5. Given a real number $x$ between 0 and 1, consider its decimal representation $0, c_{1} c_{2} c_{3} \ldots$ We call $B(x)$ the set of different subsequences of six consecutive digits that appear in the sequence $c_{1} c_{2} c_{3} \ldots$

For example, $B(1 / 22)=\{045454,454545,545454\}$.

Determine the minimum number of elements of $B(x)$ as $x$ varies among the irrational numbers between 0 and 1 (i.e., those whose decimal expansion is neither finite nor periodic from a certain point onward.)

Answer: 7


10. Given the standard equation of ellipse $C$ as $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>1)$, point $P$ is a moving point on the unit circle, points $A, F$ are the right vertex and right focus of ellipse $C$ respectively, and when $\angle P O A=\frac{\pi}{3}$, $|P F|=1,|P A|=\sqrt{3}$.
(1) Find the standard equation of ellipse $C$;
(2) Draw a line $l$ through point $P$ intersecting ellipse $C$ at points $M, N$, and satisfying $P$ as the midpoint of segment $M N$. Then draw the tangents to ellipse $C$ at points $M, N$ intersecting at point $Q$. Find the range of $|O Q|$ as point $P$ moves on the unit circle.

Answer: \dfrac{x^2}{4} + \dfrac{y^2}{3} = 1


10. (15 points) As shown in the figure, it is a water purification device, with water flowing from $A$ to $B$ first, and then to $C$. Originally, there were 10 pipes with the same flow rate between container $A-B$, and 10 pipes with the same flow rate between $B-C$. After swapping one pipe between $A-B$ and $B-C$, the flow rate increased by 40 cubic meters per hour. Question: By adjusting the pipeline layout, what is the maximum increase in flow rate from $A$ to $C$ in cubic meters?

Answer: 200


For which values of the parameter \( a \) does the equation \((x^2 - a)^2 + 2(x^2 - a) + (x - a) + 2 = 0\) have exactly one solution? Indicate the solution for the found values of the parameter \( a \).

Answer: \dfrac{3}{4}


Problem 7.4. The figure shows 5 lines intersecting at one point. One of the resulting angles is $34^{\circ}$. How many degrees is the sum of the four angles shaded in gray?

![](https://cdn.mathpix.com/cropped/2024_05_06_4d36ea080cd7b65cb574g-4.jpg?height=504&width=515&top_left_y=91&top_left_x=469)

Answer: 146


3. In a regular quadrilateral pyramid $P-A B C D$, it is known that $A_{1} 、 C_{1}$ are the midpoints of $P A 、 P C$ respectively. Then
$$
\frac{V_{\text {=floor } A_{1}}-B C_{1} D}{V_{\text {regular quadrilateral pyramid } P-A B C D}}=
$$
$\qquad$

Answer: \dfrac{1}{4}


4. Let $T$ be a set of ordered triples $(x, y, z)$, where $x, y, z$ are integers, and $0 \leqslant x, y, z \leqslant 9$. Two players, A and B, play the following game: A selects a triple $(x, y, z)$ from $T$, and B has to guess A's chosen triple using several "moves". One "move" consists of: B giving A a triple $(a, b, c)$ from $T$, and A responding with the number $|x+y-a-b|+|y+z-b-c|+|z+x-c-a|$. Find the minimum number of "moves" required for B to determine A's chosen triple.
(Bulgaria provided)

Answer: 3


A bronze lion was carved, and on it the following inscription was written: "I can pour water from my eyes, throat, and right leg. If I open my right eye, I fill my basin in 2 days, if I open my left eye, in 3 days. The water flowing from my leg fills the basin in 4 days, and the water flowing from my throat fills it in 6 hours. Tell me how long it would take to fill the basin if water flowed from my eyes, throat, and leg all at once?"

Answer: \dfrac{288}{61}


5. On the sides of a rectangle with length $8 \mathrm{~cm}$ and width $6 \mathrm{~cm}$, points are placed starting from one vertex, each $2 \mathrm{~cm}$ apart. How many different lines do these points determine?

NATIONAL MATHEMATICS COMPETITIONPrimošten, April 4-6, 2016.6th grade-elementary school

Answer: 63


(1) Any two points are not collinear;
(2) For any point \( P \) in \( S \), there are at least \( k \) points in \( S \) that are equidistant from \( P \).

Prove: \( k \leq \frac{1}{2} + \sqrt{2n} \).

Answer: k \leq \frac{1}{2} + \sqrt{2n}


5. In the equation $\overline{A B C} \times \overline{A B C}=\overline{D E F E D}$, the same letter represents the same digit, and different letters represent different digits, then the six-digit number $\overline{A B C D E F}=$ $\qquad$.

Answer: 307942


6. A circle of radius 3 touches line $P$ at point $A$ and line $Q$ at point $B$ such that the chord $A B$ subtends an arc of $60^{\circ}$. Lines $P$ and $Q$ intersect at point $F$. Point $C$ is located on ray $A F$, and point $D$ is on ray $F B$ such that $A C=B D=4$. Find the length of the median of triangle $C A D$, drawn from vertex $A$.

Answer: \dfrac{3 + 4\sqrt{3}}{2}


8-1. A beginner gardener planted daisies, buttercups, and marguerites on their plot. When they sprouted, it turned out that there were 5 times more daisies than non-daisies, and 5 times fewer buttercups than non-buttercups. What fraction of the sprouted plants are marguerites?

Answer: 0


10. If real numbers $b, c$ satisfy $b^{2}+c^{2}=1$, and
$$
f(x)=a x+b \sin x+c \cos x
$$

has two perpendicular tangent lines on its graph, then the range of values for $a$ is $\qquad$.

Answer: 0


Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

Answer: 5


Problem 7.7. In three of the six circles of the diagram, the numbers 4, 14, and 6 are recorded. In how many ways can natural numbers be placed in the remaining three circles so that the products of the triples of numbers along each of the three sides of the triangular diagram are the same?

![](https://cdn.mathpix.com/cropped/2024_05_06_d43b1f0a7bd77fffa87ag-22.jpg?height=285&width=321&top_left_y=685&top_left_x=566)

Answer: 6


3. (20 points) At the Journalism Faculty of the University of Enchanted Commonwealth, 4 chickens are applying. The faculty has 2 places in the daytime program and 3 places in the evening program. Assuming all 4 chickens will be admitted to the faculty, determine the number of outcomes in which exactly two chickens will be admitted to the evening program.

Answer: 6


4. In the city of liars and knights, there are 366 residents, all born on different days of a leap year. All residents of the city answered two questions. To the question “Were you born in February?” 100 people answered affirmatively, and to the question “Were you born on the 30th?” 60 people answered affirmatively. How many knights were born in February?

Answer: 29


An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]

Answer: 2


As shown in Figure 3, in the rectangular coordinate system $xOy$, the circle $(\odot O): x^{2}+y^{2}=4$ intersects the positive half-axis of $x$ at point $A$. Another circle $(\odot A): (x-2)^{2}+y^{2}=r^{2}$ (where $r>0$) intersects $(\odot O)$ at points $B$ and $C$.
(1) Find the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$.
(2) Let $P$ be any point on $(\odot O)$ that is different from $B$ and $C$. The lines $PB$ and $PC$ intersect the $x$-axis at points $M$ and $N$, respectively. Find the maximum value of $S_{\triangle POM} \cdot S_{\triangle PON}$.

Answer: -2


6. Let $A B C$ be a triangle with $A B=5, B C=6, C A=7$. Let $D$ be a point on ray $A B$ beyond $B$ such that $B D=7, E$ be a point on ray $B C$ beyond $C$ such that $C E=5$, and $F$ be a point on ray $C A$ beyond $A$ such that $A F=6$. Compute the area of the circumcircle of $D E F$.

Answer: \dfrac{1004}{3}\pi


Consider nine points in space, no four of which lie in a plane. We connect each pair of points with an edge (i.e., a straight line segment), and color each such edge red, blue, or leave it uncolored. Determine the smallest value of $n$ such that whenever the number of colored edges is exactly $n$, the set of colored edges necessarily contains a triangle whose edges are all the same color.

Answer: 33


At the intersection of roads \( A \) and \( B \) (straight lines) there is a settlement \( C \) (a point). Sasha walks along road \( A \) towards point \( C \), taking 40 steps per minute, with a step length of 65 cm. At the beginning of the walk, Sasha was at a distance of 260 m from point \( C \). Dania walks towards \( C \) along road \( B \) at a speed of 75 steps per minute, with a step length of 50 cm and at the beginning of their joint movement was at a distance of 350 m from \( C \). Each of them, having passed point \( C \), continues to walk along their road without stopping. Consider moments when Dania and Sasha have each taken an integer number of steps. Find the minimum possible distance between them (along the roads) at such moments of time. How many steps has each of them taken by the time this distance is minimized?

Answer: 18.1


8. Given that the three sides of $\triangle A B C$ are $10$, $12$, and $14$, respectively, and that $\triangle A B C$ is folded into a tetrahedron along its three midlines. Then the diameter of the circumscribed sphere of the tetrahedron is $\qquad$

Answer: \sqrt{55}


In the circular segment \( A M B \), a trapezoid \( A C D B \) is inscribed, where \( A C = C D \) and \( \angle C A B = 51^\circ 20^\circ \). Find the angular measure of arc \( A M B \).

Answer: 154^\circ


7. Let the dihedral angles between the three lateral faces and the base of a tetrahedron all be $60^{\circ}$. The side lengths of the base are $7, 8, 9$. Then the lateral surface area of the pyramid is

Answer: 24\sqrt{5}


14. Given a line segment $C D$ of length 6 with midpoint $M$. Two triangles $\triangle A C D$ and $\triangle B C D$ are constructed on the same side with $C D$ as one side, and both have a perimeter of 16, satisfying $\angle A M B = 90^{\circ}$. Find the minimum area of $\triangle A M B$.

Answer: \dfrac{400}{41}


## Task A-3.7.

We have ten white, and one red, blue, green, yellow, and purple card each. The white cards are indistinguishable from each other. On exactly one side of each card is the symbol $X$. In how many ways can we stack these cards one on top of the other so that no two cards are facing each other with the side that has the $X$?

Answer: 5765760


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

Answer: 8


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

Answer: 8


In rectangle $ADEH$, points $B$ and $C$ trisect $\overline{AD}$, and points $G$ and $F$ trisect $\overline{HE}$. In addition, $AH=AC=2$, and $AD=3$. What is the area of quadrilateral $WXYZ$ shown in the figure? The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 3


Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same? Express your answer in the form \(\frac{k}{m}\), and find the value of \(k + m\).

Answer: 4


Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$. The original answer is in \frac{k}{m} format, where $k$ and $m$ are integers. Please give the value of $k + m$.

Answer: 8


Circles with centers \( O \) and \( P \) have radii \( 2 \) and \( 4 \), respectively, and are externally tangent. Points \( A \) and \( B \) on the circle with center \( O \) and points \( C \) and \( D \) on the circle with center \( P \) are such that \( AD \) and \( BC \) are common external tangents to the circles. Find the area of the concave hexagon \( AOBCPD \). The original answer is in the form \( k\sqrt{m} \). Please provide the value of \( k + m \).

Answer: 26


Let \(S\) be the set of all points \((x,y)\) in the coordinate plane such that \(0 \le x \le \frac{\pi}{2}\) and \(0 \le y \le \frac{\pi}{2}\). Find the area of the subset of \(S\) for which
\[
\sin^2x-\sin x \sin y + \sin^2y \le \frac{3}{4}.
\]
The answer is in the form \(\frac{k\pi^2}{m}\). Please calculate the value of \(k + m\).

Answer: 7


The wording is just ever so slightly different, however the problem is identical.

Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.

Answer: f(n) = n


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

Answer: 6


8 There are 5 table tennis balls, among which 3 are new balls, and 2 are old balls (i.e., balls that have been used at least once). For each match, 2 balls are taken out to use, and after use, they are all put back. Let the number of new balls taken in the second match be $\xi$, then the mathematical expectation $E \xi=$ $\qquad$

Answer: \dfrac{18}{25}


In a tournament with 5 teams, there are no ties. In how many ways can the $\frac{5 \cdot 4}{2}=10$ games of the tournament occur such that there is neither a team that won all games nor a team that lost all games?

Answer: 544


A courier departs from $A$ towards $B$; at the same time, another courier departs from $B$ towards $A$. They meet after 8 hours. How many $\mathrm{km}$ does each courier travel per hour, if the distance from $A$ to $B$ is $176 \mathrm{~km}$ and it takes one courier 1 hour longer to travel $60 \mathrm{~km}$ than it takes the other courier?

Answer: 10


8. The residents of the village Razumevo, located 3 km away from the river, love to visit the village Vkusnotevo, situated 3.25 km downstream on the opposite bank of the river, 1 km away from the shore. The width of the river is 500 m, the speed of the current is 1 km/h, and the banks are parallel straight lines. The residents of Razumevo have laid out the shortest route, taking into account that they always cross the river in a direction perpendicular to the shore with their own speed of 2 km/h. How long does this route take if the speed on land does not exceed 4 km/h? Enter the answer in hours in the provided field.

Answer: \dfrac{3}{2}


10,11

The radius of the base of the cylinder is equal to $r$, and the height is equal to $5 r$. A parallelepiped is circumscribed around the cylinder, the ratio of the volume of which to the volume of the cylinder is $\frac{\tilde{5}}{\pi}$. Find the length of the segment of the larger diagonal of the parallelepiped lying inside the cylinder.

Answer: 3r


p1. Determine the number of positive integers less than $2020$ that are written as sum of two powers of $3$.

p2. A student must choose three classes among the branches of Physics, Literature, and Mathematics, to build his $7$-day calendar. Each day he must choose only one of them. The only restriction is that on four consecutive days it must have all three branches. Determine the possible number of calendars that the student can make.

[url=https://artofproblemsolving.com/community/c4h2454972p20439395]p3.[/url] In a triangle $ABC$, the medians $AM$ and $BN$ are drawn, Draw through $N$ a parallel to $BC$ and through $C$ a parallel to $BN$. These two lines intersect at $P$ and let $D$ be the midpoint of $PN$. Show that $CD$ is parallel to $MN$.

p4. A doctor prescribes his patient to take $48$ pills for $30$ days, at least one and no more than $6$ a day. Prove that no matter how the patient decides to take them, following the doctor's instructions, there are a number of consecutive days when taking exactly $11$ pills.

Answer: 28


Two circles of the same radius are placed inside a right triangle, each touching one of the legs, the hypotenuse, and the other circle. Find the radii of these circles if the legs of the triangle are $a$ and $b$.

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


1. In a $3 \times 4$ table, 12 numbers are arranged such that all seven sums of these numbers in the rows and columns of the table are distinct. What is the maximum number of numbers in this table that can be zero?

Answer: 8


23. In the figure below, $\mathrm{ADE}$ is a triangle with $\angle \mathrm{AED}=120^{\circ}$, and $\mathrm{B}$ and $\mathrm{C}$ are points on side $\mathrm{AD}$ such that $\mathrm{BCE}$ is an equilateral triangle. If $\mathrm{AB}=4 \mathrm{~cm}, \mathrm{CD}=16 \mathrm{~cm}$ and $\mathrm{BC}=x \mathrm{~cm}$, find the value of $x$.

Answer: 8


4. The base of the pyramid is a parallelogram. A plane is drawn through one side of the base and the midline of the opposite lateral face. Into what ratio does this plane divide the volume of the pyramid?

Answer: \dfrac{5}{3}


## 115. Math Puzzle $12 / 74$

A long rectangular club room is to be prepared.

It was found that the perimeter of the floor area, measured in meters, is numerically equal to its area.

How long are the sides of the room if they are a whole number of meters?

Answer: 3


\section*{Problem 1 - 061231}

In the same plane, let \(n\) points \((n>2)\) be distributed such that for each of them, there is only one nearest neighbor among the others. For each of these \(n\) points, the vector starting from it and ending at its nearest neighbor, and only this vector, is drawn.

Determine the maximum number of these vectors that can end at the same one of the \(n\) points.

Answer: 5


## Task 3 - 030513

Klaus has come up with an interesting problem for the "Puzzle Corner":

In the multiplication problem

$$
13 * \cdot \quad 7 *=1 * * * *
$$

all * should be replaced by digits so that all three numbers end with the same digit, and that in the result, the digit in the tens place is the same as the digit in the hundreds place.

What did you consider when solving this problem?

Answer: 136 \times 76 = 10336


5. An isosceles triangle \(ABC\) with vertex \(B\) is inscribed in a circle of radius \(R\). A circle of radius \(r\) is inscribed in it. Another circle of radius \(r\) is inscribed in the segment bounded by the chord \(AB\) and the smaller arc. Find the ratio \(R: r\).

For a complete solution to each problem, 7 points are awarded.

The maximum total score is 35.

Answer: \dfrac{8}{3}


## Task $1 / 89$

We are looking for all pairs $\left(p_{1} ; p_{2}\right)$ of (in the decimal system) two-digit prime numbers $p_{1}$ and $p_{2}$ with $p_{1}<p_{2}$, that satisfy the following conditions:

1. Neither the units nor the tens digits of $p_{1}$ and $p_{2}$ are prime numbers.
2. The tens digits of $p_{1}$ and $p_{2}$ are not divisible by 3.
3. The sum $p_{1}+p_{2}$ is neither divisible by 10 nor by 13.

Answer: (19; 89)


Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[Diagram in the individuals file for this exam on the Chmmc website]

Answer: 39


8. In the following vertical calculation, each letter represents a digit from $0 \sim 9$, the same letter represents the same digit, and different letters represent different digits. When the vertical calculation is correct, $A+B+C=$ $\qquad$.
$$
\begin{array}{l}
\begin{array}{llll}
A & B & B & C
\end{array} \\
\begin{array}{lll}
B & B & C
\end{array} \\
\begin{array}{r}
+\quad B \quad B \quad C \\
\hline B A B B
\end{array} \\
\end{array}
$$

Answer: 15


9. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, a sphere is constructed with vertex $A$ as the center and $\frac{2 \sqrt{3}}{3}$ as the radius. The length of the curve formed by the intersection of the sphere's surface with the cube's surface is $\qquad$.

Answer: \dfrac{5\sqrt{3}}{6}\pi


We have $2012$ sticks with integer length, and sum of length is $n$. We need to have sticks with lengths $1,2,....,2012$. For it we can break some sticks ( for example from stick with length $6$ we can get $1$ and $4$).
For what minimal $n$ it is always possible?

Answer: 2012 \times 2011 + 1


11.023. In a regular quadrilateral pyramid, the side of the base is 6 dm, and the height is 4 dm. Find the lateral surface area of the truncated pyramid cut off from the given one by a plane parallel to its base and at a distance of 1 dm from it.

Answer: \dfrac{105}{4}


3. Given that the function $f(x)$ is an increasing function defined on $[-4,+\infty)$, to ensure that for all real numbers $x$ in the domain, the inequality $f\left(\cos x-b^{2}\right) \geqslant f\left(\sin ^{2} x-b-3\right)$ always holds, the range of the real number $b$ is $\qquad$.

Answer: \left[ \dfrac{1 - 2\sqrt{2}}{2}, 1 \right]


3. Find the smallest natural number $\mathrm{n}$ such that the number of zeros at the end of the number ( $\mathrm{n}+20$ )! is exactly 2020 more than the number of zeros at the end of the number n!.

Answer: 5^{2017} - 20


Problem 3 (MAR CP 1992) :
From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ?

Answer: 358926471


Three, let the height of the regular tetrahedron $P-ABC$ be $PO$, and $M$ be the midpoint of $PO$. A plane passing through $AM$ is parallel to the edge $BC$, dividing the tetrahedron into upper and lower parts. Try to find the ratio of the volumes of these two parts.

Answer: \dfrac{4}{21}


Find all positive integers $x, y$ such that $x^{2}+3 y, y^{2}+3 x$ are both perfect squares.

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

Answer: (1, 1)


Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.

Answer: 80^\circ


Problem 10.3. Petya and Daniil are playing the following game. Petya has 36 candies. He lays out these candies in the cells of a $3 \times 3$ square (some cells may remain empty). After this, Daniil chooses four cells forming a $2 \times 2$ square and takes all the candies from there. What is the maximum number of candies that Daniil can guarantee to take?

Answer: 9


$4 \cdot 20$ There is a $\triangle A B C, a 、 b 、 c$ are the sides opposite to $\angle A 、 \angle B 、 \angle C$ respectively. If $b$ is the arithmetic mean of $a 、 c$, and $\operatorname{tg} \frac{B}{2}$ is the geometric mean of $\operatorname{tg} \frac{A}{2} 、 \operatorname{tg} \frac{C}{2}$, try to form a quadratic equation with $\operatorname{tg} \frac{A}{2}, \operatorname{tg} \frac{C}{2}$ as its roots.

Answer: 3x^2 - 2\sqrt{3}x + 1 = 0


52. In the left table below, adding 1 or subtracting 1 simultaneously to the numbers in two cells that share a common edge is called one operation. After a finite number of operations, the left table becomes the right table. What is the number at $A$ in the right table? $\qquad$ _.

Answer: 5


4. For the parabola $y^{2}=2 p x(p>0)$, the focus is $F$, the directrix is $l$, and $A, B$ are two moving points on the parabola, satisfying $\angle A F B=\frac{\pi}{3}$. Let the midpoint of segment $A B$ be $M$, and the projection of $M$ on $l$ be $N$. Then the maximum value of $\frac{|M N|}{|A B|}$ is $\qquad$

Answer: 1


For non-zero integers \( x \) and \( y \) we define \( x \nabla y = x^{y} + 4x \). For example, \( 2 \nabla 3 = 2^{3} + 4(2) = 16 \). Determine all real numbers \( x \) such that \( x \nabla 2 = 12 \).

Answer: -6 \text{ and } 2


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

Answer: 165


7. The number of apples produced by a group of farmers is less than 1000. It is known that they shared the apples in the following way. In turn, each farmer took from the collection of apples either exactly one-half or exactly one-third of the apples remaining in the collection. No apples were cut into pieces. After each farmer had taken his share, the rest was given to charity. Find the greatest number of farmers that could take part in the apple sharing.
7. A group of farmers harvested a batch of apples, fewer than 1000. They shared the apples in the following way: each person took a turn to take either exactly half or exactly one-third of the remaining apples, and no apples were cut. After all the farmers had taken their share, the rest was donated to charity. What is the maximum number of farmers who could participate in the apple sharing?

Answer: 12


${ }^{[}$The Law of Cosines Area of a triangle (through two sides and the angle between them). [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part

In triangle $A B C$, medians $A M$ and $B P$ are drawn. It is known that $\angle A P B=\angle B M A, \cos \angle A C B=0.8, B P=1$.

Find the area of triangle $A B C$.

#

Answer: \dfrac{2}{3}


On December 31st at 4:35 PM, Misha realized that he didn't have New Year gifts for his whole family. He wants to give different gifts to his mom, dad, brother, and sister. Each of the gifts is available in 4 stores: Romashka, Oduvanchik, Nezabudka, and Landysh, which close at 8:00 PM. The travel time from home to any store and between any two stores is 30 minutes.

The table shows the cost of gifts in all four stores and the time Misha will spend shopping in each store. What is the minimum amount of money Misha can spend if he absolutely needs to buy all 4 gifts?

| Store     | Mom  | Dad  | Brother | Sister | Time Spent in Store (min) |
|-----------|------|------|---------|--------|--------------------------|
| Romashka  | 1000 | 750  | 930     | 850    | 35                       |
| Oduvanchik| 1050 | 790  | 910     | 800    | 30                       |
| Nezabudka | 980  | 810  | 925     | 815    | 40                       |
| Landysh   | 1100 | 755  | 900     | 820    | 25                       |

Answer: 3435


7.3. How many seven-digit natural numbers exist, for which the product of the first three digits is 30, the product of the three digits in the center is 7, and the product of the last three digits is 15?

Answer: 4


1. Given a regular tetrahedron $P-ABC$ with a volume of $9 \sqrt{3}$, the dihedral angle between the side face $PAB$ and the base $ABC$ is $60^{\circ}$. Point $D$ is on segment $AB$ such that $AD=\frac{1}{6} AB$, point $E$ is on segment $AC$ such that $AE=\frac{1}{6} AC$, and point $F$ is the midpoint of segment $PC$. Plane $DEF$ intersects segment $PB$ at point $G$. Find the area of quadrilateral $DEFG$.

Answer: \sqrt{57}


# Task 4. (12 points)

The angle bisectors of angles $A, B$, and $C$ of triangle $A B C$ intersect the circumcircle of this triangle at points $A_{1}, B_{1}$, and $C_{1}$, respectively. Find the distances between point $A_{1}$ and the center of the inscribed circle of triangle $A B C$, given that $\angle A_{1} B_{1} C_{1}=50^{\circ}, \angle A_{1} C_{1} B_{1}=70^{\circ}, B_{1} C_{1}=\sqrt{3}$.

Answer: 1


4. Let $f(x)=(x+a)(x+b)$ (where $a, b$ are given positive real numbers), and $n \geqslant 2$ be a given integer. For non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}+x_{2}+\cdots+x_{n}=1
$$

find the maximum value of
$$
F=\sum_{1 \leq i<j \leq n} \min \left\{f\left(x_{i}\right), f\left(x_{j}\right)\right\}
$$

(Leng Gangsong)

Answer: \dfrac{n(n-1)}{2} \left( \dfrac{1}{n} + a \right) \left( \dfrac{1}{n} + b \right)


Inside a large rectangle of size \( A \times B \), there is a smaller rectangle of size \( a \times b \) (see the figure).

Find the difference between the total area of the yellow quadrilaterals and the total area of the green quadrilaterals, given that \( A = 20 \), \( B = 30 \), \( a = 4 \), \( b = 7 \).

Answer: 20


Let \( U \) and \( V \) be independent random variables, where \( U \) is uniformly distributed on the interval \( [0, 1] \), and \( \{u\} \) denotes the fractional part of the number \( u \). Show that the random variable \( \{U + V\} \) is uniformly distributed on \( [0, 1] \) and is independent of \( V \).

Answer: \{U + V\} \text{ is uniformly distributed on } [0, 1] \text{ and independent of } V


71. What are the last two digits of the number

$$
n^{a}+(n+1)^{a}+(n+2)^{a}+\ldots+(n+99)^{a}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_b937612abb2773f7abffg-018.jpg?height=46&width=972&top_left_y=892&top_left_x=89)
a) $a=4$
b) $a=8$ ?

Answer: 30


Let $a$ be the largest positive root of the equation $x^{3} - 3x^{2} + 1 = 0$. Prove:
$$
17\left|\left[a^{1788}\right], 17\right|\left[a^{1988}\right]
$$

where the notation $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$.

Answer: 17


11. (20 points) Let $\{a_n\}$ be an arithmetic sequence with a non-zero common difference, satisfying $a_3 + a_6 = a_9$ and $a_5 + a_7^2 = 6a_9$. Let the sequence $\{b_n\}$ have the sum of its first $n$ terms denoted by $S_n$, and $4S_n + 2b_n = 3$. For any $i \in \mathbf{Z}_+$, insert $i$ numbers $x_{i1}, x_{i2}, \cdots, x_{ii}$ between $b_i$ and $b_{i+1}$, such that $b_i, x_{i1}, x_{i2}, \cdots, x_{ii}, b_{i+1}$ form an arithmetic sequence. Let $T_n = \sum_{i=1}^{n} \sum_{j=1}^{i} x_{ij}$. Does there exist positive integers $m$ and $n$ such that $T_n = \frac{a_{m+1}}{2a_m}$? If so, find all such pairs of positive integers $(m, n)$; if not, explain why.

Answer: (3, 3)


A [i]calendar[/i] is a (finite) rectangular grid. A calendar is [i]valid[/i] if it satisfies the following conditions:

(i) Each square of the calendar is colored white or red, and there are exactly 10 red squares.

(ii) Suppose that there are $N$ columns of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the top row to the bottom row, and within each row from left to right, there do not exist $N$ consecutive numbers such that the squares they are in are all white.

(iii) Suppose that there are $M$ rows of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the left-most column to the right-most column, and within each column from bottom to top, there do not exist $M$ consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by $90^{\circ}$, the resulting calendar still satisfies (ii).

How many different kinds of valid calendars are there?

(Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the $10$ red squares are at different locations.)

Answer: 3628800


Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$.

[i]Proposed by Andy Xu[/i]

Answer: 52


8. Xiao Yu's family consists of five people: Dad, Mom, Brother, Sister, and Xiao Yu. From the 1st to the 5th of this month, there are two chores every day: one person cooks, and another person washes the dishes, and everyone has to cook once and wash the dishes once during these 5 days. One day, they had the following conversation:
Brother: I just washed the dishes yesterday, and I have to cook tomorrow, it's so tiring!
Sister: I'm even more tired, I have chores both tomorrow and the day after!
Xiao Yu: Dad and I always do chores on the same day.
Dad: Yes, fortunately, after cooking, I can rest for two days before it's my turn to wash the dishes again.
So, the five-digit number formed by the dates when Dad, Mom, Brother, Sister, and Xiao Yu wash the dishes, in order, is

Answer: 54132


1. As shown in the figure, in $\odot O$, $\overparen{A D B} = 90^{\circ}$, chord $A B = a$, and a circular arc with center $B$ and radius $B A$ intersects $\odot O$ at another point $C$. Then the area $S$ of the crescent shape (the shaded part in the figure) formed by the two circular arcs is $S=$ . $\qquad$

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


31. The following rectangle is formed by nine pieces of squares of different sizes. Suppose that each side of the square $\mathrm{E}$ is of length $7 \mathrm{~cm}$. Let the area of the rectangle be $x \mathrm{~cm}^{2}$. Find the value of $x$.

Answer: 1056


12. The following infinite sequence composed of 1s and 2s has an interesting characteristic: starting from the first term, group the terms with the same number, then write down the number of terms in each group in order, and the infinite sequence formed by these numbers is exactly the sequence itself. This sequence is called the Kolakoski sequence. According to this characteristic, continue to write the next 8 terms of this sequence (from the 14th term to the 21st term), if it is known that the sum of the first 50 terms of this sequence is 75, and the 50th term is 2, then the 73rd term, 74th term, 75th term, and 76th term can be known as $\qquad$

Answer: 1


The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$.

Answer: \dfrac{n+1}{n-1}


19. Among the $(3 n)!$ permutations made from $3 n$ distinct elements $a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}, c_{1}, c_{2}, \cdots, c_{n}$, how many satisfy the condition: for any natural number $k(k=1,2, \cdots, n)$, $a_{k}$ is not adjacent to $b_{k}$?

Answer: \sum_{m=0}^{n} (-1)^m \binom{n}{m} 2^m (3n - m)!


Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$.

Answer: 90^\circ


Sebestyén, starting from his third birthday, receives a triangular prism-shaped cake every birthday, with a candle at each of the top three vertices, and on top, as many candles as his age, so that the total number of candles on the cake equals his age, with no three candles lying on the same straight line. Sebestyén likes to cut the cake into triangular slices, the vertices of which are given by the positions of the candles (the interior of the triangles must not contain any candles). How many slices can the cake be divided into on his $k$-th birthday?

Answer: 2k - 5


Given a sphere of radius 1 with center at point $O$. From a point $A$ outside the sphere, four rays are drawn. The first ray intersects the sphere's surface sequentially at points $B 1$ and $C 1$, the second at points $B 2$ and $C 2$, the third at points $B 3$ and $C 3$, and the fourth at points $B 4$ and $C 4$. The lines $B 1 B 2$ and $C 1 C 2$ intersect at point $E$, and the lines $B 3 B 4$ and $C 3 C 4$ intersect at point $F$. Find the volume of the pyramid $O A E F$, if $A O=2$, $E O=F O=3$, and the angle between the faces $A O E$ and $A O F$ is $30^{\circ}$.

Answer: \dfrac{35}{24}


10.156. Two equilateral triangles are inscribed in a circle of radius $R$ such that their mutual intersection divides each side into three equal segments. Find the area of the intersection of these triangles.

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


Exercise 8. A list of numbers is said to be pretty if it consists of strictly positive integers such that the sum of these integers is equal to their product. Determine the smallest number of ones that a pretty list of 100 numbers can contain.

Answer: 95


A convex polyhedron has six square faces and eight regular hexagonal faces. It is known that no two square faces share a vertex. Determine the volume of the polyhedron, given that its edges are of unit length.

Answer: 8\sqrt{2}


\section*{Exercise 2 - 330932}

For each positive integer \(n\), imagine another number \(n^{\prime}\) formed according to the following rule: From the digit representation of \(n\) in the decimal system, the first digit is removed and appended to the end of the last digit.

Then let \(n^{\prime}\) be the number with the resulting digit representation. Investigate whether there are numbers \(n\) divisible by 7 for which \(n^{\prime}=n / 7\) holds!


Ménehould is organizing a birthday party, which will have 15 girls and 15 boys (including herself). She is preparing a personal label for each guest. In how many ways can she arrange the labels on the seats of the round table while maintaining the alternation of girl-boy?

Answer: 2 \times 15! \times 15!


3. Let's call a natural number an almost palindrome if it can be transformed into a palindrome by changing one of its digits. How many nine-digit almost palindromes exist? (20 points)

Answer: 3240000


Example 5 Let $a$ and $d$ be non-negative real numbers, $b$ and $c$ be positive numbers, and $b+c \geqslant a+d$. Find the minimum value of $\frac{b}{c+d}+$ $\frac{c}{a+b}$.

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


In the fictional country of Mahishmati, there are  $50$  cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city  $A$ , an ordered list of cities  $C_1,\ldots, C_{50}$  is called an *antitour* from  $A$  if

- every city (including  $A$ ) appears in the list exactly once, and
- for each  $k\in \{1,2,\ldots, 50\}$ , it is impossible to go from  $A$  to  $C_k$  by a sequence of exactly  $k$  (not necessarily distinct) flights.

Baahubali notices that there is an antitour from  $A$  for any city  $A$ . Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.

*Proposed by Sutanay Bhattacharya*

Answer: (25!)^2


A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits  $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$
What is the probability of having at most five different digits in the sequence?

Answer: 0.622


Let  $P_0P_5Q_5Q_0$  be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select  $4$  points on the segment  $P_0P_5$ , and immediately after selecting those points, we label those  $4$  selected points  $P_1, P_2, P_3, P_4$  from left to right. Similarly, we randomly select  $4$  points on the segment  $Q_0Q_5$ , and immediately after selecting those points, we label those  $4$  points  $Q_1, Q_2, Q_3, Q_4$  from left to right. The segments  $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$  divide the rectangular chocolate bar into  $5$  smaller trapezoidal pieces of chocolate. The probability that exactly  $3$  pieces of chocolate contain both dark and white chocolate can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
[Diagram in the individuals file for this exam on the Chmmc website]

Answer: 39


Toner Drum and Celery Hilton are both running for president. A total of  $2015$  people cast their vote, giving  $60\%$  to Toner Drum. Let  $N$  be the number of "representative'' sets of the  $2015$  voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when  $N$  is divided by  $2017$ .

*Proposed by Ashwin Sah*

Answer: 605


About pentagon  $ABCDE$  is known that angle  $A$  and angle  $C$  are right and that the sides  $| AB | = 4$ ,  $| BC | = 5$ ,  $| CD | = 10$ ,  $| DE | = 6$ . Furthermore, the point  $C'$  that appears by mirroring  $C$  in the line  $BD$ , lies on the line segment  $AE$ . Find angle  $E$ .

Answer: 90


Bororin P.A.

On the surface of a spherical planet, there are four continents separated from each other by an ocean. We will call a point in the ocean special if for it there are at least three nearest (equidistant from it) land points, and all are on different continents. What is the maximum number of special points that can exist on this planet?

Answer: 4


## Task 6/82

Given the inequality $|x|+|y| \leq n$ with $n \in N, x ; y \in G$ (where $G$ denotes the set of integers). Determine the number of ordered solution pairs $(x ; y)$.

Answer: (2n+1)^2 - 2n(n+1)


7. Let the incircle radius of equilateral $\triangle A B C$ be 2, and the incenter be
I. If point $P$ satisfies $P I=1$, then the maximum value of the ratio of the areas of $\triangle A P B$ to $\triangle A P C$ is $\qquad$.

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


2. Adding the first equation to the other two, we get:

$$
x^{2}+2 x+y^{2}+4 y+z^{2}+6 z=-14
$$

Next, we complete the square for each variable. As a result, we obtain:

$$
(x+1)^{2}+(y+2)^{2}+(z+3)^{2}=0
$$

Therefore, $\mathrm{x}=-1, y=-2, z=-3$ is the only possible solution. It remains to verify this by substituting it into the second and third equations of the system.

Answer: (-1, -2, -3)


There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

Answer: 6


2. Simplify the expression $\frac{2 \cos ^{2} x}{\sin x-\sin ^{-1} x}-\sin x$. For which values of $x$ does the expression have no meaning?

(6 points)

Answer: -3 \sin x


9.1. What is the maximum number of different rectangles that an 8 by 8 chessboard can be cut into? All cuts must follow the grid lines. Rectangles are considered different if they are not equal as geometric figures.

Answer: 12


# 2. Option 1.

Tourists Vitya and Pasha are walking from city A to city B at equal speeds, while tourists Katya and Masha are walking from city B to city A at equal speeds. Vitya met Masha at 12:00, Pasha met Masha at 15:00, and Vitya met Katya at 14:00. How many hours after noon did Pasha meet Katya?

Answer: 5


Problem. Find all functions $f:(0, \infty) \rightarrow(0, \infty)$ such that

$$
f\left(y(f(x))^{3}+x\right)=x^{3} f(y)+f(x)
$$

for all $x, y>0$.

Answer: f(x) = x


5. For a positive integer $n$, denote by $\varphi(n)$ the number of positive integers $k \leq n$ relatively prime to $n$. How many positive integers $n$ less than or equal to 100 are divisible by $\varphi(n)$ ?

Answer: 20


Example 18 Let $a_{n}$ be the number of $n$-digit numbers, formed by $1,2,3,4,5$, that have at least 3 different digits, where 1 and 2 are not adjacent. Find $a_{n}$ and $a_{5}$.

Answer: 2034


Through the midpoint of the hypotenuse of a right triangle, a perpendicular is drawn to it. The segment of this perpendicular, enclosed within the triangle, is equal to c, and the segment enclosed between one leg and the extension of the other is equal to 3c. Find the hypotenuse.

#

Answer: 4c


A $2N \times 2N$ board is covered by non-overlapping $1 \times 2$ dominoes. A rooked knight (a chess piece) traversed the board, visiting each cell exactly once (each move of the rooked knight is to a cell adjacent by side). We shall call a move a longitudinal move if it is a transition from one cell of a domino to the other cell of the same domino. What is

a) the greatest;

b) the least possible number of longitudinal moves?

Answer: 2N^2


1. Solve the equation $5^{2 x}-7 \cdot 5^{x}+10=0$.

Answer: 1


Two people, \(A\) and \(B\), need to travel from point \(M\) to point \(N\), which is 15 km away from \(M\). They can walk at a speed of 6 km/h. In addition, they have a bicycle that can be ridden at a speed of 15 km/h. \(A\) starts the journey on foot, while \(B\) rides the bicycle until they meet a pedestrian \(C\) who is walking from \(N\) to \(M\). After the meeting, \(B\) continues on foot, and \(C\) rides the bicycle to meet \(A\), and then hands over the bicycle to \(A\), who then rides it to \(N\).

a) When should pedestrian \(C\) leave \(N\) so that \(A\) and \(B\) arrive at point \(N\) at the same time (if \(C\) walks at the same speed as \(A\) and \(B\))?

b) When should pedestrian \(C\) leave \(N\) so that the travel time for \(A\) and \(B\) to point \(N\) is minimized (considering \(C\) walks at the same speed as \(A\) and \(B\) and the travel time starts from when \(A\) and \(B\) leave \(M\) and ends when the last of them arrives at \(N\))?

Answer: \dfrac{3}{11}


10.301. Find the ratio of the area of a square inscribed in a segment with an arc of $180^{\circ}$ to the area of a square inscribed in a segment of the same circle with an arc of $90^{\circ}$.

Answer: 10


13.

A box of hawthorns has more than one hundred grains. When counted in groups of 3, there is 1 extra; when counted in groups of 4, there are 2 extra; when counted in groups of 5, there are 3 extra; and when counted in groups of 6, there are 4 extra. The box of hawthorns has at most ( ) grains.

Answer: 178


Given $\triangle A B C$, extend the three sides by 1, 2, 3 times respectively, to get $\triangle A^{\prime} B^{\prime} C^{\prime}$. Ask how many times the area of $\triangle A^{\prime} B^{\prime} C^{\prime}$ is compared to the area of $\triangle A B C$.

Answer: 18


3. One day, three young men met at the library, and one of them said: "Starting today, I will come to the library once a day." The second person said: "I will come to the library once every two days." The third person stated that the library is closed on Sundays and does not open. One of the young men suggested that if any of them were to come to the library on a day it is closed, they would be allowed to skip that day and come the next day, and their schedule for visiting the library would start from that day onward. The young men agreed to this.

They met again at the library on a Monday. Try to determine on which day the conversation recorded above took place.

Answer: Saturday


6. Find the largest possible number of integers that can be selected from the set $\{1,2,3, \ldots, 100\}$ such that no two of them differ by 2 or 5.

Answer: 44


25. 8 equilateral triangles with side lengths of $1$, $2$, $3$ cm respectively cover a part of a larger equilateral triangle with a side length of 5 cm. Then, the difference between the area of the shaded part in the large triangle and the sum of the areas of the shaded parts in all the small triangles is $\qquad$ square cm.

Answer: 0


An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

[i]Author: Alex Zhu[/i]

Answer: 1197


Let the rest energy of a particle be  $E$ . Let the work done to increase the speed of this particle from rest to  $v$  be  $W$ . If  $ W = \frac {13}{40} E $ , then  $ v = kc $ , where  $ k $  is a constant. Find  $10000k$  and round to the nearest whole number.

*(Proposed by Ahaan Rungta)*

Answer: 6561


10th Australian 1989 Problem B3 Let N be the positive integers. The function f : N → N satisfies f(1) = 5, f( f(n) ) = 4n + 9 and f(2 n ) = 2 n+1 + 3 for all n. Find f(1789). Solution

Answer: 3581


3. Let the real numbers $x_{1}, x_{2}$, and $x_{3}$ be the roots of the equation $3 x^{3}-p x^{2}+28 x-p=0$, taking into account their possible multiplicities. Then $3 x^{3}-p x^{2}+28 x-p \equiv 3\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right)=$

$$
=3 x^{3}-3 x^{2}\left(x_{1}+x_{2}+x_{3}\right)+3 x\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}\right)-3 x_{1} x_{2} x_{3},
$$

from which $x_{1}+x_{2}+x_{3}=p / 3, \quad x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}=28 / 3$. Taking this into account, the target expression is transformed as follows:

$$
\begin{aligned}
&\left(x_{1}-2\right)^{2}+\left(x_{2}-2\right)^{2}+\left(x_{3}-2\right)^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4\left(x_{1}+x_{2}+x_{3}\right)+12= \\
&=\left(x_{1}+x_{2}+x_{3}\right)^{2}-2\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}\right)-4\left(x_{1}+x_{2}+x_{3}\right)+12= \\
&= \frac{p^{2}}{9}-\frac{56}{3}-\frac{4 p}{3}+12=\frac{p^{2}-12 p-60}{9} .
\end{aligned}
$$

Now let's plot the graph of the equation $3 x^{3}-p x^{2}+28 x-p=0$ on the plane $(x, p)$. Since the parameter enters this equation linearly, it is more convenient to solve it for $p$:

$$
p\left(x^{2}+1\right)=3 x^{3}+28 x=3 x\left(x^{2}+1\right)+25 x \Rightarrow p(x)=3 x+\frac{25 x}{x^{2}+1}
$$

We will find the set of values of the parameter $p$ for which the equation has three real roots. For this, we will investigate $p(x)$:

$$
p^{\prime}(x)=3+\frac{25\left(1-x^{2}\right)}{\left(x^{2}+1\right)^{2}}=\frac{3 x^{4}-19 x^{2}+28}{\left(x^{2}+1\right)^{2}}=\frac{\left(x^{2}-4\right)\left(3 x^{2}-7\right)}{\left(x^{2}+1\right)^{2}}
$$

The odd function $p(x)$ has minima at points $\bar{x}_{2}=-\frac{\sqrt{21}}{3}, \bar{x}_{4}=2$ and maxima at points $\bar{x}_{1}=-2, \bar{x}_{3}=\frac{\sqrt{21}}{3}$,

where

$$
p_{1}=p\left(\bar{x}_{1}\right)=-16, p_{2}=p\left(\bar{x}_{2}\right)=-\frac{7 \sqrt{21}}{2}
$$

$$
p_{3}=p\left(\bar{x}_{3}\right)=\frac{7 \sqrt{21}}{2}, p_{4}=p\left(\bar{x}_{4}\right)=16
$$

For all $c \in\left(p_{2} ; p_{1}\right) \cup\left(p_{4} ; p_{3}\right)$, the line $p=c$ intersects the graph at three points, and the equation $3 x^{3}-p x^{2}+28 x-p=0$ has three distinct roots. For parameter values that coincide with one of $p_{1}, p_{2}, p_{3}$, or $p_{4}$ (on the boundaries of the intervals), there are two points of intersection, which corresponds to the case of multiple roots.

The quadratic trinomial $\frac{p^{2}-12 p-60}{9}$, decreasing for $p6$, takes the smallest possible value on the set $\left[p_{2} ; p_{1}\right] \cup\left[p_{4} ; p_{3}\right]$ when $p=p_{4}=16$.

Thus, under the conditions of the problem,

$$
\min \left(\left(x_{1}-2\right)^{2}+\left(x_{2}-2\right)^{2}+\left(x_{3}-2\right)^{2}\right)=\frac{16^{2}-12 \cdot 16-60}{9}=\frac{4}{9}
$$

Answer: \dfrac{4}{9}


II. Answer Questions (Total 56 points)
9. (16 points) Given a tetrahedron $P-ABC$ with each edge of the base being 1 and the height being $\sqrt{2}$, the center of its inscribed sphere is $O$, and the radius is $r$. Find the area of the planar region within the base $ABC$ consisting of points that are no more than $2r$ away from point $O$.

Answer: \dfrac{6 + \pi}{24}


10.203. From a point on the circumference, two chords of lengths 9 and $17 \mathrm{~cm}$ are drawn. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm.

Answer: \dfrac{85}{8}


2. In the convex quadrilateral $ABCD$, $AB=CD$, $AC$ is a diagonal, $\angle DAC>\angle BCA$, and $\angle DAC$ and $\angle BCA$ are supplementary, $\angle BAC>\angle ACD$, and $\angle BAC$ and $\angle ACD$ are complementary. Then $\angle B=$

Answer: 45^\circ


4. Misha and Masha were traveling by train to Kirov. Misha was lying on the bunk, while Masha was looking out the window. "How far is it to Kirov?" Misha asked Masha at 12:00. "73 kilometers," Masha replied. To the same question asked at 12:15 and 12:45, Masha answered: "62 kilometers" and "37 kilometers." It is known that if the distance was not a whole number of kilometers, Masha rounded it to the nearest whole number each time (and if there were two such numbers, she rounded to either one at her discretion). Find the speed of the train, given that it was constant. List all possibilities and prove that there are no others.

Answer: 48


A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]

Answer: 540


In $\triangle ABC$, max $\{\angle A, \angle B \} = \angle C + 30^{\circ}$ and $\frac{R}{r} = \sqrt{3} + 1$, where $R$ is the radius of the circumcircle and $r$ is the radius of the incircle. Find $\angle C$ in degrees.

Answer: 60


Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.

Answer: 40


And it came to pass that Jeb owned over a thousand chickens. So Jeb counted his chickens. And Jeb reported the count to Hannah. And Hannah reported the count to Joshua. And Joshua reported the count to Caleb. And Caleb reported the count to Rachel. But as fate would have it, Jeb had over-counted his chickens by nine chickens. Then Hannah interchanged the last two digits of the count before reporting it to Joshua. And Joshua interchanged the first and the third digits of the number reported to him before reporting it to Caleb. Then Caleb doubled the number reported to him before reporting it to Rachel. Now it so happens that the count reported to Rachel was the correct number of chickens that Jeb owned. How many chickens was that?

Answer: 2052


The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.

Answer: 29


Jerry is mowing a rectangular lawn which is $77$ feet north to south by $83$ feet east to west. His lawn mower cuts a path $18$ inches wide. Jerry mows the grass by cutting a path from west to east across the north side of the lawn and then making a right turn cutting a path along the east side of the lawn. When he completes mowing each side of the lawn, he continues by making right turns to mow a path along the next side. How many right turns will he make?

Answer: 102


A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?

Answer: 105717501


Let $T$ be a $30-60-90$ triangle with hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.

Answer: 50


At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does the coo-coo bird coo-coo from $12:42$ PM on Monday until $3:42$ AM on Wednesday?

Answer: 279


We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?

Answer: 7


Michael is celebrating his fifteenth birthday today. How many Sundays have there been in his lifetime?

Answer: 783


There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?

Answer: 225


A cyclic  $n$ -gon is divided by non-intersecting (inside the  $n$ -gon) diagonals to  $n-2$  triangles. Each of these triangles is similar to at least one of the remaining ones. For what  $n$  this is possible?

Answer: 4


For a nonempty set  $\, S \,$  of integers, let  $\, \sigma(S) \,$  be the sum of the elements of  $\, S$ .  Suppose that  $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$  is a set of positive integers with  $\, a_1 < a_2 < \cdots < a_{11} \,$  and that, for each positive integer  $\, n\leq 1500, \,$  there is a subset   $\, S \,$  of  $\, A \,$  for which  $\, \sigma(S) = n$ .  What is the smallest possible value of  $\, a_{10}$ ?

Answer: 248


**p1.** Solve for  $x$  and  $y$  if  $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$  and  $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ **p2.** Find positive integers  $p$  and  $q$ , with  $q$  as small as possible, such that  $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$ .**p3.** Define  $a_1 = 2$  and  $a_{n+1} = a^2_n -a_n + 1$  for all positive integers  $n$ . If  $i > j$ , prove that   $a_i$  and  $a_j$  have no common prime factor.**p4.** A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it.
Prove that the number of these smaller triangular regions is always odd.**p5.** In triangle  $ABC$ , let  $\angle ABC=\angle ACB=40^o$  is extended to  $D$  such that  $AD=BC$ . Prove that  $\angle BCD=10^o$ .
![Image](https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png)

PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 10


Round 1**p1.** How many powers of  $2$  are greater than  $3$  but less than  $2013$ ?**p2.** What number is equal to six greater than three times the answer to this question? **p3.** Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on one end, with base radius  $3$  centimeters and height  $10$  centimeters. What is the exterior surface area of the styrofoam cup?

Round 2**p4.** Andrew has two  $6$ -foot-length sticks that he wishes to make into two of the sides of the entrance to his fort, with the ground being the third side. If he wants to make his entrance in the shape of a triangle, what is the largest area that he can make the entrance?**p5.** Ethan and Devin met a fairy who told them “if you have less than  $15$  dollars, I will give you cake”. If both had integral amounts of dollars, and Devin had 5 more dollars than Ethan, but only Ethan got cake, how many different amounts of money could Ethan have had?**p6.** If  $2012^x = 2013$ , for what value of  $a$ , in terms of  $x$ , is it true that  $2012^a = 2013^2$ ?

Round 3**p7.** Find the ordered triple  $(L, M, T)$  of positive integers that makes the following equation true:   $$ 1 + \dfrac{1}{L + \dfrac{1}{M+\dfrac{1}{T}}}=\frac{79}{43}. $$  **p8.** Jonathan would like to start a banana plantation so he is saving up to buy an acre of land, which costs  $\$ 600,000 $. He deposits $ \ $300,000$  in the bank, which gives  $20\%$  interest compounded at the end of each year. At this rate, how many years will Jonathan have to wait until he can buy the acre of land?**p9.** Arul and Ethan went swimming at their town pool and started to swim laps to see who was in better shape. After one hour of swimming at their own paces, Ethan completed  $32$  more laps than Arul. However, after that, Ethan got tired and swam at half his original speed while Arul’s speed didn’t change. After one more hour, Arul swam a total of  $320$  laps. How many laps did Ethan swim after two hours?

Round 4**p10.** A right triangle with a side length of  $6$  and a hypotenuse of 10 has circles of radius  $1$  centered at each vertex. What is the area of the space inside the triangle but outside all three circles?**p11.** In isosceles trapezoid  $ABCD$ ,  $\overline{AB} \parallel\overline{CD}$  and the lengths of  $\overline{AB}$  and  $\overline{CD}$  are  $2$  and  $6$ , respectively. Let the diagonals of the trapezoid intersect at point  $E$ . If the distance from  $E$  to  $\overline{CD}$  is  $9$ , what is the area of triangle  $ABE$ ?**p12.** If  $144$  unit cubes are glued together to form a rectangular prism and the perimeter of the base is  $54$  units, what is the height?

 PS. You should use hide for answers. Rounds 6-8 are [here](https://artofproblemsolving.com/community/c3h3136014p28427163) and 9-12 [here](https://artofproblemsolving.com/community/c3h3137069p28442224).  Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Answer: 2


To *dissect* a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
![Image](https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png)
An *integer-ratio* right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides  $3,4,5$  is an*integer-ratio* right triangle, and so is a triangle with sides  $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$ . On the other hand, the right triangle with sides $ \sqrt2 ,\sqrt5, \sqrt7$  is not an *integer-ratio* right triangle. Determine, with proof, all integers  $n$  for which it is possible to completely *dissect* a regular  $n$ -sided polygon into *integer-ratio* right triangles.

Answer: 4


Determine the maximum number of bishops that we can place in a  $8 \times 8$  chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.

Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the  $2$  main diagonals and the ones parallel to those ones.

Answer: 16


Let  $a$  be a positive real numbers. Let  $t,\ u\ (t<u)$  be the  $x$  coordinates of the point of intersections of the curves :  $C_1:y=|\cos x|\ (0\leq x\leq \pi),\ C_2:y=a\sin x\ (0\leq x\leq \pi).$  Denote by  $S_1$  the area of the part bounded by  $C_1,\ C_2$  and  $y$ -axis in  $0\leq x\leq t$ , and by  $S_2$  the area of the part bounded by  $C_1,\ C_2$  in  $t\leq x\leq u$ . When  $a$  moves over all positive real numbers, find the minimum value of  $S_1+S_2$ .

Answer: 2\sqrt{2}-2


We call a graph with n vertices  $k-flowing-chromatic$  if:
1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors.
2. we can choose a hamilton cycle  $v_1,v_2,\cdots , v_n$ , and move the chess on  $v_i$  to  $v_{i+1}$  with  $i=1,2,\cdots ,n$  and  $v_{n+1}=v_1$ , such that any two neighboring chess also have different colors.
3. after some action of step 2 we can make all the chess reach each of the n vertices.
Let T(G) denote the least number k such that G is k-flowing-chromatic.
If such k does not exist, denote T(G)=0.
denote  $\chi (G)$  the chromatic number of G.
Find all the positive number m such that there is a graph G with  $\chi (G)\le m$  and  $T(G)\ge 2^m$  without a cycle of length small than 2017.

Answer: 3


Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he must observe the following rules. The numbers of the houses
delivered to must follow an odd-even-odd-even pattern throughout, each house
except house 1 is visited exactly once (house 1 is visited twice) and the postman
at no time is allowed to cross the road to the house directly opposite. How
many different delivery sequences are possible?

Answer: 12


Let  $ABC$  be a triangle in the coordinate plane with vertices on lattice points and with  $AB = 1$ .  Suppose the perimeter of  $ABC$  is less than  $17$ .  Find the largest possible value of  $1/r$ , where  $r$  is the inradius of  $ABC$ .

Answer: 1+\sqrt{50}+\sqrt{65}


Triangle  $ABC$  is inscribed in a unit circle  $\omega$ . Let  $H$  be its orthocenter and  $D$  be the foot of the perpendicular from  $A$  to  $BC$ . Let  $\triangle XY Z$  be the triangle formed by drawing the tangents to  $\omega$  at  $A, B, C$ . If  $\overline{AH} = \overline{HD}$  and the side lengths of  $\triangle XY Z$  form an arithmetic sequence, the area of  $\triangle ABC$  can be expressed in the form  $\tfrac{p}{q}$  for relatively prime positive integers  $p, q$ . What is  $p + q$ ?

Answer: 11


Let  $R$  be a square region and  $n \ge 4$  an integer. A point  $X$  in the interior of  $R$  is called *n-ray partitional* if there are  $n$  rays emanating from  $X$  that divide  $R$  into  $n$  triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? $\textbf{(A)}\ 1500 \qquad
\textbf{(B)}\ 1560 \qquad
\textbf{(C)}\ 2320  \qquad
\textbf{(D)}\ 2480 \qquad
\textbf{(E)}\ 2500$

Answer: 2320


Let  $O$  be the set of odd numbers between 0 and 100.  Let  $T$  be the set of subsets of  $O$  of size  $25$ .  For any finite subset of integers  $S$ , let  $P(S)$  be the product of the elements of  $S$ .  Define  $n=\textstyle{\sum_{S \in T}} P(S)$ .  If you divide  $n$  by 17, what is the remainder?

Answer: 15


Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?

Answer: 7


A mathematician is walking through a library with twenty-six shelves, one for each letter of the alphabet. As he walks, the mathematician will take at most one book off each shelf. He likes symmetry, so if the letter of a shelf has at least one line of symmetry (e.g., M works, L does not), he will pick a book with probability  $\frac12$ . Otherwise he has a  $\frac14$  probability of taking a book. What is the expected number of books that the mathematician will take?

Answer: \frac{41}{4}


A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational period. It is at the precise distance of  $22,236$  miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to  $24$  hours. Be cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of  $22,236$  miles above the equator). Given that the radius of the Earth is  $3960$  miles, find the exact straight distance between two neighboring stations. Write your answer in the form  $a + b\sqrt{c}$ , where  $a, b, c$  are integers and  $c > 0$  is square-free.

Answer: -13098+13098\sqrt{5}


Initially on the computer screen - some prime number. Every second, the number on the screen is replaced by the number obtained from the previous one by adding its last digit, increased by 1. What is the maximum time it will take for a composite number to appear on the screen?

#

Answer: 5


8. The equation $5 z^{4}+10 z^{3}+10 z^{2}+5 z+1=0$ has four complex roots, the real part of the complex root with the largest modulus is $\qquad$ _.

Answer: -\dfrac{1}{2}


13. As shown in the figure, Jia, Yi, and Bing are three stations, with the distance from Yi station to Jia and Bing stations being equal. Xiaoming and Xiaoqiang start from Jia and Bing stations respectively at the same time and walk towards each other. Xiaoming meets Xiaoqiang 100 meters past Yi station, then they continue walking. Xiaoming reaches Bing station and immediately turns back, and 300 meters past Yi station, he catches up with Xiaoqiang again. What is the distance between Jia and Bing stations?

Answer: 600


In the quadratic equation with respect to $x$:
\[ x^{2} + z_{1} x + z_{2} + m = 0 \]
where $z_{1}, z_{2}, m$ are all complex numbers, and \( z_{1}^{2} - 4 z_{2} = 16 + 20i \). Let the two roots of this equation be $\alpha$ and $\beta$, and given that \( |\alpha - \beta| = 2 \sqrt{7} \), find the maximum and minimum values of \( |m| \).

Answer: 7 + \sqrt{41}


51 Let the set $S=\{100,101,102, \cdots, 999,1000\}, A=\left\{a_{1}, a_{2}, a_{3}, \cdots, a_{n-1}, a_{n}\right.$ $\mid a_{1}, a_{2}, \cdots, a_{n}$ are positive numbers, and $\left.\frac{a_{2}}{a_{1}}=\frac{a_{3}}{a_{2}}=\cdots=\frac{a_{n}}{a_{n-1}}=q>1\right\}$. Try to find the maximum possible number of elements in the intersection $S \cap A$.

Answer: 6


Suppose \( n \in \mathbb{Z}^{+}, n \geq 4, p \geq n-2 \) is a prime number. \( M \) is a set of \( n \) points in the plane with no three points collinear, and the mapping \( f: M \rightarrow \{0,1, \ldots, p-1\} \) satisfies the conditions:
1. Only one point maps to 0;
2. If \( A, B, C \) are three distinct points in \( M \), and \( K \) is the circumcircle of \( \triangle ABC \), then \(\sum_{P \in M \cap K} f(P) \equiv 0 \pmod{p}\).

Prove: All the points in \( M \) lie on a single circle.

Answer: \text{All points in } M \text{ lie on a single circle.}


12. Let the circle $C: x^{2}+y^{2}-\left(2 a^{2}-4\right) x-4 a^{2} y+5 a^{4}-4=0$. Then the equation of the locus of the center of circle $C$ is

Answer: 2x - y + 4 = 0


10.2. There are 2004 small boxes on the table, each containing 1 ball. It is known that some of the balls are white, and there are an even number of white balls. You are allowed to point to any 2 boxes and ask: "Do they contain at least 1 white ball?" How many times do you need to ask, at a minimum, to determine a box that contains a white ball?

Answer: 2003


105 A cube of wood with an edge length of 3 meters, has a square hole with a side length of 1 meter drilled through the center of each face, all the way through, with the edges of the hole parallel to the corresponding edges of the cube. The total surface area (including the inside) of this object is $\qquad$ square meters.

Answer: 72


5. Dima has 25 identical bricks of size $5 \times 14 \times 17$. Dima wants to build a tower from all his bricks, each time adding one more brick on top (each new brick adds 5, 14, or 17 to the current height of the tower). We will call a number $n$ constructible if Dima can build a tower of height exactly $n$. How many constructible numbers exist?

Answer: 98


Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BP D$?

Answer: 60


4. Fold a triangle with side lengths of $10$, $12$, and $14$ along its three midlines to form a tetrahedron. Then the diameter of the circumscribed sphere of the tetrahedron is $\qquad$ .

Answer: \sqrt{55}


# Task 7. (14 points)

In a certain company, no two employees have jobs of the same difficulty, and no two employees receive the same salary. On April 1, each employee made two statements:

(a) There will not be 12 employees with more difficult jobs.

(b) At least 30 employees have a higher salary. How many employees are in the company if some employees told the truth twice, and the rest lied twice.

#

Answer: 42


A game starts with seven coins aligned on a table, all with the crown face up. To win the game, you need to flip some coins in such a way that, in the end, two adjacent coins always have different faces up. The rule of the game is to flip two adjacent coins in each move. What is the minimum number of moves required to win the game?

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-016.jpg?height=143&width=842&top_left_y=1650&top_left_x=641)

Answer: 4


Let \( P_{m}=\{A_{1}, A_{2}, \cdots, A_{m}\} \) be a partition of an \( n \)-element set \( X \) (i.e., \( A_{1}, A_{2}, \cdots, A_{m} \) partition \( X \)). By splitting one \( A_{i} \) into two sets, we generate a partition \( P_{m+1} \) of \( X \), which consists of \( m+1 \) sets. \( P_{m+1} \) is called a refinement of \( P_{m} \). If
\[ P_{1}, P_{2}, \cdots, P_{n} \]

are partitions of an \( n \)-element set \( X \), and each is a refinement of the previous one (obviously \( P_{n} \) consists of \( n \) sets, and \( P_{1} \) consists of only one set, \( X \)), this sequence is called a chain of length \( n \). Determine the number of chains of length \( n \).

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


6. Circle $k_{1}$ with a radius of 8 cm lies inside circle $k$. Both circles intersect circle $k_{2}$ with a radius of 15 cm, as shown in the figure. What is the radius of $k$ if the shaded area inside $k$ but outside $k_{1}$ is equal to the total shaded area inside $k_{2}$?

![](https://cdn.mathpix.com/cropped/2024_05_06_b124656272d750e4349eg-3.jpg?height=611&width=740&top_left_y=1091&top_left_x=681)

Answer: 17


For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\} = x - \lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying:
\[\{x\} = \frac{x + \lfloor x \rfloor + \lfloor x + \frac{1}{2} \rfloor}{20}\]
If $S$ is the sum of all numbers in $A$, find $\lfloor S \rfloor$.

Answer: 21


Consider two geometric sequences $16$, $a_1$, $a_2$, $\ldots$ and $56$, $b_1$, $b_2$, $\ldots$ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.

Answer: 490


Let $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the remainder when $a_1+a_2+\dots+a_{2013}$ is divided by $1000$.

Answer: 693


For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[ \sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)}, \] where $N=6^{1999}$.

Answer: 8


Juan makes a list of $2018$ numbers. The first number is $1$. Each subsequent number is obtained by adding one of the numbers $1, 2, 3, 4, 5, 6, 7, 8,$ or $9$ to the previous number. Given that none of the numbers in the list ends in $0$, what is the largest possible value for the last number on the list?

Answer: 17929


Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$.

Answer: 292


For each positive integer $n \ge 2$, define $k(n)$ to be the largest integer $m$ such that $(n!)^m$ divides $2016!$. What is the minimum possible value of $n + k(n)$?

Answer: 89


Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). How many possible DNA sequences are there on Rthea?

Answer: 28812


The integers $1, 2, \ldots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.

Answer: 2014


Let $\triangle AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF$, respectively, such that $BD$ is parallel to $EF$. Point $C$ is chosen in the interior of $\triangle AEF$ such that quadrilateral $ABCD$ is cyclic. If $BC = 3$ and $CD = 4$, then the ratio of the areas $\frac{[ABCD]}{[AEF]}$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Answer: 5300


Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Neymar chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Neymar will never be able to write on the blackboard?

Answer: 2033


A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2 + kx + 2014$ and $x^2 + kx - 2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.

Answer: 0


Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks.

Answer: 4489


A positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $n = a^b$. How many positive integer divisors of $729000000$ are base-able?

Answer: 90


The Garfield Super Winners play  $100$  games of foosball, in which teams score a non-negative integer number of points and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of  $7$  points per game but allow an average of  $8$  points per game. Given that the Garfield Super Winners never won or lost by more than  $10$ , what is the largest possible number of games that they could win?

*2019 CCA Math Bonanza Lightning Round #4.1*

Answer: 81


Let there be a figure with  $9$  disks and  $11$  edges, as shown below.

We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write  $0$  in disk  $A$ , and  $1$  in disk  $I$ . Find the minimum sum of all real numbers written in  $11$  edges.

Answer: \frac{1}{11}


Consider a rectangular grid of  $ 10 \times 10$  unit squares. We call a *ship* a figure made up of unit squares connected by common edges. We call a *fleet* a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.

Answer: 25


We call an even positive integer  $n$  *nice* if the set  $\{1, 2, \dots, n\}$  can be partitioned into  $\frac{n}{2}$  two-element subsets, such that the sum of the elements in each subset is a power of  $3$ . For example,  $6$  is nice, because the set  $\{1, 2, 3, 4, 5, 6\}$  can be partitioned into subsets  $\{1, 2\}$ ,  $\{3, 6\}$ ,  $\{4, 5\}$ . Find the number of nice positive integers which are smaller than  $3^{2022}$ .

Answer: 2^{2022}-1


Let  $S=\{1,2,3,\ldots,280\}$ . Find the smallest integer  $n$  such that each  $n$ -element subset of  $S$  contains five numbers which are pairwise relatively prime.

Answer: 217


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

Answer: 18


We mean a traingle in  $\mathbb Q^{n}$ , 3 points that are not collinear in  $\mathbb Q^{n}$

a) Suppose that  $ABC$  is triangle in  $\mathbb Q^{n}$ . Prove that there is a triangle  $A'B'C'$  in  $\mathbb Q^{5}$  that  $\angle B'A'C'=\angle BAC$ .
b) Find a natural  $m$  that for each traingle that can be embedded in  $\mathbb Q^{n}$  it can be embedded in  $\mathbb Q^{m}$ .
c) Find a triangle that can be embedded in  $\mathbb Q^{n}$  and no triangle similar to it can be embedded in  $\mathbb Q^{3}$ .
d) Find a natural  $m'$  that for each traingle that can be embedded in  $\mathbb Q^{n}$  then there is a triangle similar to it, that can be embedded in  $\mathbb Q^{m}$ .
You must prove the problem for  $m=9$  and  $m'=6$  to get complete mark. (Better results leads to additional mark.)

Answer: 6


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

Answer: 56+\frac{112\sqrt{2}}{3}


Let  $a$  be a fixed real number. Consider the equation  $$ (x+2)^{2}(x+7)^{2}+a=0, x \in R $$  where  $R$  is the set of real numbers. For what values of  $a$ , will the equ have exactly one double-root?

Answer: -39.0625


Answer: 1 - 2^{-1/4}


Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$.
Find the minimal natural number $k$, satisfying the condition:  for any $M$, we can color all the subsets of $M$ with $k$ colors, such that whenever $A\neq f(A)$, $A$ and $f(A)$ are colored with different colors.

Answer: 2


To glaze 15 windows of various sizes and shapes, 15 pieces of glass have been prepared exactly to fit the windows (the windows are such that each window should have one piece of glass). The glazier, not knowing that the glasses are matched, works as follows: he approaches the next window and tries the unused glasses until he finds one that is large enough (i.e., either exactly fitting or one from which a fitting piece can be cut), if no such glass is found, he moves to the next window, and so on, until he has gone through all the windows. It is not allowed to assemble a piece of glass from several parts. What is the maximum number of windows that can remain unglazed?

Answer: 7


Twelve distinct natural numbers are written in a circle, one of which is 1. Any two neighboring numbers differ either by 10 or by 7. What is the largest possible value of the greatest number written?

Answer: 58


Let \( p \) be a prime number, \( n \in \mathbb{Z}_{+} \), and

\[ n = n_{0} + n_{1} p + \cdots + n_{t} p^{t}, \]

where \( n_{i} \in \mathbb{N} \), \( 0 \leq n_{i} \leq p-1 \) for \( i = 0, 1, \ldots, t \).

Let \( S_{n} \) be the set of ordered triples \((a, b, c)\) that satisfy the following conditions:
1. \( a, b, c \in \mathbb{N} \);
2. \( a + b + c = n \);
3. \( p \nmid \frac{n!}{a! \cdot b! \cdot c!} \).

How many ordered triples \((a, b, c)\) are there in the set \( S_{n} \)?

Answer: \prod_{i=0}^{t} \frac{(n_i + 1)(n_i + 2)}{2}


Let $I$ be the incenter of triangle $\triangle ABC$. The angle bisectors of angles $A, B, C$ intersect the opposite sides at points $A'$, $B'$, $C'$ respectively. Prove that:
$$
\frac{1}{4}<\frac{A I \cdot B I \cdot C I}{A A^{\prime} \cdot B B^{\prime} \cdot C C^{\prime}} \leqslant \frac{8}{27}
$$

Answer: \frac{1}{4}<\frac{A I \cdot B I \cdot C I}{A A^{\prime} \cdot B B^{\prime} \cdot C C^{\prime}} \leqslant \frac{8}{27}


12. (5 points) Three people, A, B, and C, start from point A to point B at the same time. The ratio of their speeds is 4:5:12, where A and B walk, and C rides a bicycle. C can carry one person at a time (the speed remains unchanged). To ensure that all three arrive at point B in the shortest time possible, the ratio of the distances walked by A and B is $\qquad$

Answer: 7:10


The hero of an animated cartoon faces off against his arch-enemy once again and needs to unleash his famous Reflective Beam. On the square in the figure below, the beam must, starting from \( F \), ricochet exactly once off the sides \( CD \), \( AD \), and \( AB \), in that order, before hitting the enemy at position \( E \). Whenever the beam ricochets off one of the sides of the square, the angle of incidence is equal to the angle of exit, as shown in the figure to the right. Given that \( BE = EF = FC = 2 \, \text{m} \) and that the beam travels at \( 1 \, \text{m/s} \), determine the time elapsed between the beam being fired from \( F \) and its arrival at point \( E \).

Answer: 2\sqrt{61}


A building has seven rooms numbered 1 through 7, all on one floor, and any number of doors connecting these rooms. These doors may be one-way, admitting motion in only one of the two directions, or two-way. In addition, there is a two-way door between room 1 and the outside, and a treasure in room 7. Your object is to choose the arrangement of the rooms and the locations of the doors in such a way that

(a) it is possible to enter room 1 , reach the treasure, and make it back outside,

(b) the minimum number of steps required to to this (each step consisting of walking through a door) is as large as possible.

Answer: 14


Example. Find the probability that among six cards randomly taken from a deck of 36 cards, there will be exactly three black figure cards.

Answer: \dfrac{546}{5797}


A point whose coordinates are both integers is called a lattice point.  How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$ ?
Please give the answer directly without any intermediate steps.

Answer: 98


## 19. Tram

On one tram line, trams depart regularly every 10 minutes throughout the day. A tram takes one hour to travel from one end of the line to the other. A passenger boards a tram at one terminal stop and rides to the last stop of the tram; out of boredom, he looks out the window and counts the oncoming trams of the same route. How many tram sets will he count in total.

Answer: 12


Example 2 There are $r$ people participating in a chess tournament, every 2 people play one game, the winner of each game gets 2 points, the loser gets 0 points, and in the case of a draw, each gets 1 point. After the tournament, there is exactly one person who has the fewest wins, and only he has the highest score. Find the minimum value of $r$. (16th All-Russian Mathematical Olympiad problem)

Answer: 6


1.2. In triangle $ABC$ with angle $BAC$ equal to $24^{\circ}$, points $X$ and $Y$ are taken on sides $AB$ and $AC$ respectively. The circle centered at $Y$ and passing through $A$ also passes through $X$, and the circle centered at $X$ and passing through $B$ also passes through $C$ and $Y$. Find $\angle ABC$.

Answer: 54^\circ


Let $n$ be represented as the difference of squares of two nonzero natural numbers, then there are $F(n)$ ways to do so.

For example, $15=8^{2}-7^{2}=4^{2}-1^{2}$, so $F(15)=2$; whereas 2 cannot be represented, hence $F(2)=0$. Then, the calculation result of $F(1)+F(2)+F(3)+\cdots+F(100)$ is \_\_\_\_\_.

Answer: 116


On a board, the numbers $2, 3, 4, \ldots, 29, 30$ are written. For one dollar, you can mark any number. If a number is already marked, you can mark its divisors and multiples for free. For the smallest amount of dollars, how can you mark all the numbers on the board? [6 points]

(I.V. Yashchenko)

Answer: 5


12. Let $n$ be a natural number, write $n$ as a sum of powers of $p$ (where $p$ is a positive integer greater than 1) and each power of $p$ can appear at most $p^{2}-1$ times, denote the total number of such decompositions as $C(n, p)$. For example: $8=4+4=4+2+2=4+2+1+1=2+2+2+1+1=8$, then $C(8,2)=5$. Note that in $8=4+1+1+1+1$, $1=2^{0}$ appears 4 times, which exceeds $2^{2}-1=3$, so it is not counted. Then $C(2002,17)=$ $\qquad$ .

Answer: 118


4. Each of $A$ and $B$ is a four-digit palindromic integer, $C$ is a three-digit palindromic integer, and $A-B=C$.
What are the possible values of $C$ ?
[A palindromic integer reads the same 'forwards' and 'backwards'.]

Answer: 121


3. In triangle $ABC$, side $AB$ is the longest. On side $AB$, points $D$ and $E$ are chosen such that $\overline{AC}=\overline{AD}$ and $\overline{BE}=\overline{BC}$. Determine the angle $\measuredangle ACB$, if $\measuredangle ECD=20^{\circ}$.

Answer: 140


(10 From a material site beside a road, electric poles are to be transported along a straight road in the same direction to be planted at the roadside 500 m away. At the 500 m point, one pole is planted, and then one pole is planted every 50 m along the roadside. Knowing that the transport vehicle can carry a maximum of 3 poles at a time, to complete the task of transporting and planting 20 poles, and returning to the material site, the minimum total distance traveled by the transport vehicle is $\qquad$ $\mathrm{m}$.

Answer: 14000


8. The equation $\frac{1}{x^{2}}+\frac{1}{y}+\frac{1}{z}=\frac{4}{3}$ has $\qquad$ integer solutions.

Answer: 14


10. (3 points) 60 adventure team members need to cross a river. There is only one rubber boat on the river that can carry 6 people (a round trip counts as two times), and it takes 3 minutes to cross the river once. The total time required for all team members to cross to the other side of the river is $\qquad$ minutes.

Answer: 69


B4 If all students in the class were to sit in their own desks, there would be 11 desks too few. If, however, two students were to sit at each desk, there would be 5 desks too many. How many desks and how many students are there in the class?

(7 points)

Space for solving the problems of set $B$.

![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-04.jpg?height=86&width=446&top_left_y=200&top_left_x=1456)

## PROBLEMS FOR THE SECOND YEAR

Time for solving: 90 minutes. In set A, a correct answer will be worth two points, while a wrong answer will result in a deduction of half a point. Enter the answers for set A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_c987701c9bb62194911fg-04.jpg?height=172&width=1196&top_left_y=576&top_left_x=446)

Answer: 21


3. Let's consider 20 statements:

| "I have exactly one sister." | "I have exactly one brother." |
| :--- | :--- |
| "I have exactly two sisters." | "I have exactly two brothers." |
| "I have exactly ten sisters." | "I have exactly ten brothers." |

a) Each of the four siblings made a different one of these 20 statements. Is it possible that all four were telling the truth?

b) Find the largest natural number $n$ such that each of the $n$ siblings can make a different one of these 20 statements and be telling the truth.

(Josef Tkadlec)

Answer: 4


Question 82, Given vectors $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$ have an angle of $120^{\circ}$ between them, $|\overrightarrow{\mathrm{OA}}|=2|\overrightarrow{\mathrm{OB}}|=6$. A line segment $\mathrm{PQ}$ of length $2|\overrightarrow{\mathrm{AB}}|$ is drawn through 0, and $\mathrm{P}, \mathrm{Q}$ are symmetric about 0. When the angle between vector $\overrightarrow{\mathrm{PQ}}$ and $\overrightarrow{\mathrm{AB}}$ is what, will $\overrightarrow{\mathrm{AP}} \cdot \overrightarrow{\mathrm{BQ}}$ be maximized, and find this maximum value.

Answer: -9


16. (15 points) One day, Dad drove Xiaohong to a place 1000 meters away from the school and let her walk the rest of the way. As a result, Xiaohong took 22.5 minutes to get to school that day. If Xiaohong rides a bicycle from home to school, it takes her 40 minutes. She usually walks at a speed of 80 meters per minute, and riding a bicycle is 800 meters per minute slower than Dad driving. Find the distance from Xiaohong's home to the school.

Answer: 12000


10.160. In a trapezoid with an area of $594 \mathrm{~m}^{2}$, the height is 22 m, and the difference between the parallel sides is $6 \mathrm{~m}$. Find the length of each of the parallel sides.

Answer: 30


16. The product of all natural divisors of a natural number $n$ (including $n$) ends with 120 zeros. How many zeros can the number $n$ end with? (List all possible options and prove that there are no others.)

Answer: 1,2,3,4,5


Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Answer: 16902


Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly $10$ operations.

Answer: 144


The cells of a $5\times5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through $9$ cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly $3$ red cells, exactly $3$ white cells, and exactly $3$ blue cells no matter which route he takes.

Answer: 1680


For any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\sum^{\lfloor \log_{23} n\rfloor}_{i=1}
s_{20} \left( \left\lfloor \frac{n}{23^i} \right\rfloor \right)= 103 \,\,\, \text{and} \,\,\,
\sum^{\lfloor \log_{20} n\rfloor}_{i=1} s_{23} \left( \left\lfloor \frac{n}{20^i} \right\rfloor \right)= 115$$
Compute $s_{20}(n) - s_{23}(n)$.

Answer: 81


Let $P(n)=(n-1^3)(n-2^3)\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \mid P(n)$ for any $n>2023$. If $d$ is product of $m$ not necessarily distinct primes, find $m$.

Answer: 48


The country of HMMTLand has $8$ cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly $3$ other cities via a single road, and from any pair of distinct cities, either exactly $0$ or $2$ other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

Answer: 875


Let $\zeta = \cos \frac {2pi}{13} + i \sin  \frac {2pi}{13}$ . Suppose $a > b > c > d$ are positive integers satisfying
$$|\zeta^a + \zeta^b + \zeta^c +\zeta^d| =\sqrt3.$$
Compute the smallest possible value of $1000a + 100b + 10c + d$.

Answer: 7521


Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1 : 2 : 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of this chord.

Answer: 42


In four-dimensional space, the $24$-cell of sidelength $\sqrt{2}$ is the convex hull of (smallest convex set containing) the $24$ points $(\pm 1, \pm 1, 0, 0)$ and its permutations. Find the four-dimensional volume of this region.

Answer: 8


If $p(x)$ is a polynomial with integer coeffcients, let $q(x) = \frac{p(x)}{x(1-x)}$ . If $q(x) = q\left(\frac{1}{1-x}\right)$ for every $x \ne 0$, and $p(2) = -7, p(3) = -11$, find $p(10)$.

Answer: 521


Let $x$ be the largest root of $x^4 - 2009x + 1$. Find the nearest integer to $\frac{1}{x^3-2009}$ .

Answer: -13