Example 3 Given $A \cup B \cup C=\{1,2, \cdots, 6\}$, and $A \cap B=\{1,2\},\{1,2,3,4\} \subseteq B \cup C$.
Then the number of $(A, B, C)$ that satisfy the conditions is $\qquad$ groups (different orders of $A, B, C$ are considered different groups).

Answer: 1600


10.007. An isosceles trapezoid with a side length of 17 cm is circumscribed around a circle with a diameter of 15 cm. Find the bases of the trapezoid.

Answer: 25


Answer: \frac{1}{a_{1} + a_{2}} + \frac{1}{a_{2} + a_{3}} + \frac{1}{a_{3} + a_{1}} > 1


3. In the plane, points $A_{1}, A_{2}, \ldots, A_{n}$ are given such that no three of them are collinear. Let $p_{i j}$ be the line defined by points $A_{i}$ and $A_{j}$. Determine the maximum number of intersection points of the lines $p_{i j}$ and $p_{k l}$, where $i, j, k, l$ are distinct elements of the set $\{1,2, \ldots, n\}$.

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


2. Let $\theta$ be the largest interior angle of a triangle, and satisfies $\sin 8 \theta=\sin 2 \theta$. Then the set of possible values of $\theta$ is $\qquad$ (express using the roster method).

Answer: \left\{ \dfrac{\pi}{3}, \dfrac{\pi}{2}, \dfrac{2\pi}{3}, \dfrac{7\pi}{10}, \dfrac{9\pi}{10} \right\}


## Task B-1.5.

A circle with a radius of $3 \text{~cm}$ is inscribed in a parallelogram such that it touches three of its sides. The measure of the acute angle of the parallelogram is $60^{\circ}$, and one side of the parallelogram is $2 \sqrt{3} \text{~cm}$ longer than the other side. Determine the distance from the center of the circle to the farthest vertex of the parallelogram.

Answer: 2\sqrt{21}


Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$.

Answer: 3


3. (25 points) On a circle, there are $n$ different positive integers $a_{1}$, $a_{2}, \cdots, a_{n}$ placed in a clockwise direction. If for any number $b$ among the ten positive integers $1, 2, \cdots, 10$, there exists a positive integer $i$ such that $a_{i}=b$ or $a_{i}+a_{i+1}=b$, with the convention that $a_{n+1}=a_{1}$, find the minimum value of the positive integer $n$.

Answer: 6


Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers.

[i]S. Berlov[/i]

Answer: 1


## Task A-1.3.

Let $A B C D$ be a rectangle and $k$ a circle with its center at the center of the rectangle. The circle $k$ intersects side $\overline{A B}$ at points $K$ and $L$, and side $\overline{C D}$ at points $M$ and $N$, and $L N \perp A C$. If the midpoint of segment $\overline{A N}$ lies on the circle $k$, what is the ratio of the lengths of the sides of the given rectangle?

Answer: \sqrt{3}


5. In the Cartesian coordinate system, grid points are numbered as follows: $(0,0)$ is No. 1, $(1,0)$ is No. 2, $(1,1)$ is No. 3, $(0,1)$ is No. 4, $(0,2)$ is No. 5, $(1,2)$ is No. 6, $(2,2)$ is No. 7, $(2, 1)$ is No. 8, $(2,0)$ is No. 9, (as shown in Figure 7-1). Following the order of the arrows in the figure, the coordinates of the 2006th point are . $\qquad$

Answer: (44,19)


1. Try to design a method to divide a square into 8 smaller squares without repetition or omission (the sizes of the smaller squares can be different); also, how to divide a square into 31 smaller squares under the same requirements?

Answer: 8


11. Given an integer array consisting of 121 integers, where each number is between 1 and 1000 (repetition is allowed), this array has a unique mode (i.e., the integer that appears most frequently). Let $D$ be the difference between this mode and the arithmetic mean of the array. When $D$ reaches its maximum value, what is $[D]$ equal to?

Answer: 947


3.22. The volume of the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ is $V$. Find the volume of the common part of the tetrahedra $A B_{1} C D_{1}$ and $A_{1} B C_{1} D$.

Answer: \dfrac{V}{6}


## Task A-4.2.

Let $S$ be the set of all natural numbers less than 1000 whose all digits in decimal notation are even. Let $\omega$ be a complex number such that $\omega^{2}+\omega+1=0$.

Calculate the sum $\sum_{k \in S} \omega^{k}$, i.e., the sum of the values $\omega^{k}$ for all $k$ in the set $S$.

Answer: -2


2.1. During the time it took for a slowly moving freight train to cover 1200 m, a schoolboy managed to ride his bicycle along the railway tracks from the end of the moving train to its beginning and back to the end. In doing so, the bicycle's distance meter showed that the cyclist had traveled 1800 m. Find the length of the train (in meters).

Answer: 500


Kluci found an old minefield plan, see the picture. The numbers are on the fields where there are no mines, and they indicate the number of adjacent mined fields. Determine how many mines there are in the field and where they are. (Fields are adjacent if they share a vertex or a side.)

| 1 |  | 2 |  | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  | 3 |  | 3 |  |
| 3 |  |  |  | 3 |
|  | 2 |  |  |  |
|  |  |  | 2 |  |

Answer: 8


9. As shown in the figure, in the isosceles right triangle $A B C$, $\angle B A C=90^{\circ}$, points $D, E$ are on $B C$, and $C D=B E, C N \perp A E$ at point $N, C N$ intersects $A D$ at point $M$. If $\triangle A M N$ is an isosceles right triangle, $A N=\sqrt{2-\sqrt{2}}$, then the area of $\triangle A B C$ is $\qquad$

Answer: 2


To decorate the exterior of a building, a large square was painted on the wall. They drew the circumscribed circle of the square and also semicircles extending outward from the sides of the square. The arcs of the circles define four crescent-shaped fields. What is the side length of the square if the area of one such field is $1 \mathrm{~m}^{2}$?

Answer: 2


(3) 18 football teams are participating in a round-robin tournament, which means that in each round, the 18 teams are divided into 9 groups, and each group's two teams play one match. In the next round, the teams are regrouped to play, for a total of 17 rounds, so that each team plays one match against each of the other 17 teams. After $n$ rounds of the tournament according to any feasible schedule, there always exist 4 teams that have played a total of only 1 match among themselves. Find the maximum possible value of $n$.

Answer: 7


【Question 15】Junjun starts from $\mathrm{A}$ and travels at a constant speed to $\mathrm{B}$. When Junjun starts, Aping starts from $\mathrm{B}$ and travels at a constant speed to $\mathrm{A}$. They meet at point $\mathrm{C}$ along the way. After meeting, Junjun walks another 100 meters and then turns around to chase Aping, catching up to Aping 360 meters from $\mathrm{C}$. When Junjun catches up to Aping, he immediately turns around and heads to $\mathrm{B}$. As a result, when Junjun arrives at $\mathrm{B}$, Aping also arrives at $\mathrm{A}$. What is the distance between $\mathrm{A}$ and $\mathrm{B}$ in meters?

Answer: 1656


In the image, squares $ABCD, EFCA, GHC E$, and $IJHE$ are illustrated. Points $S, B$, $F$, and $G$ are centers of these squares, respectively. The segment $AC$ is $1 \text{ cm}$ long.

Determine the area of triangle $IJS$.

(Hint: The area of the triangle is half the area of some rectangle.)

Answer: 3


Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?

Answer: 4


[ $[\quad$ Rule of product $\quad]$

a) Leshа climbs a ladder with 10 steps. In one jump, he can either go up one step or two steps. In how many ways can Leshа climb the ladder?

b) When descending the same ladder, Leshа skips some steps (he can even skip all 10). In how many ways can he descend the ladder?

#

Answer: 89


15. On a long strip of paper, the numbers $1, 2, 3, \ldots, n$ are written in sequence. The long strip is cut into five segments, each containing some consecutive natural numbers (the digits of a single number are not split between different segments). We calculated the average of the numbers in these five segments, which are $1234, 345, 128, 19$, and $9.5$ (the order of these five numbers is scrambled). Therefore, $\mathbf{n}=$ $\qquad$ -

Answer: 2014


[Systems of linear equations]

Seven coins (weighing 1, 2, ... 7 grams) are laid out in a row in some order. For each coin (except the outermost ones), the sum of the weights of its neighbors is known.

What is the maximum number of coins whose weight can be guaranteed to be known?

#

Answer: 3


$13 \cdot 60$ water is poured into $n$ cups of sufficiently large capacity, with each cup receiving an equal amount of water. Water can be poured from any cup to any other cup, but the amount of water poured must be equal to the amount of water already in the receiving cup. It is known that after a finite number of pourings, all the water can be collected into one cup. Find all possible natural numbers $n$.

Answer: 2^k


6. A circle with center $O_{1}$ and radius 2, inscribed in triangle $A B C$, touches side $B C$ at point $D$. A second circle with center $O_{2}$ and radius 4 touches the extensions of sides $A B$ and $A C$, as well as side $B C$ at point $E$. Find the area of quadrilateral $O_{1} D O_{2} E$, if the measure of angle $A C B$ is $120^{\circ}$. (20 points)

Answer: 10\sqrt{3}


In a convex quadrilateral, the sum of the distances from a vertex to the sides is the same for all vertices. Prove that this quadrilateral is a parallelogram.

Answer: \text{The quadrilateral is a parallelogram.}


8. Let $A$ be any set of 7 real numbers, and let the set $B=\{x y \mid x, y \in A$ and $x \neq y\}$, denote the number of elements in set $B$ as $|B|$, then $|B|_{\text {max }}+|B|_{\text {min }}=$ $\qquad$ .

Answer: 30


9. (5 points) Xiaohong brought 5 RMB notes of 50 yuan, 20 yuan, and 10 yuan denominations, 6 notes of 20 yuan, and 7 notes of 10 yuan. She bought a commodity worth 230 yuan. Therefore, there are $\qquad$ ways to pay.

Answer: 11


3.2. In the circle $\omega$, an isosceles triangle $ABC$ with base $AC$ is inscribed. It turns out that the radius of the inscribed circle of $ABC$ is equal to the radius of the circle that touches the smaller arc $BC$ of the circle $\omega$ and the side $BC$ at its midpoint (see Fig. 2). Find the ratio of the sides of triangle $ABC$.

Answer: 2:2:1


5. An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near planet $12 I V 1961$ are consecutive terms of a decreasing geometric progression. The three most recent intervals (in years) are the roots of the cubic equation $t^{3}-c t^{2}+350 t-1000=0$, where $c-$ is some constant. What will be the duration of the next interval until the appearance of the comet?

Answer: \dfrac{5}{2}


6. Given circle $\Gamma:(x-1)^{2}+y^{2}=r^{2}(r>0)$, parabola $G: y^{2}=4 x$. A line $l$ passing through the point $(1,0)$ intersects the circle $\Gamma$ at points $C$ and $D$ (point $C$ is above the $x$-axis), and intersects the parabola $G$ at points $A$ and $B$ (point $A$ is above the $x$-axis). If there are three lines $l$ such that $|A C|=$ $|B D|$, then the range of $r$ is $\qquad$

Answer: (2, +\infty)


Biyankov Yu.A.

In an isosceles triangle \(ABC\), a point \(M\) is marked on the lateral side \(BC\) such that the segment \(MC\) is equal to the height of the triangle dropped to this side, and a point \(K\) is marked on the lateral side \(AB\) such that the angle \(KMC\) is a right angle. Find the angle \(ACK\).

Answer: 45^\circ


1. On the first day, $2^{n}$ schoolchildren played ping-pong in a "knockout" format: first, two played, then the winner played with the third, the winner of this pair played with the fourth, and so on until the last schoolchild (there are no draws in ping-pong). On the second day, the same schoolchildren played for the cup: first, they arbitrarily split into pairs and played, the losers were eliminated, and the winners again arbitrarily split into pairs and played, and so on. It turned out that the sets of pairs that played on the first and second days were the same (possibly with different winners). Find the largest possible value of $n$.

B. Frenkin

Answer: 3


There is a stack of 52 face-down playing cards on the table. Mim takes 7 cards from the top of this stack, flips them over, and puts them back at the bottom, calling it one operation. The question is: how many operations are needed at least to make all the playing cards face-down again?

Answer: 112


For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors.
Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

Answer: 2


8. There is a pair of tightly adjacent rubber drive wheels, each with a mark line passing through the center (as shown in the figure below).

The radius of the driving wheel is 105 cm, and the radius of the driven wheel is 90 cm. When they start to rotate, the mark lines on both wheels are in a straight line. Question: How many turns does the driving wheel need to make at least so that the mark lines on both wheels are in a straight line again?

Answer: 3


4. (8 points) A cruise ship travels from upstream location $A$ to downstream location $B$ in 1 hour. On the return trip, the ship doubles its speed and still takes 1 hour. Therefore, if the cruise ship also doubles its speed when departing from $A$, it will take $\qquad$ minutes to reach $B$.

Answer: 36


In a rectangle with sides \(a\) and \(b\), the angle bisectors of all the corners are drawn until they intersect. Find the area of the quadrilateral formed by the bisectors.

Answer: \dfrac{(a - b)^2}{2}


Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?

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


Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$. Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.

Answer: 338


\section*{Task 2 - 011012}

In the USSR, in 37 minutes today, as much gas is produced as was produced in the entire year 1913 in Tsarist Russia.

Calculate the increase in percent!

Answer: 1420441


325. Tetrahedron - from progression. Twelve numbers $a_{i}$ form an arithmetic progression, so that $a_{k}+d=a_{k+1}$. Find the volume of the tetrahedron with vertices at points $\left(a_{1}^{2}, a_{2}^{2}, a_{3}^{2}\right),\left(a_{4}^{2}, a_{5}^{2}, a_{6}^{2}\right)$, $\left(a_{7}^{2}, a_{8}^{2}, a_{9}^{2}\right),\left(a_{10}^{2}, a_{11}^{2}, a_{12}^{2}\right)$.

Answer: 0


2. Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

Answer: 11


6. Given the system of equations $\left\{\begin{array}{l}\frac{x}{a}+\frac{y}{b}=1, \\ x^{2}+y^{2}=50\end{array}\right.$ has only integer solutions. Then the number of real pairs $(a, b)$ that satisfy the condition is $\qquad$ .

Answer: 60


3B. Find the largest divisor of 1001001001 that is less than 10000.

Answer: 9901


a) Helena and Gabriel jump simultaneously from a raft into a river and swim in opposite directions. Gabriel swims downstream with the current, and Helena swims upstream against the current, possibly at a different speed. After 5 minutes, they turn around and swim back to the raft, each maintaining a constant speed throughout the time. Who arrives first? (During this time, the raft was anchored, i.e., it remained stationary despite the current.)

b) At his usual speed, a man descends a river that is $15 \mathrm{~km}$ long in 5 hours less than the time he spends swimming the same river in the opposite direction. If he doubles his usual speed, he spends only 1 hour less descending the river than the time spent on the return trip. Assuming the speed of the current remains constant during the trips, what is its value in $\mathrm{km} / \mathrm{h}$?

Answer: Helena


8.1. In a batch, there are 90000 boxes of vaccine weighing 3300 grams each, and 5000 small boxes weighing 200 grams each. What is the minimum number of temperature-controlled containers needed if, according to the new rules, no more than 100 kilograms can be placed in each?

Answer: 3000


Each of the thirty sixth-graders has one pen, one pencil, and one ruler. After their participation in the Olympiad, it turned out that 26 students lost a pen, 23 - a ruler, and 21 - a pencil. Find the smallest possible number of sixth-graders who lost all three items.

Answer: 20


## Task A-3.6.

Given is a right triangle $A B C$. On the bisector of the right angle $\varangle A C B$ a point $M$ is chosen. If $\sin \varangle M A C=\frac{8}{17}$ and $\cos \varangle M B C=\frac{3}{5}$, find the ratio $|A B|:|C M|$.

Answer: \dfrac{5\sqrt{58}}{16}


Answer: 5


6. (4 points) Given three circles with radii 3, 4, and 5, each pair of which touch each other at points $A, B$, and $C$. Find the sum of the distances from the center of the circumscribed circle of triangle $A B C$ to its sides.

Answer: $\frac{5}{\sqrt{14}}+\frac{5}{\sqrt{21}}+\frac{5}{\sqrt{30}}$.

Answer: \frac{5}{\sqrt{14}}+\frac{5}{\sqrt{21}}+\frac{5}{\sqrt{30}}


Three. (50 points) In a $\left(2^{n}-1\right) \times\left(2^{n}-1\right)(n$ $\geqslant 2)$ grid, each cell is filled with 1 or -1. If the number in any cell is equal to the product of the numbers in the cells that share an edge with it, then this filling method is called "successful". Find the total number of "successful" fillings.

In a $\left(2^{n}-1\right) \times\left(2^{n}-1\right)(n$ $\geqslant 2)$ grid, each cell is filled with 1 or -1. If the number in any cell is equal to the product of the numbers in the cells that share an edge with it, then this filling method is called "successful". Find the total number of "successful" fillings.

Answer: 1


10. As shown in the figure, $\triangle A B C$ is an isosceles right triangle, $E$ is a point on side $B C$ such that $B E=3 C E$, and points $D, A, F$ are collinear. Let the height from $D$ to $B E$ in $\triangle D B E$ be $h_{1}$, and the height from $F$ to $E C$ in $\triangle F E C$ be $h_{2}$. We have $3 h_{1}+h_{2}=3$ cm. The sum of the areas of $\triangle D B E$ and $\triangle F E C$ is 6 square cm, then the area of $\triangle A B C$ is $\qquad$ square cm.

Answer: 64


2. For real numbers $x$ and $y$, the following holds:

$$
x^{3}+x^{2}+x y+x+y+2=0 \quad \text { and } \quad y^{3}-y^{2}+3 y-x=0
$$

Determine the value of the expression $x-y$.

Answer: -1


Example 6. Express $6 \sin ^{2} \frac{\alpha}{2}-8 \sin ^{4} \frac{\alpha}{2}$ as a product.

Answer: 2 \sin \frac{\alpha}{2} \sin \frac{3\alpha}{2}


38. Find the largest number by which each of the fractions $\frac{154}{195}$, $\frac{385}{156}$, and $\frac{231}{130}$ can be divided to yield natural numbers.

Answer: \dfrac{77}{780}


22nd IMO 1981 shortlist Problem 3 A sequence a n is defined as follows, a 0 = 1, a n+1 = (1 + 4a n + √(1 + 24a n ))/16 for n ≥ 0. Find an explicit formula for a n .

Answer: \dfrac{2 + 3 \cdot 2^{-n} + 4^{-n}}{6}


32nd Swedish 1992 Problem 2 The squares in a 9 x 9 grid are numbered from 11 to 99, where the first digit is the row and the second the column. Each square is colored black or white. Squares 44 and 49 are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square 99?

Answer: white


Shapovalov A.V.

The weight of each weight in the set is a non-integer number of grams. They can balance any integer weight from 1 g to 40 g (weights are placed on one pan of the scales, the weight to be measured - on the other). What is the smallest number of weights in such a set?

Answer: 6


Maxim will go to first grade on September 1, 2014, at the age of 6, and this day will not be his birthday. What is the probability that he was born in 2008?

Answer: \dfrac{244}{365}


11. (12 points) In the figure, by connecting every other vertex of a regular octagon with each side measuring 12 cm, two squares can be formed. The area of the shaded region in the figure is $\qquad$ square centimeters.

Answer: 288


7.1. What is the maximum number of L-shaped pieces

|  |
| :--- |

consisting of three $1 x 1$ squares, that can be placed in a 5x7 rectangle? (The L-shaped pieces can be rotated and flipped, but they cannot overlap).

Answer: 11


An arbitrary circle passing through \(X\) and \(B\) intersects the circumcircle of triangle \(ABC\) at points \(P\) and \(Q\), different from \(X\) and \(B\). Prove that all possible lines \(PQ\) pass through a single point.

Answer: S


Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$ , as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$ . AIME 1987 Problem 15.png
Please give the answer directly without any intermediate steps.

Answer: 462


1. Let $n$ be a natural number. Denote by $p_{k}$ the number of non-negative integer solutions to the equation $k x+(k+1) y=n-k+1$. Determine the sum $p_{1}+p_{2}+\ldots+p_{n+1}$.

Answer: n+1


II. (50 points) Let positive numbers $a, b$ satisfy $a>\frac{b}{\sqrt{2}}$ and such that the inequality about $x$
$$
\sqrt{\dot{x}-1} \geqslant a \sqrt{x+1}-b
$$

always has real solutions. Try to find the range of $f(a, b)=a^{2}-3 a b+b^{2}$.

Answer: (-\infty, 1)


6. Two circles touch each other and the sides of triangle $ABC$. The first circle with radius $\frac{1}{18}$ touches sides $AB$ and $AC$ at points $L$ and $K$, the second circle with radius $\frac{2}{9}$ touches sides $AC$ and $BC$ at points $N$ and $M$. Find the area of triangle $ABC$ if $AL=\frac{1}{9}, CM=\frac{1}{6}$.

Answer: \dfrac{3}{11}


8. Positive integers $a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}$ satisfy $2 \leq a_{i} \leq 166$ and $a_{i}^{b_{i}} \equiv a_{i+1}^{2}(\bmod 167)$ for each $1 \leq i \leq 7$ (where $a_{8}=a_{1}$ ). Compute the minimum possible value of $b_{1} b_{2} \cdots b_{7}\left(b_{1}+b_{2}+\cdots+b_{7}\right.$ ).

Answer: 675


Problem 7.2. (15 points) Given an isosceles triangle $A B C(A B=B C)$. On the sides $A B, B C, A C$ points $K, L, M$ are marked respectively such that $\angle A K M=90^{\circ}, \angle B L K=90^{\circ}$ and $K M=K L$. What is the measure of angle $C M L ?$

Answer: 90^\circ


5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $30^{\circ}$.

Answer: 14\sqrt[4]{3}


Suppose that $f(x)$ is a function defined for every real number $x$ with $0 \leq x \leq 1$ with the properties that

- $f(1-x)=1-f(x)$ for all real numbers $x$ with $0 \leq x \leq 1$,
- $f\left(\frac{1}{3} x\right)=\frac{1}{2} f(x)$ for all real numbers $x$ with $0 \leq x \leq 1$, and
- $f(a) \leq f(b)$ for all real numbers $0 \leq a \leq b \leq 1$.

What is the value of $f\left(\frac{6}{7}\right)$ ?

PART B

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

Useful Facts for Part B:

$\operatorname{gcd}(a, b c)=\operatorname{gcd}(a, b)$ for all integers $a, b$ and $c$ with $\operatorname{gcd}(a, c)=1$.

$\operatorname{gcd}(a, b)=\operatorname{gcd}(a, b-q a)$ for all integers $a, b$ and $q$.

Answer: \dfrac{3}{4}


9. (16 points) Given the parabola $C_{1}: y=x^{2}, M$ is a moving point in the plane. Two tangent lines are drawn from $M$ to the parabola $C_{1}$, touching it at points $A$ and $B$. If the area of $\triangle M A B$ is a constant $2 c^{3}$, find the equation of the locus of point $M$.

Answer: y = x^2 - c^2


11. (5 points) Xiao Hong left home close to 11 AM, at which time the hour and minute hands of the clock overlapped. She returned home close to 5 PM the same day, at which time the hour and minute hands of the clock were in opposite directions (in a straight line). How many hours did Xiao Hong stay out?

Answer: 6


As shown in the figure, in the right triangle $\triangle ABC$, $AB=3$ and $AC=4$. Points $D, E, F, G, H, I$ are all on the rectangle $KLMJ$, and $ABED$, $ACHI$, and $BCGF$ are all squares. What is the area of rectangle $KLMJ$?

Answer: 110


5.090 In a chess match between two teams of 8 people, the participants of the games and the color of the pieces for each participant are determined by a draw. What is the number of different outcomes of the draw?

Answer: 10321920


5. The numbers $1,2,3, \ldots, 999,1000$ are written one after another in this order on a circle. We cross out the number 1, and then every 15th number $(1,16,31, \ldots)$. When we complete one full circle, we count again, including the crossed-out numbers. How many numbers will remain uncrossed?

Answer: 800


## Task 1 - 090711

Cut out a rectangular piece of paper, divide it into eight congruent rectangles using straight lines, and write the same letter on both the front and back of each rectangular surface, as indicated in the diagram!

$$
\begin{array}{|c|c|c|c|}
\hline \mathrm{O} & \mathrm{N} & \mathrm{G} & \mathrm{A} \\
\hline \mathrm{W} & \mathrm{G} & \mathrm{F} & \mathrm{L} \\
\hline
\end{array}
$$

Fold the piece of paper so that the letters are stacked in the order W O L F G A N G!

The solution is the appropriately folded paper or a description of the procedure.

Answer: WOLFGANG


11.4. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length a. The ends of the segment intersecting edge $C_{1} D_{1}$ lie on the lines $A A_{1}$ and $B C$. What is the minimum length that this segment can have

Answer: 3a


1st Irish 1988 Problem 5 Let X = {1, 2, 3, 4, 5, 6, 7}. How many 7-tuples (X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 ) are there such that each X i is a different subset of X with three elements and the union of the X i is X?

Answer: 31169376000


[ Isosceles, Inscribed, and Circumscribed Trapezoids ]

[Properties and characteristics of isosceles triangles. ]

Let $M$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$, in which sides $AB$, $AD$, and $BC$ are equal to each other.

Find the angle $CMD$, given that $DM = MC$, and $\angle CAB \neq \angle DBA$.

Answer: 120^\circ


4. As shown in Figure 5, an equilateral $\triangle ABC$ with side length 26 is inscribed in a circle, and chord $DE \parallel BC$. The extension of $DE$ intersects the extensions of $AB$ and $AC$ at points $F$ and $G$, respectively. If the lengths of segments $AF$ and $DF$ are both positive integers, then the chord $DE=$ $\qquad$

Answer: 16


6. Point $K$ lies on side $A B$ of triangle $A B C$ with an angle of $120^{\circ}$ at vertex $C$. Circles are inscribed in triangles $A K C$ and $B K C$ with centers $O$ and $Q$ respectively. Find the radius of the circumcircle of triangle $O Q C$, if $O K=6, K Q=7$.

Answer: \dfrac{\sqrt{255}}{3}


[ Angles subtended by equal arcs and equal chords]

[ Auxiliary similar triangles $\quad]$

From the vertex of the obtuse angle $A$ of triangle $A B C$, a height $A D$ is dropped. From point $D$, a circle with radius equal to $A D$ is described, intersecting the sides of triangle $A B$ and $A C$ at points $M$ and $N$ respectively. Find the side $A C$, if it is known that $A B=c, A M=m$ and $A N=n$.

Answer: \dfrac{m c}{n}


## Zadatak B-4.5.

U trokutu $A B C$ je $a=|B C|=\sqrt{21} \mathrm{~cm}, b=|A C|=4 \mathrm{~cm}$ i $\alpha=\varangle B A C=120^{\circ}$. Na stranici $\overline{B C}$ odredite točku $D$ tako da obujam rotacijskog tijela nastalog rotacijom trokuta $A B D$ oko stranice $A B$ bude jednak obujmu rotacijskog tijela nastalog rotacijom trokuta $A C D$ oko stranice $A C$. U kojem omjeru točka $D$ dijeli stranicu $a$ ?

Answer: \dfrac{1}{2}


B4 In the following diagram, two lines that meet at a point $A$ are tangent to a circle at points $B$ and $C$. The line parallel to $A C$ passing through $B$ meets the circle again at $D$. Join the segments $C D$ and $A D$. Suppose $A B=49$ and $C D=28$. Determine the length of $A D$.

Answer: 63


5. As shown in the figure, the area of $\triangle B C F$ is $2022, \angle A=90^{\circ}, \angle A B C=30^{\circ} . A D=D C, C E$ bisects $\angle A C B$. Then the area of $\triangle A B C$ is $\qquad$ .

Answer: 5055


16. Let $a>0, x_{1}, x_{2}, \cdots, x_{n} \in[0, a](n \geqslant 2)$ and satisfy
$$x_{1} x_{2} \cdots x_{n}=\left(a-x_{1}\right)^{2}\left(a-x_{2}\right)^{2} \cdots\left(a-x_{n}\right)^{2}$$

Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.

Answer: \left( \frac{\sqrt{4a + 1} - 1}{2} \right)^{2n}


4. As shown in Figure 3, in $\triangle A B C$, $M$ is the midpoint of side $B C$, and $M D \perp A B, M E \perp A C$, with $D$ and $E$ being the feet of the perpendiculars. If $B D=2, C E=1$, and $D E \parallel B C$, then $D M^{2}$ equals $\qquad$

Answer: 1


Anzhans A.

There are 100 silver coins, ordered by weight, and 101 gold coins, also ordered by weight. It is known that all coins have different weights. We have a two-pan balance that allows us to determine which of any two coins is heavier. How can we find the coin that ranks 101st in weight among all the coins with the fewest number of weighings?

Answer: 8


June and Julia live 2.5 miles apart. It takes June 10 minutes to ride her bike directly to Julia's house. At the same rate, how many minutes would it take June to ride the 4 miles from her own house to Bernard's house, considering a headwind slows her return trip by 20%?

Answer: 20


Answer: 2


Gabriella created a Venn diagram showing the number of students in her physics class who like three different genres of music: Rock, Jazz, and Classical. There are 40 students in her class. Also, she knows $\frac{1}{4}$ of students like Rock, $\frac{1}{5}$ like Jazz, 8 students like other genres (not these three), and 6 students do not like any of these three genres. How many students like all three genres (Rock, Jazz, and Classical)? [asy]unitsize(50);
import graph;
pair A = (0,-1); pair B = (sqrt(3)/2,1/2); pair C = (-sqrt(3)/2,1/2);
draw(Circle(A,1.2) ^^ Circle(B,1.2) ^^ Circle(C,1.2));
label("12",A); label("4",B); label("5",C); label("$z$",(0,0)); label("$w$",(B+C)/2); label("$y$",(A+B)/2); label("$x$",(A+C)/2);
label("Classical",1.5C,C); label("Other Genres",2B,C); label("Rock", 1.7A,A);[/asy]

Answer: 2


In a class of 40 students, 20 students have dogs, $\frac{1}{4}$ of the students have cats, and 10 students have other types of pets. Five students don't have any pets at all. In a Venn diagram, the number of students who have only dogs is labeled as 15, only cats as 4, and only other pets as 5. The numbers of students who have exactly two types of pets are labeled as follows: dogs and cats ($x$), dogs and other pets ($y$), and cats and other pets ($z$). How many students have all three types of pets?

Answer: 0


An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?

Answer: 360


Monica tosses a fair 8-sided die. If the roll is a prime number, then she wins that amount in dollars. If the roll is composite, she loses that amount in dollars. If she rolls a 1, she neither wins nor loses anything, and if she rolls an 8, she is awarded an extra $5. What is the expected value of her winnings on one die toss?

Answer: \$1.50


Let $n$ be a positive integer. If $b\equiv (5^{3n}+4)^{-1}\pmod{11}$, what is the remainder when $b$ is divided by 11?

Answer: 7


Jonathan purchases a rectangular mirror and places it within a wooden frame. The overall dimensions of the framed mirror are 100 cm by 50 cm. The frame itself has a uniform width of 8 cm around the mirror. What is the area of the mirror?

Answer: 2856 \text{ cm}^2


A bridge is structured by aligning a wooden plank between two isosceles triangular supports. Here’s the diagram:
```
         A         D
        / \       / \
       /   \     /   \
      B-----C   E-----F
```
Given $AB = AC$ and $DE = DF$, and with $\angle BAC = 15^\circ$ and $\angle EDF = 45^\circ$, determine the sum $\angle DAC + \angle ADE$.

Answer: 150^\circ


Six points are placed inside or on a circle with radius $1$. Determine the smallest possible number $b$, such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$.

Answer: 1


Given six points in the plane \((-\frac{5}{2}, 2), (0,0), (0,3), (4,0), (4,3), (2,4)\), with no three points collinear, there is a unique conic section passing through all these points. Propose that this conic section is an ellipse centered at \((2, \frac{3}{2})\) with its axes parallel to the coordinate axes. Find the length of the minor axis of this ellipse.

Answer: 5


Julie has an infinite number of balls and empty boxes that can each hold up to seven balls. The boxes are arranged in a row from left to right. At the first step, she places a ball in the leftmost box. At each subsequent step, she places a ball in the first box from the left that still has room, and empties any box to its left. Determine the total number of balls in the boxes as a result of Julie's $1729$th step.

Answer: 7


Let $AB$ be a diameter of a circle centered at $O$.  Let $F$ be a point on the circle such that $F$ is not on the semicircle containing $E$. Let the tangent at $B$ intersect the tangent at $F$ and $AF$ at points $C'$ and $D'$ respectively. If $\angle BAF = 30^\circ$, find $\angle C'ED'$, in degrees.

Answer: 60^\circ


Given the property for any positive integer $n$, the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}^n$ is equal to $\begin{pmatrix} G_{n + 1} & 2G_n \\ 2G_n & G_{n - 1} \end{pmatrix}$, where $G_n$ corresponds to a sequence defined by $G_{n} = 2G_{n - 1} + G_{n - 2}$ with $G_1 = 1$ and $G_2 = 2$. Calculate $G_{100} G_{102} - 4G_{101}^2$.

Answer: -3^{101}


Let $f(n)$ be the sum of all the divisors of a positive integer $n$. Determine the number of positive integers $n$ for which $f(f(n)) = n + 3$.

Answer: 0


In rectangle $ADEH$, points $B$ and $C$ bisect $\overline{AD}$, and points $G$ and $F$ bisect $\overline{HE}$. In addition, $AH=AC=3$. What is the area of quadrilateral $WXYZ$?

Answer: \frac{9}{4}


For all integers $x$ and $y$, define the operation $\diamond$ such that $x \diamond 0 = x$, $x \diamond y = y \diamond x$, and $(x + 2) \diamond y = (x \diamond y) + 2y + 3$. Calculate the value of $8 \diamond 3$.

Answer: 39


Let $\triangle ABC$ have side lengths $AB = 40$, $BC = 24$, and $AC = 32$. Point $Y$ lies in the interior of $\overline{AC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABY$ and $\triangle BCY$, respectively. Find the minimum possible area of $\triangle BI_1I_2$ as $Y$ varies along $\overline{AC}$.

Answer: 96


100 bulbs were hung in a row on the New Year's tree. Then the bulbs began to switch according to the following algorithm: all bulbs were lit initially, then after one second, every second bulb was turned off, and after another second, every third bulb switched: if it was on, it was turned off, and if it was off, it was turned on. Again, after one second, every fourth bulb switched, after another second - every fifth bulb, and so on. This procedure continued for 100 seconds. Determine the probability that a randomly chosen bulb is lit after all this (bulbs do not burn out or break).

Answer: 0.1


A weight supported by a helical spring is lifted by a distance $b$ and then released. It begins to fall, with its acceleration described by the equation $g = -p^2 s$, where $p$ is a constant and $s$ is the distance from the equilibrium position. Find the equation of motion.

Answer: s(t) = b \cos(pt)


Several children purchase two types of items priced at 3 yuan and 5 yuan each. Each child buys at least one item, but the total amount spent by each child must not exceed 15 yuan. Xiaomin stated that among the children, there are definitely at least three children who buy the same quantity of each type of item. What is the minimum number of children?

Answer: 25


In a convex quadrilateral, the lengths of the diagonals are 2 cm and 4 cm. Find the area of the quadrilateral, given that the lengths of the segments connecting the midpoints of the opposite sides are equal.

Answer: 4 \text{ cm}^2


From home to work and back. If a person walks to work and takes transport back, it takes him a total of one and a half hours. If he takes transport both ways, the entire journey takes him 30 minutes. How much time will it take for the person to walk both to work and back?

Answer: 2.5 \text{ hours}


In the figure, \(ABCD\) is a square with an area that is \(\frac{7}{32}\) of the area of triangle \(XYZ\). What is the ratio between \(XA\) and \(XY\)?

Answer: \frac{1}{8}


The tangent line drawn to the point \( A \) on the cubic parabola \( y = ax^3 \) intersects the \( Y \)-axis at point \( B \) and intersects the curve again at another point \( C \). Determine the geometric locus of the midpoints of the distances \( AB \) and \( BC \), considering that the location of point \( A \) varies.

Answer: y = 5 a x^3


4. In the maze diagram in Fig. 7, each segment (link) is a corridor, and each circle is a small room. In some rooms, there are beacons that hum - each with its own voice. When in any room, the robot hears the signal from each beacon and determines the distance to it by the attenuation, i.e., the number of links on the shortest path to the beacon. For example, if a beacon is in room $A$ (in chess notation, this is room a4), and the robot is in room $B$ (this is room g4), the distance between them is 5 links. The robot has the maze diagram and the beacons are marked on it. For what minimum number of beacons can the robot, when in any room, uniquely determine its location by sound?

a) Indicate how such a number of beacons can be placed. Why will the robot be able to determine its room by them?

b) Prove that it is impossible to manage with fewer beacons.

![](https://cdn.mathpix.com/cropped/2024_05_06_7a6694d067306ba85dedg-06.jpg?height=505&width=525&top_left_y=770&top_left_x=777)

Fig. 7: Maze

Answer: 3


Alex is a software engineer working on privacy controls for a tech company. This week, Alex needs to update the privacy settings for a new feature. The feature is used by 150 employees and each employee requires 5 different privacy settings. For each privacy setting, Alex has to write 3 lines of code and spend an additional 10 minutes reviewing the code. If Alex works 8 hours a day and spends half the time coding and the other half reviewing, how many days will it take Alex to complete the update?

Answer: 32


A fashion photographer is planning an editorial shoot and wants to use a specific color grading technique that involves balancing the primary colors: red, green, and blue. To achieve the perfect look, she needs to mix her current color palette with additional colors. She has a base palette containing 180 units of red, 225 units of green, and 195 units of blue.

For the shoot's theme, each color needs to be increased by a specific percentage: red by 20%, green by 15%, and blue by 10%. After the increase, she also wants to add an equal number of additional units to each color to further enhance the overall vibrancy. If she decides to add 30 additional units to each color after the percentage increase, how many units of red, green, and blue will she have in total for the shoot?

Answer: 779.25


A group of climate researchers is studying how different weather patterns affect the growth of corn in a region. They have collected data over 4 weeks. In the first week, the temperature was 70°F and 5 inches of rain fell. In the second week, the temperature was 75°F and 3 inches of rain fell. In the third week, the temperature was 80°F and 4 inches of rain fell. In the fourth week, the temperature was 85°F and 2 inches of rain fell. The researchers observed that for every degree increase in temperature, the corn yield decreases by 2 bushels, and for every inch of rain, the yield increases by 3 bushels.

If the initial yield without any changes in temperature or rain is 200 bushels, what is the total corn yield at the end of the 4 weeks?

Answer: 182


A retired firefighter named Sam, who is fascinated by extreme weather, decides to help his community prepare for the upcoming stormy season. He volunteers to distribute sandbags to protect homes from potential flooding. Sam plans to give each household 15 sandbags. There are 12 neighborhoods in the community, and each neighborhood has 20 households. However, after distributing sandbags to 7 neighborhoods, Sam receives a report that another storm is approaching, and he needs to distribute an additional 5 sandbags to each household. How many sandbags does Sam need to distribute in total to all the households in the community?

Answer: 4800


Alex is a superfan podcast listener who listens to a variety of social issue-focused podcasts. Each week, Alex listens to 3 different podcasts that cover topics such as equality, mental health, and environmental awareness. Each podcast releases 2 new episodes per week. Alex has set a goal to share insights from these episodes with 5 friends to help raise awareness about important social issues.

This week, Alex listened to all the new episodes from each podcast and shared insights from 60% of the episodes with their friends. How many episodes did Alex share insights from with their friends this week?

Answer: 4


A journalist from a local news agency in Segno is covering a community event. The event starts at 9:00 AM and ends at 2:30 PM. The journalist plans to interview 3 people every hour throughout the event's duration. Each interview lasts about 10 minutes. During one of the hours, the journalist takes a 20-minute break. How many people does the journalist interview by the end of the event?

Answer: 16


A social justice advocate from another state is organizing online campaigns to raise awareness about a discriminatory ordinance. They manage to recruit 8 volunteers, each of whom reaches out to 50 people every week. If each of these people shares the campaign with 5 additional people, how many people in total are reached after 4 weeks?

Answer: 9600


Alex, an Australian expatriate living in Canada, loves listening to comedy podcasts. Every week, he listens to 5 different comedy podcasts, each with a new episode released weekly. Each podcast episode is about 45 minutes long. In a typical week, Alex spends 3 times as much time listening to podcasts as he does watching Australian comedy shows online. If Alex watches 2 Australian comedy shows online each week, and each show is 30 minutes long, how many total minutes does Alex spend on comedy, both podcasts and shows, in one week?

Answer: 285


Eddie, a seasoned software developer with a knack for witty remarks, is working on a new WPF application. To ensure his application runs smoothly, he calculates that he needs to implement a total of 48 features. Each feature takes him approximately 2 hours to develop. However, being a .NET expert, he can optimize his code and save 15 minutes per feature. On top of that, Eddie spends 1 hour every day reviewing and improving his witty remarks in the application, which adds up to 7 hours in a week. If Eddie works 5 days a week and wants to complete all features within 2 weeks, how many hours per day should Eddie work on developing the features, excluding the time spent on his witty remarks?

Answer: 8.4


Judge Johnson, known for her traditional views on the justice system, believes in the importance of incarceration for rehabilitation. Each month, she handles a certain number of cases, half of which result in incarceration. In January, she handled 24 cases. In February, she managed 30 cases. In March, the number of cases increased by 10% compared to February.

Judge Johnson is curious about how many cases led to incarceration over these three months. Calculate the total number of cases that resulted in incarceration during this period.

Answer: 44


An aspiring actor wants to practice his acting skills by watching all of Ethan Hawke's films. He discovers that Ethan Hawke has appeared in 45 movies. The actor decides to watch 3 movies per week. After 5 weeks of watching, he realizes he missed a week and plans to catch up by watching 5 movies each week for the next 2 weeks. How many movies will the aspiring actor have watched in total by the end of these 2 weeks?

Answer: 22


A local hunter named Alex is interested in participating in wildlife conservation efforts. To help with the conservation of deer in the nearby forest, Alex decides to monitor the deer population over several months. In January, Alex counts 120 deer in the forest. By March, due to conservation efforts, the deer population increases by 25%. However, in April, a natural event causes the population to decrease by 15%. How many deer are in the forest at the end of April?

Answer: 128


Jim, an overzealous baseball fan, believes there's a conspiracy in the number of hot dogs sold at baseball games. At his favorite team's stadium, he notices that the number of hot dogs sold increases by 20% whenever the home team wins. On a particular weekend, the team played 3 games and won 2 of them. On Friday's game, 1,000 hot dogs were sold. How many hot dogs were sold in total over the weekend if the number of hot dogs sold is the same for each win and loss, except for the increase after a win?

Answer: 3400


Dr. Mindwell is a psychology professor who specializes in mental health and provides guidance and resources to help people navigate their challenges. She hosts a weekly mental health workshop where she distributes helpful resources to attendees. This week, she prepared 120 pamphlets and 60 stress-relief journals to give out.

During the workshop, she noticed that each attendee took 2 pamphlets and 1 journal. By the end of the workshop, Dr. Mindwell had 20 pamphlets and 15 journals left.

How many attendees were at Dr. Mindwell's workshop this week?

Answer: 50


Alex is a young American soccer enthusiast who loves learning about the history of the game. He recently discovered that the United States hosted the FIFA World Cup in 1994, a year before he was born. Alex learned that during the 1994 World Cup, a total of 52 matches were played, and the total attendance for all matches combined was 3,587,538 spectators. Alex wonders about the average number of spectators per match during that tournament. Calculate the average number of spectators per match by dividing the total attendance by the number of matches played.

Answer: 68991


Jamie is an ambitious news reporter who strives to provide the quickest updates on world politics. One day, Jamie received news alerts from three different time zones. The first alert came from London at 9:00 AM local time, the second alert came from New York at 4:00 AM local time, and the third alert came from Tokyo at 6:00 PM local time. Jamie wants to calculate the total number of hours between the first alert and the last alert, considering the time differences between the cities: London is 5 hours ahead of New York and 9 hours behind Tokyo. How many hours are there between the first alert from London and the last alert from Tokyo?

Answer: 0


At the veterans' hospital, the aged but lively administrator is organizing a special event for 120 veterans. She is known for her efficiency and excellent organizational skills, so she has decided to arrange the veterans into groups for different activities. Each activity requires a specific number of participants: painting needs 12 veterans, gardening needs 15 veterans, and storytelling needs 18 veterans. The administrator wants to ensure that all veterans participate in at least one activity. How many veterans will be left without an assigned activity if she assigns the maximum possible number to each of these three activities?

Answer: 0


The South African ambassador in Kenya is organizing a special event to promote trade relationships between South Africa and Kenya. She decides to invite 120 South African business delegates and 80 Kenyan business delegates. During the event, each South African delegate exchanges contact information with 3 Kenyan delegates, and each Kenyan delegate exchanges contact information with 2 South African delegates. How many total exchanges of contact information occur during the event?

Answer: 240


Alex, a dedicated professional who analyzes and predicts cyber threats, is tasked with protecting an organization's network from potential attacks. To strengthen the network, Alex decides to implement a series of protective measures. First, he identifies 12 potential vulnerabilities. For each vulnerability, he estimates that it will take 3 hours to analyze and another 4 hours to implement a fix. After analyzing and fixing the vulnerabilities, Alex initiates a monitoring system that checks each vulnerability every 6 hours. If Alex works 8 hours a day, how many total days will it take him to complete the analysis, implement all fixes, and monitor each vulnerability once?

Answer: 12


Maria is a young immigrant who has just arrived in a new country. She's excited about making new friends and values the support and friendship she can share with others. On her first day at school, she meets 4 new friends: Alex, Jamie, Sam, and Lee. Each of her new friends gives her a small welcome gift. Alex gives Maria 3 colorful pencils, Jamie gives her 2 notebooks, Sam gives her 5 stickers, and Lee gives her 4 erasers. If Maria wants to give each of her 4 friends an equal number of gifts in return, how many gifts should she prepare in total?

Answer: 16


Maria, a fellow probate court survivor, organizes group meetings to discuss strategies and share resources. She has scheduled 5 meetings this month, and she expects an average of 8 participants per meeting. Each participant brings 3 resource materials to share, such as pamphlets, brochures, or articles. If Maria wants to prepare a folder for each participant that includes one copy of each unique resource material shared during the meetings, how many total resource materials does Maria need to prepare for all the participants across all meetings?

Answer: 4800


A podcast host records an episode with Dave Franco, which lasts for 90 minutes. During the first 30 minutes, they discuss Dave's early career, followed by a 15-minute segment about his creative process. The remaining time is dedicated to discussing his recent projects. If the host wants to include 3 commercial breaks of equal duration throughout the episode, how long is each commercial break in minutes?

Answer: 5


Jenna is a production line operator who assembles gadgets at a factory. Each day, she relies on Sam, the materials coordinator, to deliver the raw materials needed for her work. Jenna can assemble 15 gadgets per hour when she has enough materials. One day, Sam delivers 120 gadgets' worth of materials in the morning and another 90 gadgets' worth in the afternoon. If Jenna works for 8 hours that day, how many more gadgets could she have assembled if Sam had delivered all the materials in the morning?

Answer: 0


A positive integer $n$ is considered "funny" if for every positive divisor $d$ of $n$, the number $d+2$ is a prime number. Determine all funny numbers that have the maximum possible number of divisors.

Answer: 135


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

Answer: 256


A set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each with at least two elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers.

Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.

Answer: 30


Two three-letter strings, $aaa$ and $bbb$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a $1/3$ chance of being received incorrectly, as an $a$ when it should have been a $b$ , or as a $b$ when it should be an $a$ . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a^{}$ be the three-letter string received when $aaa$ is transmitted and let $S_b^{}$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a^{}$ comes before $S_b^{}$ in alphabetical order. When $p$ is written as a fraction in lowest terms , what is its numerator ?

Answer: 532


Twelve congruent disks are placed on a circle $C$ of radius $1$ in such a way that the twelve disks cover $C$ , no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$ , where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$ .

Answer: 135


5. Let $K$ be the incenter of $\triangle A B C$, and let points $C_{1}, B_{1}$ be the midpoints of sides $A B, A C$, respectively. Line $A C$ intersects $C_{1} K$ at point $B_{2}$, and line $A B$ intersects $B_{1} K$ at point $C_{2}$. If $S_{\triangle A B_{2} c_{2}}=S_{\triangle A B C}$, find $\angle C A B$.

Answer: 60^\circ


6. As shown in Figure 1, in the right triangle $\triangle ABC$, $\angle ACB=90^{\circ}$, $CA=4$, $P$ is the midpoint of the semicircular arc $\overparen{AC}$, connect $BP$, the line segment $BP$ divides the figure $APCB$ into two parts. The absolute value of the difference in the areas of these two parts is $\qquad$.

Answer: 4


3. A five-digit number ABCDE, consisting of different digits, is divisible by both the three-digit number CDE and the two-digit number AB. Find the smallest possible value of ABCDE.

Answer: 12480


Four brothers inherited a plot of land in the shape of a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three brothers received plots of land measuring 360, 720, and $900 \mathrm{~m}^{2}$ respectively. How much land did the fourth brother receive?

Answer: 540


Problem 4. Determine all non-zero natural numbers $p$ for which there exists $n \in \mathbb{N}^{*}$ such that $p^{n}+3^{n}$ divides the number $p^{n+1}+3^{n+1}$.

Answer: 3


8. Arrange $1,2, \cdots, n^{2}$ in a clockwise spiral pattern into an $n$ by $n$ table $T_{n}$, with the first row being $1,2, \cdots, n$. For example: $T_{3}=\left[\begin{array}{lll}1 & 2 & 3 \\ 8 & 9 & 4 \\ 7 & 6 & 5\end{array}\right]$.
Let 2018 be in the $i$-th row and $j$-th column of $T_{100}$, then $(i, j)=$ . $\qquad$

Answer: (34,95)


3. Arrange $1,2, \cdots, k$ in a row so that each number is strictly greater than all the numbers preceding it, or strictly less than all the numbers preceding it. Let the number of different arrangements be $a_{k}(k=1,2, \cdots)$. Then $a_{n}=$ $\qquad$ .

Answer: 2^{n-1}


Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$.

Answer: \dfrac{(7 - \sqrt{7})(1 + \sqrt{7})^{2003} + (7 + \sqrt{7})(1 - \sqrt{7})^{2003}}{14}


A six-digit number $N=\overline{a_1a_2...a_{6}}$ composed of non-repeating digits from 1 to 6 satisfies the condition $|a_{k+1}-a_{k}| \neq 1, (k\in \{1,2, \cdots, 5\})$. Then the number of such six-digit numbers is \_\_\_\_\_.

Answer: 90


8. Given the function $f(x)=\log _{\frac{1}{2}} x$, let $x=\frac{a}{f(a)}$, $y=\frac{b}{f(b)}, z=\frac{c}{f(c)}(0<c<b<a<1)$. Then, the order of $x 、 y 、 z$ in terms of size is $\qquad$ .

Answer: z < y < x


6.108. $\left\{\begin{array}{l}3(2-\sqrt{x-y})^{-1}+10(2+\sqrt{x+y})^{-1}=5 . \\ 4(2-\sqrt{x-y})^{-1}-5(2+\sqrt{x+y})^{-1}=3 .\end{array}\right.$

## Solution.

Rewrite the system of equations as

$\left\{\begin{array}{l}\frac{3}{2-\sqrt{x-y}}+\frac{10}{2+\sqrt{x+y}}=5 \\ \frac{4}{2-\sqrt{x-y}}-\frac{5}{2+\sqrt{x+y}}=3 .\end{array}\right.$

Domain of definition: $\left\{\begin{array}{l}x-y \geq 0 \\ x+y \geq 0 \\ x-y \neq 4\end{array}\right.$

Let $\left\{\begin{array}{l}\frac{1}{2-\sqrt{x-y}}=u, \\ \frac{1}{2+\sqrt{x+y}}=v .\end{array}\right.$ Relative to $u$ and $v$, the system of equations takes the form

$\left\{\begin{array}{l}3 u+10 v=5, \\ 4 u-5 v=3\end{array}\right.$

From this, we get $u=1 ; v=\frac{1}{5}$. Then

$$
\left\{\begin{array}{l}
\frac{1}{2-\sqrt{x-y}}=1, \\
\frac{1}{2+\sqrt{x+y}}=\frac{1}{5}
\end{array},\left\{\begin{array} { l }
{ 2 - \sqrt { x - y } = 1 , } \\
{ 2 + \sqrt { x + y } = 5 }
\end{array} \Leftrightarrow \left\{\begin{array} { l }
{ \sqrt { x - y } = 1 , } \\
{ \sqrt { x + y } = 3 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x-y=1 \\
x+y=9
\end{array}\right.\right.\right.\right.
$$

from which

$\left\{\begin{array}{l}x=5 \\ y=4\end{array}\right.$

Answer: $(5 ; 4)$ )

6:109. $\left\{\begin{array}{l}\sqrt[3]{x}+\sqrt[3]{y}=4 \\ x+y=28\end{array}\right.$

Answer: (27, 1)


13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).

Upon entering the station, the Scientist boards the first train that arrives. It is known that trains run at approximately equal intervals in both directions, and that the train traveling on the northern route (through "Belorusskaya") takes 17 minutes to travel from "Kievskaya" to "Taganskaya" or vice versa, while the train on the southern route (through "Paveletskaya") takes 11 minutes.

Out of habit, the Scientist always calculates everything. Once, he calculated that over many years of observation:

- the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the train traveling clockwise arrives at the same station. The same is true for "Taganskaya";

- the average travel time from home to work is 1 minute less than the travel time from work to home.

Find the expected interval between trains traveling in the same direction.

![](https://cdn.mathpix.com/cropped/2024_05_06_addd5bb914f503d18f1fg-08.jpg?height=785&width=785&top_left_y=1606&top_left_x=1138)

Answer: 3


5. In quadrilateral $A B C D$, the diagonals intersect at point $O$. It is known that $S_{A B O}=S_{C D O}=\frac{3}{2}$, $B C=3 \sqrt{2}$, and $\cos \angle A D C=\frac{3}{\sqrt{10}}$. Find the sine of the angle between the diagonals of this quadrilateral, if its area takes the smallest possible value under the given conditions.

Answer: \dfrac{6\sqrt{37}}{37}


A photo archive is arranged in identical albums, with exactly 4 photos on each page. The 81st photo ends up on the 5th page of one album, and the 171st photo ends up on the 3rd page of another. How many photos does each album hold?

Answer: 32


In the garden, there were three boxes of apples. Altogether, there were more than 150 apples, but less than 190. Maruška moved apples from the first box to the other two boxes so that the number of apples in each of these two boxes doubled compared to the previous state. Similarly, Marta moved apples from the second box to the first and third. Finally, Šárka moved apples from the third box to the first and second according to the same rules. When Vojta came to the garden, he was surprised that there were the same number of apples in each box.

How many apples were there in each of the boxes originally?

(L. Hozová)

Hint. In which box were there the most apples after the second move?

Answer: 91


3 [ [sphere, circumscribed around a pyramid]

A regular quadrilateral pyramid is inscribed in a sphere. The radius of the sphere is 1. The plane angle at the vertex of the pyramid is $45^{\circ}$. Find the lateral surface area of the pyramid.

#

Answer: 4


## Task A-1.7.

How many five-digit natural numbers are there whose product of digits is equal to 900?

Answer: 210


In the inscribed quadrilateral $\mathrm{ABCD}$, the circle circumscribed around triangle $\mathrm{COD}$ (where $\mathrm{O}$ is the intersection point of the diagonals) passes through the center of the circumscribed circle of quadrilateral $\mathrm{ABCD}$. Prove that quadrilateral $\mathrm{ABCD}$ is a trapezoid.

Answer: \text{ABCD is a trapezoid}


12. (11 points) Class 3 (1) has a total of 45 students who have signed up for the school sports meet. Apart from the tug-of-war competition, in which the entire class participates, each student must participate in at least one of the other three events. It is known that 39 students have signed up for the shuttlecock kicking competition, and 28 students have signed up for the basketball shooting competition. How many students have signed up for all three events?

Answer: 22


13. (3 points) $A, B, C, D, E$ five boxes contain 9, 5, 3, 2, 1 balls respectively. The first student finds the box with the fewest balls, then takes 1 ball from each of the other boxes and puts it into this box. The second student finds the box with the fewest balls, then takes 1 ball from each of the other boxes and puts it into this box $\cdots$; after the 199th student has finished, $A, B, C, D, E$ five boxes will each have $\qquad$ balls, $\qquad$ balls, $\qquad$ balls, $\qquad$ balls, $\qquad$ balls.

Answer: 4


The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ so that $\sin^2\theta  = \tfrac{m}{n}$. Find $m + n$.

[asy]
import graph; size(3.2cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle);
draw((-1,3)--(-1,2));
draw((-1,2)--(-0.13,1.5));
draw((-0.13,1.5)--(0.73,2));
draw((0.73,2)--(0.73,3));
draw((0.73,3)--(-0.13,3.5));
draw((-0.13,3.5)--(-1,3));
draw((-1,3.5)--(0.73,3.5));
draw((0.73,3.5)--(0.73,1.5));
draw((-1,1.5)--(0.73,1.5));
draw((-1,3.5)--(-1,1.5));
label("$ A $",(-1.4,3.9),SE*labelscalefactor);
label("$ B $",(-1.4,3.28),SE*labelscalefactor);
label("$ C $",(-1.4,2.29),SE*labelscalefactor);
label("$ D $",(-1.4,1.45),SE*labelscalefactor);
label("$ E $",(-0.3,1.4),SE*labelscalefactor);
label("$ F $",(0.8,1.45),SE*labelscalefactor);
label("$ G $",(0.8,2.24),SE*labelscalefactor);
label("$ H $",(0.8,3.26),SE*labelscalefactor);
label("$ I $",(0.8,3.9),SE*labelscalefactor);
label("$ J $",(-0.25,3.9),SE*labelscalefactor); [/asy]

Answer: 55


Alex, a young playwright living next door, is working on a new play and wants to involve his neighbors in the creative process. He decides to host a series of writing workshops to brainstorm ideas. He plans to hold these workshops over 3 weeks, with each week featuring 2 sessions. During each session, Alex invites 5 neighbors to join him, including himself.

In the first week, 2 of the neighbors bring a friend to each session, increasing the number of participants. In the second week, 1 neighbor cannot attend any session, but the rest still participate. In the third week, Alex decides to invite 1 additional neighbor to each session, but only 3 sessions actually occur due to scheduling conflicts.

How many total participant attendances does Alex have across all the sessions over the three weeks?

Answer: 40


Alex is a Java software engineer who specializes in health information exchange. In a week, Alex needs to exchange health data records between 4 hospitals. Each hospital has 15,000 records to be exchanged. For each record, Alex's software takes 0.2 seconds to process and validate before the exchange.

If Alex works 5 days a week and 8 hours each day, how many complete records can Alex's software process and validate in a single week?

Answer: 720000


A musician who specializes in vintage analog instruments spends 3 hours each day practicing on a classic synthesizer and 2 hours each day playing a vintage drum machine. If the musician dedicates 2 additional hours on the weekends (Saturday and Sunday) to performing live with these instruments, how many total hours does the musician spend creating and performing music in a week?

Answer: 39


Alex is a programmer with expertise in Python and pandas, and they enjoy automating and improving workflows. One day, Alex decided to automate a process that involves handling data files. Initially, Alex had 12 data files that needed to be processed manually, which took 15 minutes each. After automating the process using Python and pandas, Alex reduced the processing time to 5 minutes per file.

If Alex spends 2 hours writing the automation script, how much total time, in hours, will Alex save by using the automation to process all 12 files compared to doing it manually?

Answer: 0


A fisherman named Sam is famous for providing the secret ingredient for a popular street food delicacy. Every day, he catches fish and collects a special type of seaweed that enhances the flavor of the dish.

On Monday, Sam caught 15 fish and gathered 4 bundles of seaweed. Each bundle of seaweed adds a unique flavor to 3 plates of the delicacy. On Tuesday, he caught 20 fish and gathered 5 bundles of seaweed. Finally, on Wednesday, he caught 10 fish and gathered 6 bundles of seaweed.

Each fish can be used to prepare 2 plates of the delicacy. Based on the fish and seaweed he gathered from Monday to Wednesday, how many plates of the street food delicacy can Sam prepare in total, assuming he uses all the fish and seaweed he collected?

Answer: 45


Jamie, an event planner, is organizing a large banquet and decides to use digital invitations to streamline the RSVP process. She sends out 250 digital invitations. After one week, she receives 60% of the responses. Of those who responded, 75% confirmed their attendance. How many people have confirmed they will attend the banquet after the first week?

Answer: 113


Dr. Riley, an archaeologist studying ancient artifacts, has discovered a collection of 120 bronze coins buried in an old excavation site. Over the years, corrosion has affected 45% of these coins, making them difficult to analyze and categorize. Dr. Riley plans to clean the corroded coins to restore them for further study.

After cleaning each corroded coin, Dr. Riley finds that 60% of them can be fully restored, while the rest are too damaged to be of any use.

How many of the original 120 bronze coins are successfully restored and usable for further study after the cleaning process?

Answer: 32


The cunning criminal mastermind has devised a plan that involves four different heists, each with its own twist. In the first heist, he steals 150 gold coins. For the second heist, he challenges a novelist to come up with a plot twist, and for every clever idea, he adds 25 more gold coins to his loot, ultimately collecting a total of 200 gold coins. In the third heist, he doubles his total collection from the first two heists. In the final heist, he decides to leave behind 15% of his total collection for the authorities to find, as a hint to his cleverness. How many gold coins does the mastermind have left after the final heist?

Answer: 595


Emma and Alex are co-founders of a tech startup. Together, they are dedicated to growing their company and ensuring it excels. In a bid to increase their company's excellence, they decide to conduct a series of workshops over the next month. Emma plans to organize 3 workshops per week, while Alex plans to organize 2 additional workshops per week. If each workshop requires 5 hours of preparation and they both commit to equally sharing all the preparation time, how many hours will each of them need to dedicate to workshop preparation by the end of the month, assuming the month has 4 weeks?

Answer: 50


A songwriter is trying to write a new song with unique and compelling lyrics. They decide that each verse of the song should have 8 lines. The songwriter plans to write 5 verses for the song. If they have a total of 120 unique words they can use, how many times will each word be used, on average, throughout the entire song?

Answer: 1


Dr. Penny Fathom, a historical researcher and monetary mechanism enthusiast, is studying the evolution of coinage systems over the centuries. She discovers a fascinating pattern related to currency issued in the 18th century. In her research, she finds that in the year 1700, a kingdom issued 1,200 coins. Each subsequent year, the number of coins issued increased by a fixed amount. By the year 1710, the kingdom had issued a total of 1,650 coins.

Dr. Fathom wants to calculate the yearly increase in the number of coins issued. Can you help her determine how many more coins were issued each year?

Answer: 45


Anna is a loyal supporter of the political party "Latvian Farmers' Union and Union of Greens and Farmers". She decided to organize a community event to promote sustainable farming practices. She invited local farmers and supporters to join the event. Anna prepared 5 different types of seeds to give away, each type in packets of 8. Additionally, she plans to distribute 3 eco-friendly reusable bags to every participant. If 12 participants attended the event, how many seed packets and reusable bags did Anna distribute in total?

Answer: 76


Mr. Baker, the understanding owner of "Sweet Loyalty Bakery," values his regular customers for their loyalty. On Monday, he sold a total of 150 pastries. Out of these, 60% were sold to his loyal customers, who each bought 3 pastries on average. On Tuesday, Mr. Baker noticed an increase in sales and sold 180 pastries, with 70% going to his loyal customers, each buying 4 pastries on average. How many loyal customers purchased pastries from Mr. Baker on Monday and Tuesday combined?

Answer: 62


Mike wants to purchase a rectangular mirror (the shaded region) that fits exactly inside a frame. The outer perimeter of the frame measures 100 cm by 140 cm. The width of each side of the frame is 12 cm. What is the area of the mirror?

Answer: 8816 \text{ cm}^2


Julien and his Canadian friend Lucas are at a café in Lyon that accepts both euros and Canadian dollars. They want to buy a pie, but neither of them has enough money. If the pie costs 12 euros and Lucas has a Canadian ten-dollar bill, how many euros does Julien need to contribute to the cost of the pie if 1 euro = 1.5 CAD?

Answer: 5.33 \text{ euros}


Molly flips a fair coin eight times, each time surprisingly landing on heads. After the coin flips, she rolls a fair six-sided die. Calculate the probability that:
a) Molly will flip tails on her next coin flip, and
b) the die roll results in either a 1 or a 2.

Answer: \frac{1}{6}


A torus is formed by revolving a circle in the $xz$-plane centered at $(4, 0, 2)$ with radius $2$ around the $z$-axis. This torus rests on a flat table represented by the $xy$-plane. Determine the radius of the largest spherical ball that can be placed atop the center of the torus while still touching the horizontal plane.

Answer: 2


Chewbacca has 15 pieces of cherry gum, 25 pieces of grape gum, and 10 pieces of apple gum. Some pieces are loose, and others are in complete packs, with each pack containing exactly $y$ pieces of gum. If Chewbacca loses one pack of cherry gum and finds one pack of apple gum, then the ratio of cherry gum he has to the number of grape gum will be the same as if he finds 2 extra packs of grape gum. Find $y$.

Answer: \frac{5}{2}


Eleven stones are arranged in a straight line. They are counted left to right as $1,2,3, \ldots, 11$, and then from right to left, so that the stone previously counted as 10 is now 12. The pattern continues to the left until the stone previously counted as 1 is counted as 21. The pattern then reverses so that the stone originally counted as 2 is counted as 22, and so on. This counting continues. Which of the original stones is counted as 123?

Answer: 3


In Mr. Green's 30-student physics class, no two students have the same initials, and each student's first name and last name begin with the same letter. Including "Y" as a vowel, what is the probability that a randomly picked student's initials are vowels, assuming all 26 letters of the alphabet are used as initials? Express your answer as a common fraction.

Answer: \frac{3}{13}


Two vertices of a right triangle are located at \((3, 3)\) and \((0,0)\). The third vertex is on the positive branch of the \(x\)-axis. What are the coordinates of the third vertex if the area of the triangle is 18 square units?

Answer: (12,0)


Triangles $PQR$ and $PRS$ are isosceles with $PQ=QR$ and $PR=RS$. Point $S$ is inside $\triangle PQR$, $\angle PQR = 50^\circ$, and $\angle PRS = 110^\circ$. What is the degree measure of $\angle
PQS$?

Answer: 30^\circ


The matrix
\[\begin{pmatrix} \frac{1}{13} & -\frac{12}{13} \\ -\frac{3}{13} & \frac{10}{13} \end{pmatrix}\]
is used to project onto a certain vector $\begin{pmatrix} u \\ v \end{pmatrix}$. Find $\frac{v}{u}$.

Answer: -1


12 people attend two separate meetings. At the first meeting, 7 people attend and shake hands with each other. At the next meeting, 5 new people attend along with 2 people from the previous meeting, and everyone shakes hands with each other without anyone from the first meeting shaking hands again. How many total handshakes occur at both meetings?

Answer: 41


Initially, an amoeba is placed in a puddle, and from that day on, each amoeba splits into two amoebas each subsequent day. However, a second amoeba is introduced into the puddle on the third day. Calculate the total number of amoebas at the end of fourteen days.

Answer: 10240


A rectangular prism has a volume of 720 cubic units, where $a, b,$ and $c$ are integers and $1 < c < b < a$. Find the maximum possible value of $b$ if $c=3$, while ensuring $a$, $b$, and $c$ are distinct prime factors.

Answer: 5


In a weekend volleyball tournament, there are six teams: A, B, C, D, E, and F. On Saturday, Team A plays Team E, Team B plays Team C, and Team D plays Team F. Then on Sunday, the three Saturday winners play round-robin matches (each team plays every other team once) for first, second, and third places, while the three losers play for fourth, fifth, and sixth places. There are no ties in any game. Given that a possible ranking of the teams from first to sixth place at the end of the tournament is AECBDF, determine the total number of possible six-team ranking sequences at the end of the tournament.

Answer: 288


In the diagram, rectangle $ABCD$ is divided into four identical squares. If $ABCD$ has a perimeter of 160 cm, what is its area, in square centimeters?

Answer: 1600 \text{ square centimeters}


The ratio of the areas of two squares is $\frac{75}{98}$. After rationalizing the denominator, what is the simplified form of the ratio of their side lengths, and calculate the sum of all integers involved.

Answer: 25


25.2. (USA, 82). In a society consisting of 1982 people, among any four people, at least one can be chosen who is acquainted with the other three. What is the minimum possible number of people who are acquainted with everyone?

Answer: 1979


(2) On a certain section of railway, there are three stations $A$, $B$, and $C$ in sequence, with $A B=5 \mathrm{~km}$ and $B C=3 \mathrm{~km}$. According to the train schedule, the train is supposed to depart from station $A$ at 8:00, arrive at station $B$ at 8:07 and stop for 1 minute, and arrive at station $C$ at 8:12. In actual operation, assume the train departs from station $A$ on time, stops for 1 minute at station $B$, and travels at a constant speed of $v \mathrm{~km} / \mathrm{h}$ while moving. The absolute value of the difference between the actual arrival time of the train at a station and the scheduled time is called the operation error of the train at that station.
(1) Write down the operation errors of the train at stations $B$ and $C$;
(2) If the sum of the operation errors at stations $B$ and $C$ is required to be no more than 2 minutes, find the range of values for $v$.

Answer: [39, \frac{195}{4}]


Let $I$ be the center of the incircle of triangle $ABC$ and let $A', B'$ and $C'$ be the symmetrics of $I$ with respect to the lines $(BC), (CA)$ and $(AB)$ respectively. The circumcircle of $A'B'C'$ passes through $B$. Find $\widehat{A B C}$.

Answer: 60^\circ


## Task 2 - 210912

Mr. Schulze meets Mr. Lehmann after a long time and invites him to his home. On the way, he tells Mr. Lehmann that he is the father of three children. Mr. Lehmann wants to know how old they are; he is satisfied with age statements in full years.

Mr. Schulze replies: "The product of the three age statements is 72. The sum of the three age statements is my house number. We have just arrived at our house; you can see my number." To this, Mr. Lehmann responds: "But one cannot uniquely determine the three age statements from this information." "That's true," Mr. Schulze agrees, "but sometime between the birth of the second and the third child, this house was built. We worked on the house for one year and one day." "Thank you! Now the age statements can be uniquely determined," Mr. Lehmann concludes the conversation.

Investigate whether there is a combination of three age statements that satisfy all the statements in this conversation! Investigate whether there is only one such combination! If that is the case, determine this!

Answer: 2, 6, 6


## Task A-1.2.

Determine all real numbers $a$ for which the equation

$$
|| x-2|-x+a|=x+3
$$

has exactly two real solutions.

Answer: -3


An equilateral triangle with side length \( n \) is divided into \( n^2 \) smaller equilateral triangles with side length 1 by lines parallel to its sides. The \( \frac{n(n+1)}{2} \) vertices of the smaller triangles are each labeled with a real number, such that if \( ABC \) and \( BCD \) are small triangles, then the sum of the labels on \( A \) and \( D \) equals the sum of the labels on \( B \) and \( C \). The labels on the vertices of the original triangle are \( a \), \( b \), and \( c \). Determine:

1. The shortest distance between a vertex with the largest label and a vertex with the smallest label.
2. The sum of all the labels.

Answer: n


Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)
$$

for all real number $x$ and $y$.
Answer: The possible functions are $f(x)=0$ for all $x$ and $f(x)=x^{2}$ for all $x$.

Answer: 0


1. On the coordinate plane, plot the set of points $L(x, y)$ that are solutions to the equation $\frac{x^{2}+x}{y^{2}+y}=1$.

Answer: y = x


Proposition 1 As shown in Figure 2, place three rectangles $X_{1}, X_{2}, X_{3}$ in an acute triangle $T$. Let $A(X)$ denote the area of rectangle $X$. Question: Does the maximum value of $\frac{A\left(X_{1}\right)+A\left(X_{2}\right)+A\left(X_{3}\right)}{A(T)}$ exist? If it exists, find the maximum value.

Answer: \dfrac{3}{4}


8. In the figure shown in Figure 3, on both sides of square $P$, there are $a$ and $b$ squares to the left and right, and $c$ and $d$ squares above and below, where $a$, $b$, $c$, and $d$ are positive integers, satisfying
$$
(a-b)(c-d)=0 \text {. }
$$

The shape formed by these squares is called a "cross star".
There is a grid table consisting of 2014 squares, forming a $38 \times 53$ grid. Find the number of cross stars in this grid table.
(Tao Pingsheng, provided)

Answer: 13483236


p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$.

p2. If $a, b$, and $c$ satisfy the system of equations
$$  \frac{ab}{a+b}=\frac12$$
$$\frac{bc}{b+c}=\frac13 $$
$$ \frac{ac}{a+c}=\frac17 $$
Determine the value of $(a- c)^b$.

p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of the quadrilateral $PMCN$.
[img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img]

p4. Given the rule of motion of a particle on a flat plane $xy$ as following:
$N: (m, n)\to (m + 1, n + 1)$
$T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers.
How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ?

p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

Answer: \dfrac{1}{2} [ABC]


[ Processes and Operations ]

Between neighboring camps, it takes 1 day to travel. The expedition needs to transfer 1 can of food to the camp located 5 days away from the base camp and return. At the same time:

- each member of the expedition can carry no more than 3 cans of food;
- in 1 day, he consumes 1 can of food;
- food can only be left in camps.

What is the minimum number of cans of food that will need to be taken from the base camp for this purpose?

Answer: 243


4. Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that

$$
f(x+f(y)+f(f(z)))=y f(1+f(f(y))(x+z)) \quad \text { for all } x, y, z \in \mathbb{R}^{+}
$$

Answer: f(x) = \dfrac{1}{x}


6.2. A person has less than 1 Ruble (the currency unit of Russia, 1 Ruble = 100 Kopecks). He buys 5 pieces of candy, leaving 4 Kopecks; buys 6 pencils, leaving 3 Kopecks; buys 7 exercise books, leaving 1 Kopeck. Please answer: How much does each piece of candy cost? How much money does the person have in total? Try to provide all possible answers.

Answer: 19


Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$?

Answer: Yes


Let $A B C D$ and $E F G H$ be squares with side lengths 33 and 12, respectively, with $E F$ lying on side $D C$ (as shown in the figure below). Let $X$ be the intersection point of segments $H B$ and $D C$. Suppose that $\overline{D E}=18$.

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-17.jpg?height=754&width=542&top_left_y=1209&top_left_x=839)

a) Calculate the length of the segment $\overline{E X}$.

b) Prove that the points $A, X$, and $G$ are collinear.

Answer: 4


13. As shown in the figure, $ABCD$ is a square. Two ants $\mathbf{P}$ and $\mathbf{Q}$ start from point $\mathbf{C}$ at the same time, both moving at a speed of $0.25 CD /$ second. $\mathbf{P}$ moves along the ray $CD$ (the ray means that after the ant reaches point $\mathbf{D}$, it continues to move in the $CD$ direction), and $\mathbf{Q}$ moves along the ray $CB$. After moving for 2 seconds, the area of the overlapping part of $\triangle QPC$ and square $ABCD$ is 100 square centimeters. Then, after moving for $\mathbf{5}$ seconds, what is the area of the overlapping part of $\triangle QPC$ and square $ABCD$?

Answer: 575


78. Chaochao used 24 matchsticks to form a 3x3 grid, which contains 14 squares. Qiqi took away some matchsticks, and as a result, there were no complete squares left in the remaining figure. Therefore, Qiqi took away at least $\qquad$ matchsticks.

Answer: 6


6. Find the smallest natural number $n$, such that in any simple graph with 10 vertices, $n$ edges, and two-colored edges, there always exists a monochromatic triangle or a monochromatic quadrilateral.

Answer: 31


1. The Chinese Football League A has 12 football clubs participating, with a home and away double round-robin system, meaning any two teams play each other twice, once at home and once away. A win earns 3 points, a draw earns 1 point each, and a loss earns 0 points. At the end of the league, teams are ranked according to their points. The maximum points difference between two teams with adjacent positions on the points table can be $\qquad$ points.

Answer: 46


Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number  of primes that can occur before encoutering the first composite term?

Answer: 5


14. As shown in the figure, the area of right trapezoid $\mathrm{ABCD}$ is $52, D E=F C, \mathrm{AF}$ intersects $\mathrm{BE}$ at $\mathrm{O}$, and the area of triangle $\mathrm{AOB}$ is 17. Therefore, the sum of the areas of triangles $\mathrm{AOE}$ and $\mathrm{BOF}$ is $\qquad$.

Answer: 18


A card shuffling machine always rearranges the cards in the same way relative to their initial order in one step. We placed the 13 spade cards in ascending order from 2 to Ace. The machine shuffled the cards once, and we put the resulting deck back in, which the machine shuffled again, then dealt out the 13 cards in the following order:

Jack, 10, King, 9, Ace, $4,5,2,6$, Queen, $7,3,8$.

What was the order of the cards after the first shuffle?

Answer: 10, 2, 5, King, Queen, 8, 4, 3, Jack, 6, Ace, 9, 7


Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows:

$a_0 = 0$,

$a_1 = 1$, and

$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.

Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.

Answer: 2


3. Ten consecutive natural numbers are written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square?

Answer: 4


8.4. In the record $*+*+*+*+*+*+*+*=* *$, replace the asterisks with different digits so that the equality is true. Justify that the given example is unique up to the order of the addends in the left part of the equation.

Answer: 0 + 1 + 2 + 4 + 5 + 7 + 8 + 9 = 36


C2. We call a positive integer sunny if it consists of four digits and if, moreover, each of the two outer digits is exactly 1 greater than the digit next to it. The numbers 8723 and 1001 are, for example, sunny, but 1234 and 87245 are not.

(a) How many sunny numbers are there whose double is also a sunny number?

(b) Prove that every sunny number greater than 2000 is divisible by a three-digit number that has a 9 in the middle.

Answer: 16


4. Two squares $A B C D$ and $B E F G$ share the vertex $B$, with $E$ on the side $B C$ and $G$ on the side $A B$, as shown.
The length of $C G$ is $9 \mathrm{~cm}$ and the area of the shaded region is $47 \mathrm{~cm}^{2}$.
Calculate the perimeter of the shaded region.

Answer: 32


4. As shown in Figure 3, let $\odot O$ be the circumcircle of the unit square $ABCD$, and point $P$ lies on the minor arc $AB$ (excluding $A$ and $B$). $DP$ intersects $AO$ at point $Q$. Then the range of $k=\frac{S_{\triangle AQD}}{S_{\triangle OPC}}$ is

Answer: [2\sqrt{2} - 2, 1)


Let $n$ be a natural number, with the prime factorisation
    \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define
    \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

Answer: \frac{1}{k}x


27. 9 racing cars have different speeds, and they need to compete to determine who is faster, but there are no timing tools, so they can only race on the track to see who comes first, and each time a maximum of 3 cars can race. Therefore, the minimum number of races needed to guarantee selecting the 2 fastest cars is $\qquad$.

Answer: 5


Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction, the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?

Answer: 55


A pool has three inlets and one outlet. When the outlet and two of the inlets are open, it takes 6 hours, 5 hours, and 4 hours respectively to fill the pool. When the outlet and all three inlets are open, it takes 3 hours to fill the pool. If all three inlets are open without the outlet, how many hours will it take to fill the pool?

Answer: 2\frac{14}{23}


Magnus and Viswanathan play a game against each other three times:
- In each game, each player's score is a positive integer. The players' scores at the end of the three games are six different integers.
- In each game, the winner's score is at least 25 points. If the winner's score is 25, their opponent's score must be at most 23. If the winner's score is greater than 25, their opponent's score must be exactly 2 less than the winner's score.
- Viswanathan wins either the first game or the second game, but not both.
- Viswanathan wins the third game with a score of 25.
- Each player's score in the second game is the average of their scores in the first and third games.
What was Magnus's score in the third game?

Answer: 19


Two circles with a radius of two units intersect at points \( A \) and \( B \). One of their common tangents touches the circles at points \( E \) and \( F \). What can be the radius of the circle that passes through points \( E, F \), and \( A \)?

Answer: 1


Through point \( A \) located on a circle, a diameter \( AB \) and a chord \( AC \) are drawn, where \( AC = 8 \) and \( \angle BAC = 30^\circ \).

Find the length of the chord \( CM \), which is perpendicular to \( AB \).

Answer: 8


1. Let $S(x)$ denote the sum of the digits of the natural number $x$. Solve the equations:

a) $x + S(x) + S(S(x)) = 1993$

b) $x + S(x) + S(S(x)) + S(S(S(x))) = 1993$

2. It is known that the number $n$ is the sum of the squares of three natural numbers. Show that the number $n^{2}$ is also the sum of the squares of three natural numbers.

Answer: 1963


On weekdays, a scatterbrained scientist takes the Moscow metro's circular line from the "Taganskaya" station to the "Kievskaya" station, and back in the evening. He boards the first arriving train. Trains run at approximately equal intervals in both directions. The northern route (via "Belorusskaya") takes 17 minutes between "Kievskaya" and "Taganskaya" or vice-versa, and the southern route (via "Paveletskaya") takes 11 minutes.

The scientist has calculated that:

- A counterclockwise train arrives at "Kievskaya" on average 1 minute and 15 seconds after a clockwise train. The same is true for "Taganskaya";
- The average journey from home to work takes 1 minute less than the return journey.

Find the expected interval between trains going in the same direction.

Answer: 3


Lattice points are points on a rectangular coordinate plane having both \( x \)- and \( y \)-coordinates being integers. A moving point \( P \) is initially located at \((0,0)\). It moves 1 unit along the coordinate lines (in either direction) in a single step.

1. If \( P \) moves 1 step, then \( P \) can reach \( a \) different lattice points. Find \( a \).

2. If \( P \) moves not more than 2 steps, then \( P \) can reach \( b \) different lattice points. Find \( b \).

3. If \( P \) moves 3 steps, then \( P \) can reach \( c \) different lattice points. Find \( c \).

4. If \( d \) is the probability that \( P \) lies on the straight line \( x + y = 9 \) when \( P \) advances 9 steps, find \( d \).

Answer: \frac{1}{10}


In a regular hexagonal pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the ratio of the radius of the circumscribed sphere to the radius of the inscribed sphere.

Answer: 1+\sqrt{\frac{7}{3}}


As shown in the figure, the side length of the square $ABCD$ is $18$. Point $E$ is the midpoint of $CD$, and $ABFH$ is a rectangle. The areas of the two shaded triangles are equal. What is the area of the quadrilateral $AEFB$?

Answer: 216


Frank's teacher asks him to write down five integers such that the median is one more than the mean, and the mode is one greater than the median. Frank is also told that the median is 10. What is the smallest possible integer that he could include in his list?
A 3
B 4
C 5
D 6
E 7

Answer: 4


In the fourth annual Swirled Series, the Oakland Alphas are playing the San Francisco Gammas. The first game is played in San Francisco and succeeding games alternate in location. San Francisco has a 50% chance of winning their home games, while Oakland has a probability of 60% of winning at home. Normally, the series will stretch on forever until one team gets a three-game lead, in which case they are declared the winners. However, after each game in San Francisco, there is a 50% chance of an earthquake, which will cause the series to end with the team that has won more games declared the winner. What is the probability that the Gammas will win?

Answer: \frac{34}{73}


For a three-digit natural number \( n \), let \( S(n) \) represent the digit sum of \( n \). How many numbers \( a \) are there among the three-digit numbers such that \( S(a) = S(2a) \)?

Answer: 80


Inside triangle \(ABC\), a point \(D\) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). Given that \(AB = 15\) and \(CD = 8\), find the length of segment \(AD\). If necessary, round the answer to 0.01 or express the answer as a common fraction.

Answer: 3.5


Let's assume that Olga Pavlovna has \( x \) liters, and Maria Petrovna has \( y \) liters of jam left. The numbers \( x \) and \( y \) are randomly and independently chosen from the interval from 0 to 1. Consider a random point with coordinates \((x, y)\) from the unit square \( F \). The event \( A \) "Olga Pavlovna and Maria Petrovna together have at least 1 liter, but less than 1.5 liters of jam" is represented by the inequality \( 1 \leq x + y < 1.5 \) and is depicted as a trapezoid bounded by the lines \( x + y = 1 \) and \( x + y = 1.5 \). Thus,

$$
\mathrm{P}(A) = \frac{S_{A}}{S_{F}} = \frac{3}{8} = 0.375
$$

Answer: 0.375


The diameter \( AB \) of a circle is extended beyond point \( B \), and point \( C \) is marked on this extension. From point \( C \), a secant is drawn at an angle of \( 7^\circ \) to \( AC \), intersecting the circle at points \( D \) and \( E \), starting from point \( C \). It is given that \( DC = 3 \) and the angle \( DAC \) is \( 30^\circ \). Find the diameter of the circle.

Answer: 4\sqrt{3}\sin(7)


Roger the ant is traveling on a coordinate plane, starting at \((0,0)\). Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point \(P\) for which he could have reached \(P\) more quickly had he taken a different route. For example, if he goes from \((0,0)\) to \((1,0)\) to \((1,1)\) to \((1,2)\) to \((0,2)\), he stops at \((0,2)\) because he could have gone from \((0,0)\) to \((0,1)\) to \((0,2)\) in only 2 seconds. The expected number of steps Roger takes before he stops can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100 a + b\).

Answer: 1103


Anton went to the dairy store. He had no money, but he had empty bottles: six 1-liter bottles (worth 20 kopeks each) and six 0.5-liter bottles (worth 15 kopeks each). In the store, there was bulk milk being sold at 22 kopeks per liter. What is the maximum amount of milk he could bring home? He has no other containers besides the empty bottles.

Answer: 5


a) Consider a $6 \times 6$ chessboard. What is the minimum number of kings that can be placed on the board so that all squares are either occupied or attacked by one of the kings?

b) Still considering the $6 \times 6$ chessboard, what is the maximum number of kings that can be placed on the board such that no kings attack each other?

Answer: 9


## PROBLEM 44. COWS AND SHEEP

Two men jointly owned $x$ cows, which they sold for $x$ dollars a head. With the money they received, they bought sheep at 12 dollars a head. Since the proceeds from the sale of the cows did not divide by 12, they used the remaining money to buy a lamb. Then they divided the flock so that each of them had the same number of animals. The man with the lamb was therefore somewhat shortchanged. To make up for it, the second man gave him his harmonica. How much is the harmonica worth?

Answer: 4


5. Let the set of $n(2<n<6)$ complex numbers satisfy the following 2 conditions to form a set $S$.
(1) $1 \in S$;
(2) If $x_{1} \in S, x_{2} \in S$, then $x_{1}-2 x_{2} \cos \theta \in S$, where $\theta$ $=\arg \frac{z_{1}}{x_{2}}$.
Then the set $S=$

Answer: \{\,1, -1, i, -i\,\}


## Task 2.

Let $a \geqslant 2018$ be a real number. In each of 2018 jars, there is a finite number of balls, with the mass of each ball being of the form $a^{k}$, where $k \in \mathbb{Z}$. The total mass of the balls in each jar is the same. What is the minimum number of balls of the same mass among the balls in the jars?

Answer: 2018


5.1. A grenade lying on the ground explodes into a multitude of small identical fragments, which scatter in a radius of $L=90$ m. Determine the time interval (in seconds) between the moments of impact on the ground of the first and the last fragment, if such a grenade explodes in the air at a height of $H=10 \mathrm{m}$. The acceleration due to gravity is considered to be $g=10$ m/s ${ }^{2}$. Air resistance is negligible.

Answer: 6