 unit. If the expedition has a total of 16 units of colonization effort available, how many different combinations of planets can be colonized given that all planets are unique?

Answer: 1141


A cone has a uniform circular base of radius $6$ feet and a slant height of $13$ feet. The cone’s side breaks, causing the height from the base to the tip along the cone's side to reduce by $2$ feet while maintaining a right angle at the vertex. Determine the new height, $H$, of the cone from the base to the tip.

Answer: H = \sqrt{85}


A cylindrical tank with a radius of $6$ feet and a height of $15$ feet is filled with water. A solid cube with a side length of $12$ feet is placed into the tank such that one of its body diagonals is vertical. Calculate the volume of water displaced by the cube, $w$, and find $w^2$.

Answer: 2985984


A biased die lands on 1, 2, or 3 with probability $\frac{1}{6}$ each, and on 4, 5, or 6 with probability $\frac{1}{2}$. Landing on numbers 1 through 3 gains $\$2$, but landing on 4, 5, or 6 loses $\$5$. Calculate the expected value of a single roll of this die.

Answer: \$-1.50


In a right triangle $XYZ$, medians are drawn from points $X$ and $Y$ to segments $\overline{YZ}$ and $\overline{XZ}$, respectively, dividing these segments in half. The lengths of the medians from $X$ and $Y$ are 5 units and $3\sqrt{5}$ units, respectively. The area of triangle $XYZ$ is 30 square units. Determine the length of segment $\overline{XY}$.

Answer: 2\sqrt{14}


A number $s$ can be expressed as a four-place decimal $0.wxyz$, where $w, x, y, z$ are digits. We want to approximate $s$ by a fraction whose numerator is 1, 2, or 3, and whose denominator is an integer. Find how many possible values for $s$ exist if the closest such fraction to $s$ is $\frac{3}{11}$.

Answer: 179


Given that $\angle A = 30^\circ$ and $\angle AFG = \angle AGF$, calculate the sum of $\angle B + \angle D$ in the pentagon shown. Assume that $\angle BFD + \angle AFG = 180^\circ$, and that $\angle AFG + \angle AGF + \angle GAF = 180^\circ$ as in a triangle.

Answer: 75^\circ


A circle is circumscribed around quadrilateral $WXYZ$. The inscribed angles $\angle WXY$ and $\angle YZW$ are given as $50^\circ$ and $70^\circ$ respectively. Calculate the sum of the angles $\angle WYZ + \angle XYZ$.

Answer: 60^\circ


In a race, all runners start at point $A$, touch any part of a 1000-meter long wall, and stop at point $B$. The length from $A$ to the start of the wall is 400 meters, and from the end of the wall to $B$ is 600 meters. What is the minimum distance a participant must run? Assume the wall, $A$, and $B$ are aligned vertically, and the race is entirely in a vertical plane. Express your answer to the nearest meter.

Answer: 1414


Triangle $PQR$ is an isosceles triangle with $PQ=QR$. Point $S$ is the midpoint of both $\overline{QR}$ and $\overline{PT}$, and $\overline{RT}$ is 15 units long. What is the length of $\overline{PS}$?  Express your answer as a decimal to the nearest tenth.

Answer: 7.5


In a geometry problem, Isabella has drawn a pair of concentric circles. She starts drawing chords $\overline{CD}$, $\overline{DE}$, and so on, of the larger circle, each tangent to the smaller one. If $m\angle CDE=60^\circ$, how many chords will Isabella draw before she returns to her starting point at $C$?

Answer: 3


Define $n!!$ similarly as in the original problem. We are interested in the sum $\sum_{i=1}^{2010} \frac{(2i)!!}{(2i+1)!!}$ when it is expressed as a fraction in lowest terms. The denominator is $2^a b$ with $b$ odd. Find $\dfrac{ab}{10}$.

Answer: 0


Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, suppose that 0, $z_1,$ and $z_2$ form the vertices of a right-angled triangle with the right angle at $z_1$. Find $\frac{a^2}{b}.$

Answer: 2


Triangles $\triangle ABC$ and $\triangle PQR$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,10)$, $C(20,0)$, $P(30,0)$, $Q(30,20)$, $R(50,0)$. A rotation of $n$ degrees counterclockwise around the point $(x,y)$ where $0<n<180$, will transform $\triangle ABC$ to $\triangle PQR$. Find $n+x+y$.

Answer: 120


Let $f(n)$ denote the sum of all divisors of a positive integer $n$. Define $n$ as superdeficient if $f(f(n)) = n + 3$. Determine how many superdeficient integers exist under this definition.

Answer: 0


Square $WXYZ$ has an area of $144$. Point $P$ lies on side $\overline{WY}$ such that $WP:PY = 2:1$. Points $Q$ and $R$ are the midpoints of $\overline{WP}$ and $\overline{ZP}$, respectively. Quadrilateral $WQRP$ has area $25$. What is the area of triangle $RYZ$?

Answer: 12


A large rectangle has area 168 square units and is divided into four rectangles by two segments, with one segment parallel to the width and the other not necessarily parallel to the length. Given three of these resulting rectangles have areas 33, 45, and 20, find the area of the fourth rectangle.

Answer: 70


Three years ago, Tom was twice as old as his sister Sara, and five years before that, Tom was three times as old as Sara.

Let $t$ and $s$ represent Tom's current age and Sara's current age, respectively. Determine the number of years until the ratio of their ages will be 3:2.

Answer: 7


In a physics lab experiment, a certain fixed natural constant $k$ is measured as $3.56897$ with an accuracy of $\pm 0.00145$. The researcher aims to declare a value of $k$ where every reported digit is significant. The announced value must precisely reflect $k$ when it is rounded to that number of digits. What is the most accurate value that the researcher can announce for $k$?

Answer: 3.57


A frog makes $4$ jumps, each exactly $0.5$ meters long. Determine the probability that the frog's final position is no more than $0.5$ meters from its starting position.

Answer: \frac{1}{8}


Jasmine now has access to three sizes of shampoo bottles: a small bottle that holds 25 milliliters, a medium bottle that holds 75 milliliters, and a large bottle that holds 600 milliliters. Determine the minimum total number of bottles Jasmine must buy to almost completely fill a large bottle, using small bottles for the remaining volume.

Answer: 8


Emily a cell phone plan costs $30 each month, including unlimited calls up to the first 50 hours, and each additional hour is charged at $15. Additionally, each text message costs $0.10, but after the first 150 messages, the cost per message doubles. In February, Emily sent 200 text messages and talked for 52 hours. Calculate the total amount Emily had to pay.

Answer: 85


Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 420 miles, bought 10 gallons of gas, and continued driving until his gas tank was three-quarters full. Determine the total distance Karl drove that day.

Answer: 420


Three years ago, Tom was four times as old as his brother Jerry, and five years before that, Tom was five times as old as Jerry. Determine how many years from now the ratio of their ages will be $3 : 1$.

Answer: 7


It is known that \(\frac{C+E+Д+b+M+O+\breve{И}}{K+Л+A+C+C}=\frac{22}{23}\),

where different letters represent different digits, and the same letters represent the same digits. What values can \(C\) take?

Answer: 0


Mr. and Mrs. Seventh have 7 children, all born on April 1, actually within six consecutive April 10. This year, for their birthdays, Mrs. Seventh made a cake with candles for each one - the number of candles equal to the number of years of each. João Seventh, the child who likes Mathematics the most, noticed that this year the total number of candles is double what it was 2 years ago and that there are 2 more cakes. How many candles will be lit this year?

Answer: 26


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


A point \( P \) is chosen randomly inside a triangle with sides 13, 20, and 21. Find the probability that the circle centered at \( P \) with radius 1 will intersect at least one of the sides of the triangle.

Answer: \frac{75}{196}


Let \( \dot{E} = \{1, 2, 3, \ldots, 200\} \), \( G = \left\{ a_1, a_2, \ldots, a_{100} \right\} \subset E \), and \( G \) satisfies the following properties:
1. For any \( 1 \leq i < j \leq 100 \), we have \( a_i + a_j \neq 201 \);
2. \( \sum_{i=1}^{100} a_i = 10080 \).

Prove that the number of odd numbers in \( G \) is a multiple of 4 and that the sum of the squares of all elements in \( G \) is a fixed value.

Answer: 1349380


Apprentices A and B are discussing who should cooperate with their master to process a batch of parts. A says, "If I cooperate with the master, I will complete 20% of all the work." B says, "That's not a big deal. If I cooperate with the master, I will complete 40% of all the work." At this moment, the master arrives and tells B, "If A joins to help us, you can process 60 fewer parts." If what they said is correct, how many parts are in this batch?

Answer: 1150


Five friends - Katya, Polina, Alena, Lena, and Sveta - gather every day in the park after buying ice cream at the shop around the corner. One day, they had a conversation:

Polina: I was standing next to Alena.

Alena: I was the first in line!

Lena: No one was behind me.

Katya: There were five people in front of me.

Sveta: There was only one person behind me.

The girls are friends, so they do not lie to each other. How many people were there between Katya and Polina?

Answer: 0


Rectangle \( R_{0} \) has sides of lengths 3 and 4. Rectangles \( R_{1}, R_{2} \), and \( R_{3} \) are formed such that:
- all four rectangles share a common vertex \( P \),
- for each \( n=1,2,3 \), one side of \( R_{n} \) is a diagonal of \( R_{n-1} \),
- for each \( n=1,2,3 \), the opposite side of \( R_{n} \) passes through a vertex of \( R_{n-1} \) such that the center of \( R_{n} \) is located counterclockwise of the center of \( R_{n-1} \) with respect to \( P \).

Compute the total area covered by the union of the four rectangles.

Answer: 30


As shown in the figure, fill each cell of a $5 \times 5$ grid with a natural number (natural numbers include 0) such that the average of the four numbers in every $2 \times 2$ sub-square is 100. What is the minimum average of the 25 natural numbers in the entire $5 \times 5$ grid?

Answer: 64


Eva and Marek were playing badminton while Viktor tallied the rallies. After every 10 rallies, Viktor drew an X. Later, he replaced every 10 Xs with an O and erased the corresponding 10 Xs. When Eva and Marek finished the game, Viktor had drawn the following:

## OOOXXXXXXX

Determine the minimum and maximum number of rallies Eva and Marek could have played.

Answer: 379


Given a triangle \(ABC\) such that the median, altitude, and angle bisector from \(A\) divide the angle \(\widehat{A}\) into four equal angles \(\alpha\). Express all the angles in the figure in terms of \(\alpha\) and calculate \(\alpha\).

Answer: 22.5


A set $S$ is constructed as follows. To begin, $S = \{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some nonzero polynomial $a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0$ for some $n\geq{1}$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have?

Answer: 9


The trisectors of angles $B$ and $C$ of scalene triangle $ABC$ meet at points $P$ and $Q$ as shown. Angle $A$ measures 39 degrees and angle $QBP$ measures 14 degrees. What is the measure of angle $BPC$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(4,5)--(0,0)--(2,1.5)--(3,0)--(1.7,0.5)--(0,0));
label("$P$", (1.7,0.5), S); label("$Q$", (2,1.5), N); label("$B$", (0,0),W); label("$A$", (4,5), N); label("$C$", (3,0), E);
[/asy]

Answer: 133


In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6.  Each of the points $W$, $X$, $Y$, and $Z$ is a vertex of one of the small squares.  Square $ABCD$ can be constructed with sides passing through $W$, $X$, $Y$, and $Z$. What is the maximum possible distance from $A$ to $P$? [asy]
path square = scale(2) * unitsquare;

draw(square); draw(shift(4) * square); draw(shift(4, 4) * square); draw(shift(0, 4) * square);
draw((2, 0)--(4, 0)); draw((0, 2)--(0, 4)); draw((6, 2)--(6, 4)); draw((2, 6)--(4, 6));

pair a = shift(3, 4) * dir(135);
pair b = shift(4, 3) * dir(45);
pair c = shift(3, 2) * dir(-45);
pair d = shift(2, 3) * dir(-135);
draw(a--b--c--d--cycle);

label("$2$", (1, 6), N); label("$2$", (3, 6), N); label("$2$", (5, 6), N);

label("$2$", (6, 5), E); label("$2$", (6, 3), E); label("$2$", (6, 1), E);

label("$W$", (2, 4), NW); label("$X$", (4, 4), NE); label("$Y$", (4, 2), SE); label("$Z$", (2, 2), SW);
label("$A$", a, N); label("$B$", b, E); label("$C$", c, S); label("$D$", d, W); label("$P$", (6, 0), SE);
[/asy]

Answer: 6


Lupe went to the store and paid for her purchase with a $\$ 10$ bill. She found that the digits making the amount of her purchase could be rearranged to make the amount she received back in change. If her purchase amount and her change amount were different and each amount was at least $\$1 $, how many possible amounts of change could she have received?

Answer: 8


77. Heat Flow. The temperature of three sides of a square metal sheet is maintained at $0^{\circ} \mathrm{C}$, while the temperature of the fourth side is maintained at $100^{\circ} \mathrm{C}$. Neglecting heat loss due to radiation, find the temperature at the center of the sheet.

Answer: 25


\section*{Problem 3 - 081043}

In a plane \(\epsilon\), let \(k\) be a circle with a given radius \(r\); furthermore, let a natural number \(n>2\) be given.

A diameter \(P Q\) of \(k\) is divided into \(n\) equal parts; the division points are \(R_{1}, R_{2}, \ldots, R_{n-1}\), such that

\[
P R_{1}=R_{1} R_{2}=\ldots=R_{n-2} R_{n-1}=R_{n-1} Q
\]

holds.

One of the two half-planes into which \(\epsilon\) is divided by the line \(g_{P Q}\) is called \(H\), the other \(H^{\prime}\). Then let \(c_{i}\) be the semicircle in \(H\) over \(P R_{i}\), and \(c_{i}^{\prime}\) the semicircle in \(H^{\prime}\) over \(R_{i} Q\), and finally \(b_{i}\) the curve composed of \(c_{i}\) and \(c_{i}^{\prime}\) \((i=1,2,3, \ldots, n-1)\).

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1267.jpg?height=671&width=805&top_left_y=795&top_left_x=634)

Calculate the areas of the regions into which the circular area is divided by any two adjacent curves \(b_{1}, \ldots, b_{n-1}\) or by \(b_{1}\) or \(b_{n-1}\) and the respective semicircle!

Answer: \dfrac{\pi r^2}{n}


The points \( K, L, M \), and \( N \) are the midpoints of the sides \( AB, BC, CD, \) and \( AD \) respectively of a parallelogram \( ABCD \) with area \( s \). Find the area of the quadrilateral formed by the intersection of the lines \( AL, AM, CK, \) and \( CN \).

Answer: \dfrac{s}{3}


Problem 1. Calculate:
a) $\lim _{x \rightarrow \infty}\left(\sqrt{\frac{1}{4} x^{2}+x+2}+\sqrt{\frac{1}{4} x^{2}+2 x+5}-\sqrt{x^{2}+x+1}\right)$;
b) $\lim _{n \rightarrow \infty} \sin \left(\pi \sqrt{n^{2}+3}\right)$.

Answer: \dfrac{5}{2}


Three. (20 points) As shown in Figure 2, $\triangle A O B$ is an equilateral triangle, and the coordinates of point $B$ are $(2,0)$. A line $l$ is drawn through point $C$ $(-2,0)$, intersecting $A O$ at $D$ and $A B$ at $E$, such that the areas of $\triangle A D E$ and $\triangle D C O$ are equal. Find the function expression of line $l$.

Answer: y = \dfrac{\sqrt{3}}{7}(x + 2)


1.59 (1) $f(x)$ is a monovariate quartic polynomial with integer coefficients, i.e.,
$$
f(x)=c_{4} x^{4}+c_{3} x^{3}+c_{2} x^{2}+c_{1} x+c_{0},
$$

where $c_{0}, c_{1}, c_{2}, c_{3}, c_{4}$ are all integers. Prove that if $a, b$ are integers and $a > b$, then $a - b$ divides $f(a) - f(b)$.
(2) Two people, A and B, are doing homework in the classroom.
A asks B: “How many problems have you finished?”
B replies: “The number of problems I have finished is a positive integer and is a root of a monovariate quartic polynomial with integer coefficients. Can you guess it?”
Then, A tries substituting 7 into the polynomial and gets a value of 77. At this point, B says: “The number of problems I have finished is more than 7.”
A says: “Alright, I will try a larger integer.” So A substitutes $B$ into the polynomial and gets a value of 85.
B looks and says: “The number of problems I have finished is more than $B$.” After some thought and calculation, A finally determines the number of problems B has finished.
Try to find the number of problems B has finished based on the conversation.
(China Hebei Province Mathematics Competition, 1979)

Answer: 14


It is known that the polynomial \( f(x) = 8 + 32x - 12x^2 - 4x^3 + x^4 \) has 4 distinct real roots \(\{x_1, x_2, x_3, x_4\}\). The polynomial of the form \( g(x) = b_0 + b_1 x + b_2 x^2 + b_3 x^3 + x^4 \) has roots \(\{x_1^2, x_2^2, x_3^2, x_4^2\}\). Find the coefficient \( b_1 \) of the polynomial \( g(x) \).

Answer: -1216


A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Answer: 1330


Let \( X = (X_1, X_2, \ldots) \) be a Gaussian random sequence, and let \( L \) be an arbitrary linear subspace of \( \mathbb{R}^{\infty} \).

Prove that the "zero-one" law holds for Gaussian systems:

\[ \mathrm{P}(X \in L) = 0 \text{ or } 1. \]

In particular, show that \( \mathrm{P}(\sup_{n} |X_{n}| < \infty) = 0 \) or 1.

Answer: 0


12. As shown in Figure 5, in $\triangle A B C$, it is given that $A C=B C, \angle C=$ $20^{\circ}, D$ and $E$ are points on sides $B C$ and $A C$ respectively. If $\angle C A D=20^{\circ}$, $\angle C B E=30^{\circ}$, find the size of $\angle A D E$.

Answer: 30


Let \(a\), \(b\), and \(c\) be the side lengths of a triangle whose perimeter does not exceed \(2\pi\). Prove that \(\sin a\), \(\sin b\), and \(\sin c\) can also form the side lengths of a triangle.

Answer: \sin a, \sin b, \sin c \text{ can form a triangle}


[ Tournaments and tournament tables ] Counting in two ways
[ Examples and counterexamples. Constructions ]

In the competition, 10 figure skaters participate. The competition is judged by three judges in the following way: each judge independently distributes places (from first to tenth) among the skaters, after which the winner is the skater with the smallest sum of places. What is the maximum value that this sum can take for the winner (the winner is unique)?

Answer: 15


20. (2002 Shanghai Spring College Entrance Examination) Triangular prism $O A B-O A_{1} B_{1}$, plane $O B B_{1} O_{1} \perp O A B, \angle O_{1} O B=$ $60^{\circ}, \angle A O B=90^{\circ}$, and $O B=O O_{1}=2, O A=\sqrt{3}$. Find:
(1) The size of the dihedral angle $O_{1}-A B-O$;
(2) The distance $d$ between the skew lines $A_{1} B$ and $O A$ (this question is adapted from the original).

Answer: \arccos \dfrac{\sqrt{2}}{4}


1. Determine all ordered pairs $(a, b)$ of integers for which $\frac{1}{a}-\frac{1}{b}=\frac{1}{5}$.

Answer: (4, 20)


Example 8 A deck of tri-color cards consists of 32 cards in total, with 10 cards of each color: red, yellow, and blue, numbered $1, 2, \cdots, 10$; there are also a big joker and a small joker, each numbered 0. Several cards are drawn from this deck, and the score is calculated according to the following rule: each card with number $k$ is worth $2^{k}$ points. If the total score of these cards is 2004, they are called a "good" card group. Find the number of "good" card groups.

Answer: 1006009


4. Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_{1}, x_{2}, \ldots, x_{n}$. Find all solutions for each such $n$.
$$
\begin{aligned}
x_{1}+x_{2}+\cdots+x_{n} & =16 \\
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}} & =1
\end{aligned}
$$

Answer: 4


$$
\begin{array}{l}
\text { 13. Given the function } f(x)=-2 x+4 \text {, let } \\
S_{n}=f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f(1) \\
\left(n \in \mathbf{N}_{+}\right) \text {. }
\end{array}
$$

If the inequality $\frac{a^{n}}{S_{n}}<\frac{a^{n+1}}{S_{n+1}}$ always holds, find the range of real number $a$.

Answer: (\frac{5}{2}, \infty)


A box contains 111 balls: red, blue, green, and white. It is known that if you draw 100 balls without looking into the box, you will definitely get at least four balls of different colors. What is the minimum number of balls that need to be drawn, without looking into the box, to ensure that you get at least three balls of different colors?

Answer: 88


Fill integers $1, 2, 3, \cdots, 2016^{2}$ into a $2016 \times 2016$ grid, with each cell containing one number, and all numbers being different. Connect the centers of any two cells with a vector, with the direction from the smaller number to the bigger number. If the sum of numbers in each row and each column of the grid is equal, find the sum of all such vectors.

Answer: 0


Which fraction \( \frac{p}{q} \), where \( p \) and \( q \) are positive integers less than 100, is closest to \( \sqrt{2} \)? Find all digits after the decimal point in the decimal representation of this fraction that coincide with digits in the decimal representation of \( \sqrt{2} \) (without using any tables).

Answer: \frac{99}{70}


Let \( A, B, C \) be points in that order along a line, such that \( AB = 20 \) and \( BC = 18 \). Let \( \omega \) be a circle of nonzero radius centered at \( B \), and let \( \ell_{1} \) and \( \ell_{2} \) be tangents to \( \omega \) through \( A \) and \( C \), respectively. Let \( K \) be the intersection of \( \ell_{1} \) and \( \ell_{2} \). Let \( X \) lie on segment \(\overline{KA}\) and \( Y \) lie on segment \(\overline{KC}\) such that \( XY \parallel BC \) and \( XY \) is tangent to \( \omega \). What is the largest possible integer length for \( XY \)?

Answer: 35


A convex polyhedron has \( n \) faces that are all congruent triangles with angles \( 36^{\circ} \), \( 72^{\circ} \), and \( 72^{\circ} \). Determine, with proof, the maximum possible value of \( n \).

Answer: 36


Three individuals, $A$, $B$, and $C$, visit teacher $D$'s home, where $D$ gives each person a hat with a four-digit number on it. It is known that all three four-digit numbers are perfect squares (e.g., $4 = 2^2$, $100 = 10^2$; such numbers are called perfect squares). Additionally, each of these four-digit numbers has a tens digit of 0 and a units digit that is not 0. Each person can see the numbers on the others' hats but not their own. The following conversation occurs among them:

$A$ says, "The units digits of the numbers on $B$'s and $C$'s hats are the same."
$B$ and $C$ simultaneously say, "After hearing what $A$ said, I know my own number."
$A$ says, "After hearing what $B$ and $C$ said, I also know my own number, and the units digit of my number is an even number."

Determine the sum of the numbers on $A$'s, $B$'s, and $C$'s hats.

Answer: 14612


The base of the pyramid $ABCD$ is an equilateral triangle $ABC$ with side length 12. The edge $BD$ is perpendicular to the base plane and equals $10\sqrt{3}$. All vertices of this pyramid lie on the lateral surface of a right circular cylinder whose axis intersects the edge $BD$ and the plane $ABC$. Determine the radius of the cylinder.

Answer: \frac{20\sqrt{3}}{\sqrt{17}}


The Wrestling Federation assigned each participant in the competition a qualification number. It is known that in matches between wrestlers whose qualification numbers differ by more than 2, the wrestler with the smaller number always wins. The tournament for 256 wrestlers is conducted using an Olympic system: at the beginning of each day, the fighters are paired, and the loser is eliminated from the competition (there are no ties). What is the highest qualification number that the winner can have?

Answer: 16


Four spheres are placed inside a cone such that they touch each other externally and the lateral surface of the cone. Three of the spheres have a radius of 3 and also touch the base of the cone. Find the radius of the fourth sphere if the angle between the slant height and the base of the cone is $\frac{\pi}{3}$.

Answer: 9-4\sqrt{2}


Given \(n (n>1)\) as a positive integer, let \(a_{k}=\left(\frac{n-k}{k}\right)^{k-\frac{n}{2}}\) for \(0 < k \leq n\), \(k \in \mathbf{N}\). Find the maximum value of the sequence \(\{a_{k}\}\).

Answer: 1


Let \( E = \{1,2,3, \cdots, 200\} \) and \( G = \{a_{1}, a_{2}, a_{3}, \cdots, a_{100}\} \subset E \). The set \( G \) possesses the following two properties:
I) For any \( 1 \leq i < j \leq 100 \), it holds that \( a_{i} + a_{j} \neq 201 \);
II) \( \sum_{i=1}^{100} a_{i} = 10080 \).

Prove that the number of odd numbers in \( G \) is a multiple of 4, and that the sum of the squares of all numbers in \( G \) is a constant.

Answer: 1349380


Gary and Deep play a game in which there are no ties. Each player is equally likely to win each game. The first player to win 4 games becomes the champion, and no further games are played. Gary wins the first two games. What is the probability that Deep becomes the champion?

Answer: \frac{3}{16}


The function \( f(x) \) is a strictly increasing function defined on \(\mathbf{N}\) and taking values in \(\mathbf{N}\) (a function \( f(x) \) is called a strictly increasing function on a set \(A\) if for any \( x_1, x_2 \in A \), when \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \)). Additionally, it satisfies the condition \( f(f(k)) = 3k \). Find the value of \( f(1) + f(9) + f(96) \).

Answer: 197


Let $AB$, multiplied by $C$, equal $DE$. If $DE$ is subtracted from $FG$, the result is $HI$.

Each letter represents a distinct digit (1, 2, 3, 4, 5, 6, 7, 8, or 9). The digit 0 does not appear in the problem.

Answer: 93-68=25


Given 2000 points on a circle, starting from one point labeled with the number 1, move clockwise to the second point and label it with the number 2. Continue this process such that from the point labeled with number \( n \), move clockwise \( n \) points to label the next point with the number \( n+1 \). Continue this process until the points are labeled from 1 to 1993. Some points may be labeled with more than one number, and some points may not be labeled at all. What is the smallest number present on the point labeled with 1993?
(11th Annual American Invitational Mathematics Examination, 1993)

Answer: 118


A cylinder \( C \) and a cone \( K \) are circumscribed around a sphere; the volumes of the cylinder and cone are denoted by \( V_{1} \) and \( V_{2} \), respectively. What is the minimum possible value of \( v \) if \( V_{1} = 1 \)? What is the angle at the apex of the axial section of the circumscribed cone \( K \) when its volume is \(\boldsymbol{0}\)?

Answer: \frac{4}{3}


In a space with 1989 points, no three of which are collinear, divide these points into 30 groups with distinct numbers of points in each group. If a triangle is formed by choosing one point from each of any three different groups, what should the number of points in each group be to maximize the total number of such triangles?

Answer: 51, 52, 53, 54, 55, 56, 58, 59, \ldots, 81


Consider an acute-angled triangle \( A B C \) with \(|A B|<|A C|\). Let \( M \) and \( N \) be the midpoints of segments \(\overline{B C}\) and \(\overline{A B}\), respectively. The circle with diameter \(\overline{A B}\) intersects the lines \(B C, A M\) and \(A C\) at \(D\), \(E\), and \(F\), respectively. Let \(G\) be the midpoint of \(\overline{F C}\). Prove that the lines \(N F, D E\) and \(G M\) are concurrent.

Answer: \text{The lines } NF, DE, \text{ and } GM \text{ are concurrent.}


8. If one element is deleted from the set $\{1!, 2!, \cdots, 100!\}$, and the product of the remaining elements is exactly a perfect square, then the deleted element is $\qquad$ .

Answer: 50!


 Touching Circles

Two circles touch each other externally at point $A$. Through point $B$ on their common tangent $A B$, two lines are drawn, one intersecting the first circle at points $M$ and $N$, and the other intersecting the second circle at points $P$ and $Q$. It is known that $A B=6, B M=9, B P=5$. Find the ratio of the areas of triangles $M N O$ and $P Q O$, where point $O$ is the intersection of lines $M P$ and $N Q$.

Answer: \dfrac{625}{121}


IMO 1974 Problem A1 Three players play the following game. There are three cards each with a different positive integer. In each round the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, one player has received 20, another 10 and the third 9 counters. In the last round the player with 10 received the largest number of counters. Who received the middle number on the first round?

Answer: 9


2. In the quadrilateral $A B C D$, it holds that $\measuredangle A B C=104^{\circ}, \measuredangle A D C=128^{\circ}$, and $A B=B C=2$. Calculate the length of the diagonal $B D$.

Answer: 2


6. Points $A_{1}, \ldots, A_{12}$ are the vertices of a regular 12-gon. How many different 11-segment open broken lines without self-intersections with vertices at these points exist? Broken lines that can be transformed into each other by rotation are considered the same.

Answer: 1024


Each edge of the tetrahedron \(PABC\) is equal to \(1\); \(BD\) is the height of the triangle \(ABC\). The equilateral triangle \(BDE\) lies in a plane forming an angle \(\phi\) with the edge \(AC\), and the points \(P\) and \(E\) lie on the same side of the plane \(ABC\). Find the distance between the points \(P\) and \(E\).

Answer: \dfrac{\sqrt{5 - 2\sqrt{6} \sin \phi}}{2}


6. $A, B$ two people are playing a table tennis match, with the rule that the first to win 3 games more than the opponent wins. After 13 games, $A$ finally wins with a record of 8 wins and 5 losses. Then the number of all different possible outcomes of these 13 games is $\qquad$

Answer: 243


2. Eight knights are randomly placed on a chessboard (not necessarily on distinct squares). A knight on a given square attacks all the squares that can be reached by moving either (1) two squares up or down followed by one squares left or right, or (2) two squares left or right followed by one square up or down. Find the probability that every square, occupied or not, is attacked by some knight.

Answer: 0


8. In the school Spartakiad, teams from classes $8^{\mathrm{A}}$, $8^{\mathrm{E}}$, and $8^{\mathrm{B}}$ participated. In each of the competitions, one of these teams took 1st place, another took 2nd place, and another took 3rd place. After the Spartakiad, points were tallied: $x$ points were awarded for 1st place, $y$ for 2nd place, and $z$ for 3rd place $\left(x>y>z>0\right.$ - integers). In the end, team $8^{\mathrm{A}}$ received 22 points, and teams $8^{\text {B }}$ and $8^{\text {E }}$ each received 9 points. How many competitions were there in total, and which team took 2nd place in the grenade throwing competition, given that the team $8^{\mathrm{B}}$ took 1st place in the vault over the "buck"?

Answer: 5


9. Wang Qiang has four colors of small cylindrical rods, Table 1 lists the lengths of the rods of different colors.
Table 1
\begin{tabular}{|c|c|c|c|c|}
\hline Color & Green & Yellow-Red & Purple & Red \\
\hline Length & $3 \mathrm{~cm}$ & $4 \mathrm{~cm}$ & $8 \mathrm{~cm}$ & $9 \mathrm{~cm}$ \\
\hline
\end{tabular}

Now, several small rods are to be taken and joined together to form a long rod of length $2010 \mathrm{~cm}$, and each of the four colors of small rods must be used at least 81 times. Then the number of different ways to do this is.

Answer: 91


On a $4 \times 4$ board, the numbers from 1 to 16 must be placed in the cells without repetition, such that the sum of the numbers in each row, column, and diagonal is the same. We call this sum the Magic Sum.

a) What is the Magic Sum of this board?
b) If the sum of the cells marked with $X$ in the board below is 34, what is the sum of the cells marked with $Y$?

| $Y$ |  |  | $Y$ |
| :--- | :--- | :--- | :--- |
|  | $X$ | $X$ |  |
|  | $X$ | $X$ |  |
| $Y$ |  |  | $Y$ |

c) If we fill in with consecutive natural numbers from $k$ to $(k+15)$, such that the Magic Sum is 50, what is the value of $k$?

Answer: 34


One side of an equilateral triangle is equal to \( a \). A circle with a radius of \( a/3 \) is drawn from its center. Determine the area of the part of the triangle lying outside the circle.

Answer: \dfrac{a^2 (3\sqrt{3} - \pi)}{18}


15.1. $[9.3$ (15 points)] In trapezoid $ABCD$, diagonal $AC$ is equal to 1 and is also its height. Perpendiculars $AE$ and $CF$ are drawn from points $A$ and $C$ to sides $CD$ and $AB$ respectively. Find $AD$, if $AD=CF$ and $BC=CE$.

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


5.66 A deck of cards has a total of 54 cards. The magician divides them into several piles, and the audience writes a natural number on each card, which is equal to the number of cards in the pile where the card is located. Then the magician mixes the cards in a special way and divides them into several piles again. The audience writes a natural number on each card again, which is also equal to the number of cards in the pile where the card is now located. This process continues. How many times does this process of dividing piles and writing numbers need to be repeated at least, so that the (unordered) number arrays written on each card are all different?

Answer: 3


There are $2n$ passengers on a flight, and the airline has loaded $n$ chicken meals and $n$ fish meals for them. It is known that a passenger prefers chicken with a probability of 0.5, and prefers fish with a probability of 0.5. We consider a passenger dissatisfied if they do not get what they prefer.

a) Find the most likely number of dissatisfied passengers.

b) Find the expected number of dissatisfied passengers.

c) Find the variance of the number of dissatisfied passengers.

Answer: 1


2. Transform each equation by bringing the fractions to a common denominator. We obtain

$$
\left\{\begin{array}{l}
(\sqrt{2}+1)(x-y)=\sqrt{2}\left(x^{2}+y^{2}\right)-3 x y \\
(\sqrt{2}-1)(x-y)=\sqrt{2}\left(x^{2}+y^{2}\right)+3 x y
\end{array}\right.
$$

if $x \neq \pm \sqrt{2} y$ and $y \neq \pm \sqrt{2} x$. Adding and subtracting these equations, we find

$$
\left\{\begin{array}{c}
x-y=x^{2}+y^{2} \\
y-x=3 x y
\end{array}\right.
$$

Therefore, $x^{2}+3 x y+y^{2}=0$, from which either $x=y=0$ (this option does not fit due to the conditions stated above), or $y=\frac{-3 \pm \sqrt{5}}{2} x \neq 0$. Then from the second equation, we find $\frac{-3 \pm \sqrt{5}}{2} \cdot 3 x^{2}=\frac{-5 \pm \sqrt{5}}{2} x$, so $x=$ $\frac{-5 \pm \sqrt{5}}{3 \cdot(-3 \pm \sqrt{5})}=\frac{5 \pm \sqrt{5}}{6}$, and then $y=\frac{-5 \pm \sqrt{5}}{6}$ respectively (here the signs are taken either both upper or both lower).

Answer: \left(\frac{5 + \sqrt{5}}{6}, \frac{-5 + \sqrt{5}}{6}\right)


Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$ , and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$ . Find the sum of all possible values of $|b|$ .
Please give the answer directly without any intermediate steps.

Answer: 420


Let $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{2n+1}$ be vectors of length 1. Prove that in the sum $\mathbf{c}= \pm \mathbf{a}_{1} \pm \mathbf{a}_{2} \pm \ldots \pm \mathbf{a}_{2n+1}$, the signs can be chosen such that $|\mathbf{c}| \leq 1$.

Answer: |\mathbf{c}| \leq 1


Problem 10.4. (15 points) Once, 45 friends living in different parts of the world decided to exchange news with each other. To do this, they plan to arrange $k$ video meetings, at each of which every person will share their news, as well as all the news from other people they have learned previously.

For the video meetings, 10 days were proposed, but it turned out that each friend can only be present on 8 of them. What is the smallest natural number $k$ that guarantees we can select $k$ days for the video meetings from the proposed 10 so that everyone learns the news from everyone else?

(Between the proposed days, no new news arises among the people, and they do not communicate with each other in any other way. On each of the proposed days, one video meeting is held, which is attended by everyone who can be present on that day.)

Answer: 5


62nd Putnam 2001 Problem B3 Let @n denote the closest integer to √n. Find ∑ (2 @n + 2 -@n )/2 n , where the sum is taken from n = 1 to n = infinity. Solution

Answer: 3


\section*{Problem 1 - 101211}

In a parent-teacher meeting, exactly 18 fathers and exactly 24 mothers were present, with at least one parent of each student in the class attending.

Of exactly 10 boys and exactly 8 girls, both parents were present for each. For exactly 4 boys and exactly 3 girls, only the mother was present, while for exactly 1 boy and exactly 1 girl, only the father was present.

Determine the number of all those children in this class who have siblings in the same class! (There are no children in this class who have step-parents or step-siblings.)

Answer: 4


90. In the morning, Xiao Wang rode a bike from place A to place B. At noon 12 o'clock, Xiao Li drove a car from place A to place B as well. At 1:30 PM, the distance between the two was 18 kilometers, and at 2:30 PM, the distance between them was again 18 kilometers. Xiao Li arrived at place B at 4 PM, and Xiao Wang arrived at place B at 6 PM. Xiao Wang set off in the morning at $\qquad$ o'clock.

Answer: 10


For each positive integer $n$, define the function $f(n)=\left\{\begin{array}{ll}0 & \text { when } n \text { is a perfect square, } \\ {\left[\frac{1}{\{\sqrt{n}\}}\right]} & \text { when } n \text { is not a perfect square. }\end{array}\right.$ (where $[x]$ denotes the greatest integer not exceeding $x$, and $\{x\}=x-[x])$, find: $\sum_{k=1}^{240} f(k)$.

---

The function $f(n)$ is defined as follows:
- $f(n) = 0$ if $n$ is a perfect square.
- $f(n) = \left[\frac{1}{\{\sqrt{n}\}}\right]$ if $n$ is not a perfect square.

Here, $[x]$ represents the greatest integer less than or equal to $x$, and $\{x\}$ represents the fractional part of $x$, which is $x - [x]$.

We need to find the value of $\sum_{k=1}^{240} f(k)$.

Answer: 768


4. An isosceles triangle can be cut into two isosceles triangles (not necessarily equal). Find the values that the smallest angle of such a triangle can take. In your answer, specify the smallest of these values in degrees, multiplied by 6006.

ANSWER: 154440.

Answer: 154440


First, a boat traveled $a$ km across a lake and then half of that distance along a river flowing into the lake. The entire trip lasted 1 hour. Find the boat's speed in still water, given that the speed of the river's current is $с$ km/h.

Answer: \frac{3a + 2c + \sqrt{9a^2 - 4ac + 4c^2}}{4}


On a bookcase there are $ n \geq 3$ books side by side by different authors. A librarian considers the first and second book from left and exchanges them iff they are not alphabetically sorted. Then he is doing the same operation with the second and third book from left etc. Using this procedure he iterates through the bookcase three times from left to right. Considering all possible initial book configurations how many of them will then be alphabetically sorted?

Answer: 6 \cdot 4^{n-3}


1. For what smallest natural $k$ does the number 2016 $\cdot$ 20162016 $\cdot$ 201620162016 $\cdot$... 20162016...2016( $k$ factors) divide without remainder by $3^{67}$?

Answer: 27


Two circles with radii \( R \) and \( r \) are tangent to each other either internally or externally (\( R > r \)). Prove that the radius \( r_{0} \) of a circle that is tangent to both given circles and their line of centers is determined by the formula:

\[ r_{0} = \frac{4Rr(R \mp r)}{(R \pm r)^{2}} \]

where the upper signs correspond to internal tangency.

Answer: r_0 = \frac{4Rr(R \mp r)}{(R \pm r)^2}


During the Teachers' Day, 200 teachers at a school sent text messages to greet each other, with each teacher sending exactly 1 text message to another teacher. Now, from them, the maximum number of $k$ teachers can be selected to attend an award ceremony, such that none of them has sent a text message to any of the others. Try to find the minimum value of $k$.

Answer: 67


2. Given the vertices of a regular 120-gon $A_{1}, A_{2}, A_{3}, \ldots, A_{120}$. In how many ways can three vertices be chosen from them to form an obtuse triangle? (10 points)

Answer: 205320


7. A turtle and a rabbit are in a 1000-meter race. The rabbit took a nap 200 meters from the finish line. When it woke up, it found that the turtle was 10 meters from the finish line, so it immediately chased after and eventually both reached the finish line at the same time. If the turtle crawled at a constant speed throughout the race, and the rabbit's speed before and after the nap was the same, then, during the rabbit's nap, the turtle crawled $\qquad$ meters.

Answer: 950


Let \( a, b \), and \( c \) be complex numbers such that \( |a|=|b|=|c|=1 \). If
\[ \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=1 \]
as well, determine the product of all possible values of \( |a+b+c| \).

Answer: 2


Let $\mathbb{R}^{+}=(0, \infty)$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0)=0$ which satisfy the equality

$$
f(f(x)+P(y))=f(x-y)+2 y
$$

for all real numbers $x>y>0$.

Answer: f(x) = x


Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.

Answer: 665


Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$. Find the maximum possible value of $AB$.

Answer: 69


105(974). Determine the value of $b$, if the graph of the function defined by the formula passes through the point with coordinates $(3 ; 10)$:

1) $y=x+b$;
2) $y=3 x+b$
3) $y=-\frac{1}{3} x+b$;
4) $y=-\frac{1}{2} x+b$.

A n s w e r: 1) $b=7$; 2) $b=1$; 3) $b=11$; 4) $b=11.5$.

106(975). Define the function by a formula whose graph is a straight line passing through points $A$ and $B$:

1) $A(-6 ;-3), B(2 ;-3)$
2) $A(-4 ;-4), B(3,3)$
3) $A(2 ; 2), B(0 ; 4)$;
4) $A(3 ;-8), B(-5 ; 32)$.

A n s w e r: 1) $y=-3$; 2) $y=x$; 3) $y=-x+4$; 4) $y=-5 x+7$.

N o t e. During club sessions, students can be introduced to the equation of a straight line passing through two points with given coordinates. Otherwise, solving such types of problems reduces to solving a system of two linear equations with two unknowns, which we obtain by substituting the coordinates of the given points into the equation of the line $y=k x+b$. Thus, solving problem (4) we get the system:

$$
\left\{\begin{array}{l}
-8=3 k+b \\
32=-5 k+b
\end{array}\right.
$$

solving which we find $k=-5, b=7$, hence $y=-5 x+7$.

$\triangle$ 107(976). A cylindrical helical spring of height 122 mm is to be made from steel wire with a diameter of 5 mm. Find the number of turns of the spring, if the gap between the turns of the unloaded spring should be 8 mm.

Answer: 10


G2.1 On a $3 \times 3$ grid of 9 squares, each squares is to be painted with either Red or Blue. If $\alpha$ is the total number of possible colouring in which no $2 \times 2$ grid consists of only Red squares, determine the value of $\alpha$.

Answer: 417


8.1. In a circle, 58 balls of two colors - red and blue - are arranged. It is known that the number of triples of consecutive balls, among which there are more red ones, is the same as the number of triples with a majority of blue ones. What is the smallest number of red balls that could be present?

![](https://cdn.mathpix.com/cropped/2024_05_06_6c94238ddc8917ebef5dg-8.jpg?height=891&width=897&top_left_y=231&top_left_x=591)

Answer: 20


## Task A-4.1.

A natural number is called Babylonian if it is greater than 9 and if its representation in base 60 is equal to its decimal representation without the leading digit. For example, the number 123 is Babylonian because $123=(23)_{60}$. How many Babylonian numbers are there less than 10000?

Answer: 50


Four. (15 points) As shown in Figure 3, in the Cartesian coordinate system $x O y$, $\odot O: x^{2}+y^{2}=4$ intersects the positive half-axis of the $x$-axis at point $A$, $\odot A:(x-2)^{2}+y^{2}=r^{2}(r>0)$ intersects $\odot O$ at points $B$ and $C$.
(1) Find the minimum value of $\overrightarrow{A B} \cdot \overrightarrow{A C}$;
(2) Let $P$ be any point on $\odot O$ other than $B$ and $C$, and let the lines $P B$ and $P C$ intersect the $x$-axis at points $M$ and $N$, respectively. Find the maximum value of $S_{\triangle P O M} S_{\triangle P O N}$.

Answer: -2


1. (mathematics) The probabilities of finding Kuzia the flea at vertices $A, B, C$ and $D$ of the square $A B C D$ are $p_{1}, p_{2}, p_{3}$ and $p_{4}$ respectively $\left(p_{1}+p_{2}+p_{3}+p_{4}=1\right)$. The flea jumps to the nearest vertex of the square, and which of the two it jumps to is completely random and equally likely. Find the probability of finding Kuzia at vertex $B$ after her 2019th jump.

Answer: \dfrac{p_1 + p_3}{2}


20. In the figure below, $A B C$ is a triangle with $A B=10 \mathrm{~cm}$ and $B C=40 \mathrm{~cm}$. Points $D$ and $E$ lie on side $A C$ and point $F$ on side $B C$ such that $E F$ is parallel to $A B$ and $D F$ is parallel to $E B$. Given that $B E$ is an angle bisector of $\angle A B C$ and that $A D=13.5 \mathrm{~cm}$, fmd the length of $C D$ in $\mathrm{cm}$.

Answer: 24


Let \( f(x) \) be an even function defined on \(\mathbf{R}\) with a period of 2 and strictly decreasing on the interval \([0,1]\), such that \( f(\pi) = 1 \) and \( f(2\pi) = 2 \). Determine the solution set of the system of inequalities \(\left\{\begin{array}{l}1 \leqslant x \leqslant 2, \\ 1 \leqslant f(x) \leqslant 2\end{array}\right.\).

Answer: [\pi - 2, 8 - 2\pi]


Three, (50 points) Find the maximum value of the positive integer $r$ such that: for any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ elements in common.

Find the maximum value of the positive integer $r$ such that: for any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ elements in common.

Answer: 200


## Task A-1.3.

Dino, Pino, and Tino go to the same kindergarten. For the game, each boy needs two dice of the same color, but it is not necessary that the dice of different boys be of different colors. The caregiver has red, blue, and green dice in one box. If she draws without looking, how many dice does she need to draw from the box to be sure that each boy can take two dice of the same color?

Answer: 8


There are 400 tons of cargo stored in a warehouse, each with a weight that is a multiple of a quintal and does not exceed 10 tons. It is known that any two pieces of cargo have different weights. What is the minimum number of trips a 10-ton truck must make to guarantee the transportation of all this cargo from the warehouse?

Answer: 51


8. Among the following four numbers $1307674368000$, $1307674368500$, $1307674368200$, $1307674368010$, only one is exactly the product of the integers from 1 to 15. This number is ( ).

Answer: 1307674368000


Consider three points $P(0,-\sqrt{2}),\ Q(0,\ \sqrt{2}),\ A(a,\ \sqrt{a^2+1})\ (0\leq a\leq 1)$.

(1) Show that the difference of two line segments $PA-AQ$ is constant regardless of $a$, then find the value.

(2) Let $B$ be the point of intersection between the half-line passing through $A$ with the end point $Q$ and the parabola $y=\frac{\sqrt{2}}{8}x^2$, and let $C$ be the point of intersection between the perpendicular line drawn from the point $B$ to the line $y=2$.
Show that the sum of the line segments $PA+AB+BC$ is constant regardless of $a$, then find the value.

Answer: 2


$\underline{\text { Frankin B.R. }}$ 5.

A billiard table has the shape of a rectangle $2 \times 1$, with pockets located at the corners and at the midpoints of the longer sides. What is the smallest number of balls that need to be placed inside the rectangle so that each pocket lies on a line with some two balls?

Answer: 4


$$
\begin{array}{l}
\text { 2. Compare the sizes: } \frac{1}{2}(\sqrt{1998}-\sqrt{2000}) \\
\sqrt{1998}-\sqrt{1999} .
\end{array}
$$
3. Given that $\alpha$ is a root of the equation $x^{2}-3 x-1=0$. Then $\alpha^{6}+120 \alpha^{-2}=$ $\qquad$ .

Answer: \frac{1}{2}(\sqrt{1998} - \sqrt{2000}) > \sqrt{1998} - \sqrt{1999}


A convoy of trucks is delivering supplies to a disaster relief center. Each truck has a capacity of 10 tons. If each tent receives 1.5 tons of supplies, there are supplies left that are less than one truckload. If each tent receives 1.6 tons of supplies, there is a shortage of supplies equating to more than 2 truckloads. What is the minimum number of tents at this relief center?

Answer: 213


The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron.

Answer: \dfrac{168 + 112\sqrt{2}}{3}


A circle is circumscribed around a right triangle. Another circle with the same radius touches the legs of this triangle, with one of the points of tangency being the vertex of the triangle. Find the ratio of the area of the triangle to the area of the common part of the two given circles.

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


There are 100 distinct points on a plane and \( n \) different lines \( l_{1}, l_{2}, \cdots, l_{n} \). Let \( a_{k} \) denote the number of points through which the line \( l_{k} \) passes. If \( a_{1} + a_{2} + \cdots + a_{n} = 250 \), find the smallest possible value of \( n \).

Answer: 21


In the cells of a \(75 \times 75\) table, pairwise distinct natural numbers are placed. Each of them has no more than three different prime divisors. It is known that for any number \(a\) in the table, there exists a number \(b\) in the same row or column such that \(a\) and \(b\) are not coprime. What is the maximum number of prime numbers that can be in the table?

Answer: 4218


Given a trapezoid \(ABCD\) with \(BC \parallel AD\). Point \(H\) on side \(AB\) is such that \(\angle DHA = 90^\circ\). It is known that \(CH = CD = 13\) and \(AD = 19\). Find the length of segment \(BC\).

Answer: 9.5


Petya wants to place 99 coins on a $2 \times 100$ board such that no two coins are in adjacent cells with a common side, and no more than one coin is placed in each cell. In how many ways can he place the coins?

Answer: 396


A $4 \times 8$ paper rectangle is folded along the diagonal as shown in the figure. What is the area of the triangle that is covered twice?

Answer: 10


On the first rank of a chessboard, there are 8 black queens, and on the last rank, there are 8 white queens. What is the minimum number of moves required for the white queens to switch places with the black queens? White and black queens move alternately, one queen per turn.

Answer: 23


A total of 99 people participated in a math competition, which was divided into three sections: geometry, number theory, and combinatorics. Xiao Ming ranked 16th in the number theory exam, 30th in the combinatorics exam, and 23rd in the geometry exam. Xiao Ming did not tie with anyone in any of the exams. The final overall ranking is determined by summing the scores of the three exams and then ranking from high to low. If we use the rank $\mathrm{A}$ to represent the best possible overall ranking Xiao Ming could achieve (the smaller the $\mathrm{A}$, the better the ranking), and the rank $\mathrm{B}$ to represent the worst possible overall ranking Xiao Ming could achieve, then $100 \cdot A + B = (\quad)$.

Answer: 167


A ladybird is free to fly between the \(1 \times 1\) cells of a \(10 \times 10\) square grid. She may begin in any \(1 \times 1\) cell of the grid. Every second she flies to a different \(1 \times 1\) cell that she has not visited before. Find the smallest number of cells the ladybird must visit, including her starting cell, so that you can be certain that there is a \(2 \times 2\) grid of adjacent cells, each of which she has visited.

Answer: 76


Vanya wrote the number 1 on the board and then added several more numbers. Each time Vanya writes a new number, Mitya calculates the median of the existing set of numbers and writes it down in his notebook. At a certain moment, the following numbers are recorded in Mitya's notebook: $1 ; 2 ; 3 ; 2.5 ; 3 ; 2.5 ; 2 ; 2 ; 2 ; 5$.

a) What number was written on the board fourth?

b) What number was written on the board eighth?

Answer: 2


Arrange the elements of the set \(\left\{2^{x}+2^{y} \mid x, y \in \mathbf{N}, x < y\right\}\) in ascending order. What is the 60th number? (Answer with a number).

Answer: 2064


A copper wire with cross-sectional area $A$ carries a current $I$ during a short circuit. Initially, the wire's temperature was $T_{0}$. Estimate how long it will take for the wire to melt. Ignore radiation losses. The resistivity of the material depends on temperature: $\varrho=\varrho_{0}\left[1+\alpha\left(T-T_{0}\right)\right]$, where $\varrho_{0}$ is the resistivity at temperature $T_{0}$ and $\alpha$ is the temperature coefficient of the material. The temperature $T$ is given in Kelvin. Data: $\alpha=\frac{1}{235} \ {}^{\circ} \mathrm{C}^{-1}$, $\varrho_{0}=1.65 \cdot 10^{-8} \Omega \mathrm{m}$, $A=2 \mathrm{~mm}^{2}$, $I=120 \mathrm{~A}$, $T_{0}=313 \mathrm{~K}$.

Answer: 23


The bank issued a loan to citizen \( N \) on September 9th in the amount of 200 thousand rubles. The loan repayment date is November 22nd of the same year. The interest rate is 25% per annum. Determine the amount (in thousand rubles) that citizen \( N \) must return to the bank. Assume that the year has 365 days, and the bank accrues daily interest on the loan amount, excluding the days of loan issuance and repayment.

Answer: 210


In a \(2000 \times 2000\) grid, each cell contains either a 1 or a -1, and the sum of all the numbers in the grid is non-negative. Prove that it is possible to find 1000 rows and 1000 columns such that the sum of the numbers in the cells at their intersections is at least 1000.

Answer: 1000


Let \( f(n) \) be a function satisfying the following three conditions for all positive integers \( n \) :
(a) \( f(n) \) is a positive integer,
(b) \( f(n+1) > f(n) \),
(c) \( f(f(n)) = 3n \).
Find \( f(2001) \).

Answer: 3816


At a certain school, there are 6 subjects offered, and a student can take any combination of them. It is noticed that for any two subjects, there are fewer than 5 students taking both of them and fewer than 5 students taking neither. Determine the maximum possible number of students at the school.

Answer: 20


Place two letters $a$ and two letters $b$ in the 16 cells shown in the figure, with at most one letter per cell. If the same letters are neither in the same row nor in the same column, find the number of different ways to do this.

Answer: 3960


You are allowed to cut out any 18 squares from a $20 \times 20$ chessboard, and after that, place several rooks on the remaining squares such that no two rooks attack each other. Rooks attack each other if they are in the same row or column of the board and there are no cut-out squares between them. What is the maximum number of rooks that can be placed under these conditions?

Answer: 38


Given a cube \( ABCD-A_1B_1C_1D_1 \) with edge length 2, point \( P \) lies on the incircle \( \odot I \) of square \( ABCD \), and point \( Q \) lies on the circumcircle \( \odot O \) of square \( AA_1D_1D \). Find the maximum and minimum values of \( |PQ| \).

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


Two wise men play the following game. The numbers $0,1,2, \ldots, 1024$ are written down. The first wise man crosses out 512 numbers (of his choice), the second wise man then crosses out 256 of the remaining numbers, and then the first wise man crosses out 128 numbers, and so on. On the tenth turn, the second wise man crosses out one number; two numbers remain. After this, the second wise man pays the first wise man the difference between these two numbers. How should the first wise man play to his advantage? How should the second? How much will the second wise man pay the first if both play optimally?

Answer: 32


As shown in the figure, three equally sized square sheets of paper in red, yellow, and blue are stacked in a large square box. Given that the exposed areas are 25 for the blue sheet, 19 for the red sheet, and 11 for the yellow sheet, what is the area of the large square box?

Answer: 64


There are 7 safes and 7 codes for them, but it is unknown which code matches which safe. What is the minimum number of attempts needed to accurately match all the codes to the safes?

Answer: 21


In a regular triangular prism \(P-ABC\) where all edge lengths are \(1\), points \(L\), \(M\), and \(N\) are the midpoints of edges \(PA\), \(PB\), and \(PC\) respectively. Determine the area of the cross-section of the circumscribed sphere intersected by the plane \(LMN\).

Answer: \frac{\pi}{3}


On the island of Unfortune, there live knights who always tell the truth and liars who always lie. One day, 2022 natives gathered at a round table, and each of them made the statement:

"Next to me sit a knight and a liar!"

It is known that three knights made a mistake (i.e., unintentionally lied). What is the maximum number of knights that could have been at the table?

Answer: 1349


What is the greatest number of subsequences of the form \( n, n+1, n+2 \) that a sequence of 2007 natural numbers can have? For example, the sequence \( 1,2,2,3,3 \) has 4 such subsequences.

Answer: 669^3


Three people, A, B, and C, are sawing wooden sticks of the same thickness. They receive sticks of 8 meters, 10 meters, and 6 meters in length respectively, and are required to saw them into 2-meter sections. After completing the task, A, B, and C have sawn 24, 25, and 27 sections respectively. How many more times did the fastest sawyer saw compared to the slowest sawyer?

Answer: 2


16 children of different heights stood in a circle, all facing the center. Each of them said: "My right neighbor is taller than my left neighbor." What is the minimum number?

Answer: 2


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

Answer: 3


Two natural numbers \( a \) and \( b \), their least common multiple is 60. Then, the number of possible values for the difference between these two natural numbers is _____.

Answer: 23


The numbers $112,121,123,153,243,313$, and 322 are among the rows, columns, and diagonals of a $3 \times 3$ square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list?

Answer: 524


4. Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring of the squares of a $2 n \times k$ chessboard with $n$ colours, there are 2 columns and 2 rows such that the 4 squares in their intersections have the same colour.

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


1. Let $\alpha, \beta$ be a pair of conjugate complex numbers. If $|\alpha-\beta|=2 \sqrt{3}$ and $\frac{a}{\beta^{2}}$ is a real number, then $|\alpha|=$ $\qquad$ .

Answer: 2


5.1. (14 points) A guard has detained a stranger and wants to drive him away. But the person caught said that he had made a bet with his friends for 100 coins that the guard would not drive him away (if the guard does, he has to pay his friends 100 coins, otherwise they pay him), and, deciding to bribe the guard, offered him to name a sum. What is the largest number of coins the guard can ask for so that the stranger, guided solely by his own profit, would definitely pay the guard?

Answer: 199


2. Determine the number of 3-digit numbers in base 10 having at least one 5 and at most one 3.

Answer: 249


Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.

---

The translation preserves the original text's formatting and structure.

Answer: 2020


9. Si Si wrote $1, 2, 3, \cdots \cdots, 99$ on the blackboard. Rui Rui doesn't like the number 2, so he erased the following numbers:
(1) multiples of 2;
(2) numbers containing the digit 2 (such as 92);
(3) numbers whose sum of digits is divisible by 2 (such as 31).
Rui Rui erased $\qquad$ numbers in total.

Answer: 79


4.1. For triangle $A B C$, the following is known: $A B=12, B C=10, \angle A B C=120^{\circ}$. Find $R^{2}$, where $R-$ is the radius of the smallest circle in which this triangle can be placed.

![](https://cdn.mathpix.com/cropped/2024_05_06_6bdc1b7f380f62058621g-04.jpg?height=397&width=589&top_left_y=201&top_left_x=745)

Answer: 91


1. Given 50 quadratic trinomials. The graphs of these trinomials have their vertices on the $x$-axis. No three graphs intersect at the same point. What is the minimum number of intersection points they can form?

Answer: 600


3 [ Text problems (other) $\quad]$

Four photographs can be arranged to form three different rectangles (see fig.). The perimeter of one of them is 56 cm. Find the perimeters of the other two rectangles, if the perimeter of a photograph is 20 cm.
![](https://cdn.mathpix.com/cropped/2024_05_06_53798a1e86d520071242g-07.jpg?height=362&width=958&top_left_y=2380&top_left_x=548)

Answer: 40


2. (8 points) One day, Xiao Ming was chatting with his brother. Xiao Ming said to his brother: "I am particularly looking forward to the year 2013, because 2, 0, 1, and 3 are four different digits. I have never experienced such a year before." His brother smiled and said: "Yes, we can call such years 'lucky years'. If we count this way, next year will be the 2nd 'lucky year' I have experienced." So, the brother was born in the year $\qquad$.

Answer: 1987


4. Petya wrote down $n$ consecutive two-digit numbers $(n \geqslant 2)$ in a row on the board, the first of which does not contain the digit 4, and the last one does not contain the digit 7. Vasya thought that this was the decimal representation of a natural number $x$, and he factored $x$ into prime factors. It turned out that there were only two of them and they differ by 4. What is written on the board?

Answer: 2021


## Task 3 - 150933

For a number $x$, the following four pairs $\left(A_{1}, A_{2}\right),\left(B_{1}, B_{2}\right),\left(C_{1}, C_{2}\right),\left(D_{1}, D_{2}\right)$ of statements are made, of which exactly one is true and exactly one is false.

Investigate whether there is a number $x$ that satisfies this requirement! Determine, if this is the case, every such number $x$!

A1) There is no number other than $x$ that satisfies the requirement of this task.

A2) $x$ is a natural number, in whose (decimal) representation a digit appears twice.

B1) $x-5$ is an integer divisible by 6.

B2) $x+1$ is an integer divisible by 12.

C1) $x$ is a natural number whose (decimal) representation begins with the digit 3.

C2) $x$ is the number 389.

D1) $x$ is a three-digit prime number with $300<x<399$, i.e., one of the numbers $307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397$.

D2) $x$ is a natural number consisting of three identical digits.

Answer: 317


A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1, 6, \dots, 96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

Answer: 3003


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

Answer: 132


A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 AM, also with equal probability. Let $\frac{p}{q}$ for $p$, $q$ coprime be the probability that they fall in love. Find $p + q$.

Answer: 10


Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as:
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
where $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Answer: 4


Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE = \angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.

Answer: 115


In a triangle $ABC$ ($\angle BCA = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference having a 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


A robot is standing on the bottom left vertex $(0,0)$ of a $5 \times 5$ grid and wants to go to $(5,5)$, only moving to the right $(a,b) \mapsto (a+1,b)$ or upward $(a,b) \mapsto (a,b+1)$. However, this robot is not programmed perfectly and sometimes takes the upper-left diagonal path $(a,b) \mapsto (a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0 \leq a,b \leq 5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.

Answer: 1650


About the pentagon $ABCDE$, we know that $AB = BC = CD = DE$, $\angle C = \angle D = 108^\circ$, $\angle B = 96^\circ$. Find the value in degrees of $\angle E$.

Answer: 102


Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times 30 \times 30$ cube. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.

Answer: 4500


For any positive integer, if the number of $2$'s in its digits is greater than the number of $3$'s in its digits, we call that a **good** number. Conversely, if the number of $3$'s in its digits is greater than the number of $2$'s, we call that a **bad** number. For example, the number $2023$ has two $2$'s and one $3$, so $2023$ is a good number. However, in the number $123$, both the number of $2$'s and $3$'s are one, so $123$ is neither a good number nor a bad number.

Find the difference between the number of good numbers and bad numbers among the positive integers not greater than $2023$.

Answer: 22


Let $E$ and $F$ be the respective midpoints of $BC$ and $CD$ in a convex quadrilateral $ABCD$. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.

Answer: 6


Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1 \leq n \leq 100$, find the maximum value of $\phi(n^2+2n) - \phi(n^2)$.

Answer: 72


Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.

Answer: 17


Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?

Answer: 2019


A set $S$ consisting of $2019$ different positive integers has the following property: the product of every $100$ elements of $S$ is a divisor of the product of the remaining $1919$ elements. What is the maximum number of prime numbers that $S$ can contain?

Answer: 1820


I1.1 Three cubes with volumes $1,8,27$ are glued together at their faces. If $a$ is the smallest possible surface area of the resulting polyhedron, find the value of $a$.

Answer: 72


Answer: 72


9.3. Find the smallest natural number $n$ such that in any set of $n$ distinct natural numbers, not exceeding 1000, it is always possible to select two numbers, the larger of which does not divide the smaller one.

Answer: 11


$$
x(t)=5(t+1)^{2}+\frac{a}{(t+1)^{5}},
$$

where $a$ is a positive constant. Find the minimum value of $a$ that satisfies $x(t) \geqslant 24$ for all $t \geqslant 0$.

Answer: 2 \left( \dfrac{24}{7} \right)^{7/2}


69. Faulty Locomotive. We set off by railway from Angchester to Clinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\frac{3}{5}$ of the original. As a result, we arrived in Clinkerton 2 hours late, and the driver said that if the malfunction had occurred 50 miles further, the train would have arrived 40 minutes earlier.

What is the distance from Angchester to Clinkerton?

Answer: 200


60. In what ratio does a plane, parallel to two skew edges of a triangular pyramid and dividing one of the other edges in the ratio $2: 1$, divide the volume of the pyramid?

Answer: \dfrac{20}{7}


Define $\triangle ABC$ with incenter $I$ and $AB=5$, $BC=12$, $CA=13$. A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$'s area divided by $\pi.$

Answer: \dfrac{16}{3}


18 scientists arrive at a conference, and exactly 10 of them know a sensational piece of news. During the coffee break, all scientists form random pairs, and in each pair, any scientist who knows the news shares it with the other, if they do not already know.

a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.

b) Find the probability that after the coffee break, the number of scientists who know the news will be 14.

c) Let $X$ be the number of scientists who know the sensational news after the coffee break. Find the expected value of $X$.

Answer: 0


XL OM - II - Task 3

Given is a trihedral angle $ OABC $ with vertex $ O $ and a point $ P $ inside it. Let $ V $ be the volume of the parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ OA^{\rightarrow} $, $ OB^{\rightarrow} $, $ OC^{\rightarrow} $. Calculate the minimum volume of the tetrahedron, whose three faces are contained in the faces of the trihedral angle $ OABC $, and the fourth face contains the point $ P $.

Answer: \dfrac{9}{2}V


10. Teacher Yangyang's city has 4 subway lines, each of which intersects with every other line at exactly one point, serving as the transfer station for those two lines. One day, Teacher Yangyang had a sudden idea to start from the subway station near her home (which is not a transfer station), make at least one transfer at each transfer station, and finally return home. How many transfers does Teacher Yangyang need to make to achieve her goal? $\qquad$ times.

Answer: 8


The base of the tetrahedron \( S-ABCD \) is a parallelogram \( ABCD \). A plane passes through vertex \( A \) and the midpoint \( K \) of the edge \( SC \), intersecting edges \( SB \) and \( SD \) at points \( M \) and \( N \) respectively. Prove that \( \frac{1}{3} \leqslant \frac{V_{1}}{V} \leqslant \frac{3}{8} \), where \( V \) is the volume of the tetrahedron \( S-ABCD \), and \( V_{1} \) is the volume of the tetrahedron \( S-AMKN \).

Answer: \dfrac{1}{3} \leqslant \dfrac{V_{1}}{V} \leqslant \dfrac{3}{8}


Exercise 7. Let $k$ be a strictly positive integer. For any real number $x$, the real number $|x|$ is the absolute value of $x$, which is $x$ if $x$ is positive, $-x$ if $x$ is negative. Find the number of triplets $(x, y, z)$ where $x, y, z$ are integers such that $x+y+z=0$ and $|x|+|y|+|z|=2k$.

Answer: 6k


Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak closes two adjacent digits with a black disc. Then Arutyun comes and identifies both closed digits and their order. What is the minimal $N$ for which they can perform such a trick?

Answer: 101


Let $a$ and $b$ be prime numbers such that $a + b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

Answer: 10000


Let $ABC$ be a triangle with $BC = 49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\frac{AB \cdot AC}{-BC + AB + AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Answer: 250049


Let $a > b > c > d$ be positive integers satisfying $a + b + c + d = 502$ and $a^2 - b^2 + c^2 - d^2 = 502$. Calculate how many possible values of $a$ are there.

Answer: 124


A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

1. If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
2. If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b \le 50$ such that if $a, b \in S$ then $S = \mathbb{N}$.

Answer: 2068


For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$.

Answer: 256


In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2 + 10$, $10 + 2$, and $3 + 2 + 2 + 5$ are three such ways.

Answer: 70


Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX \cdot TY$.

Answer: 460


Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?

Answer: 69


Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy:
\[
x + y + z = -\frac{\sqrt{3}}{2} - i\sqrt{5}
\]
and
\[
xyz = \sqrt{3} + i\sqrt{5}.
\]
If $x = x_1 + ix_2$, $y = y_1 + iy_2$, and $z = z_1 + iz_2$ for real $x_1, x_2, y_1, y_2, z_1$, and $z_2$, then:
\[
(x_1x_2 + y_1y_2 + z_1z_2)^2
\]
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Answer: 1516


A pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \, \text{kg}$, slides and hits the pulley, coming to a stop. The initial distance between the block and the pulley is $1.0 \, \text{m}$. During the slide, the rope makes a constant angle of $10.0^\circ$ with the horizontal. The hanging block has a mass of $M = 2.0 \, \text{kg}$. The table has a coefficient of friction of $0.50$ with the block. The pulley is frictionless, and the block remains on the ground throughout. Let $t$ be the time in seconds for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures.

Answer: 62


Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^{\circ}$ and $\angle DBA = 70^{\circ}$, then find the area of $ABCD$.

Answer: 85


Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$. What is the maximum possible value of $a + b + c + d + e$?

Answer: 153


Answer: 64


A positive integer $n$ is defined as a \textit{stepstool number} if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.

Answer: 687


Let $S = \{1, \cdots, 6\}$ and $\mathcal{P}$ be the set of all nonempty subsets of $S$. Let $N$ equal the number of functions $f:\mathcal{P} \to S$ such that if $A, B \in \mathcal{P}$ are disjoint, then $f(A) \neq f(B)$. Determine the number of positive integer divisors of $N$.

Answer: 9792


Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively. Furthermore, denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$, respectively. Find the circumradius of $\triangle XYZ$.

Answer: 30


Let $ABCD$ be a square with vertices $A=(0,0)$ and $B=(1,1)$. A point $P\left(\frac{2}{7},\frac{1}{4}\right)$ is located inside the square. An ant begins its journey from point $P$, touches three sides of the square, and returns to point $P$. The shortest possible distance the ant can travel is expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers, and $a$ is not divisible by any square number other than $1$. Determine the value of $(a+b)$.

Answer: 19


In the following diagram, two sides of a square are tangent to a circle with a diameter of $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ such that the area of the square is $m + \sqrt{n}$. Find $m + n$.

Answer: 536


An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.

Answer: 130


The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance of 2 units apart. Find the area of the irregularly shaped ten-sided figure shown.

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Answer: 66


It is given a polygon with $2013$ sides labeled as $A_{1}A_{2}\ldots A_{2013}$. The vertices are marked with numbers such that the sum of the numbers on any $9$ consecutive vertices is constant and equal to $300$. Given that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine the number with which $A_{2013}$ is marked.

Answer: 67


As shown in the figure, the graph \(\Gamma\) of the function \(y = \cos x + 1\) for \(0 \leq x \leq 2\pi\) is drawn inside the rectangle \(OABC\). When \(AB\) coincides with \(OC\), forming a cylinder, find the eccentricity of the curve \(\Gamma\) on the surface of the cylinder.

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


As shown in the diagram, \( AB \) is a path 28 meters long, and \( M \) is the midpoint of \( AB \). A dog starts running from a point on the left side of \( M \) along the path. The first run is 10 meters, the second run is 14 meters; the odd-numbered runs are 10 meters each, and the even-numbered runs are 14 meters each. The dog changes direction as follows: if after finishing a run \( M \) is to its right, it runs to the right next; if \( M \) is to its left, it runs to the left next. After running 20 times, the dog is 1 meter to the left of point \( B \). How far was the dog from point \( A \) when it started?

Answer: 7


Six numbers are randomly selected from the integers 1 to 45 inclusive. Let \( p \) be the probability that at least three of the numbers are consecutive. Find the value of \( \lfloor 1000 p \rfloor \). (Note: \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \) ).

Answer: 56


Ali Baba went to a cave where there are gold and diamonds. He had with him one large bag. It is known that a full bag of gold weighs 200 kg, and if the bag is filled only with diamonds, it weighs 40 kg (the empty bag weighs nothing). A kilogram of gold costs 20 dinars, and a kilogram of diamonds costs 60 dinars. What is the maximum amount of money Ali Baba can earn if he can carry no more than 100 kg in this bag? Provide justification for your answer.

Answer: 3000


On a circle with a radius of 5, around which an equilateral triangle is described, a point \( D \) is taken. It is known that the distance from point \( D \) to one of the vertices of the triangle is 9. Find the sum of the distances from point \( D \) to the other two vertices of the triangle.

Answer: 9


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

Answer: -\frac{1}{8}


There are infinitely many cards, each displaying a real number. For each real number \( x \), there is exactly one card displaying the number \( x \). Two players each select a distinct set of 100 cards, \( A \) and \( B \), from these. Establish a rule that satisfies the following conditions to determine the winner between the two players:
1. The winner only depends on the relative order of these 200 cards: If these 200 cards are placed face down in increasing order, with the audience knowing which card belongs to which player but not the actual numbers, the audience can still decide who the winner is;
2. If the elements of the two sets, written in increasing order, are
   \[
   A = \{a_1, a_2, \ldots, a_{100}\}, \quad B = \{b_1, b_2, \ldots, b_{100}\}
   \]
   such that for all \( i \in \{1,2, \ldots, 100\} \), \( a_i > b_i \), then \( A \) defeats \( B \).
3. If three players each select a distinct set of 100 cards, \( A \), \( B \), and \( C \), and \( A \) defeats \( B \), \( B \) defeats \( C \), then \( A \) defeats \( C \).

Question: How many such rules are there?
(Note: Two rules are considered different if for some sets \( A \) and \( B \), one rule states \( A \) defeats \( B \), while another rule states \( B \) defeats \( A \).)

Answer: 100


In the shown diagram, fill the numbers \( 1, 2, \ldots, 10 \) into the ten circles such that the sum of the numbers in any five consecutive adjacent circles does not exceed a certain integer \( M \). Find the minimum value of \( M \) and complete your diagram.

Answer: 28


Vasya is furiously cutting a rectangular piece of paper with scissors. Every second he cuts the first piece he sees with a random straight cut into two parts.

a) Find the expected number of sides of the polygon that Vasya will randomly encounter after an hour of such work.

b) Solve the same problem if the initial paper had the shape of an arbitrary polygon.

Answer: 4


Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shylock any two coins of different colors in exchange for one coin of the other color; for example, he may give Shylock one red coin and one blue coin, and receive one yellow coin in return. Bassanio wishes to end with coins that are all the same color, and he wishes to do this while having as many coins as possible. How many coins will he end up with, and what color will they be?

Answer: 7


A fair coin is tossed 5 times. The probability of getting exactly one heads is non-zero and is equal to the probability of getting exactly two heads. Let the simplified fraction \(\frac{i}{j}\) represent the probability of getting exactly 3 heads out of 5 tosses. Find the value of \(i + j\).

Answer: 283


Let $\triangle ABC$ have three internal angles $\angle A, \angle B, \angle C$, and the side lengths opposite to these angles are $a, b, c$ respectively. Given that $a < b < c$, and
\[
\begin{cases}
\frac{b}{a} = \frac{\left|b^{2} + c^{2} - a^{2}\right|}{bc} \\
\frac{c}{b} = \frac{\left|c^{2} + a^{2} - b^{2}\right|}{ca} \\
\frac{a}{c} = \frac{\left|a^{2} + b^{2} - c^{2}\right|}{ab}
\end{cases}
\]
find the ratio of the radian measures of the angles $\angle A, \angle B, \angle C$.

Answer: 1:2:4


Given a triangle \(ABC\) and a rhombus \(BDEF\), all vertices of which lie on the sides of the triangle \(ABC\), and the angle at \(E\) is obtuse.

Find the area of triangle \(ABC\), given \(AE = 3\), \(CE = 7\), and the radius of the circle inscribed in the rhombus is 1.

Answer: 5\sqrt{5}


A mathematician left point A for point B. After some time, a physicist left point A as well. The physicist caught up with the mathematician after 20 km, and without stopping, continued to point B and then turned back. They met for the second time 20 km away from point B. Each then continued in their respective directions. Upon reaching points A and B respectively, they turned back and headed towards each other again. At how many kilometers from point A will they meet for the third time, if the distance between points A and B is 100 km?

Answer: 45


In triangle $PQR$, $\angle Q$ is a right angle, and the altitude from $Q$ meets $\overline{PR}$ at $S$. The lengths of the sides of triangle $PQR$ are integers, and $PR=3^{5}, PS=3^3$. If $\tan P = m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Answer: 3


A regular octagon is inscribed in a square such that each side of the square is trisected by the vertices of the octagon. The perimeter of the square is now 144 centimeters. What is the area of the octagon inscribed in the square?

Answer: 1008


A quadrilateral in the plane has vertices at \((0, 0)\), \((0, 2)\), \((3, 2)\), and \((5, 5)\). Calculate the area of this quadrilateral in square units.

Answer: 5.5


9. Let $a, b, c$ be positive real numbers, and $a b c + a + c = b$. Determine the maximum value of
$$
p=\frac{2}{a^{2}+1}-\frac{2}{b^{2}+1}+\frac{3}{c^{2}+1}
$$

(1999, Vietnam Mathematical Olympiad)

Answer: \dfrac{10}{3}


6. In convex quadrilateral $A B C D$ we have $A B=15, B C=16, C D=12, D A=25$, and $B D=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle A B D$. Line $C B$ meets $\gamma$ again at $F$, line $A F$ meets $M C$ at $G$, and line $G D$ meets $\gamma$ again at $E$. Determine the area of pentagon $A B C D E$.

Answer: 396


6. There is a batch of parts, with the smallest diameter being $12 \mathrm{~mm}$ and the largest being $12.5 \mathrm{~mm}$. If $x$ parts are randomly selected, there will always be 2 parts with a diameter difference less than $0.01 \mathrm{~mm}$, then the minimum value of $x$ is

Answer: 52


10. (12 points) There are two docks, $A$ and $B$, on a river, with $A$ upstream and $B$ downstream. There are two boats, Boat A and Boat B, where Boat A's speed in still water is twice that of Boat B. Boat A and Boat B start from docks $A$ and $B$ respectively at the same time, heading towards each other. When Boat A starts, it leaves a box that can float on the water surface. After 20 minutes, the two boats meet, and at this moment, Boat A leaves another box on the water surface. After some time, Boat A realizes it has lost its cargo and turns back to search for it. When Boat A finds the second box, Boat B happens to encounter the first box left by Boat A. How many minutes did it take for Boat A to realize it had lost its cargo from the time it started? (Neglect the time for turning around.)

Answer: 40


(BXMO 2014) Let $a, b, c$ and $d$ be strictly positive integers. Determine the smallest value that the expression:

$$
S=\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+c+d}{b}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor
$$

can take.

Answer: 9


Find all real numbers $x, y, z$ such that

$$
x+y+z=3, \quad x^{2}+y^{2}+z^{2}=3, \quad x^{3}+y^{3}+z^{3}=3
$$

Distinction between polynomial and polynomial function Here, we explain why it is necessary to make this distinction, starting by defining a polynomial in another way. Here, $\mathbb{K}=\mathbb{Q}, \mathbb{R}, \mathbb{C}$ or $\mathbb{Z} / p \mathbb{Z}$ equipped with the usual addition and multiplication laws.

Definition. A polynomial with coefficients in $\mathbb{K}$ is an infinite sequence of elements of $\mathbb{K}$ that is zero from a certain rank.

Example. For example, $(0,1,2,3,0,0, \ldots, 0, \ldots)$ is a polynomial, as is $(0,0, \ldots, 0, \ldots)$. However, $(1,1, \ldots, 1, \ldots)$ is not one.

Definition. Let $P=\left(u_{n}\right)_{n}$ and $Q=\left(v_{n}\right)_{n}$ be two polynomials. We define the polynomial $P+Q$ by the sequence $w_{n}=u_{n}+v_{n}$ (which is indeed zero from a certain rank) and the polynomial $P \times Q$ by the sequence $\left(z_{n}\right)$, where $z_{n}=\sum_{i+j=n} u_{n} v_{n}$ (verify that $\left(z_{n}\right)$ is zero from a certain rank). We identify the elements of $\mathbb{K}$ with the constant polynomials via the application that associates to an element $\lambda \in \mathbb{K}$ the polynomial $(\lambda, 0,0, \ldots, 0, \ldots)$. Note that this is consistent with the intuitive notion of multiplying a polynomial by an element of $\mathbb{K}$: if $\left(u_{n}\right)$ is a polynomial and $\lambda \in \mathbb{K}$, then the polynomial $\lambda \times\left(u_{n}\right)$ is the polynomial $\left(\lambda u_{n}\right)$.

We now introduce the indeterminate $X$.

Definition. Let us denote by $X$ the polynomial $(0,1,0,0, \ldots)$.

Proposition. Every polynomial $P$ can be expressed in the form $P=\sum_{i=0}^{n} a_{n} X^{n}$. We denote $P$ or $P(X)$ indifferently to remind that we denote $X$ the indeterminate (we could very well denote it $Y!$ ).

Proof. If $P=\left(a_{0}, a_{1}, a_{2}, \ldots\right)$, let $N$ be an integer such that $i \geq N$ implies $a_{i}=0$. Then $P(X)=a_{0}+a_{1} X+\cdots+a_{N} x^{N}$. This is an immediate consequence of the definition of $X$ and the multiplication between polynomials.

Here is now the link between polynomial and associated polynomial function. Recall that, for now, a polynomial is just a sequence of numbers that is zero from a certain rank and is not seen as an application.

Proposition. Let $P(X)=a_{0}+a_{1} X+\cdots+a_{n} X^{n} \in \mathbb{K}[X]$ be a polynomial. We denote by $\widetilde{P}$ the application defined by $\widetilde{P}(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ for $x \in \mathbb{K}$, which we call the polynomial function associated with $P$. The application $P \mapsto \widetilde{P}$ is injective if $\mathbb{K}$ is infinite. If $\mathbb{K}$ is finite, this application is not necessarily injective.

Proof. First, consider the case where $\mathbb{K}$ is infinite. Let $P, Q \in \mathbb{K}[X]$ such that $\widetilde{P}=\widetilde{Q}$. Write $P(X)=\sum_{i} a_{i} X^{i}$ and $Q(X)=\sum_{i} b_{i} X^{i}$. Then the polynomial $P(x)-Q(x)$, in the sense of the previous sections, has an infinite number of roots, so it is zero. Therefore, $a_{i}=b_{i}$ for all $i$.

However, in the case where $\mathbb{K}$ is finite, the previous reasoning does not apply. Moreover, let us exhibit a counterexample. Consider $\mathbb{K}=\mathbb{Z} / p \mathbb{Z}$ and $P(X)=X^{p}-X$. According to Fermat's little theorem, for all $x \in \mathbb{K}$, we have $P(x)=0$. Thus, $P$ is not the zero polynomial, but the two associated polynomial functions are the same.

In other words, when $\mathbb{K}$ is infinite (for example, $\mathbb{K}=\mathbb{Q}, \mathbb{R}, \mathbb{C}$, which explains why we did not lose generality in the first sections), we can talk without distinction about a polynomial or the associated polynomial function. However, in other cases, one must be very careful!

Elements of the answer to the exercises

Answer: 1


Task B-3.4. In trapezoid $A B C D$ with perpendicular diagonals, the lengths of the bases $|A B|=a=4,|D C|=c=3$ are known. If the leg $\overline{B C}$ with base $a$ forms an angle of $60^{\circ}$, what is its length?

Answer: \dfrac{1 + \sqrt{193}}{4}


Rumcajs teaches Cipísek to write numbers. They started from one and wrote consecutive natural numbers. Cipísek pleaded to stop, and Rumcajs promised that they would stop writing when Cipísek had written a total of 35 zeros. What is the last number Cipísek writes?

Answer: 204


Problem 5.6. A three-digit number and two two-digit numbers are written on the board. The sum of the numbers that contain a seven in their notation is 208. The sum of the numbers that contain a three in their notation is 76. Find the sum of all three numbers.

Answer: 247


4. Find the last non-zero digit of $50!=1 \times 2 \times \cdots \times 50$.

Answer: 2


5. In triangle $ABC$ (angle B is obtuse), the altitude $BH$ and the bisector $AK$ are drawn. Find the angle $AKB$, if the angle $KHC$ is $45^{\circ}$.

Answer: 45^\circ


5. As shown in Figure 2, in $\triangle A B C$, $\angle A B C=120^{\circ}$,
$A B=B C=2$. Take a point $D$ on side
$A C$ (excluding $A$ and $C$), and fold $\triangle A B D$
along line segment $B D$, to get
$\triangle P B D$. When plane
$P B D \perp$ plane $A B C$, the maximum distance from point $P$ to plane $A B C$ is $\qquad$

Answer: 2