If  $0 \le p \le 1$  and  $0 \le q \le 1$ , define  $F(p, q)$  by
\[
  F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
\]
Define  $G(p)$  to be the maximum of  $F(p, q)$  over all  $q$  (in the interval  $0 \le q \le 1$ ).  What is the value of  $p$  (in the interval  $0 \le p \le 1$ ) that minimizes  $G(p)$ ?

Solution: \boxed{ p = \frac{7}{12} }


Let  $a_1$ ,  $a_2$ ,  $\ldots$  be an infinite sequence of (positive) integers such that  $k$  divides  $\gcd(a_{k-1},a_k)$  for all  $k\geq 2$ .  Compute the smallest possible value of  $a_1+a_2+\cdots+a_{10}$ .


The  $n^{th}$  derivative of a function  $f(x)$  (if it exists) is denoted by  $f^{(n)}(x) $ . Let  $f(x)=\frac{e^x}{x}$ .
Suppose  $f$  is differentiable infinitely many times in  $(0,\infty) $ . Then find  $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

Solution: \boxed{ 1 }


Consider a unit circle with center  $O$ . Let  $P$  be a point outside the circle such that the two line segments passing through  $P$  and tangent to the circle form an angle of  $60^\circ$ . Compute the length of  $OP$ .


Let  $AB$  be a diameter of a circle and let  $C$  be a point on the segement  $AB$  such that  $AC : CB = 6 : 7$ . Let  $D$  be a point on the circle such that  $DC$   is perpendicular to  $AB$ . Let  $DE$  be the diameter through  $D$ . If  $[XYZ]$  denotes the area of the triangle  $XYZ$ , find  $[ABD]/[CDE]$  to the nearest integer.


Find all triplets  $(a,b,p)$  where  $a,b$  are positive integers and  $p$  is a prime number satisfying:  $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

Solution: \boxed{ (2, 2, 2) }


1. Let  $f(n) = n^2+6n+11$  be a function defined on positive integers. Find the sum of the first three prime values  $f(n)$  takes on.

*Proposed by winnertakeover*


In his spare time, Thomas likes making rectangular windows. He builds windows by taking four  $30\text{ cm}\times20\text{ cm}$  rectangles of glass and arranging them in a larger rectangle in wood. The window has an  $x\text{ cm}$  wide strip of wood between adjacent glass pieces and an  $x\text{ cm}$  wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is  $x$ ?

[center]<see attached>[/center]


Let  $ABC$  be a triangle with sides  $AB=19$ ,  $BC=21$ , and  $AC=20$ . Let  $\omega$  be the incircle of  $ABC$  with center  $I$ . Extend  $BI$  so that it intersects  $AC$  at  $E$ . If  $\omega$  is tangent to  $AC$  at the point  $D$ , then compute the length of  $DE$ .


Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is  $\dfrac{7}{2}$  times the original one.


Let  $\mathcal S$  be the sphere with center  $(0,0,1)$  and radius  $1$  in  $\mathbb R^3$ .  A plane  $\mathcal P$  is tangent to  $\mathcal S$  at the point  $(x_0,y_0,z_0)$ , where  $x_0$ ,  $y_0$ , and  $z_0$  are all positive.  Suppose the intersection of plane  $\mathcal P$  with the  $xy$ -plane is the line with equation  $2x+y=10$  in  $xy$ -space.  What is  $z_0$ ?


Assume integer  $m \geq 2.$  There are  $3m$  people in a meeting, any two of them either shake hands with each other once or not.We call the meeting " $n$ -interesting", only if there exists  $n(n\leq 3m-1)$  people of them, the time everyone of whom shakes hands with other  $3m-1$  people is exactly  $1,2,\cdots,n,$  respectively. If in any " $n$ -interesting" meeting, there exists  $3$  people of them who shake hands with each other, find the minimum value of  $n.$

Solution: \boxed{ 2m+1 }


Let  $P(x)$  be a polynomial with integer coefficients, leading coefficient 1, and  $P(0) = 3$ . If the polynomial  $P(x)^2 + 1$  can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of  $P$  is as small as possible, compute the largest possible value of  $P(10)$ .

*2016 CCA Math Bonanza Individual #13*


Square  $1$  is drawn with side length  $4$ . Square  $2$  is then drawn inside of Square  $1$ , with its vertices at the midpoints of the sides of Square  $1$ . Given Square  $n$  for a positive integer  $n$ , we draw Square  $n+1$  with vertices at the midpoints of the sides of Square  $n$ . For any positive integer  $n$ , we draw Circle  $n$  through the four vertices of Square  $n$ . What is the area of Circle  $7$ ?

*2019 CCA Math Bonanza Individual Round #2*


Two congruent equilateral triangles  $\triangle ABC$  and  $\triangle DEF$  lie on the same side of line  $BC$  so that  $B$ ,  $C$ ,  $E$ , and  $F$  are collinear as shown. A line intersects  $\overline{AB}$ ,  $\overline{AC}$ ,  $\overline{DE}$ , and  $\overline{EF}$  at  $W$ ,  $X$ ,  $Y$ , and  $Z$ , respectively, such that  $\tfrac{AW}{BW} = \tfrac29$  ,  $\tfrac{AX}{CX} = \tfrac56$  , and  $\tfrac{DY}{EY} = \tfrac92$ . The ratio  $\tfrac{EZ}{FZ}$  can then be written as  $\tfrac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
[asy]
size(200);
defaultpen(linewidth(0.6));
real r = 3/11, s = 0.52, l = 33, d=5.5;
pair A = (l/2,l*sqrt(3)/2), B = origin, C = (l,0), D = (3*l/2+d,l*sqrt(3)/2), E = (l+d,0), F = (2*l+d,0);
pair W = r*B+(1-r)*A, X = s*C+(1-s)*A, Y = extension(W,X,D,E), Z = extension(W,X,E,F);
draw(E--D--F--B--A--C^^W--Z);
dot(" $A$ ",A,N);
dot(" $B$ ",B,S);
dot(" $C$ ",C,S);
dot(" $D$ ",D,N);
dot(" $E$ ",E,S);
dot(" $F$ ",F,S);
dot(" $W$ ",W,0.6*NW);
dot(" $X$ ",X,0.8*NE);
dot(" $Y$ ",Y,dir(100));
dot(" $Z$ ",Z,dir(70));
[/asy]


In the following figure, the bigger wheel has circumference  $12$ m and the inscribed wheel has circumference  $8 $ m.  $P_{1}$  denotes a point on the bigger wheel and  $P_{2}$  denotes a point on the smaller wheel. Initially  $P_{1}$  and  $P_{2}$  coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?


Let  $p(x)$  be the monic cubic polynomial with roots  $\sin^2(1^{\circ})$ ,  $\sin^2(3^{\circ})$ , and  $\sin^2(9^{\circ})$ . Suppose that  $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$ , where  $0 <a,b \le 90$  and  $a,b,n$  are positive integers.  What is  $a+b+n$ ?

*Proposed by Andrew Yuan*


Let  $A,B,C$  be points in that order along a line, such that  $AB=20$  and  $BC=18$ . Let  $\omega$  be a circle of nonzero radius centered at  $B$ , and let  $\ell_1$  and  $\ell_2$  be tangents to  $\omega$  through  $A$  and  $C$ , respectively. Let  $K$  be the intersection of  $\ell_1$  and  $\ell_2$ . Let  $X$  lie on segment  $\overline{KA}$  and  $Y$  lie on segment  $\overline{KC}$  such that  $XY\|BC$  and  $XY$  is tangent to  $\omega$ . What is the largest possible integer length for  $XY$ ?


Let  $n$  be an integer.  If the tens digit of  $n^2$  is 7, what is the units digit of  $n^2$ ?


Rob is helping to build the set for a school play.  For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo.  He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle.  Three more lengths of bamboo are then cut to connect the other ends of the first three rods.  Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece.  These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are  $60$ ,  $20$ , and  $15$  respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.


There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly  $1200$ . What is the maximum possible number of cups in the kitchen?


Let all possible  $2023$ -degree real polynomials:  $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$ ,
where  $P(0)+P(1)=0$ , and the polynomial has 2023 real roots  $r_1, r_2,\cdots r_{2023}$  [not necessarily distinct] so that  $0\leq r_1,r_2,\cdots r_{2023}\leq1$ . What is the maximum value of  $r_1 \cdot r_2 \cdots r_{2023}?$


Solution: \boxed{x = 0}


Triangle  $ABC$  is equilateral.  Points  $D$  and  $E$  are the midpoints of segments  $BC$  and  $AC$  respectively.   $F$  is the point on segment  $AB$  such that  $2BF=AF$ .  Let  $P$  denote the intersection of  $AD$  and  $EF$ ,  The value of  $EP/PF$  can be expressed as  $m/n$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .

[asy]
import olympiad;
size(150);
defaultpen(linewidth(0.7) + fontsize(11pt));
pair A = origin, B = (1,0), C = dir(60), D = (B+C)/2, E = C/2, F = 2*B/3, P = intersectionpoint(A--D,E--F);
draw(A--B--C--A--D^^E--F);
label(" $A$ ",A,SW);
label(" $B$ ",B,SE);
label(" $C$ ",C,N);
label(" $D$ ",D,NE);
label(" $E$ ",E,NW);
label(" $F$ ",F,S);
label(" $P$ ",P,N);
[/asy]


Aurick throws  $2$  fair  $6$ -sided dice labeled with the integers from  $1$  through  $6$ . What is the probability that the sum of the rolls is a multiple of  $3$ ?


Jonathan has a magical coin machine which takes coins in amounts of  $7, 8$ , and  $9$ . If he puts in  $7$  coins, he gets  $3$  coins back; if he puts in  $8$ , he gets  $11$  back; and if he puts in  $9$ , he gets  $4$  back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with  $15$  coins, what is the minimum number of entries he can make to end up with  $4$  coins?


Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks.  Considering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like?


A prime number $ q $ is called***'Kowai'***number if  $ q = p^2 + 10$  where  $q$ ,   $p$ ,  $p^2-2$ ,   $p^2-8$ ,   $p^3+6$  are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers.


A gardener plants three maple trees, four oak trees, and five birch trees in a row.  He plants them in random order, each arrangement being equally likely.  Let  $\frac{m}{n}$  in lowest terms be the probability that no two birch trees are next to one another.  Find  $m + n$ .


Let  $f$  be a differentiable real-valued function defined on the positive real numbers. The tangent lines to the graph of  $f$  always meet the  $y$ -axis 1 unit lower than where they meet the function. If  $f(1)=0$ , what is  $f(2)$ ?


Suppose that  $a$  and  $ b$  are distinct positive integers satisfying  $20a + 17b = p$  and   $17a + 20b = q$  for certain primes  $p$  and  $ q$ . Determine the minimum value of  $p + q$ .


Let  $ABCD$  be a square with side length  $100$ . Denote by  $M$  the midpoint of  $AB$ . Point  $P$  is selected inside the square so that  $MP = 50$  and  $PC = 100$ . Compute  $AP^2$ .

*Based on a proposal by Amogh Gaitonde*


The familiar  $3$ -dimensional cube has  $6$   $2$ -dimensional faces,  $12$   $1$ -dimensional edges, and  $8$   $0$ -dimensional vertices. Find the number of  $9$ -dimensional sub-subfaces in a  $12$ -dimensional cube.


Starting with a positive integer  $M$  written on the board , Alice plays the following game: in each move, if  $x$  is the number on the board, she replaces it with  $3x+2$ .Similarly, starting with a positive integer  $N$  written on the board, Bob plays the following game: in each move, if  $x$  is the number on the board, he replaces it with  $2x+27$ .Given that Alice and Bob reach the same number after  playing  $4$  moves each, find the smallest value of  $M+N$


There are  $n$  cards such that for each  $i=1,2, \cdots n$ , there are exactly one card labeled  $i$ . Initially the cards are piled with increasing order from top to bottom. There are two operations:

- $A$  : One can take the top card of the pile and move it to the bottom;
- $B$  : One can remove the top card from the pile.

The operation  $ABBABBABBABB \cdots $  is repeated until only one card gets left. Let  $L(n)$  be the labeled number on the final pile. Find all integers  $k$  such that  $L(3k)=k$ .

Solution: \boxed{ k = 1 }


Let  $\Gamma$  be the maximum possible value of  $a+3b+9c$  among all triples  $(a,b,c)$  of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If  $\Gamma = \frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers, then find  $p+q$ .


For any positive integer  $x$ , let  $f(x)=x^x$ . Suppose that  $n$  is a positive integer such that there exists a positive integer  $m$  with  $m  \neq 1$  such that  $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of  $n$ .

*Proposed by Luke Robitaille*


A mason has bricks with dimensions  $2\times5\times8$  and other bricks with dimensions  $2\times3\times7$ . She also has a box with dimensions  $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.


Find the smallest positive integer that cannot be expressed in the form  $\frac{2^a - 2^b}{2^c - 2^d}$ , where  $a$ ,  $ b$ ,  $c$ ,  $d$  are non-negative integers.


A number  $ x$  is uniformly chosen on the interval  $ [0,1]$ , and  $ y$  is uniformly randomly chosen on  $ [\minus{}1,1]$ . Find the probability that  $ x>y$ .


The sequence  $a_1, a_2, \ldots$  is geometric with  $a_1=a$  and common ratio  $r$ , where  $a$  and  $r$  are positive integers.  Given that  $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$  find the number of possible ordered pairs  $(a,r)$ .


Let  $O_1$  and  $O_2$  be concentric circles with radii 4 and 6, respectively.  A chord  $AB$  is drawn in  $O_1$  with length  $2$ .  Extend  $AB$  to intersect  $O_2$  in points  $C$  and  $D$ .  Find  $CD$ .


The rodent control task force went into the woods one day and caught  $200$  rabbits and  $18$  squirrels. The next day they went into the woods and caught  $3$  fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught  $3$  fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?


fantasticbobob is proctoring a room for the SiSiEyMB with  $841$  seats arranged in  $29$  rows and  $29$  columns. The contestants sit down, take part  $1$  of the contest, go outside for a break, and come back to take part  $2$  of the contest. fantasticbobob sits among the contestants during part  $1$ , also goes outside during break, but when he returns, he finds that his seat has been taken. Furthermore, each of the  $840$  contestants now sit in a chair horizontally or vertically adjacent to their original chair. How many seats could fantasticbobob have started in?

*2019 CCA Math Bonanza Team Round #8*


Consider a  $2 \times 2$  grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?


Shirley has a magical machine.  If she inputs a positive even integer  $n$ , the machine will output  $n/2$ , but if she inputs a positive odd integer  $m$ , the machine will output  $m+3$ . The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most  $100$ . What number should she input?


Let  $A_n$  denote the answer to the  $n$ th problem on this contest ( $n=1,\dots,30$ ); in particular, the answer to this problem is  $A_1$ . Compute  $2A_1(A_1+A_2+\dots+A_{30})$ .

*Proposed by Yang Liu*


Find all positive integers  $n$  such that  $n+ \left[ \frac{n}{6} \right] \ne \left[ \frac{n}{2} \right] + \left[ \frac{2n}{3} \right]$

Solution: \boxed{ n = 6k + 1 }


In a triangle  $ABC$ , let  $H, I$  and  $O$  be the orthocentre, incentre and circumcentre, respectively. If the points  $B, H, I, C$  lie on a circle, what is the magnitude of  $\angle BOC$  in degrees?


Find all positive integers  $a,b,c$  satisfying  $(a,b)=(b,c)=(c,a)=1$  and \[ \begin{cases} a^2+b\mid b^2+c b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of  $a^2+b$  are congruent to  $1$  modulo  $7$

Solution: \boxed{ (1, 1, 1) }


The work team was working at a rate fast enough to process  $1250$  items in ten hours. But after working for six hours, the team was given an additional  $150$  items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?


Let  $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$  . Find  $[S]$ .

You may use the fact that  $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$  for all integers  $n \ge 1$ .


Each of the  $10$  dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?


The squadron of  $10$  powerful destroyers and  $20$  small boats is about to attack the island. All ships are positioned on the straight line, and are equally spaced.

Two torpedo boats with  $10$  torpedoes each want to protect the island. However, the first torpedo boat can shoot only  $10$  successive boats, whereas the second one can shoot  $10$  targets which are next by one. Note that they have to shoot at the same moment, so that some targets may be hit by both torpedoes.

What is the biggest number of destroyers that can avoid the torpedoes no matter which targets the torpedo boats choose?

*Proposed by Bohdan Rublyov*


Let G, O, D, I, and T be digits that satisfy the following equation:

\begin{tabular}{ccccc}
&G&O&G&O
+&D&I&D&I
\hline
G&O&D&O&T
\end{tabular}

(Note that G and D cannot be  $0$ , and that the five variables are not necessarily different.)

Compute the value of GODOT.


What is the largest possible number of edges in a graph on  $2n$  nodes, if there exists exactly one way to split its nodes into  $n$  pairs so that the nodes from each pair are connected by an edge?

*Proposed by Anton Trygub*

Solution: \boxed{ n^2 }


Let  $G=10^{10^{100}}$  (a.k.a. a googolplex). Then \[\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\] can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Determine the sum of the digits of  $m+n$ .

*Proposed by Yannick Yao*


Find the product of all possible real values for  $k$  such that the system of equations  $$ x^2+y^2= 80 $$    $$ x^2+y^2= k+2x-8y $$  has exactly one real solution  $(x,y)$ .

*Proposed by Nathan Xiong*


Given is a acute angled triangle  $ABC$ . The lengths of the altitudes from  $A, B$  and  $C$  are successively  $h_A, h_B$  and  $h_C$ . Inside the triangle is a point  $P$ . The distance from  $P$  to  $BC$  is  $1/3 h_A$  and the distance from  $P$  to  $AC$  is  $1/4 h_B$ . Express the distance from  $P$  to  $AB$  in terms of  $h_C$ .

Solution: \boxed{ \frac{5}{12} h_C }


Let  $T$  be a triangle whose vertices have integer coordinates, such that each side of  $T$  contains exactly  $m$  points with integer coordinates. If the area of  $T$  is less than  $2020$ , determine the largest possible value of  $m$ .


Let  $n > 1$  be an integer. In a circular arrangement of  $n$  lamps  $L_0, \ldots, L_{n-1},$  each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps,  $Step_0, Step_1, \ldots .$   If  $L_{j-1}$  ( $j$  is taken mod  $n$ ) is ON then  $Step_j$  changes the state of  $L_j$  (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If  $L_{j-1}$  is OFF then  $Step_j$  does not change anything at all. Show that:

(i) There is a positive integer  $M(n)$  such that after  $M(n)$  steps all lamps are ON again,

(ii) If  $n$  has the form  $2^k$  then all the lamps are ON after  $n^2-1$  steps,

(iii) If  $n$  has the form  $2^k + 1$  then all lamps are ON after  $n^2 - n + 1$  steps.

Solution: \boxed{n^2 - n + 1}


Positive real numbers are arranged in the form:  $ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$  $ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$  $ 4 \ \ \ 8 \ \ \ 13 ...$  $ 7 \ \ \ 12 ...$  $ 11 ...$ Find the number of the line and column where the number 2002 stays.

Solution: \boxed{(15, 49)}


Sequences  $(x_n)_{n\ge1}$ ,  $(y_n)_{n\ge1}$  satisfy the relations  $x_n=4x_{n-1}+3y_{n-1}$  and  $y_n=2x_{n-1}+3y_{n-1}$  for  $n\ge1$ . If  $x_1=y_1=5$  find  $x_n$  and  $y_n$ .

Calculate  $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$ .

Solution: \boxed{ \frac{3}{2} }


$6$  cats can eat  $6$  fish in  $1$  day, and  $c$  cats can eat  $91$  fish in  $d$  days. Given that  $c$  and  $d$  are both whole numbers, and the number of cats,  $c$ , is more than  $1$  but less than  $10$ , find  $c + d$ .


$a, b, c, d$  are integers with  $ad \ne bc$ . Show that  $1/((ax+b)(cx+d))$  can be written in the form  $ r/(ax+b) + s/(cx+d)$ . Find the sum  $1/1\cdot  4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$ .


The venusian prophet Zabruberson sent to his pupils a  $ 10000$ -letter word, each letter being  $ A$  or  $ E$ : the *Zabrubic word*. Their pupils consider then that for  $ 1 \leq k \leq 10000$ , each word comprised of  $ k$  consecutive letters of the Zabrubic word is a *prophetic word* of length  $ k$ . It is known that there are at most  $ 7$  prophetic words of lenght  $ 3$ . Find the maximum number of prophetic words of length  $ 10$ .


It is known that  $a^{2005} + b^{2005}$  can be expressed as the polynomial of  $a + b$  and  $ab$ . Find the coefficients' sum of this polynomial.


Let $C=\{ (i,j)|i,j$  integers such that  $0\leq i,j\leq 24\}$ How many squares can be formed in the plane all of whose vertices are in  $C$  and whose sides are parallel to the  $X-$ axis and  $Y-$  axis?


When  $126$  is added to its reversal,  $621,$  the sum is  $126 + 621 = 747.$  Find the greatest integer which when added to its reversal yields  $1211.$


Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started?


There are two prime numbers  $p$  so that  $5p$  can be expressed in the form  $\left\lfloor \dfrac{n^2}{5}\right\rfloor$  for some positive integer  $n.$  What is the sum of these two prime numbers?


Evaluate

\[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\]

where  $ [x] $  is the integer equal to  $ x $  or less than  $ x $ .


For which  $k$  the number  $N = 101 ... 0101$  with  $k$  ones is a prime?

Solution: \boxed{ k = 2 }


Let  $a,b$  be the smaller sides of a right triangle. Let  $c$  be the hypothenuse and  $h$  be the altitude from the right angle. Fint the maximal value of  $\frac{c+h}{a+b}$ .


For each integer  $n\geq 1$ , let  $S_n$  be the set of integers  $k > n$  such that  $k$  divides  $30n-1$ .  How many elements of the set \[\mathcal{S} = \bigcup_{i\geq 1}S_i = S_1\cup S_2\cup S_3\cup\ldots\] are less than  $2016$ ?


Let  $m$  and  $n$  be positive integers. Fuming Zeng gives James a rectangle, such that  $m-1$  lines are drawn parallel to one pair of sides and  $n-1$  lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an  $m\times n$  grid of smaller rectangles. Fuming Zeng chooses  $m+n-1$  of the  $mn$  smaller rectangles and then tells James the area of each of the smaller rectangles. Of the  $\dbinom{mn}{m+n-1}$  possible combinations of rectangles and their areas Fuming Zeng could have given, let  $C_{m,n}$  be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than  $1000A$ .

[i]Proposed by James Lin


Let  $(a_{n})_{n\geq 1}$  be a sequence defined by  $a_{n}=2^{n}+49$ . Find all values of  $n$  such that  $a_{n}=pg, a_{n+1}=rs$ , where  $p,q,r,s$  are prime numbers with  $p<q, r<s$  and  $q-p=s-r$ .

Solution: \boxed{ n = 7 }


Mad scientist Kyouma writes  $N$  positive integers on a board. Each second, he chooses two numbers  $x, y$  written on the board with  $x > y$ , and writes the number  $x^2-y^2$  on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of N.

Solution: \boxed{ N = 252 }


A permutation of  $\{1, 2, \dots, 7\}$  is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation  $(3, 4, 2, 1, 6, 5, 7)$  can be partitioned correctly into the blocks  $[3, 4, 2, 1]$  and  $[6, 5, 7]$ , since when these blocks are sorted, the permutation becomes  $(1, 2, 3, 4, 5, 6, 7)$ . Find the expected value of the maximum number of blocks into which the permutation can be partioned correctly.


Suppose  $\alpha,\beta,\gamma\in\{-2,3\}$  are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note:  $\mathbb{R}_{\ge0}$  is the set of nonnegative real numbers). What is the sum of the possible values of  $M$ ?


Let  $n$  be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer.  Compute the remainder when  $3^n$  is divided by  $1000$ .


Given that  $a,b,c$  are integers with  $abc = 60$ , and that complex number  $\omega \neq 1$  satisfies  $\omega^3=1$ , find the minimum possible value of  $|a + b\omega + c\omega^2|$ .


The chord  $[CD]$  is parallel to the diameter  $[AB]$  of a circle with center  $O$ . The tangent line at  $A$  meet  $BC$  and  $BD$  at  $E$  and  $F$ . If  $|AB|=10$ , calculate  $|AE|\cdot |AF|$ .


Define a \emph{crossword puzzle} to be a  $15 \times 15$  grid of squares, each of which is either black or white. In a crossword puzzle, define a \emph{word} to be a sequence of one or more consecutive white squares in a row or column such that the squares immediately before and after the sequence both are either black or nonexistent. (The latter case would occur if an end of a word coincides with an end of a row or column of the grid.)  A crossword puzzle is \emph{tasty} if every word consists of an even number of white squares. Compute the sum of all nonnegative integers  $n$  such that there exists a tasty crossword puzzle with exactly  $n$  white squares.

*Proposed by Luke Robitaille*


If  $a, b, c \ge 4$  are integers, not all equal, and  $4abc = (a+3)(b+3)(c+3)$  then what is the value of  $a+b+c$  ?


Determine the number of  $8$ -tuples  $(\epsilon_1, \epsilon_2,...,\epsilon_8)$  such that  $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$  and  $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$  is a multiple of  $3$ .


Consider the set of all ordered  $6$ -tuples of nonnegative integers  $(a,b,c,d,e,f)$  such that \[a+2b+6c+30d+210e+2310f=2^{15}.\] In the tuple with the property that  $a+b+c+d+e+f$  is minimized, what is the value of  $c$ ?

*2021 CCA Math Bonanza Tiebreaker Round #1*


An ant is walking on the edges of an icosahedron of side length  $1$ . Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.

[center]<see attached>[/center]


Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.

Solution: \boxed{ ABCD }


Higher Secondary P4

If the fraction  $\dfrac{a}{b}$  is greater than  $\dfrac{31}{17}$  in the least amount while  $b<17$ , find  $\dfrac{a}{b}$ .


What is the biggest shadow that a cube of side length  $1$  can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

Solution: \boxed{ \sqrt{3} }


Numbers  $1,\frac12,\frac13,\ldots,\frac1{2001}$  are written on a blackboard. A student erases two numbers  $x,y$  and writes down the number  $x+y+xy$  instead. Determine the number that will be written on the board after  $2000$  such operations.


$A,B,C$  and  $D$  are points on the parabola  $y = x^2$  such that  $AB$  and  $CD$  intersect on the  $y$ -axis. Determine the  $x$ -coordinate of  $D$  in terms of the  $x$ -coordinates of  $A,B$  and  $C$ , which are  $a, b$  and  $c$  respectively.

Solution: \boxed{\frac{ab}{c}}


Find all positive integers  $n$  for which both  $837 + n$  and  $837 - n$  are cubes of positive integers.


Find all natural numbers  $n$  such that  $n$ ,  $n^2+10$ ,  $n^2-2$ ,  $n^3+6$ , and  $n^5+36$  are all prime numbers.

Solution: \boxed{ n = 7 }


Let  $V$  be the set of all continuous functions  $f\colon [0,1]\to \mathbb{R}$ , differentiable on  $(0,1)$ , with the property that  $f(0)=0$  and  $f(1)=1$ . Determine all  $\alpha \in \mathbb{R}$  such that for every  $f\in V$ , there exists some  $\xi \in (0,1)$  such that
\[f(\xi)+\alpha = f'(\xi)\]

Solution: \boxed{\frac{1}{e - 1}}


Let  $r_1$ ,  $r_2$ ,  $\ldots$ ,  $r_{20}$  be the roots of the polynomial  $x^{20}-7x^3+1$ .  If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form  $\tfrac mn$  where  $m$  and  $n$  are positive coprime integers, find  $m+n$ .


Find the largest integer  $ n$  satisfying the following conditions:
(i)  $ n^2$  can be expressed as the difference of two consecutive cubes;
(ii)  $ 2n\plus{}79$  is a perfect square.


Find the largest positive integer  $N$  so that the number of integers in the set  $\{1,2,\dots,N\}$  which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).


We color all vertexs of a convex polygon with  $10$  vertexs by  $2$  colors: red and blue  $($ each vertex is colored by  $1$  color $).$ How many ways to color all the vertexs such that there are no  $2$  adjacent vertex that are both colored red?


Find the smallest positive real  $k$  satisfying the following condition: for any given four DIFFERENT real numbers  $a,b,c,d$ , which are not less than  $k$ , there exists a permutation  $(p,q,r,s)$  of  $(a,b,c,d)$ , such that the equation  $(x^{2}+px+q)(x^{2}+rx+s)=0$  has four different real roots.


Determine the least possible value of the natural number  $n$  such that  $n!$  ends in exactly  $1987$  zeros.

NoteNote. Here (and generally in MathLinks) natural numbers supposed to be positive.


Square  $CASH$  and regular pentagon  $MONEY$  are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?


Suppose that  $a_1 = 1$ , and that for all  $n \ge 2$ ,  $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$  Suppose furthermore that  $b_n = a_1 + a_2 + \ldots + a_n$  for all  $n$ . If  $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$  for some  $k$ , find  $k$ .

*Proposed by Andrew Wu*


Let  $m$  be a positive integer less than  $2015$ . Suppose that the remainder when  $2015$  is divided by  $m$  is  $n$ . Compute the largest possible value of  $n$ .

*Proposed by Michael Ren*


Determine all functions  $f:\mathbb R\to\mathbb R$  such that equality  $$ f(x + y + yf(x)) = f(x) + f(y) + xf(y) $$  holds for all real numbers  $x$ ,  $y$ .

Proposed by Athanasios Kontogeorgis

Solution: \boxed{f(x) \equiv x}


Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form  $\tfrac{p}{q}$  for positive integers  $p, q$  satisfying  $\gcd(p, q) = 1$ , what is  $p + q$ ?

Solution: \boxed{ 8 }


Today there are  $2^n$  species on the planet Kerbin, all of which are exactly n steps from an original species. In an evolutionary step, One species split into exactly two new species and died out in the process. There were already  $2^n-1$  species in the past, which are no longer present today can be found, but are only documented by fossils. The famous space pioneer Jebediah Kerman once suggested reducing the biodiversity of a planet by doing this to measure how closely two species are on average related, with also already extinct species should be taken into account. The degree of relationship is measured two types, of course, by how many evolutionary steps before or back you have to do at least one to get from one to the other. What is the biodiversity of the planet Kerbin?


Determine the sum of absolute values for the complex roots of  $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$


A container is shaped like a square-based pyramid where the base has side length  $23$  centimeters and the height is  $120$  centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon  $5$  centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.


Show that the number  $r(n)$  of representations of  $n$  as a sum of two squares has  $\pi$  as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

Solution: \boxed{ \pi }


Suppose the polynomial  $f(x) = x^{2014}$  is equal to  $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$  for some real numbers  $a_0,... , a_{2014}$ . Find the largest integer  $m$  such that  $2^m$  divides  $a_{2013}$ .


We only know that the password of a safe consists of  $7$  different digits. The safe will open if we enter  $7$  different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than  $7$  attempts?

*(5 points for Juniors and 4 points for Seniors)*

Solution: \boxed{ 6 }


Hello Everyone,

i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level

Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.

it will be preferred to write the source of the problem.
please , show detailed solutions , and please post some challenging Pre-Olympiad problems..
remember ,, different approaches are most welcome :)

now .. let's get our hands dirty :lol: :

let  $ f(n)$  denote the sum of the digits of  $ n$ . Now let  $ N \equal{} {4444^{4444}}$ .
Find  $ f\left( {f\left( {f\left( N \right)} \right)} \right)$ .


Find all functions  $f : \mathbb{N} \to \mathbb{N}$  satisfying the following conditions:

- For every  $n \in \mathbb{N}$ ,  $f^{(n)}(n) = n$ . (Here  $f^{(1)} = f$  and  $f^{(k)} = f^{(k-1)} \circ f$ .)
- For every  $m, n \in \mathbb{N}$ ,  $\lvert f(mn) - f(m) f(n) \rvert < 2017$ .

Solution: \boxed{ f(n) = n }


Given a square  $ABCD,$  with  $AB=1$  mark the midpoints  $M$  and  $N$  of  $AB$  and  $BC,$  respectively. A lasar beam shot from  $M$  to  $N,$  and the beam reflects of  $BC,CD,DA,$  and comes back to  $M.$  This path encloses a smaller area inside square  $ABCD.$  Find this area.


Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his  $20$  friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over  $500$  candies, what is the fewest number of candies the bag can contain?


If  $x_{1}, x_{2},\ldots ,x_{n}$  are positive real numbers with  $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$ , ﬁnd the minimum value of  $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$ .

Solution: \boxed{ \frac{1}{n(n-1)} }


Determine all positive integers  $n$  such that  $n$  divides  $\phi(n)^{d(n)}+1$  but  $d(n)^5$  does not divide  $n^{\phi(n)}-1$ .

Solution: \boxed{ n = 2 }


Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score.

Let  $p$  be the probability that his score is a multiple of 13. There exists relatively prime positive integers  $a$  and  $b,$  with  $b$  as small as possible, such that  $|p - a/b| < 10^{-10}.$  What is  $a/b?$ *Proposed by Dilhan Salgado, Daniel Li*


Let  $I$  be the set of points  $(x,y)$  in the Cartesian plane such that   $$ x>\left(\frac{y^4}{9}+2015\right)^{1/4} $$   Let  $f(r)$  denote the area of the intersection of  $I$  and the disk  $x^2+y^2\le r^2$  of radius  $r>0$  centered at the origin  $(0,0)$ . Determine the minimum possible real number  $L$  such that  $f(r)<Lr^2$  for all  $r>0$ .


There is number  $N$  on the board. Every minute Ivan makes next operation: takes any number  $a$  written on the board,
 erases it, then writes all divisors of  $a$  except  $a$ ( Can be same numbers on the board). After some time on the board there are  $N^2$  numbers.
For which  $N$  is it possible?

Solution: \boxed{ N = 1 }


Let  $F$  be the family of all sets of positive integers with  $2010$  elements that satisfy the following condition:
The difference between any two of its elements is never the same as the difference of any other two of its elements. Let  $f$  be a function defined from  $F$  to the positive integers such that  $f(K)$  is the biggest element of  $K \in F$ . Determine the least value of  $f(K)$ .


Let  $ABC$  be a triangle with  $AB=5$ ,  $BC=6$ ,  $CA=7$ . Let  $D$  be a point on ray  $AB$  beyond  $B$  such that  $BD=7$ ,  $E$  be a point on ray  $BC$  beyond  $C$  such that  $CE=5$ , and  $F$  be a point on ray  $CA$  beyond  $A$  such that  $AF=6$ . Compute the area of the circumcircle of  $DEF$ .

*Proposed by James Lin.*

Solution: \boxed{\frac{251}{3} \pi}


You drop a 7 cm long piece of mechanical pencil lead on the ﬂoor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is  $\frac{m}n$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m + n$ .

*Proposed by Aaron Lin*

Solution: \boxed{ 1007 }


Let  $n(n\geq2)$  be a  natural number and  $a_1,a_2,...,a_n$  natural positive real numbers. Determine the least possible value of the expression    $$ E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n} $$

Solution: \boxed{ 4^{n+1} }


Show that we cannot form more than  $4096$  binary sequences of length  $24$  so that any two differ in at least  $8$  positions.


Determine all non-constant monic polynomials  $f(x)$  with integer coefficients for which there exists a natural number  $M$  such that for all  $n \geq M$ ,  $f(n)$  divides  $f(2^n) - 2^{f(n)}$
*Proposed by Anant Mudgal*

Solution: \boxed{ f(x) = x }


Find all pairs  $\left(m,n\right)$  of positive integers, with  $m,n\geq2$ , such that  $a^n-1$  is divisible by  $m$  for each  $a\in \left\{1,2,3,\ldots,n\right\}$ .

Solution: \boxed{ (m, n) = (p, p-1) }


Numbers  $a, b$  and  $c$  are positive integers and  $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$  Show that \[\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.\]

Solution: \boxed{ \frac{41}{42} }


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

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


The plane is partitioned into congruent regular hexagons. Of these hexagons, some  $1998$  are marked. Show that one can select  $666$  of the marked hexagons in such a way that no two of them share a vertex.


Triangle  $ABC$  has  $\angle{A}=90^{\circ}$ ,  $AB=2$ , and  $AC=4$ . Circle  $\omega_1$  has center  $C$  and radius  $CA$ , while circle  $\omega_2$  has center  $B$  and radius  $BA$ . The two circles intersect at  $E$ , different from point  $A$ . Point  $M$  is on  $\omega_2$  and in the interior of  $ABC$ , such that  $BM$  is parallel to  $EC$ . Suppose  $EM$  intersects  $\omega_1$  at point  $K$  and  $AM$  intersects  $\omega_1$  at point  $Z$ . What is the area of quadrilateral  $ZEBK$ ?


In square  $ABCD$  with side length  $2$ , let  $M$  be the midpoint of  $AB$ . Let  $N$  be a point on  $AD$  such that  $AN = 2ND$ . Let point  $P$  be the intersection of segment  $MN$  and diagonal  $AC$ . Find the area of triangle  $BPM$ .

*Proposed by Jacob Xu*


In a rectangle  $ABCD$ , two segments  $EG$  and  $FH$  divide it into four smaller rectangles.  $BH$  intersects  $EG$  at  $X$ ,  $CX$  intersects  $HF$  and  $Y$ ,  $DY$  intersects  $EG$  at  $Z$ . Given that  $AH=4$ ,  $HD=6$ ,  $AE=4$ , and  $EB=5$ , find the area of quadrilateral  $HXYZ$ .


Suppose  $(a,b)$  is an ordered pair of integers such that the three numbers  $a$ ,  $b$ , and  $ab$  form an arithmetic progression, in that order. Find the sum of all possible values of  $a$ .

*Proposed by Nathan Xiong*


Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.

Solution: \boxed{ 1988 }


Let  $ABC$  be an equilateral triangle with side length  $2$ , and let  $M$  be the midpoint of  $\overline{BC}$ .  Points  $X$  and  $Y$  are placed on  $AB$  and  $AC$  respectively such that  $\triangle XMY$  is an isosceles right triangle with a right angle at  $M$ .  What is the length of  $\overline{XY}$ ?

Solution: \boxed{3 - \sqrt{3}}


Triangle  $ABC$  satisfies  $AB=104$ ,  $BC=112$ , and  $CA=120$ .  Let  $\omega$  and  $\omega_A$  denote the incircle and  $A$ -excircle of  $\triangle ABC$ , respectively.  There exists a unique circle  $\Omega$  passing through  $A$  which is internally tangent to  $\omega$  and externally tangent to  $\omega_A$ .  Compute the radius of  $\Omega$ .


In parallelogram  $ABCD,$  let  $O$  be the intersection of diagonals  $\overline{AC}$  and  $\overline{BD}.$   Angles  $CAB$  and  $DBC$  are each twice as large as angle  $DBA,$  and angle  $ACB$  is  $r$  times as large as angle  $AOB.$   Find the greatest integer that does not exceed  $1000r.$


Let  $\mathcal{A}$  be the set of finite sequences of positive integers  $a_1,a_2,\dots,a_k$  such that  $|a_n-a_{n-1}|=a_{n-2}$  for all  $3\leqslant n\leqslant k$ . If  $a_1=a_2=1$ , and  $k=18$ , determine the number of elements of  $\mathcal{A}$ .


Find the least number of elements of a finite set  $A$  such that there exists a function  $f : \left\{1,2,3,\ldots \right\}\rightarrow A$  with the property: if  $i$  and  $j$  are positive integers and  $i-j$  is a prime number, then  $f(i)$  and  $f(j)$  are distinct elements of  $A$ .


Given are sheets and the numbers  $00, 01, \ldots, 99$  are written on them. We must put them in boxes  $000, 001, \ldots, 999$  so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?


The numbers  $1,2,\ldots,49,50$  are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers  $a,b,c$  from the board, replaces them by their sum  $a+b+c$  and writes  $(a+b)(b+c)(c+a)$  to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all  $24$  numbers written in the notebook. Let  $A$  and  $B$  be the maximum and the minimum possible sums that Ann san obtain.
Find the value of  $\frac{A}{B}$ .

*(I. Voronovich)*


Find the smallest natural number nonzero n so that it exists in real numbers  $x_1, x_2,..., x_n$  which simultaneously check  the conditions:
1)  $x_i \in [1/2 , 2]$  ,  $i = 1, 2,... , n$
2)   $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3)  $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$


Let the sequence  $\{x_n\}$  be defined by  $x_1 \in \{5, 7\}$  and, for  $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$ . For example, the possible values of  $x_3$  are  $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and  $7^{7^7}$ . Determine the sum of all possible values for the last two digits of  $x_{2012}$ .


Find all sequences of integer  $x_1,x_2,..,x_n,...$  such that  $ij$  divides  $x_i+x_j$  for any distinct positive integer  $i$ ,  $j$ .

Solution: \boxed{ x_i = 0 }


Juan chooses a five-digit positive integer.  Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is  $52,713$ . What can the sum of the five digits of the original number be?


A permutation  $(a_1, a_2, a_3, \dots, a_{100})$  of  $(1, 2, 3, \dots, 100)$  is chosen at random. Denote by  $p$  the probability that  $a_{2i} > a_{2i - 1}$  for all  $i \in \{1, 2, 3, \dots, 50\}$ . Compute the number of ordered pairs of positive integers  $(a, b)$  satisfying  $\textstyle\frac{1}{a^b} = p$ .

*Proposed by Aaron Lin*


Given real numbers  $b_0,b_1,\ldots, b_{2019}$  with  $b_{2019}\neq 0$ , let  $z_1,z_2,\ldots, z_{2019}$  be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let  $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$  be the average of the distances from  $z_1,z_2,\ldots, z_{2019}$  to the origin.  Determine the largest constant  $M$  such that  $\mu\geq M$  for all choices of  $b_0,b_1,\ldots, b_{2019}$  that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]


Find all sets  $X$  consisting of at least two positive integers such that for every two elements  $m,n\in X$ , where  $n>m$ , there exists an element  $k\in X$  such that  $n=mk^2$ .

Solution: \boxed{ X = \{a, a^3\} }


Suppose  $f:\mathbb{R} \to \mathbb{R}$  is a function given by  $$ f(x) =\begin{cases} 1 & \mbox{if} \ x=1  e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases} $$  (a) Find  $f'(1)$ (b) Evaluate  $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$ .

Solution: \boxed{ -50500 }


Find all positive integers  $n$  such that the number  $$ n^6 + 5n^3 + 4n + 116 $$  is the product of two or more consecutive numbers.

Solution: \boxed{ n = 3 }


Let  $S$  be the set of positive integer divisors of  $20^9.$  Three numbers are chosen independently and at random from the set  $S$  and labeled  $a_1,a_2,$  and  $a_3$  in the order they are chosen. The probability that both  $a_1$  divides  $a_2$  and  $a_2$  divides  $a_3$  is  $\frac mn,$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m.$


You are trying to maximize a function of the form  $f(x, y, z) = ax + by + cz$ , where  $a$ ,  $b$ , and  $c$  are constants. You know that  $f(3, 1, 1) > f(2, 1, 1)$ ,  $f(2, 2, 3) > f(2, 3, 4)$ , and  $f(3, 3, 4) > f(3, 3, 3)$ . For  $-5 \le x,y,z \le 5$ , what value of  $(x,y,z)$  maximizes the value of  $f(x, y, z)$ ? Give your answer as an ordered triple.

Solution: \boxed{(5, -5, 5)}


Suppose that  $x$  and  $y$  are real numbers that satisfy the system of equations $2^x-2^y=1$  $4^x-4^y=\frac{5}{3}$ Determine  $x-y$


Let  $A$  be the area of the locus of points  $z$  in the complex plane that satisfy  $|z+12+9i| \leq 15$ .  Compute  $\lfloor A\rfloor$ .

Solution: \boxed{ 706 }


Find the largest integer  $n$  for which  $2^n$  divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \]*Proposed by Evan Chen*


Find all pairs of natural numbers  $(k, m)$  such that for any natural  $n{}$  the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by  $k!{}$ .

*Proposed by P. Kozhevnikov*

Solution: \boxed{ (k, m) }


Polynomial  $ P(t)$  is such that for all real  $ x$ ,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?

Solution: \boxed{ 0 }


Find the smallest positive constant  $c$  satisfying: For any simple graph  $G=G(V,E)$ , if  $|E|\geq c|V|$ , then  $G$  contains  $2$  cycles with no common vertex, and one of them contains a chord.

Note: The cycle of graph  $G(V,E)$  is a set of distinct vertices  ${v_1,v_2...,v_n}\subseteq V$ ,  $v_iv_{i+1}\in E$  for all  $1\leq i\leq n$   $(n\geq 3, v_{n+1}=v_1)$ ; a cycle containing a chord is the cycle  ${v_1,v_2...,v_n}$ , such that there exist  $i,j, 1< i-j< n-1$ , satisfying  $v_iv_j\in E$ .


How many solutions of the equation  $\tan x = \tan \tan x$  are on the interval  $0 \le x \le \tan^{-1} 942$ ?  (Here  $\tan^{-1}$  means the inverse tangent function, sometimes written  $\arctan$ .)


There are  $n+1$  containers arranged in a circle. One container has  $n$  stones, the others are empty. A move is to choose two containers  $A$  and  $B$ , take a stone from  $A$  and put it in one of the containers adjacent to  $B$ , and to take a stone from  $B$  and put it in one of the containers adjacent to  $A$ . We can take  $A = B$ . For which  $n$  is it possible by series of moves to end up with one stone in each container except that which originally held  $n$  stones.

Solution: \boxed{ n }


Find the least positive integer which is a multiple of  $13$  and all its digits are the same.

*(Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)*


Let $f(x)=\frac{ax+b}{cx+d}$  $F_n(x)=f(f(f...f(x)...))$  (with  $n\ f's$ )

Suppose that  $f(0) \not =0$ ,  $f(f(0)) \not = 0$ , and for some  $n$  we have  $F_n(0)=0$ ,
show that  $F_n(x)=x$  (for any valid x).

Solution: \boxed{ F_n(x) = x }


Let  $n$  be an integer. We consider  $s (n)$ , the sum of the  $2001$  powers of  $n$  with the exponents  $0$  to  $2000$ . So  $s (n) = \sum_{k=0}^{2000}n ^k$  . What is the unit digit of  $s (n)$  in the decimal system?


Find the sum of all the real values of x satisfying  $(x+\frac{1}{x}-17)^2$   $= x + \frac{1}{x} + 17.$


A Yule log is shaped like a right cylinder with height  $10$  and diameter  $5$ . Freya cuts it parallel to its bases into  $9$  right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by  $a\pi$ . Compute  $a$ .


A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?


We say an integer  $n$  is naoish if  $n \geq 90$  and the second-to-last digit of  $n$  (in decimal notation) is equal to  $9$ . For example,  $10798$ ,  $1999$  and  $90$  are naoish, whereas  $9900$ ,  $2009$  and  $9$  are not. Nino expresses 2020 as a sum:
    \[
    2020=n_{1}+n_{2}+\ldots+n_{k}
    \]
    where each of the  $n_{j}$  is naoish.

    What is the smallest positive number  $k$  for which Nino can do this?


Let  $n$  be a positive integer. We are given a  $3n \times 3n$  board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a  $2 \times 2$  square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all  $n$  for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.

Proposed by *Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina*

Solution: \boxed{ n }


$P(x)$  is a polynomial with real coefficients such that  $P(a_1) = 0, P(a_{i+1}) = a_i$  ( $i = 1, 2,\ldots$ ) where  $\{a_i\}_{i=1,2,\ldots}$  is an infinite sequence of distinct natural numbers. Determine the possible values of degree of  $P(x)$ .

Solution: \boxed{ \deg P = 1 }


Square  $ABCD$  is divided into four rectangles by  $EF$  and  $GH$ .  $EF$  is parallel to  $AB$  and  $GH$  parallel to  $BC$ .   $\angle BAF = 18^\circ$ .  $EF$  and  $GH$  meet at point  $P$ . The area of rectangle  $PFCH$  is twice that of rectangle  $AGPE$ . Given that the value of  $\angle FAH$  in degrees is  $x$ , find the nearest integer to  $x$ .
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55.
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $C$ ",D2(C),SE);
label(" $D$ ",D2(D),NE);
label(" $E$ ",D2(E),plain.N);
label(" $F$ ",D2(F),S);
label(" $G$ ",D2(G),W);
label(" $H$ ",D2(H),plain.E);
label(" $P$ ",D2(P),SE);
[/asy]


In equality  $$ 1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023 $$  Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?


Let  $\{\epsilon_n\}^\infty_{n=1}$  be a sequence of positive reals with  $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$ . Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right)  \]

Solution: \boxed{ -1 }


Find the functions  $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$  such that

a)  $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$  for all integers  $x,y,z$ ;

b)  $f(x+1,x)=2$  for all integers  $x$ .

Solution: \boxed{ f(x,y) = 2^{x-y} }


Gabriela found an encyclopedia with  $2023$  pages, numbered from  $1$  to  $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?


Let the medians of the triangle  $ABC$  meet at  $G$ . Let  $D$  and  $E$  be different points on the line  $BC$  such that  $DC=CE=AB$ , and let  $P$  and  $Q$  be points on the segments  $BD$  and  $BE$ , respectively, such that  $2BP=PD$  and  $2BQ=QE$ . Determine  $\angle PGQ$ .


In an acute triangle  $ABC$ , the segment  $CD$  is an altitude and  $H$  is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle  $DHB$ , determine all possible values of  $\angle CAB$ .


Let  $\Delta ABC$  be an acute-angled triangle and let  $H$  be its orthocentre. Let  $G_1, G_2$  and  $G_3$  be the centroids of the triangles  $\Delta HBC , \Delta HCA$  and  $\Delta HAB$  respectively. If the area of  $\Delta G_1G_2G_3$  is  $7$  units, what is the area of  $\Delta ABC $ ?


Determine all integers  $ n > 3$  for which there exist  $ n$  points  $ A_{1},\cdots ,A_{n}$  in the plane, no three collinear, and real numbers  $ r_{1},\cdots ,r_{n}$  such that for  $ 1\leq i < j < k\leq n$ , the area of  $ \triangle A_{i}A_{j}A_{k}$  is  $ r_{i} \plus{} r_{j} \plus{} r_{k}$ .

Solution: \boxed{ n = 4 }


In an exotic country, the National Bank issues coins that can take any value in the interval  $[0, 1]$ . Find the smallest constant  $c > 0$  such that the following holds, no matter the situation in that country:

*Any citizen of the exotic country that has a finite number of coins, with a total value of no more than  $1000$ , can split those coins into  $100$  boxes, such that the total value inside each box is at most  $c$ .*

Solution: \boxed{ \frac{1000}{91} }


Let  $f(x)=x^2-2x+1.$  For some constant  $k, f(x+k) = x^2+2x+1$  for all real numbers  $x.$  Determine the value of  $k.$


$n$  is a fixed natural number. Find the least  $k$  such that for every set  $A$  of  $k$  natural numbers, there exists a subset of  $A$  with an even number of elements which the sum of it's members is divisible by  $n$ .

Solution: \boxed{ k = 2n }


For some complex number  $\omega$  with  $|\omega| = 2016$ , there is some real  $\lambda>1$  such that  $\omega, \omega^{2},$  and  $\lambda \omega$  form an equilateral triangle in the complex plane. Then,  $\lambda$  can be written in the form  $\tfrac{a + \sqrt{b}}{c}$ , where  $a,b,$  and  $c$  are positive integers and  $b$  is squarefree. Compute  $\sqrt{a+b+c}$ .


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.


Anders is solving a math problem, and he encounters the expression  $\sqrt{15!}$ . He attempts to simplify this radical as  $a\sqrt{b}$  where  $a$  and  $b$  are positive integers. The sum of all possible values of  $ab$  can be expressed in the form  $q\cdot 15!$  for some rational number  $q$ . Find  $q$ .

Solution: \boxed{ 4 }


let  $p$ and  $q=p+2$  be twin primes. consider the diophantine equation  $(+)$  given by
                  $n!+pq^2=(mp)^2$   $m\geq1$ ,  $n\geq1$ i. if   $m=p$ ,find the value of  $p$ .
ii. how many solution quadruple  $(p,q,m,n)$  does  $(+)$  have ?


Let  $ p,q,r$  be distinct real numbers that satisfy:  $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$  Find all possible values of  $ p\plus{}q\plus{}r$ .


$ABCD$  is a rectangle. The segment  $MA$  is perpendicular to plane  $ABC$  .  $MB= 15$  ,  $MC=24$   ,  $MD=20$ . Find the length of  $MA$  .


In how many ways can one arrange seven white and five black balls in a line in such a way that there are no two neighboring black balls?


Let’s call a positive integer *interesting* if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting?


The equation of line  $\ell_1$  is  $24x-7y = 319$  and the equation of line  $\ell_2$  is  $12x-5y = 125$ . Let  $a$  be the number of positive integer values  $n$  less than  $2023$  such that for both  $\ell_1$  and  $\ell_2$  there exists a lattice point on that line that is a distance of  $n$  from the point  $(20,23)$ . Determine  $a$ .

*Proposed by Christopher Cheng*

Solution*Solution.*  $\boxed{6}$ Note that  $(20,23)$  is the intersection of the lines  $\ell_1$  and  $\ell_2$ . Thus, we only care about lattice points on the the two lines that are an integer distance away from  $(20,23)$ . Notice that  $7$  and  $24$  are part of the Pythagorean triple  $(7,24,25)$  and  $5$  and  $12$  are part of the Pythagorean triple  $(5,12,13)$ . Thus, points on  $\ell_1$  only satisfy the conditions when  $n$  is divisible by  $25$  and points on  $\ell_2$  only satisfy the conditions when  $n$  is divisible by  $13$ . Therefore,  $a$  is just the number of positive integers less than  $2023$  that are divisible by both  $25$  and  $13$ . The LCM of  $25$  and  $13$  is  $325$ , so the answer is  $\boxed{6}$ .


Given a positive integer  $ k$ , there is a positive integer  $ n$  with the property that one can obtain the sum of the first  $ n$  positive integers by writing some  $ k$  digits to the right of  $ n$ . Find the remainder of  $ n$  when dividing at  $ 9$ .


Find all functions  $f:\mathbb{N} \rightarrow \mathbb{N}$  such that for every prime number  $p$  and natural number  $x$ ,  $$ \{ x,f(x),\cdots f^{p-1}(x) \}  $$  is a complete residue system modulo  $p$ . With  $f^{k+1}(x)=f(f^k(x))$  for every natural number  $k$  and  $f^1(x)=f(x)$ .

*Proposed by IndoMathXdZ*

Solution: \boxed{ f(x) = x + 1 }


Three students  $A, B$  and  $C$  are traveling from a location on the National Highway No. $5$  on direction to Hanoi for participating the HOMC  $2018$ . At beginning,  $A$  takes  $B$  on the motocycle, and at the same time  $C$  rides the bicycle. After one hour and a half,  $B$  switches to a bicycle and immediately continues the trip to Hanoi, while  $A$  returns to pick up  $C$ . Upon meeting,  $C$  continues the travel on the motocycle to Hanoi with  $A$ . Finally, all three students arrive in Hanoi at the same time. Suppose that the average speed of the motocycle is  $50$  km per hour and of the both bicycles are  $10$  km per hour. Find the distance from the starting point to Hanoi.

Solution: \boxed{100 \text{ km}}


$2016$  bugs are sitting in different places of  $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of contacts, which can be reached by bugs?


Let  $O$  be the centre of the square  $ABCD$ . Let  $P,Q,R$  be respectively on the segments  $OA,OB,OC$  such that  $OP=3,OQ=5,OR=4$ . Suppose  $S$  is on  $OD$  such that  $X=AB\cap PQ,Y=BC\cap QR$  and  $Z=CD\cap RS$  are collinear. Find  $OS$ .


Determine the smallest natural number  $n$  having the following property: For every integer  $p, p \geq n$ , it is possible to subdivide (partition) a given square into  $p$  squares (not necessarily equal).

Solution: \boxed{ n = 6 }


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

Solution: \boxed{ k = 2 }


Let  $A_1B_1C_1$  be a triangle with  $A_1B_1 = 16, B_1C_1 = 14,$  and  $C_1A_1 = 10$ . Given a positive integer  $i$  and a triangle  $A_iB_iC_i$  with circumcenter  $O_i$ , define triangle  $A_{i+1}B_{i+1}C_{i+1}$  in the following way:

(a)  $A_{i+1}$  is on side  $B_iC_i$  such that  $C_iA_{i+1}=2B_iA_{i+1}$ .
(b)  $B_{i+1}\neq C_i$  is the intersection of line  $A_iC_i$  with the circumcircle of  $O_iA_{i+1}C_i$ .
(c)  $C_{i+1}\neq B_i$  is the intersection of line  $A_iB_i$  with the circumcircle of  $O_iA_{i+1}B_i$ .

Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]

Note:  $[K]$  denotes the area of  $K$ .

*Proposed by Yang Liu*


In a small town, there are  $n \times n$  houses indexed by  $(i, j)$  for  $1 \leq i, j \leq n$  with  $(1, 1)$  being the house at the top left corner, where  $i$  and  $j$  are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by  $(1, c)$ , where  $c \leq \frac{n}{2}$ . During each subsequent time interval  $[t, t+1]$ , the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended *neighbors* of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters?
A house indexed by  $(i, j)$  is a *neighbor* of a house indexed by  $(k, l)$  if  $|i - k| + |j - l|=1$ .

Solution: \boxed{n(n-c) + c(c-1)}


Points  $P$  and  $Q$  are chosen on diagonal  $AC$  of square  $ABCD$  such that  $AB=AP=CQ=1$ . What is the measure of  $\angle{PBQ}$  in degrees?

*2019 CCA Math Bonanza Lightning Round #1.3*


What is the smallest possible value of  $\left|12^m-5^n\right|$ , where  $m$  and  $n$  are positive integers?


In a group of  $2020$  people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends?

*2020 CCA Math Bonanza Tiebreaker Round #1*


Let  $ABC$  be a triangle with circumradius  $17$ , inradius  $4$ , circumcircle  $\Gamma$  and  $A$ -excircle  $\Omega$ . Suppose the reflection of  $\Omega$  over line  $BC$  is internally tangent to  $\Gamma$ .  Compute the area of  $\triangle ABC$ .


A train car held  $6000$  pounds of mud which was  $88$  percent water. Then the train car sat in the sun, and some of the water evaporated so that now the mud is only  $82$  percent water. How many pounds does the mud weigh now?


Point  $P_1$  is located  $600$  miles West of point  $P_2$ . At  $7:00\text{AM}$  a car departs from  $P_1$  and drives East at a speed of  $50$ mph. At  $8:00\text{AM}$  another car departs from  $P_2$  and drives West at a constant speed of  $x$  miles per hour. If the cars meet each other exactly halfway between  $P_1$  and  $P_2$ , what is the value of  $x$ ?


Compute the number of nonempty subsets  $S$  of  $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$  such that  $\frac{\max \,\, S + \min \,\,S}{2}$  is an element of  $S$ .


Let  $f(x)=x^3+ax^2+bx+c$  and  $g(x)=x^3+bx^2+cx+a$ , where  $a,b,c$  are integers with  $c\not=0$ . Suppose that the following conditions hold:

[list=a][*] $f(1)=0$ ,
[*]the roots of  $g(x)=0$  are the squares of the roots of  $f(x)=0$ .[/list]

Find the value of  $a^{2013}+b^{2013}+c^{2013}$ .


Find all real numbers  $a$  for which there exists a non-constant function  $f :\Bbb R \to \Bbb R$  satisfying the following two equations for all  $x\in \Bbb R:$ i)  $f(ax) = a^2f(x)$  and
ii)  $f(f(x)) = a f(x).$

Solution: \boxed{0, 1}


Find the area of a triangle with angles  $\frac{1}{7} \pi$ ,  $\frac{2}{7} \pi$ , and  $\frac{4}{7} \pi $ , and radius of its circumscribed circle  $R=1$ .


Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people  $(A,B,C)$  is a *trio* if one of the following is true:

- A shakes hands with B, and B shakes hands with C, or
- A doesn't shake hands with B, and B doesn't shake hands with C.

If we consider  $(A,B,C)$  and  $(C,B,A)$  as the same trio, find the minimum possible number of trios.


Determine all functions  $f:\mathbb{Z} \rightarrow \mathbb{Z}$  which satisfy the following equations:
a)	 $f(f(n))=4n+3$   $\forall$   $n \in \mathbb{Z}$ ;
b)	 $f(f(n)-n)=2n+3$   $\forall$   $n \in \mathbb{Z}$ .

Solution: \boxed{f(n) = 2n + 1}


What is the least positive integer  $n$  for which  $9n$  is a perfect square and  $12n$  is a perfect cube?


There are  $4$  distinct codes used in an intelligence station, one of them applied in each week. No two codes used in two adjacent weeks are the same code. Knowing that code  $A$  is used in the first week, find the probability that code  $A$  is used in the seventh week.


How many real numbers  $x$  are solutions to the equation  $|x - 2| - 4 =\frac{1}{|x - 3|}$  ?


The sequence  $(x_n)_{n\geqslant 0}$  is defined as such:  $x_0=1, x_1=2$  and  $x_{n+1}=4x_n-x_{n-1}$ , for all  $n\geqslant 1$ . Determine all the terms of the sequence which are perfect squares.

*George Stoica, Canada*

Solution: \boxed{ x_0 = 1 }


Find all prime numbers  $p$  which satisfy the following condition: For any prime  $q < p$ , if  $p = kq + r, 0 \leq r < q$ , there does not exist an integer  $q > 1$  such that  $a^{2} \mid r$ .


Answer the questions as below.

(1) Find the local minimum of  $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the  $x$ -axis.

Solution: \boxed{e - 2}


Aisha went shopping. At the first store she spent  $40$  percent of her money plus four dollars. At the second store she spent  $50$  percent of her remaining money plus  $5$  dollars. At the third store she spent  $60$  percent of her remaining money plus six dollars. When Aisha was done shopping at the three stores, she had two dollars left. How many dollars did she have with her when she started shopping?


11. Let  $f(z)=\frac{a z+b}{c z+d}$  for  $a, b, c, d \in \mathbb{C}$ . Suppose that  $f(1)=i, f(2)=i^{2}$ , and  $f(3)=i^{3}$ . If the real part of  $f(4)$  can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m, n$ , find  $m^{2}+n^{2}$ .


In duck language, only letters  $q$ ,  $a$ , and  $k$  are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there?
In duck language, too, the letter  $a$  is a vowel, while  $q$  and  $k$  are consonants.


Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly  $50$  of her first  $100$  shots?


Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by  $x$  square units. What is the smallest possible positive value of  $x$ ?

*Proposed by Ray Li*


How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths  $1, 2, 3, 4, 5$  and  $6,$  in any order?


Find, with proof, all positive integers  $n$  such that the equation  $x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}$  has a solution in positive integers.

Solution: \boxed{ n = 3 }


Given that there are  $24$  primes between  $3$  and  $100$ , inclusive, what is the number of ordered pairs  $(p, a)$  with  $p$  prime,  $3 \le p < 100$ , and  $1 \le a < p$  such that  $p \mid (a^{p-2} - a)$ ?


Label the vertices of a trapezoid  $T$  inscribed in the unit circle as  $A,B,C,D$  counterclockwise with  $AB\parallel CD$ . Let  $s_1,s_2,$  and  $d$  denote the lengths of  $AB$ ,  $CD$ , and  $OE$ , where  $E$  is the intersection of the diagonals of  $T$ , and  $O$  is the center of the circle. Determine the least upper bound of  $\frac{s_1-s_2}d$  over all  $T$  for which  $d\ne0$ , and describe all cases, if any, in which equality is attained.

Solution: \boxed{ 2 }


Let  $P_1$  be a regular  $n$ -gon, where  $n\in\mathbb{N}$ . We construct  $P_2$  as the regular  $n$ -gon whose vertices are the midpoints of the edges of  $P_1$ . Continuing analogously, we obtain regular  $n$ -gons  $P_3,P_4,\ldots ,P_m$ . For  $m\ge n^2-n+1$ , find the maximum number  $k$  such that for any colouring of vertices of  $P_1,\ldots ,P_m$  in  $k$  colours there exists an isosceles trapezium  $ABCD$  whose vertices  $A,B,C,D$  have the same colour.

*Radu Ignat*

Solution: \boxed{n - 1}


Find how many positive integers  $k\in\{1,2,\ldots,2022\}$  have the following property: if  $2022$  real numbers are written on a circle so that the sum of any  $k$  consecutive numbers is equal to  $2022$  then all of the  $2022$  numbers are equal.


Compute
    \[
        \left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right).
    \]

*2020 CCA Math Bonanza Team Round #4*


Let  $x$  be the length of one side of a triangle and let  $y$  be the height to that side.  If  $x+y=418$ , find the maximum possible  $\textit{integral value}$  of the area of the triangle.


Find the positive integers  $n$  that are not divisible by  $3$  if the number  $2^{n^2-10}+2133$  is a perfect cube.

NoteThe wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation  $x^3=2^{n^2-10}+2133$  where  $x,n \in \mathbb{N}$  and  $3\nmid n$ .

Solution: \boxed{ n = 4 }


One face of a tetrahedron has sides of length  $3$ ,  $4$ , and  $5$ . The tetrahedron’s volume is  $24$  and surface area is  $n$ . When  $n$  is minimized, it can be expressed in the form  $n = a\sqrt{b} + c$ , where  $a$ ,  $b$ , and  $c$  are positive integers and b is not divisible by the square of any prime. Evaluate  $a + b + c$ .


A circle passes through the vertex of a rectangle  $ABCD$  and touches its sides  $AB$  and  $AD$  at  $M$  and  $N$  respectively. If the distance from  $C$  to the line segment  $MN$  is equal to  $5$  units, find the area of rectangle  $ABCD$ .


For all real numbers  $x$ , let  $P(x)=16x^3 - 21x$ .  What is the sum of all possible values of  $\tan^2\theta$ , given that  $\theta$  is an angle satisfying \[P(\sin\theta) = P(\cos\theta)?\]


Suppose  $ n$  is a positive integer and 3 arbitrary numbers numbers are chosen from the set  $ 1,2,3,...,3n+1$  with their sum equal to  $ 3n+1$ . What is the largest possible product of those 3 numbers?

Solution: \boxed{n^3 + n^2}


Given acute triangle  $\triangle ABC$  in plane  $P$ , a point  $Q$  in space is defined such that  $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$   Point  $X$  is the point in plane  $P$  such that  $QX$  is perpendicular to plane  $P$ . Given  $\angle ABC = 40^\circ$  and  $\angle ACB = 75^\circ,$  find  $\angle AXC.$


A *substring* of a number  $n$  is a number formed by removing some digits from the beginning and end of  $n$  (possibly a different number of digits is removed from each side). Find the sum of all prime numbers  $p$  that have the property that any substring of  $p$  is also prime.


How many ways are there to arrange the letters  $A,A,A,H,H$  in a row so that the sequence  $HA$  appears at least once?

*Author: Ray Li*


The first term of a sequence is  $2014$ . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the  $2014$  th term of the sequence?


A cubic block with dimensions  $n$  by  $n$  by  $n$  is made up of a collection of  $1$  by  $1$  by  $1$  unit cubes. What is the smallest value of  $n$  so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain?


In a basketball tournament every two teams play two matches. As usual, the winner of a match gets  $2$  points, the loser gets  $0$ , and there are no draws. A single team wins the tournament with  $26$  points and exactly two teams share the last position with  $20$  points. How many teams participated in the tournament?


Let  $ABC$  be a triangle.  Point  $P$  lies in the interior of  $\triangle ABC$  such that  $\angle ABP = 20^\circ$  and  $\angle ACP = 15^\circ$ .  Compute  $\angle BPC - \angle BAC$ .


The point  $D$  at the side  $AB$  of triangle  $ABC$  is given. Construct points  $E,F$  at sides  $BC, AC$  respectively such that the midpoints of  $DE$  and  $DF$  are collinear with  $B$  and the midpoints of  $DE$  and  $EF$  are collinear with  $C.$

Solution: \boxed{ E }


If  $x=\sqrt2+\sqrt3+\sqrt6$  is a root of  $x^4+ax^3+bx^2+cx+d=0$  where  $a,b,c,d$  are integers, what is the value of  $|a+b+c+d|$ ?


The *Olimpia* country is formed by  $n$  islands. The most populated one is called *Panacenter*, and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions:

a) No pair of islands is joined by more than one bridge.

b) Using the bridges we can reach every island from Panacenter.

c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing.

Determine the number of ways we can build the bridges.


If  $x$  and  $y$  are positive integers, and  $x^4+y^4=4721$ , find all possible values of  $x+y$


For every positive integer  $n$  determine the least possible value of the expression
\[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\]
given that  $x_{1}, x_{2}, \dots , x_{n}$  are real numbers satisfying  $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$ .

Solution: \boxed{ 2^{1-n} }


Find the maximum value of real number  $A$  such that   $$ 3x^2 + y^2 + 1 \geq A(x^2 + xy + x) $$   for all positive integers  $x, y.$


Let  $n$  be a positive integer. Alice writes  $n$  real numbers  $a_1, a_2,\dots, a_n$  in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ( $a_n$  is not a neighbor of  $a_1$ , nor vice versa). A number *changes sign* if it changes from being nonnegative to negative or vice versa. In terms of  $n$ , determine the maximum number of times that  $a_1$  can change sign, across all possible values of  $a_1,a_2,\dots, a_n$  and all possible sequences of moves Alice may make.


Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?


Find the least positive integer  $ a$  such that  $ 2001$  divides  $ 55^n\plus{}a \cdot 32^n$  for some odd  $ n$ .


How many gallons of a solution which is  $15\%$  alcohol do we have to mix with a solution that is  $35\%$  alcohol to make  $250$  gallons of a solution that is  $21\%$  alcohol?

Solution: \boxed{ 175 }


Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80.  From this, John was able to determine the number of problems Mary solved correctly.  If Mary's score had been any lower, but still over 80, John could not have determined this.  What was Mary's score?  (Recall that the AHSME consists of 30 multiple-choice problems and that one's score,  $s$ , is computed by the formula  $s = 30 + 4c - w$ , where  $c$  is the number of correct and  $w$  is the number of wrong answers; students are not penalized for problems left unanswered.)


Let  $V$  be a 10-dimensional real vector space and  $U_1,U_2$  two linear subspaces such that  $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$ . Let  $\varepsilon$  be the set of all linear maps  $T: V\rightarrow V$  which have  $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$ . Calculate the dimension of  $\varepsilon$ . (again, all as real vector spaces)


A *smooth number* is a positive integer of the form  $2^m 3^n$ , where  $m$  and  $n$  are nonnegative integers.  Let  $S$  be the set of all triples  $(a, b, c)$  where  $a$ ,  $b$ , and  $c$  are smooth numbers such that  $\gcd(a, b)$ ,  $\gcd(b, c)$ , and  $\gcd(c, a)$  are all distinct.  Evaluate the infinite sum  $\sum_{(a,b,c) \in S} \frac{1}{abc}$ .  Recall that  $\gcd(x, y)$  is the greatest common divisor of  $x$  and  $y$ .


a) Show that the product of all differences of possible couples of six given positive integers is divisible by  $960$ b) Show that the product of all differences of possible couples of six given positive integers is divisible by  $34560$

PS. a) original from Albania
b) modified by problem selecting committee

Solution: \boxed{ 34560 }


For some positive integer  $n$ , there exists  $n$  different positive integers  $a_1, a_2, ..., a_n$  such that $(1)$   $a_1=1, a_n=2000$  $(2)$   $\forall i\in \mathbb{Z}$    $s.t.$    $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$ Determine the maximum value of n.


For some positive integer  $n$ , the sum of all odd positive integers between  $n^2-n$  and  $n^2+n$  is a number between  $9000$  and  $10000$ , inclusive. Compute  $n$ .

*2020 CCA Math Bonanza Lightning Round #3.1*


The function  $g\left(x\right)$  is defined as  $\sqrt{\dfrac{x}{2}}$  for all positive  $x$ .  $ $   $$ g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right) $$    $ $ can be expressed as  $\cos(b)$  using degrees, where  $0^\circ < b < 90^\circ$  and  $b = p/q$  for some relatively prime positive integers  $p, q$ . Find  $p+q$ .


All of the roots of  $x^3+ax^2+bx+c$  are positive integers greater than  $2$ , and the coefficients satisfy  $a+b+c+1=-2009$ . Find  $a$


Given that in a triangle  $ABC$ ,  $AB=3$ ,  $BC=4$  and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of  $AC$ .


Find the limit, when  $n$  tends to the infinity, of   $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$


**a)** Show that the expression  $ x^3-5x^2+8x-4 $  is nonegative, for every  $ x\in [1,\infty ) . $ **b)** Determine  $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $


Let  $\{x\}$  be a sequence of positive reals  $x_1, x_2, \ldots, x_n$ , defined by:  $x_1 = 1, x_2 = 9, x_3=9, x_4=1$ . And for  $n \geq 1$  we have:

\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]

Show that this sequence has a finite limit. Determine this limit.


In a school tennis tournament with  $ m \ge 2$  participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants m for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.


What is the largest positive integer  $n$  such that   $$ \frac{a^2}{\frac{b}{29} + \frac{c}{31}}+\frac{b^2}{\frac{c}{29} + \frac{a}{31}}+\frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c) $$  holds for all positive real numbers  $a,b,c$ .


On the table there are  $k \ge 3$  heaps of  $1, 2, \dots , k$  stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove  $1$  stone from this new heap. Thereafter, in the  $i$ -th step ( $i \ge 2$ ) we merge some three heaps containing more than  $i$  stones in total and remove  $i$  stones from the new heap. Assume that after a number of steps a single heap of  $p$  stones remains on the table. Show that the number  $p$  is a perfect square if and only if so are both  $2k + 2$  and  $3k + 1$ . Find the least  $k$  with this property.


Find  $a+b+c$ , where  $a,b,$  and  $c$  are the hundreds, tens, and units digits of the six-digit number  $123abc$ , which is a multiple of  $990$ .


Peter Pan and Crocodile are each getting hired for a job. Peter wants to get paid 6.4 dollars daily, but Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, 80 cents on day 4, and so on. After how many whole days will Crocodile's total earnings exceed that of Peter's?


Let  $ a$  be a positive real numbers. In the coordinate plane denote by  $ S$  the area of the figure bounded by the curve  $ y=\sin x\ (0\leq x\leq \pi)$  and the  $ x$ -axis and denote  $ T$  by the area of the figure bounded by the curves  $ y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\　y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$  and the  $ x$ -axis. Find the value of  $ a$  such that  $ S: T=3: 1$ .

Solution: \boxed{ a = \frac{4}{3} }


Find all positive integers   $k$  such that for the first  $k$  prime numbers  $2, 3, \ldots, p_k$  there exist positive integers  $a$  and  $n>1$ , such that  $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$ .

*V. Senderov*

Solution: \boxed{ k = 1 }


For a positive integer  $n$ , we dene  $D_n$  as the largest integer that is a divisor  of  $a^n + (a + 1)^n + (a + 2)^n$  for all positive integers  $a$ .
1. Show that for all positive integers  $n$ , the number  $D_n$  is of the form  $3^k$  with  $k \ge 0$  an integer.
2. Show that for all integers  $k \ge 0$  there exists a positive integer n such that  $D_n = 3^k$ .

Solution: \boxed{ D_n = 3^k }


Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?


Triangle  $ABC$  has side lengths  $AB=18$ ,  $BC=36$ , and  $CA=24$ . The circle  $\Gamma$  passes through point  $C$  and is tangent to segment  $AB$  at point  $A$ .
Let  $X$ , distinct from  $C$ , be the second intersection of  $\Gamma$  with  $BC$ . Moreover, let  $Y$  be the point on  $\Gamma$  such that segment  $AY$  is an angle bisector of  $\angle XAC$ .
Suppose the length of segment  $AY$  can be written in the form  $AY=\frac{p\sqrt{r}}{q}$  where  $p$ ,  $q$ , and  $r$  are positive integers such that  $gcd(p, q)=1$  and  $r$  is square free.
Find the value of  $p+q+r$ .


Done with her new problems, Wendy takes a break from math.  Still without any fresh reading material, she feels a bit antsy.  She starts to feel annoyed that Michael's loose papers clutter the family van.  Several of them are ripped, and bits of paper litter the floor.  Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash.  "That seems fair to me," confirms Hannah encouragingly.

While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it.  There is a monic polynomial of degree  $n$ , with real coefficients.  The first two terms after  $x^n$  are  $a_{n-1}x^{n-1}$  and  $a_{n-2}x^{n-2}$ , but the rest of the polynomial is cut off where Michael's page is ripped.  Wendy barely makes out a little of Michael's scribbling, showing that  $a_{n-1}=-a_{n-2}$ .  Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial.  Wendy knows neither the value of  $n$ , nor the value of  $a_{n-1}$ , but still she finds a [greatest] lower bound for the answer to the problem.  Find the absolute value of that lower bound.


Given a connected graph with  $n$  edges, where there are no parallel edges. For any two cycles  $C,C'$  in the graph, define its *outer cycle* to be
\[C*C'=\{x|x\in (C-C')\cup (C'-C)\}.\]
(1) Let  $r$  be the largest postive integer so that we can choose  $r$  cycles  $C_1,C_2,\ldots,C_r$  and for all  $1\leq k\leq r$  and  $1\leq i$ ,  $j_1,j_2,\ldots,j_k\leq r$ , we have
\[C_i\neq C_{j_1}*C_{j_2}*\cdots*C_{j_k}.\]
(Remark: There should have been an extra condition that either  $j_1\neq i$  or  $k\neq 1$ )
(2) Let  $s$  be the largest positive integer so that we can choose  $s$  edges that do not form a cycle.
(Remark: A more precise way of saying this is that any nonempty subset of these  $s$  edges does not form a cycle)
Show that  $r+s=n$ .

Note: A cycle is a set of edges of the form  $\{A_iA_{i+1},1\leq i\leq n\}$  where  $n\geq 3$ ,  $A_1,A_2,\ldots,A_n$  are distinct vertices, and  $A_{n+1}=A_1$ .

Solution: \boxed{ r + s = n }


Find the largest positive integer  $n$  such that the following is true:
There exists  $n$  distinct positive integers  $x_1,~x_2,\dots,x_n$  such that whatever the numbers  $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$  are, not all null, the number  $n^3$  do not divide  $\sum_{k=1}^n a_kx_k$ .


Find number of solutions in non-negative reals to the following equations:
\begin{eqnarray*}x_1 + x_n ^2 = 4x_n  x_2 + x_1 ^2 = 4x_1  ...  x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

Solution: \boxed{ 2^n }


Maria and Joe are jogging towards each other on a long straight path. Joe is running at  $10$  mph and Maria at  $8$  mph. When they are  $3$  miles apart, a fly begins to fly back and forth between them at a constant rate of  $15$  mph, turning around instantaneously whenever it reachers one of the runners. How far, in miles, will the fly have traveled when Joe and Maria pass each other?


Consider the set  $A=\{1,2,3\ldots ,2^n\}, n\ge 2$ . Find the number of subsets  $B$  of  $A$  such that for any two elements of  $A$  whose sum is a power of  $2$  exactly one of them is in  $B$ .

*Aleksandar Ivanov*

Solution: \boxed{ 2^{n+1} }


There were three times as many red candies as blue candies on a table. After Darrel took the same number of red candies and blue candies, there were four times as many red candies as blue candies left on the table. Then after Cloe took  $12$  red candies and  $12$  blue candies, there were five times as many red candies as blue candies left on the table. Find the total number of candies that Darrel took.


The graph of  $y^2+2xy+40|x|=400$  partitions the plane into several regions.  What is the area of the bounded region?


Find minimum of  $x+y+z$  where  $x$ ,  $y$  and  $z$  are real numbers such that  $x \geq 4$ ,  $y \geq 5$ ,  $z \geq 6$  and  $x^2+y^2+z^2 \geq 90$


Consider  $13$  marbles that are labeled with positive integers such that the product of all  $13$  integers is  $360$ . Moor randomly picks up  $5$  marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?


Find the number of pairs of positive integers  $(m; n)$ , with  $m \le n$ , such that the ‘least common multiple’ (LCM) of  $m$  and  $n$  equals  $600$ .


Real numbers  $x,y,a$  satisfy the equations  $x+y = x^3 +y^3 = x^5 +y^5 = a$ .
Find all possible values of  $a$ .

Solution: \boxed{a = 0, \pm 1, \pm 2}


Kevin is trying to solve an economics question which has six steps. At each step, he has a probability  $p$  of making a sign error. Let  $q$  be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of  $0 \le p \le 1$  is it true that  $p+q=1$ ?

*Proposed by Evan Chen*


Let  $2000 < N < 2100$  be an integer. Suppose the last day of year  $N$  is a Tuesday while the first day of year  $N+2$  is a Friday. The fourth Sunday of year  $N+3$  is the  $m$ th day of January. What is  $m$ ?

*Based on a proposal by Neelabh Deka*


Let  $ABC$  be a triangle and  $\Gamma$  the  $A$ - exscribed circle whose center is  $J$  . Let  $D$  and  $E$  be the touchpoints of  $\Gamma$   with the lines  $AB$  and  $AC$ , respectively. Let  $S$  be the area of the quadrilateral  $ADJE$ , Find the maximum value that  $\frac{S}{AJ^2}$  has and when equality holds.


**a)** Solve the equation  $ x^2-x+2\equiv 0\pmod 7. $ **b)** Determine the natural numbers  $ n\ge 2 $  for which the equation  $ x^2-x+2\equiv 0\pmod n $  has an unique solution modulo  $ n. $

Solution: \boxed{ n = 7 }


A clueless ant makes the following route: starting at point  $ A $  goes  $ 1$  cm north, then  $ 2$  cm east, then  $ 3$  cm south, then  $ 4$  cm west, immediately  $ 5$  cm north, continues  $ 6$  cm east, and so on, finally  $ 41$  cm north and ends in point  $ B $ . Calculate the distance between  $ A $  and  $ B $  (in a straight line).


A net for hexagonal pyramid is constructed by placing a triangle with side lengths  $x$ ,  $x$ , and  $y$  on each side of a regular hexagon with side length  $y$ . What is the maximum volume of the pyramid formed by the net if  $x+y=20$ ?


Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers  $a, b, c$ , respectively. They then compute the polynomial  $f(x)=x^{3}+p x^{2}+q x+r$  with roots  $a, b, c$ . If the expected value of the sum of the squares of the coefficients of  $f(x)$  is  $\frac{m}{n}$  for relatively prime positive integers  $m, n$ , find the remainder when  $m+n$  is divided by 1000 .


Adam, Bendeguz, Cathy, and Dennis all see a positive integer  $n$ . Adam says, " $n$  leaves a remainder of  $2$  when divided by  $3$ ." Bendeguz says, "For some  $k$ ,  $n$  is the sum of the first  $k$  positive integers." Cathy says, "Let  $s$  be the largest perfect square that is less than  $2n$ . Then  $2n - s = 20$ ." Dennis says, "For some  $m$ , if I have  $m$  marbles, there are  $n$  ways to choose two of them." If exactly one of them is lying, what is  $n$ ?


The  $42$  points  $P_1,P_2,\ldots,P_{42}$  lie on a straight line, in that order, so that the distance between  $P_n$  and  $P_{n+1}$  is  $\frac{1}{n}$  for all  $1\leq n\leq41$ . What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)


For  $n > 1$ , let  $a_n$  be the number of zeroes that  $n!$  ends with when written in base  $n$ . Find the maximum value of  $\frac{a_n}{n}$ .


Petya gave Vasya a number puzzle. Petya chose a digit  $X$  and said, "I am thinking of a three digit number that is divisible by 11. The hundreds digit is  $X$  and the tens digit is 3. Find the units digit." Vasya was excited because he knew how to solve this problem, but then realized that the problem Petya gave did not have an answer. What digit  $X$  did Petya chose?

*Ray Li.*

Solution: \boxed{ X = 4 }


Faces  $ABC$  and  $BCD$  of tetrahedron  $ABCD$  meet at an angle of  $30^\circ$ . The area of face  $ABC$  is  $120$ , the area of face  $BCD$  is  $80$ , and  $BC=10$ . Find the volume of the tetrahedron.


Let  $P(x) = x^3 + ax^2 + bx + 1$  be a polynomial with real coefficients and three real roots  $\rho_1$ ,  $\rho_2$ ,  $\rho_3$  such that  $|\rho_1| < |\rho_2| < |\rho_3|$ . Let  $A$  be the point where the graph of  $P(x)$  intersects  $yy'$  and the point  $B(\rho_1, 0)$ ,  $C(\rho_2, 0)$ ,  $D(\rho_3, 0)$ . If the circumcircle of  $\vartriangle ABD$  intersects  $yy'$  for a second time at  $E$ , find the minimum value of the length of the segment  $EC$  and the polynomials for which this is attained.

*Brazitikos Silouanos, Greece*

Solution: \boxed{ \sqrt{2} }


Roy is starting a baking company and decides that he will sell cupcakes. He sells  $n$  cupcakes for  $(n + 20)(n + 15)$  cents. A man walks in and buys  $\$ 10.50 $ worth of cupcakes. Roy bakes cupcakes at a rate of $ 10$ cupcakes an hour. How many minutes will it take Roy to complete the order?

Solution: \boxed{90 \text{ minutes}}


How many sequences of nonnegative integers  $a_1,a_2,\ldots, a_n$  ( $n\ge1$ ) are there such that  $a_1\cdot a_n > 0$ ,  $a_1+a_2+\cdots + a_n = 10$ , and  $\prod_{i=1}^{n-1}(a_i+a_{i+1}) > 0$ ?

*Ray Li.*

Clarifications[list=1][*]If you find the wording of the problem confusing, you can use the following, equivalent wording: "How many finite sequences of nonnegative integers are there such that (i) the sum of the elements is 10; (ii) the first and last elements are both positive; and (iii) among every pair of adjacent integers in the sequence, at least one is positive."[/list]


We call a monic polynomial  $P(x) \in \mathbb{Z}[x]$  *square-free mod n* if there <u>dose not</u> exist polynomials  $Q(x),R(x) \in \mathbb{Z}[x]$  with  $Q$  being non-constant and  $P(x) \equiv Q(x)^2 R(x) \mod n$ . Given a prime  $p$  and integer  $m \geq 2$ . Find the number of monic *square-free mod p*  $P(x)$  with degree  $m$  and coeeficients in  $\{0,1,2,3,...,p-1\}$ .

*Proposed by Masud Shafaie*

Solution: \boxed{ p^m - p^{m-1} }


The Mictlán is an  $n\times n$  board and each border of each  $1\times 1$  cell is painted either purple or orange. Initially, a catrina lies inside a  $1\times 1$  cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple?

*Proposed by CDMX*


Each of  $2k+1$  distinct 7-element subsets of the 20 element set intersects with exactly  $k$  of them. Find the maximum possible value of  $k$ .


The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her school. Betsy had 15 pieces of candy left over. Find the least number of pieces of candy the Tasty Candy Company could have placed in each one-pound bag.


A function  $f$  is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1,  f(3) &=& 3,  f(2n) &=& f(n),  f(4n+1) &=& 2f(2n+1)-f(n),  f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers  $n$ .  Determine the number of positive integers  $n$ , less than or equal to 1988, for which  $f(n) = n$ .


Let  $S$  be a subset of  $\{0,1,2,\dots ,9\}$ . Suppose there is a positive integer  $N$  such that for any integer  $n>N$ , one can find positive integers  $a,b$  so that  $n=a+b$  and all the digits in the decimal representations of  $a,b$  (expressed without leading zeros) are in  $S$ . Find the smallest possible value of  $|S|$ .

*Proposed by Sutanay Bhattacharya*

Original WordingAs pointed out by Wizard_32, the original wording is:

Let  $X=\{0,1,2,\dots,9\}.$  Let  $S \subset X$  be such that any positive integer  $n$  can be written as  $p+q$  where the non-negative integers  $p, q$  have all their digits in  $S.$  Find the smallest possible number of elements in  $S.$


A square has been divided into  $2022$  rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?


	 Determine the smallest positive integer  $n \ge 3$  for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where  $A$  denotes the result when the numbers  $2^{10}$ ,  $2^{20}$ ,  $\dots$ ,  $2^{10n}$  are written in decimal notation and concatenated (for example, if  $n=2$  we have  $A = 10241048576$ ).


An integer is square-free if it is not divisible by  $a^2$  for any integer  $a>1$ . Let  $S$  be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where  $[x]$  denote the greatest integer less than or equal to  $x$


Let  $a$ ,  $b$ ,  $c$  be positive integers such that  $29a + 30b + 31c = 366$ . Find  $19a + 20b + 21c$ .


Find all integer triples  $(a, b, c)$  satisfying the equation  $$     5 a^2 + 9 b^2 = 13 c^2. $$

Solution: \boxed{ (0, 0, 0) }