In the diagram below, square  $ABCD$  with side length 23 is cut into nine rectangles by two lines parallel to  $\overline{AB}$  and two lines parallel to  $\overline{BC}$ . The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
		 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
		 label(" $A$ ", (0,23), NW);
		 label(" $B$ ", (23, 23), NE);
		 label(" $C$ ", (23,0), SE);
		 label(" $D$ ", (0,0), SW);
		 draw((0,6)--(23,6));
		 draw((0,19)--(23,19));
		 draw((5,0)--(5,23));
		 draw((12,0)--(12,23));
		 label("13", (17/2, 21));
		 label("111",(35/2,25/2));
		 label("37",(17/2,3));
		 label("123",(2.5,12.5));[/asy]

*Proposed by Lewis Chen*


Determine the range of  $w(w + x)(w + y)(w + z)$ , where  $x, y, z$ , and  $w$  are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]


Real numbers  $X_1, X_2, \dots, X_{10}$  are chosen uniformly at random from the interval  $[0,1]$ . If the expected value of  $\min(X_1,X_2,\dots, X_{10})^4$  can be expressed as a rational number  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , what is  $m+n$ ?

*2016 CCA Math Bonanza Lightning #4.4*

Solution: \boxed{ 1002 }


Let  $(x_n)_{n\ge 0}$  a sequence of real numbers defined by  $x_0>0$  and  $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$ . Compute  $\lim_{n\to \infty}x_n$  and  $\lim_{n\to \infty} \frac{x_n^3}{n^2}$ .

Solution: \boxed{ \frac{9}{4} }


The number  $201212200619$  has a factor  $m$  such that  $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find  $m$ .


How many  $4-$ digit numbers  $\overline{abcd}$  are there such that  $a<b<c<d$  and  $b-a<c-b<d-c$  ?


Let  $\mathcal{F}$  be the set of continuous functions  $f : [0, 1]\to\mathbb{R}$  satisfying  $\max_{0\le x\le 1} |f(x)| = 1$  and let  $I : \mathcal{F} \to \mathbb{R}$ ,

\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]

a) Show that  $I(f) < 3$ , for any  $f \in \mathcal{F}$ .

b) Determine  $\sup\{I(f) \mid f \in \mathcal{F}\}$ .


Let  $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$ . For all points  $(a, b)$ , let  $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$ . Kathy constructs a set  $T$  by adding  $n$  distinct points from  $S$  to  $T$  at random. If the expected value of  $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$  is 4, then compute  $n$ .

*Proposed by Lewis Chen*


Let  $1=d_1<d_2<\ldots<d_k=n$  be all natural divisors of a natural number  $n$ .
Find all possible values of  $k$  if  $n=d_2d_3+d_2d_5+d_3d_5$ .

Solution: \boxed{ k = 9 }


In one of the hotels of the wellness planet Oxys, there are  $2019$  saunas. The managers have decided to accommodate  $k$  couples for the upcoming long weekend. We know the following about the guests: if two women know each other then their husbands also know each other, and vice versa. There are several restrictions on the usage of saunas. Each sauna can be used by either men only, or women only (but there is no limit on the number of people using a sauna at once, as long as they are of a single gender). Each woman is only willing to share a sauna with women whom she knows, and each man is only willing to share a sauna with men whom he does not know. What is the greatest possible  $k$  for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?


Suppose  $a,b,c,x,y,z$  are pairwisely different real numbers. How many terms in the following can be  $1$  at most:   $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$


Two circles touch in  $M$ , and lie inside a rectangle  $ABCD$ . One of them touches the sides  $AB$  and  $AD$ , and the other one touches  $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in  $M$  divides  $AB$  and  $CD$ .

Solution: \boxed{ 1:1 }


Let  $T$  be an acute triangle. Inscribe a rectangle  $R$  in  $T$  with one side along a side of  $T.$  Then inscribe a rectangle  $S$  in the triangle formed by the side of  $R$  opposite the side on the boundary of  $T,$  and the other two sides of  $T,$  with one side along the side of  $R.$  For any polygon  $X,$  let  $A(X)$  denote the area of  $X.$  Find the maximum value, or show that no maximum exists, of  $\tfrac{A(R)+A(S)}{A(T)},$  where  $T$  ranges over all triangles and  $R,S$  over all rectangles as above.


Let  $ a_1, a_2,\ldots  ,a_8$  be  $8$  distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every  $4$  of the  $8$  points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the  $8$  points, intersect, the point of intersection is called a *bullet*. Suppose some of the bullets are coloured red. For each pair  $(i j)$ , with  $ 1 \le i < j \le 8$ , let  $r(i,j)$  be the number of quads, each containing  $ a_i, a_j$  as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer  $n$  such that it is possible to colour  $n$  of the bullets red so that  $r(i,j)$  is a constant for all pairs  $(i,j)$ .

Solution: \boxed{ n = 14 }


Find the smallest positive integer  $n$  such that the decimal representation of  $n!(n+1)!(2n+1)! - 1$  has its last  $30$  digits all equal to  $9$ .


Let  $a$ ,  $b$ ,  $c$  be positive reals for which
\begin{align*}
  (a+b)(a+c) &= bc + 2
  (b+c)(b+a) &= ca + 5
  (c+a)(c+b) &= ab + 9
\end{align*}
If  $abc = \frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by Evan Chen*


For any integer  $a$ , let  $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of  $f(a)$  that are prime?

*Proposed by Alexander Wang*


 $k$  marbles are placed onto the cells of a  $2024 \times 2024$  grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).

a) Assume that  $k=2024$ . Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions.
b) Determine the largest value of  $k$  such that for all arrangements of  $k$  marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.

Solution: \boxed{ k = 2023 }


Let  $ABCD$  be a convex quadrilateral with side lengths satisfying the equality:  $$  AB \cdot CD = AD \cdot BC = AC \cdot BD. $$  Determine the sum of the acute angles of quadrilateral  $ABCD$ .

*Proposed by Zaza Meliqidze, Georgia*


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


When submitting problems, Steven the troll likes to submit silly names rather than his own. On day  $1$ , he gives no
name at all. Every day after that, he alternately adds  $2$  words and  $4$  words to his name. For example, on day  $4$  he
submits an  $8\text{-word}$  name. On day  $n$  he submits the  $44\text{-word name}$  “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute  $n$ .


$a$  and  $b$  are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of  $2ab.$ *Author: Ray Li*


rainbow is the name of a bird. this bird has  $n$  colors and it's colors in two consecutive days are not equal. there doesn't exist  $4$  days in this bird's life like  $i,j,k,l$  such that  $i<j<k<l$  and the bird has the same color in days  $i$  and  $k$  and the same color in days  $j$  and  $l$  different from the colors it has in days  $i$  and  $k$ . what is the maximum number of days rainbow can live in terms of  $n$ ?

Solution: \boxed{2n - 1}


In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the  $n$  -th switch the  $n$  -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from the first switch until he has to turn on a bulb; as soon as he turns a bulb on, he leaves the room. For example the first person goes in, turns the first switch on and leaves. Then the second man goes in, seeing that the first switch on turns it off and then lights the second bulb. Then the third person goes in, finds the first switch off and turns it on and leaves the room. Then the fourth person enters and switches off the first and second bulbs and switches on the third. The process continues in this way. Finally we find out that first 10 bulbs are off and the 11 -th bulb is on. Then how many people were involved in the entire process?


Given that  $x\ge0$ ,  $y\ge0$ ,  $x+2y\le6$ , and  $2x+y\le6$ , compute the maximum possible value of  $x+y$ .


Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If  $\tfrac{A}{B} = \tfrac{m}{n}$  where  $m,n$  are relatively prime positive integers, determine  $100m+n$ .

*Proposed by Lewis Chen*


Every of  $n$  guests invited to a dinner has got an invitation denoted by a number from  $1$  to  $n$ . The guests will be sitting around a round table with  $n$  seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all  $n$  for which he can select the guests in such an order to serve all the guests.

Solution: \boxed{ n }


Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  be integers selected from the set  $\{1,2,\dots,100\}$ , uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when  $M$  is divided by  $64$ ?


Let  $S$  be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in  $S$  is  $L$ . Find the number of elements in  $S$  that do not divide  $\frac{L}{2016}$ .

*Proposed by Yannick Yao*


The smallest three positive proper divisors of an integer n are  $d_1 < d_2 < d_3$  and they satisfy  $d_1 + d_2 + d_3 = 57$ . Find the sum of the possible values of  $d_2$ .


Let  $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$ .

(1) Show that  $ f(x) \equal{} 0$  has a unique solution in  $ 0 < x < \pi$ .

(2) Let  $ J \equal{} \int_0^{\pi} |f(x)|dx$ . Denote by  $ \alpha$  the solution in (1), express  $ J$  in terms of  $ \sin \alpha$ .

(3) Compare the size of  $ J$  defined in (2) with  $ \sqrt {2}$ .

Solution: \boxed{ J > \sqrt{2} }


Determine the primes  $p$  for which the numbers  $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$  are all quadratic residues modulo  $p.$ *Vlad Matei*

Solution: \boxed{ p = 2 }


Consider the set  $S$  of permutations of  $1, 2, \dots, 2022$  such that for all numbers  $k$  in the
permutation, the number of numbers less than  $k$  that follow  $k$  is even.

For example, for  $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If  $|S| = (a!)^b$  where  $a, b \in \mathbb{N}$ , then find the product  $ab$ .


Let  $\alpha$  denote  $\cos^{-1}(\tfrac 23)$ .  The recursive sequence  $a_0,a_1,a_2,\ldots$  satisfies  $a_0 = 1$  and, for all positive integers  $n$ ,   $$ a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}. $$   Suppose that the series   $$ \sum_{k=0}^\infty\dfrac{a_k}{2^k} $$   can be expressed uniquely as  $\tfrac{p\sqrt q}r$ , where  $p$  and  $r$  are coprime positive integers and  $q$  is not divisible by the square of any prime.  Find the value of  $p+q+r$ .


Find all triplets  $(a, b, c)$  of positive integers, such that  $a+bc, b+ac, c+ab$  are primes and all divide  $(a^2+1)(b^2+1)(c^2+1)$ .

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


The graph of  ${(x^2 + y^2 - 1)}^3 = x^2 y^3$  is a heart-shaped curve, shown in the figure below.
[asy]
  import graph;
  unitsize(10);

  real f(real x)
  {
    return sqrt(cbrt(x^4) - 4 x^2 + 4);
  }

  real g(real x)
  {
    return (cbrt(x^2) + f(x))/2;
  }

  real h(real x)
  {
    return (cbrt(x^2) - f(x)) / 2;
  }

  real xmax = 1.139028;

  draw(graph(g, -xmax, xmax) -- reverse(graph(h, -xmax, xmax)) -- cycle);

  xaxis(" $x$ ", -1.5, 1.5, above = true);
  yaxis(" $y$ ", -1.5, 1.5, above = true);
[/asy]
For how many ordered pairs of integers  $(x, y)$  is the point  $(x, y)$  inside or on this curve?


Say a positive integer  $n$  is *radioactive* if one of its prime factors is strictly greater than  $\sqrt{n}$ . For example,  $2012 = 2^2 \cdot 503$ ,  $2013 = 3 \cdot 11 \cdot 61$  and  $2014 = 2 \cdot 19 \cdot 53$  are all radioactive, but  $2015 = 5 \cdot 13 \cdot 31$  is not. How many radioactive numbers have all prime factors less than  $30$ ?

*Proposed by Evan Chen*


For every positive integer  $n$ , define  $S_n$  to be the sum
\[
  S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n .
\]
As  $n$  approaches infinity, what value does  $S_n$  approach?


Among all ordered pairs of real numbers  $(a, b)$  satisfying  $a^4 + 2a^2b + 2ab + b^2 = 960$ , find the smallest possible value for  $a$ .


Let  $ (x_1,x_2,\cdots)$  be a sequence of positive numbers such that  $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$  and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]

Determine real number  $ a$  such that if  $ x_1 > a$ , then the sequence is monotonically decreasing, and if  $ 0 < x_1 < a$ , then the sequence is not monotonic.

Solution: \boxed{ a = 8^{1/8} }


Let  $(a_n)\subset (\frac{1}{2},1)$ . Define the sequence  $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$ . Is this sequence convergent? If yes find the limit.

Solution: \boxed{ 1 }


$n$  students take a test with  $m$  questions, where  $m,n\ge 2$  are integers. The score given to every question is as such: for a certain question, if  $x$  students fails to answer it correctly, then those who answer it correctly scores  $x$  points, while those who answer it wrongly scores  $0$ . The score of a student is the sum of his scores for the  $m$  questions. Arrange the scores in descending order  $p_1\ge p_2\ge \ldots \ge p_n$ . Find the maximum value of  $p_1+p_n$ .

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


Triangle  $ABC$  has  $AB=BC=10$  and  $CA=16$ . The circle  $\Omega$  is drawn with diameter  $BC$ .  $\Omega$  meets  $AC$  at points  $C$  and  $D$ . Find the area of triangle  $ABD$ .


Let  $ P(x) \in \mathbb{Z}[x]$  be a polynomial of degree  $ \text{deg} P \equal{} n > 1$ . Determine the largest number of consecutive integers to be found in  $ P(\mathbb{Z})$ .

*B. Berceanu*

Solution: \boxed{ n }


Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.


How many integers  $n>1$  are there such that  $n$  divides  $x^{13}-x$  for every positive integer  $x$ ?


Find all positive integer bases  $b \ge 9$  so that the number
\[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \]
is a perfect cube in base 10 for all sufficiently large positive integers  $n$ .

*Proposed by Yang Liu*

Solution: \boxed{ b = 10 }


$n$  is a natural number. Given   $3n \cdot 3n$  table, the unit cells are  colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen  $2 \cdot 2$  square's white cells are colored orange, orange are colored black and black are colored white. Find all  $n$   such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

Solution: \boxed{ n }


In triangle  $ABC$ , let  $M$  be the midpoint of  $BC$ ,  $H$  be the orthocenter, and  $O$  be the circumcenter. Let  $N$  be the reflection of  $M$  over  $H$ . Suppose that  $OA = ON = 11$  and  $OH = 7.$  Compute  $BC^2$ .


We say that a group of  $k$  boys is  $n-acceptable$  if removing any boy from the group one can always find, in the other  $k-1$  group,  a group of  $n$  boys such that everyone knows each other. For each  $n$ , find the biggest  $k$  such that in any group of  $k$  boys that is  $n-acceptable$  we must always have a group of  $n+1$  boys such that everyone knows each other.

Solution: \boxed{ k = 2n-1 }


We have  $n$  positive integers greater than  $1$  and less than  $10000$  such that neither of them is prime but any two of them are relative prime. Find the maximum value of  $n $ .

Solution: \boxed{ 25 }


For a positive integer  $N>1$  with unique factorization  $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$ , we define
\[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\]
Let  $a_1,a_2,\dotsc, a_n$  be positive integers and  $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$  such that for all positive integers  $k$ ,  $\Omega(P(k))$  is even. Show that  $n$  is an even number.

Solution: \boxed{ n }


Two sequences  $\{a_i\}$  and  $\{b_i\}$  are defined as follows:  $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$  and  $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $ . If both sequences are defined with  $i$  ranging across the natural numbers, how many numbers belong to both sequences?

*Proposed by Isabella Grabski*


The formula for converting a Fahrenheit temperature  $F$  to the corresponding Celsius temperature  $C$  is  $C=\frac{5}{9}(F-32)$ .  An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer.  For how many integer Fahrenheit temperatures  $T$  with  $32 \leq T \leq 1000$  does the original temperature equal the final temperature?


Let  $ABC$  be a right triangle with  $\angle{ACB}=90^{\circ}$ .  $D$  is a point on  $AB$  such that  $CD\perp AB$ . If the area of triangle  $ABC$  is  $84$ , what is the smallest possible value of   $$ AC^2+\left(3\cdot CD\right)^2+BC^2? $$  *2016 CCA Math Bonanza Lightning #2.3*


Find the values of  $a\in [0,\infty)$  for which there exist continuous functions  $f:\mathbb{R}\rightarrow \mathbb{R}$ , such that  $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$ .

Solution: \boxed{ a = 0 }


Let  $ABC$  be a right isosceles triangle with right angle at  $A$ . Let  $E$  and  $F$  be points on A $B$  and  $AC$  respectively such that  $\angle ECB = 30^o$  and  $\angle FBC = 15^o$ . Lines  $CE$  and  $BF$  intersect at  $P$  and line  $AP$  intersects side  $BC$  at  $D$ . Calculate the measure of angle  $\angle FDC$ .


Suppose  $ABCD$  is a rectangle whose diagonals meet at  $E$ . The perimeter of triangle  $ABE$  is  $10\pi$  and the perimeter of triangle  $ADE$  is  $n$ . Compute the number of possible integer values of  $n$ .


Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased.

*Proposed  by Ankan Bhattacharya*


Consider the graph on  $1000$  vertices  $v_1, v_2, ...v_{1000}$  such that for all  $1 \le i < j  \le 1000$ ,  $v_i$  is connected to  $v_j$  if and only if  $i$  divides  $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.


Consider  $4n$  points in the plane, with no three points collinear. Using these points as vertices, we form  $\binom{4n}{3}$  triangles. Show that there exists a point  $X$  of the plane that belongs to the interior of at least  $2n^3$  of these triangles.


Compute the largest integer  $N$  such that one can select  $N$  different positive integers, none of which is larger than  $17$ , and no two of which share a common divisor greater than  $1$ .


Find all pairs of positive integers  $(a, b)$  such that  $4b - 1$  is divisible by  $3a + 1$  and  $3a - 1$  is divisible by  $2b + 1$ .

Solution: \boxed{(2, 2)}


In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square  $\sqrt{5}$  away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability  $p$ . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of  $p$  that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

*Proposed by Lewis Chen*


$p(n) $  is a product of all digits of n.Calculate: $ p(1001) + p(1002) + ... + p(2011) $


In  $10\times 10$  square we choose  $n$  cells. In every chosen cell we draw one arrow from the angle to opposite angle.  It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of  $n$ ?


The integers from  $1$  to  $n$  are written, one on each of  $n$  cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
What is th smallest value of  $n$  for which the second player can ensure that he competes both his moves?


Let  $P(x)$  be a real quadratic trinomial, so that for all  $x\in \mathbb{R}$  the inequality  $P(x^3+x)\geq P(x^2+1)$  holds. Find the sum of the roots of  $P(x)$ .

*Proposed by A. Golovanov, M. Ivanov, K. Kokhas*


Given  $2014$  points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?


Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the  $2014$ th smallest perimeter of all such right triangles.


At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction  $\tfrac{m}{n}$ . Find  $m+n$ .


We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If  $a$  is a ridiculous number, then  $\sqrt{a}$  and  $1+\sqrt{a}$  are also ridiculous numbers.
[/list]
A closed interval  $I$  is ``boring'' if

- $I$  contains no ridiculous numbers, and
- There exists an interval  $[b,c]$  containing  $I$  for which  $b$  and  $c$  are both ridiculous numbers.

The smallest non-negative  $l$  such that there does not exist a boring interval with length  $l$  can be represented in the form  $\dfrac{a + b\sqrt{c}}{d}$  where  $a, b, c, d$  are integers,  $\gcd(a, b, d) = 1$ , and no integer square greater than 1 divides  $c$ . What is  $a + b + c + d$ ?


Let  $ABCD$  be a square of side length  $4$ . Points  $E$  and  $F$  are chosen on sides  $BC$  and  $DA$ , respectively, such that  $EF = 5$ . Find the sum of the minimum and maximum possible areas of trapezoid  $BEDF$ .

*Proposed by Andrew Wu*


Let  $P$  be a non-zero polynomial with non-negative real coefficients, let  $N$  be a positive integer, and let  $\sigma$  be a permutation of the set  $\{1,2,...,n\}$ . Determine the least value the sum
\[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as  $x_1,x_2,...,x_n$  run through the set of positive real numbers.

*Fedor Petrov*

Solution: \boxed{n}


The expressions  $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$  and  $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$  are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers  $A$  and  $B$ .


In the triangle  $ABC$ ,  $| BC | = 1$  and there is exactly one point  $D$  on the  side   $BC$   such that  $|DA|^2 =  |DB| \cdot |DC|$ . Determine all possible values of the perimeter of  the triangle  $ABC$ .

Solution: \boxed{\sqrt{2} + 1}


For real constant numbers  $ a,\ b,\ c,\ d,$  consider the function  $ f(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$  such that  $ f( \minus{} 1) \equal{} 0,\ f(1) \equal{} 0,\ f(x)\geq 1 \minus{} |x|$  for  $ |x|\leq 1.$

Find  $ f(x)$  for which  $ \int_{ \minus{} 1}^1 \{f'(x) \minus{} x\}^2\ dx$ 　is minimized.

Solution: \boxed{ f(x) = -x^2 + 1 }


Given that  $A,B,$  and  $C$  are noncollinear points in the plane with integer coordinates such that the distances  $AB,AC,$  and  $BC$  are integers, what is the smallest possible value of  $AB?$


In a massive school which has  $m$  students, and each student took at least one subject. Let  $p$  be an odd prime. Given that:

(i) each student took at most  $p+1$  subjects.
(ii) each subject is taken by at most  $p$  students.
(iii) any pair of students has at least  $1$  subject in common.

Find the maximum possible value of  $m$ .

Solution: \boxed{ p^2 }


In a rectangular  $57\times 57$  grid of cells,  $k$  of the cells are colored black. What is the smallest positive integer  $k$  such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?

[i]Proposed by James Lin


Let  $\alpha$  be a solution satisfying the equation  $|x|=e^{-x}.$  Let  $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find  $\lim_{n\to\infty} n^2I_n.$


Find the minimum positive value of  $ 1*2*3*4*...*2020*2021*2022$  where you can replace  $*$  as  $+$  or  $-$


Suppose  $F$  is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose  $S(F),P(F)$  are area and perimeter of  $F$ .
Find the smallest k that:
 $S(F) \leq k.P(F)^2$


Let  $T = (a,b,c)$  be a triangle with sides  $a,b$  and  $c$  and area  $\triangle$ .  Denote by  $T' = (a',b',c')$  the triangle whose sides are the altitudes of  $T$  (i.e.,  $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by  $\triangle '$ .  Similarly, let  $T'' = (a'',b'',c'')$  be the triangle formed from the altitudes of  $T'$ , and denote its area by  $\triangle ''$ . Given that  $\triangle ' = 30$  and  $\triangle '' = 20$ , find  $\triangle$ .


Let  $k$  be a positive integer. Lexi has a dictionary  $\mathbb{D}$  consisting of some  $k$ -letter strings containing only the letters  $A$  and  $B$ . Lexi would like to write either the letter  $A$  or the letter  $B$  in each cell of a  $k \times k$  grid so that each column contains a string from  $\mathbb{D}$  when read from top-to-bottom and each row contains a string from  $\mathbb{D}$  when read from left-to-right.
What is the smallest integer  $m$  such that if  $\mathbb{D}$  contains at least  $m$  different strings, then Lexi can fill her grid in this manner, no matter what strings are in  $\mathbb{D}$ ?


Define the sequence  $x_1, x_2, ...$  inductively by  $x_1 = \sqrt{5}$  and  $x_{n+1} = x_n^2 - 2$  for each  $n \geq 1$ . Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$ .


Let  $S$  be the set of triples  $(a,b,c)$  of non-negative integers with  $a+b+c$  even. The value of the sum
    \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
    can be expressed as  $\frac{m}{n}$  for relative prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Nathan Xiong*


What is the smallest positive integer  $n$  such that  $n=x^3+y^3$  for two different positive integer tuples  $(x,y)$ ?


Find the number of  $12$ -digit "words" that can be formed from the alphabet  $\{0,1,2,3,4,5,6\}$  if neighboring digits must differ by exactly  $2$ .


Let  $k$  and  $a$  are positive constants. Denote by  $V_1$  the volume of the solid generated by a rotation of the figure enclosed
by the curve  $C: y=\frac{x}{x+k}\ (x\geq 0)$ , the line  $x=a$  and the  $x$ -axis around the  $x$ -axis, and denote by  $V_2$  that of
the solid by a rotation of the figure enclosed by the curve  $C$ , the line  $y=\frac{a}{a+k}$  and the  $y$ -axis around the  $y$ -axis.
Find the ratio  $\frac{V_2}{V_1}.$

Solution: \boxed{k}


Find all triples  $(a, b, c)$  of positive integers for which   $$ \begin{cases} a + bc=2010  b + ca = 250\end{cases} $$

Solution: \boxed{ (3, 223, 9) }


Let  $a_1$ ,  $a_2$ ,  $\ldots\,$ ,  $a_{2019}$  be a sequence of real numbers.  For every five indices  $i$ ,  $j$ ,  $k$ ,  $\ell$ , and  $m$  from 1 through 2019, at least two of the numbers  $a_i$ ,  $a_j$ ,  $a_k$ ,  $a_\ell$ , and  $a_m$  have the same absolute value.  What is the greatest possible number of distinct real numbers in the given sequence?


The equation  $166\times 56 = 8590$  is valid in some base  $b \ge 10$  (that is,  $1, 6, 5, 8, 9, 0$  are digits in base  $b$  in the above equation). Find the sum of all possible values of  $b \ge 10$  satisfying the equation.


In English class, you have discovered a mysterious phenomenon -- if you spend  $n$  hours on an essay, your score on the essay will be  $100\left( 1-4^{-n} \right)$  points if  $2n$  is an integer, and  $0$  otherwise. For example, if you spend  $30$  minutes on an essay you will get a score of  $50$ , but if you spend  $35$  minutes on the essay you somehow do not earn any points.

It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?

*Proposed by Evan Chen*


Find all triples  $(a,b,c)$  of real numbers all different from zero that satisfies:

\begin{eqnarray}  a^4+b^2c^2=16a\nonumber  b^4+c^2a^2=16b \nonumber c^4+a^2b^2=16c \nonumber \end{eqnarray}

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


How many ordered pairs of real numbers  $(x, y)$  are there such that  $x^2+y^2 = 200$  and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer?


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


Compute the sum of all 2-digit prime numbers  $p$  such that there exists a prime number  $q$  for which  $100q + p$  is a perfect square.


Find the greatest possible value of  $ sin(cos x) \plus{} cos(sin x)$  and determine all real
numbers x, for which this value is achieved.

Solution: \boxed{ \sin(1) + 1 }


Let  $S$  be a set of  $2020$  distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]

Find the least possible value of the number of points in  $M$ .

Solution: \boxed{ 4037 }


Find all pairs of integers  $(c, d)$ , both greater than 1, such that the following holds:

For any monic polynomial  $Q$  of degree  $d$  with integer coefficients and for any prime  $p > c(2c+1)$ , there exists a set  $S$  of at most  $\big(\tfrac{2c-1}{2c+1}\big)p$  integers, such that
\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]
contains a complete residue system modulo  $p$  (i.e., intersects with every residue class modulo  $p$ ).

Solution: \boxed{ (c, d) }


Let  $c_i$  denote the  $i$ th composite integer so that  $\{c_i\}=4,6,8,9,...$  Compute
\[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\]
(Hint:  $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$ )


Let  $ABC$  be an equilateral triangle.  $A $  point  $P$  is chosen at random within this triangle. What is the probability that the sum of the distances from point  $P$  to the sides of triangle  $ABC$  are measures of the sides of a triangle?


Decomposition of number  $n$  is showing  $n$  as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer  $n$  show that number of decompositions is  $2^{n-1}$

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


**a)** Solve in  $ \mathbb{R} $  the equation  $ 2^x=x+1. $ **b)** If a function  $ f:\mathbb{R}\longrightarrow\mathbb{R} $  has the property that   $$  (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} ,  $$  then  $ f(0)+f(1)=1. $

Solution: \boxed{ f(0) + f(1) = 1 }


Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,l)\) have distinct colours. What is the least possible number of colours needed?


Integers a, b, c, d, and e satisfy the following three properties:
(i)  $2 \le a < b <c <d <e <100$
(ii) $ \gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence.
What is the value of c?


Let  $a>1$  be an odd positive integer. Find the least positive integer  $n$  such that  $2^{2000}$  is a divisor of  $a^n-1$ .

*Mircea Becheanu*


Suppose  $z_1, z_2 , \cdots z_n$  are  $n$  complex numbers such that  $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$ . Find the maximum possible value of  $n$ . Further characterise all such maximal configurations.

Solution: \boxed{ n = 7 }


Let  $ABCD$  be a cyclic quadrilateral, and suppose that  $BC = CD = 2$ . Let  $I$  be the incenter of triangle  $ABD$ . If  $AI = 2$  as well, find the minimum value of the length of diagonal  $BD$ .


$2000$  people are standing on a line. Each one of them is either a *liar*, who will always lie, or a *truth-teller*, who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.

Solution: \boxed{ 1000 }


Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than  $11$ .
P.s. the current year in the problem is  $2010$ .

Solution: \boxed{ 24 }


Alice writes differents real numbers in the board, if  $a,b,c$  are three numbers in this board, least one of this numbers  $a + b, b + c, a + c$  also is a number in the board. What's the largest quantity of numbers written in the board???

Solution: \boxed{ 7 }


The difference between the maximal and the minimal diagonals of the regular  $n$ -gon equals to its side (  $n > 5$  ).  Find  $n$ .

Solution: \boxed{ n = 9 }


Each of the spots in a  $8\times 8$  chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?**Remark:** A black horse could attack another black horse.


Let  $n,k$ ,  $1\le k\le n$  be fixed integers. Alice has  $n$  cards in a row, where the card has position  $i$  has the label  $i+k$  (or  $i+k-n$  if  $i+k>n$ ). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card  $i$  is at position  $i$ ).

NOTE: edited from original phrasing, which was ambiguous.

Solution: \boxed{ n - \gcd(n, k) }


Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences)


A rope of length 10 *m* is tied tautly from the top of a flagpole to the ground 6 *m* away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 *cm/min*. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)


Compute the area of the trapezoid  $ABCD$  with right angles  $BAD$  and  $ADC$  and side lengths of  $AB=3$ ,  $BC=5$ , and  $CD=7$ .


Let  $C$  and  $D$  be points on the semicircle with center  $O$  and diameter  $AB$  such that  $ABCD$  is a convex quadrilateral. Let  $Q$  be the intersection of the diagonals  $[AC]$  and  $[BD]$ , and  $P$  be the intersection of the lines tangent to the semicircle at  $C$  and  $D$ . If  $m(\widehat{AQB})=2m(\widehat{COD})$  and  $|AB|=2$ , then what is  $|PO|$ ? $
\textbf{(A)}\ \sqrt 2
\qquad\textbf{(B)}\ \sqrt 3
\qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2
\qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2}
\qquad\textbf{(E)}\ \frac{2\sqrt 3} 3
$


Let  $m > n$  be positive integers such that  $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find  $3m + n$ .


Given an interger  $n\geq 2$ , determine the maximum value the sum  $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$  may achieve, and the points at which the maximum is achieved, as  $a_1,a_2,...a_n$  run over all positive real numers subject to  $a_k\geq a_1+a_2...+a_{k-1}$ , for  $k=2,...n$

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


The graph of the equation  $y = 5x  + 24$  intersects the graph of the equation  $y = x^2$  at two points. The two points are a distance  $\sqrt{N}$  apart. Find  $N$ .


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

*2022 CCA Math Bonanza Individual Round #5*


Suppose  $p < q < r < s$  are prime numbers such that  $pqrs + 1 = 4^{p+q}$ . Find  $r + s$ .


In how many ways can one select eight integers  $a_1,a_2, ... ,a_8$ , not necesarily distinct, such that  $1 \le  a_1 \le ... \le a_8 \le 8$ ?


Graph  $G_1$  of a quadratic trinomial  $y = px^2 + qx + r$  with real coefficients intersects the graph  $G_2$  of a quadratic trinomial  $y = x^2$  in points  $A$ ,  $B$ . The intersection of tangents to  $G_2$  in points  $A$ ,  $B$  is point  $C$ . If  $C \in G_1$ , find all possible values of  $p$ .

Solution: \boxed{ p = 2 }


Find, with explanation, the maximum value of  $f(x)=x^3-3x$  on the set of all real numbers  $x$  satisfying  $x^4+36\leq 13x^2$ .


Let  $(a_n)_{n \geq 0}$  be the sequence of integers defined recursively by  $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$  for  $n \geq 0.$  Find the common divisors of  $a_{1986}$  and  $a_{6891}.$


Determine the number of pairs of positive integers  $x,y$  such that  $x\le y$ ,  $\gcd (x,y)=5!$  and  $\text{lcm}(x,y)=50!$ .


Determine the largest natural number  $r$  with the property that among any five subsets with  $500$  elements of the set  $\{1,2,\ldots,1000\}$  there exist two of them which share at least  $r$  elements.

Solution: \boxed{ r = 200 }


Let  $ABC$  be a triangle with side lengths  $AB=6, AC=7,$  and  $BC=8.$  Let  $H$  be the orthocenter of  $\triangle ABC$  and  $H'$  be the reflection of  $H$  across the midpoint  $M$  of  $BC.$   $\tfrac{[ABH']}{[ACH']}$  can be expressed as  $\frac{p}{q}$ . Find  $p+q$ .

*2022 CCA Math Bonanza Individual Round #14*


The sequence  $ (a_n)$  is given by  $ a_1\equal{}1,a_2\equal{}0$  and:

 $ a_{2k\plus{}1}\equal{}a_k\plus{}a_{k\plus{}1}, a_{2k\plus{}2}\equal{}2a_{k\plus{}1}$  for  $ k \in \mathbb{N}.$

Find  $ a_m$  for  $ m\equal{}2^{19}\plus{}91.$


Let  $k$  be a fixed positive integer. The  $n$ th derivative of  $\tfrac{1}{x^k-1}$  has the form  $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$ , where  $P_n(x)$  is a polynomial. Find  $P_n(1)$ .

Solution: \boxed{ (-1)^n n! k^n }


A good approximation of  $\pi$  is  $3.14.$  Find the least positive integer  $d$  such that if the area of a circle with diameter  $d$  is calculated using the approximation  $3.14,$  the error will exceed  $1.$


You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 ﬁves. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?

Solution: \boxed{ \frac{1}{7} }


Let  $ d_n$  be the determinant of the  $ n\times n$  matrix whose entries, from left to right and then from top to bottom, are  $ \cos 1,\cos 2,\dots,\cos n^2.$  (For example,  $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3
\cos4 & \cos5 & \cos 6
\cos7 & \cos8 & \cos 9\end{vmatrix}.$  The argument of   $ \cos$   is always in radians, not degrees.)

Evaluate  $ \lim_{n\to\infty}d_n.$


Compute the sum of  $x^2+y^2$  over all four ordered pairs  $(x,y)$  of real numbers satisfying  $x=y^2-20$  and  $y=x^2+x-21$ .

*2021 CCA Math Bonanza Lightning Round #3.4*


Suppose  $\{ x_n \}_{n\geq 1}$  is a sequence of positive real numbers such that  $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$ , and for all  $n$  \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 .  \] Show that for all  $k$  \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3.  \]


Let  $ P(x)$  be a nonzero polynomial such that, for all real numbers  $ x$ ,  $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$ . Determine the maximum possible number of real roots of  $ P(x)$ .


Find all triples  $ \left(p,x,y\right)$  such that  $ p^x\equal{}y^4\plus{}4$ , where  $ p$  is a prime and  $ x$  and  $ y$  are natural numbers.

Solution: \boxed{(5, 1, 1)}


Let  $ n>1$  and for  $ 1 \leq k \leq n$  let  $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$  be the sum of the products of all possible combinations of k of the numbers  $ a_1,a_2,...,a_n$ . Furthermore let  $ P \equal{} P(a_1, a_2, . . . , a_n)$  be the sum of all  $ p_k$  with odd values of  $ k$  less than or equal to  $ n$ .

How many different values are taken by  $ a_j$  if all the numbers  $ a_j (1 \leq j \leq n)$  and  $ P$  are prime?

Solution: \boxed{ 2 }


From the vertex  $ A$  of the equilateral triangle  $ ABC$  a line is drown that intercepts the segment  $ [BC]$  in the point  $ E$ . The point  $ M \in (AE$  is such that  $ M$  external to  $ ABC$ ,  $ \angle AMB \equal{} 20 ^\circ$  and  $ \angle AMC \equal{} 30 ^ \circ$ . What is the measure of the angle  $ \angle MAB$ ?


Define the sequence  $a_0,a_1,\dots$  inductively by  $a_0=1$ ,  $a_1=\frac{1}{2}$ , and
\[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\]
Show that the series  $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$  converges and determine its value.

*Proposed by Christophe Debry, KU Leuven, Belgium.*


An exam at a university consists of one question randomly selected from the $ n$  possible questions. A student knows only one question, but he can take the exam  $n$  times. Express as a function of  $n$  the probability  $p_n$  that the student will pass the exam. Does  $p_n$  increase or decrease as  $n$  increases? Compute  $lim_{n\to \infty}p_n$ . What is the largest lower bound of the probabilities  $p_n$ ?

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


You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with ﬁfteen cards?


Find all  $x \in  R$  such that    $$  x - \left[ \frac{x}{2016} \right]= 2016 $$  , where  $[k]$  represents the largest smallest integer or equal to  $k$ .


The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by  $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-ﬁfth as large if measured in Celsius degrees?

Solution: \boxed{ F = -4 }


A computer generates all pairs of real numbers  $x, y \in (0, 1)$  for which the numbers  $a = x+my$  and  $b = y+mx$  are both integers, where  $m$  is a given positive integer. Finding one such pair  $(x, y)$  takes  $5$  seconds. Find  $m$  if  the computer needs  $595$  seconds to find all possible ordered pairs  $(x, y)$ .

Solution: \boxed{m = 11}


Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Nathan Xiong*


In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30.  Find the sum of the four terms.


In the diagram below  $ \angle CAB, \angle CBD$ , and  $\angle CDE$  are all right angles with side lengths  $AC = 3$ ,  $BC = 5$ ,  $BD = 12$ , and  $DE = 84$ . The distance from point  $E$  to the line  $AB$  can be expressed as the ratio of two relatively prime positive integers,  $m$  and  $n$ . Find  $m + n$ .

[asy]
size(300);
defaultpen(linewidth(0.8));
draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));
label(" $A$ ",origin,SW);
label(" $B$ ",(0,4),dir(160));
label(" $C$ ",(3,0),S);
label(" $D$ ",(6,8),dir(80));
label(" $E$ ",(30,-4),E);[/asy]


In square  $ABCD$ ,  $\overline{AC}$  and  $\overline{BD}$  meet at point  $E$ .
Point  $F$  is on  $\overline{CD}$  and  $\angle CAF = \angle FAD$ .
If  $\overline{AF}$  meets  $\overline{ED}$  at point  $G$ , and if  $\overline{EG} = 24$  cm, then find the length of  $\overline{CF}$ .


Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a circle. Let  $AC$  and  $BD$  intersect at point  $E$  inside the circle. If  $[ABE]\cdot[CDE]=36$ , what is the value of  $[ADE]\cdot[BCE]$ ? (Given a triangle  $\triangle ABC$ ,  $[ABC]$  denotes its area.)


Let  $a$  and  $b$  be real numbers such that  $a+b=\log_2( \log_2 3)$ . What is the minimum value of  $2^a + 3^b$  ?

Solution: \boxed{\log_2 6}


Find all positive integers  $n$  that have 4 digits, all of them perfect squares, and such that  $n$  is divisible by 2, 3, 5 and 7.


Let  $ABCD$  be a quadrilateral with  $\overline{AB}\parallel\overline{CD}$ ,  $AB=16$ ,  $CD=12$ , and  $BC<AD$ . A circle with diameter  $12$  is inside of  $ABCD$  and tangent to all four sides. Find  $BC$ .


In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of  $\alpha$ ,  $\beta$ , and  $\alpha + \beta$  radians, respectively, where  $\alpha + \beta < \pi$ .  If  $\cos \alpha$ , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Solution: \boxed{ 49 }


Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009
& x^3 + 5x^2-49x-245
& x^3 + 39x^2 - 117x - 1435\end{align*}


Find all complex-valued functions  $f$  of a complex variable such that   $$ f(z)+zf(1-z)=1+z $$  for all  $z\in \mathbb{C}$ .

Solution: \boxed{ f(z) = 1 }


There are  $800$  marbles in a bag. Each marble is colored with one of  $100$  colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.

Suppose Anna has not stopped after drawing  $699$  marbles. Compute the probability that she stops immediately after drawing the  $700$ th marble.


A set of 10 distinct integers  $S$  is chosen. Let  $M$  be the number of nonempty subsets of  $S$  whose elements have an even sum. What is the minimum possible value of  $M$ ?
<details><summary>Clarifications</summary>

- $S$  is the ``set of 10 distinct integers'' from the first sentence.
</details>

*Ray Li*


Find all real numbers  $x$  such that  $-1 < x \le 2 $  and  $$ \sqrt{2 - x}+\sqrt{2 + 2x} =\sqrt{\frac{x^4 + 1}{x^2 + 1}}+ \frac{x + 3}{x + 1}. $$  .

Solution: \boxed{ x = 1 }


The function  $f : \mathbb{R}\to\mathbb{R}$  satisfies  $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$  for all real  $x$ .  Given that  $f(1)=1$  and  $f^{\prime\prime\prime}(1)=8$ , determine  $f^\prime (1)+f^{\prime\prime}(1)$ .


For all real numbers  $r$ , denote by  $\{r\}$  the fractional part of  $r$ , i.e. the unique real number  $s\in[0,1)$  such that  $r-s$  is an integer.  How many real numbers  $x\in[1,2)$  satisfy the equation  $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$


Find the largest possible value of  $k$  for which  $3^{11}$  is expressible as the sum of  $k$  consecutive positive integers.


Bob and Bill's history class has  $32$  people in it, but only  $30$  people are allowed per class. Two people will be randomly selected for transfer to a random one of two history classes. What is the probability that Bob and Bill are both transferred, and that they are placed in the same class? Write your answer as a fraction in lowest terms.


Find all polynomials of the form   $$ P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1) $$   with integer coefficients, having  $n$  real roots  $x_1,\dots,x_n$  satisfying  $k \leq x_k \leq k+1$  for  $k=1, \dots,n$ .

Solution: \boxed{ P_1(x) = x - 2 }


While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at  $ 10 \text{m/s}$ , and you manage a three-second head start over the raptor. If the raptor runs at  $ 15\sqrt{2} \text{m/s}$ , but only runs either north or east at any given time, how many seconds do you have until it devours you?

Solution: \boxed{6 \text{ seconds}}


Let  $f(x) = x^2 + 6x + 7$ . Determine the smallest possible value of  $f(f(f(f(x))))$  over all real numbers  $x.$


Two cars travel along a circular track  $n$  miles long, starting at the same point. One car travels  $25$  miles along the track in some direction. The other car travels  $3$  miles along the track in some direction. Then the two cars are once again at the same point along the track. If  $n$  is a positive integer, find the sum of all possible values of  $n.$


How many integers  $n$  are there such that  $0  \le n \le 720$  and  $n^2 \equiv 1$  (mod  $720$ )?


Let  $ABC$  be a triangle with area  $92$  square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle  $ABC$ .


Two circles are said to be *orthogonal* if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles  $\omega_1$  and  $\omega_2$  with radii  $10$  and  $13$ , respectively, are externally tangent at point  $P$ . Another circle  $\omega_3$  with radius  $2\sqrt2$  passes through  $P$  and is orthogonal to both  $\omega_1$  and  $\omega_2$ . A fourth circle  $\omega_4$ , orthogonal to  $\omega_3$ , is externally tangent to  $\omega_1$  and  $\omega_2$ . Compute the radius of  $\omega_4$ .


There is a rectangle  $ABCD$  such that  $AB=12$  and  $BC=7$ .  $E$  and  $F$  lie on sides  $AB$  and  $CD$  respectively such that  $\frac{AE}{EB} = 1$  and  $\frac{CF}{FD} = \frac{1}{2}$ . Call  $X$  the intersection of  $AF$  and  $DE$ . What is the area of pentagon  $BCFXE$ ?

Proposed by Minseok Eli Park (wolfpack)


Let  $AB$  be diameter of a circle  $\omega$  and let  $C$  be a point on  $\omega$ , different from  $A$  and  $B$ . The perpendicular from  $C$  intersects  $AB$  at  $D$  and  $\omega$  at  $E(\neq C)$ . The circle with centre at  $C$  and radius  $CD$  intersects  $\omega$  at  $P$  and  $Q$ . If the perimeter of the triangle  $PEQ$  is  $24$ , find the length of the side  $PQ$


Two circles, both with the same radius  $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points  $A,B$  and the other at points  $C,D$ , so that  $|AB|=|BC|=|CD|=14\text{cm}$ . Another line intersects the circles at  $E,F$ , respectively  $G,H$  so that  $|EF|=|FG|=|GH|=6\text{cm}$ . Find the radius  $r$ .


The roots of a monic cubic polynomial  $p$  are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to  $10$ . Under these conditions, the largest possible value of  $|p(-1)|$  can be written as  $\frac{m}{n}$ , where  $m$ ,  $n$  are relatively prime integers. Find  $m + n$ .


The bisector of angle  $A$  of triangle  $ABC$  meet its circumcircle  $\omega$  at point  $W$ . The circle  $s$  with diameter  $AH$  ( $H$  is the orthocenter of  $ABC$ ) meets  $\omega$  for the second time at point  $P$ . Restore the triangle  $ABC$  if the points  $A$ ,  $P$ ,  $W$  are given.

Solution: \boxed{ ABC }


Let  $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$ . What is the largest integer that is a divisor of  $P(n)$  for all positive even integers  $n$ ?


Let  $n$  a positive integer. We call a pair  $(\pi ,C)$  composed by a permutation  $\pi$  $:$  { $1,2,...n$ } $\rightarrow$ { $1,2,...,n$ } and a binary function  $C:$  { $1,2,...,n$ } $\rightarrow$ { $0,1$ } "revengeful" if it satisfies the two following conditions: $1)$ For every  $i$   $\in$  { $1,2,...,n$ }, there exist  $j$   $\in$   $S_{i}=$ { $i, \pi(i),\pi(\pi(i)),...$ } such that  $C(j)=1$ . $2)$  If  $C(k)=1$ , then  $k$  is one of  the  $v_{2}(|S_{k}|)+1$  highest elements of  $S_{k}$ , where  $v_{2}(t)$  is the highest nonnegative integer such that  $2^{v_{2}(t)}$  divides  $t$ , for every positive integer  $t$ .

Let  $V$  the number of revengeful pairs and  $P$  the number of partitions of  $n$  with all parts powers of  $2$ .
Determine  $\frac{V}{P}$ .

Solution: \boxed{ n! }


Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with  $x_1$ ,  $x_2$ ,  $\cdots$ ,  $x_n$  are distinct postive integers. Find the maximum value of  $n$ .

*Proposed by Le Duc Minh*


An triangle with coordinates  $(x_1,y_1)$ ,  $(x_2, y_2)$ ,  $(x_3,y_3)$  has centroid at  $(1,1)$ . The ratio between the lengths of the sides of the triangle is  $3:4:5$ . Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\]
    the area of the triangle can be expressed as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

*2021 CCA Math Bonanza Individual Round #11*


In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?


Find the maximum number of planes in the space, such there are  $ 6$  points, that satisfy to the following conditions:
**1.**Each plane contains at least  $ 4$  of them
**2.**No four points are collinear.


For a finite set of naturals  $(C)$ , the product of its elements is going to be noted  $P(C)$ . We are going to define  $P (\phi) = 1$ . Calculate the value of the expression   $$ \sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)} $$


For each positive integer  $n$  let  $A_n$  be the  $n \times n$  matrix such that its  $a_{ij}$  entry is equal to  ${i+j-2 \choose j-1}$  for all  $1\leq i,j \leq n.$  Find the determinant of  $A_n$ .


(1) For  $ a>0,\ b\geq 0$ , Compare

 $ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$ .

(2) Find  $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$ .

Solution: \boxed{ 1 }


Find the maximal  $x$  such that the expression  $4^{27} + 4^{1000} + 4^x$  is the exact square.

Solution: \boxed{ x = 1972 }


Suppose  $\mathcal{T}=A_0A_1A_2A_3$  is a  tetrahedron with  $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$ ,  $A_0A_1=5, A_0A_2=12$  and  $A_0A_3=9$ . A cube  $A_0B_0C_0D_0E_0F_0G_0H_0$  with side length  $s$  is inscribed inside  $\mathcal{T}$  with  $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$ , and  $G_0\in \triangle A_1A_2A_3$ ; what is  $s$ ?


Find the smallest possible positive integer n with the following property:
For all positive integers  $x, y$  and  $z$  with  $x | y^3$  and  $y | z^3$  and  $z | x^3$  always to be true that  $xyz| (x + y + z) ^n$ .

(Gerhard J. Woeginger)


Find the sum of all distinct possible values of  $x^2-4x+100$ , where  $x$  is an integer between 1 and 100, inclusive.

*Proposed by Robin Park*


Let  $z=z(x,y)$  be implicit function with two variables from  $2sin(x+2y-3z)=x+2y-3z$ . Find  $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$ .


Let  $f: \mathbb{R} \to \mathbb{R}$  be a differentiable function such that  $f'(x) > f(x)>0$  for all real numbers  $x$ . Show that  $f(8) > 2022f(0)$ .

*Proposed by Ethan Tan*

Solution: \boxed{ f(8) > 2022 f(0) }


Determine all functions  $f : \mathbb{R}^2 \to\mathbb {R}$  for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever  $A,B,C,D$  are the vertices of a square with side-length one.

*Ilir Snopce*

Solution: \boxed{ f(x, y) = 0 }


What is the smallest perfect square larger than  $1$  with a perfect square number of positive integer factors?

*Ray Li*


Let  $M$  be the midpoint of the base  $AC$  of an acute-angled isosceles triangle  $ABC$ . Let  $N$  be the reflection of  $M$  in  $BC$ . The line parallel to  $AC$  and passing through  $N$  meets  $AB$  at point  $K$ . Determine the value of  $\angle AKC$ .

(A.Blinkov)


At least  $ n - 1$  numbers are removed from the set  $\{1, 2, \ldots, 2n - 1\}$  according to the following rules:
(i) If  $ a$  is removed, so is  $ 2a$ ;
(ii) If  $ a$  and  $ b$  are removed, so is  $ a \plus{} b$ .
Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.

Solution: \boxed{ n^2 }


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


A *strip* is the region between two parallel lines.  Let  $A$  and  $B$  be two strips in a plane.  The intersection of strips  $A$  and  $B$  is a parallelogram  $P$ .  Let  $A'$  be a rotation of  $A$  in the plane by  $60^\circ$ .  The intersection of strips  $A'$  and  $B$  is a parallelogram with the same area as  $P$ .  Let  $x^\circ$  be the measure (in degrees) of one interior angle of  $P$ .  What is the greatest possible value of the number  $x$ ?


Let  $a_1, a_2, \cdots, a_{2022}$  be nonnegative real numbers such that  $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs  $(i, j)$ ,  $1\leq i,j\leq 2022$ , satisfying   $$ a_i^2+a_j\ge \frac 1{2021}. $$

Solution: \boxed{2021 \times 2022}


5. Let  $S$  denote the set of all positive integers whose prime factors are elements of  $\{2,3,5,7,11\}$ . (We include 1 in the set  $S$ .) If  $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$  can be written as  $\frac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ , find  $a+b$ . (Here  $\varphi$  denotes Euler's totient function.)


The expressions  $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$  and  $ abc$  are called the elementary symmetric expressions on the three letters  $ a, b, c;$  symmetric because if we interchange any two letters, say  $ a$  and  $ c,$  the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let  $ S_k(n)$  denote the elementary expression on  $ k$  different letters of order  $ n;$  for example  $ S_4(3) \equal{} abc \plus{} abd \plus{} acd \plus{} bcd.$  There are four terms in  $ S_4(3).$  How many terms are there in  $ S_{9891}(1989)?$  (Assume that we have  $ 9891$  different letters.)


This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant  $X$ , let  $t(X)$  be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$  For any two friends  $X'$  and  $X''$ , we have  $t(X') \neq t(X''),$  $2)$  For every contestant  $X$ , the set  $\{ t(Y) : Y \text{ is a friend of } X \}$  consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?


An ant is on one face of a cube.  At every step, the ant walks to one of its four neighboring faces with equal probability.  What is the expected (average) number of steps for it to reach the face opposite its starting face?


Find all real numbers  $a$  for which the equation  $x^2a- 2x + 1 = 3 |x|$  has exactly three distinct real solutions in  $x$ .


The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction).


Compute the sum of all positive integers  $n$  such that  $n^n$  has 325 positive integer divisors. (For example,  $4^4=256$  has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)


Compute the number of nonempty subsets  $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$   that satisfy   $$ |S| +\ min(S) \cdot \max (S) = 0. $$


Given are  $100$  positive integers whose sum equals their product. Determine the minimum number of  $1$ s that may occur among the  $100$  numbers.

Solution: \boxed{ 95 }


Let  $ y_0$  be chosen randomly from  $ \{0, 50\}$ , let  $ y_1$  be chosen randomly from  $ \{40, 60, 80\}$ , let  $ y_2$  be chosen randomly from  $ \{10, 40, 70, 80\}$ , and let  $ y_3$  be chosen randomly from  $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let  $ P$  be the unique polynomial of degree less than or equal to  $ 3$  such that  $ P(0) \equal{} y_0$ ,  $ P(1) \equal{} y_1$ ,  $ P(2) \equal{} y_2$ , and  $ P(3) \equal{} y_3$ . What is the expected value of  $ P(4)$ ?


An equilateral triangle  $ABC$  is divided by nine lines parallel to  $BC$  into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is  $20$   $\text{cm}^2$ .

What is the area of triangle  $ABC$ ?


To each pair of nonzero real numbers  $a$  and  $b$  a real number  $a*b$  is assigned so that  $a*(b*c) = (a*b)c$  and  $a*a = 1$  for all  $a,b,c$ . Solve the equation  $x*36 = 216$ .


How many ways are there to insert  $+$ 's between the digits of  $111111111111111$  (fifteen  $1$ 's) so that the result will be a multiple of  $30$ ?


In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with  $2018$  links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating  $2$  or  $3$  new chains. For example, if she cuts the  $1111$ th link out of her chain first, then she will have  $3$  chains, of lengths  $1110$ ,  $1$ , and  $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from  $1$  to  $2018$  links using some combination of her chains?

*2018 CCA Math Bonanza Individual Round #10*


Let  $x_n=2^{2^{n}}+1$  and let  $m$  be the least common multiple of  $x_2, x_3, \ldots, x_{1971}.$  Find the last digit of  $m.$


In an acute scalene triangle  $ABC$ , points  $D,E,F$  lie on sides  $BC, CA, AB$ , respectively, such that  $AD \perp BC, BE \perp CA, CF \perp AB$ . Altitudes  $AD, BE, CF$  meet at orthocenter  $H$ . Points  $P$  and  $Q$  lie on segment  $EF$  such that  $AP \perp EF$  and  $HQ \perp EF$ . Lines  $DP$  and  $QH$  intersect at point  $R$ . Compute  $HQ/HR$ .

*Proposed by Zuming Feng*


For a positive integer  $n>1$ , let  $g(n)$  denote the largest positive proper divisor of  $n$  and  $f(n)=n-g(n)$ . For example,  $g(10)=5, f(10)=5$  and  $g(13)=1,f(13)=12$ . Let  $N$  be the smallest positive integer such that  $f(f(f(N)))=97$ . Find the largest integer not exceeding  $\sqrt{N}$


Find a costant  $C$ , such that   $$    \frac{S}{ab+bc+ca}\le  C $$   where  $a,b,c$  are the side lengths of an arbitrary triangle, and  $S$  is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.)


Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?


Find the maximum value of  $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$  ,  $x,y, z > 0$


Evaluate  $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$ , where  $\mathrm{Arccot}\,t$  for  $t \geq 0$  denotes the number  $\theta$  in the interval  $0 < \theta \leq \pi/2$  with  $\cot \theta = t$ .


Construct the  $ \triangle ABC$ , given  $ h_a$ ,  $ h_b$  (the altitudes from  $ A$  and  $ B$ ) and  $ m_a$ , the median from the vertex  $ A$ .

Solution: \boxed{ \triangle ABC }


Let  $a_1=24$  and form the sequence  $a_n$ ,  $n\geq 2$  by  $a_n=100a_{n-1}+134$ . The first few terms are  $$ 24,2534,253534,25353534,\ldots $$  What is the least value of  $n$  for which  $a_n$  is divisible by  $99$ ?


Find the number of ordered quadruples of positive integers  $(a,b,c, d)$  such that  $ab + cd = 10$ .


4. Suppose that  $\overline{A2021B}$  is a six-digit integer divisible by  $9$ . Find the maximum possible value of  $A \cdot B$ .

5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?

6. Suppose that  $a$  and  $b$  are positive integers such that  $\frac{a}{b-20}$  and  $\frac{b+21}{a}$  are positive integers. Find the maximum possible value of  $a + b$ .


For  $\{1, 2, 3, \dots, n\}$  and each of its nonempty subsets a unique **alternating sum** is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers.  (For example, the alternating sum for  $\{1, 2, 4, 6, 9\}$  is  $9 - 6 + 4 - 2 + 1 = 6$  and for  $\{5\}$  it is simply 5.)  Find the sum of all such alternating sums for  $n = 7$ .


A four-digit positive integer is called *virtual* if it has the form  $\overline{abab}$ , where  $a$  and  $b$  are digits and  $a \neq 0$ . For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form  $n^2+1$ , for some positive integer  $n$ .


Let  $S$  be the set of all ordered triples  $\left(a,b,c\right)$  of positive integers such that  $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$  and  $a+b+c\leq M$  for some positive integer  $M$ . Given that  $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$ , what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of  $k$ ?

*2018 CCA Math Bonanza Lightning Round #4.1*


On a blackboard the product  $log_{( )}[ ]\times\dots\times log_{( )}[ ]$  is written (there are 50 logarithms in the product). Donald has  $100$  cards:  $[2], [3],\dots, [51]$  and  $(52),\dots,(101)$ . He is replacing each  $()$   with some card of form  $(x)$  and each  $[]$  with some card of form  $[y]$ . Find the difference between largest and smallest values Donald can achieve.


Let  $p$  be a prime number and let  $\mathbb{F}_p$  be the finite field with  $p$  elements. Consider an automorphism  $\tau$  of the polynomial ring  $\mathbb{F}_p[x]$  given by
\[\tau(f)(x)=f(x+1).\]
Let  $R$  denote the subring of  $\mathbb{F}_p[x]$  consisting of those polynomials  $f$  with  $\tau(f)=f$ . Find a polynomial  $g \in \mathbb{F}_p[x]$  such that  $\mathbb{F}_p[x]$  is a free module over  $R$  with basis  $g,\tau(g),\dots,\tau^{p-1}(g)$ .

Solution: \boxed{ g = x^{p-1} }


The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points  $A (1, 2)$  and  $B (3,1)$ . Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point  $O (0, 0)$ ? Point A on the board located above and to the left of point  $B$ .

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


Determine all positive integers  $n$  for which there exist positive integers  $a_1,a_2, ..., a_n$

with  $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$  and  $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

Solution: \boxed{ n = 3 }


Pick out three numbers from  $0,1,\cdots,9$ , their sum is an even number and not less than  $10$ . We have________different ways to pick numbers.


If  $a$  and  $b$  are each randomly and independently chosen in the interval  $[-1, 1]$ , what is the probability that  $|a|+|b|<1$ ?


a) Show that  $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$ .
b) Show that  $\det(A)>0, \forall A$  fulfilling the above condition.

Solution: \boxed{\det(A) > 0}


The coefficients of the polynomial  $P(x)$  are nonnegative integers, each less than 100. Given that  $P(10) = 331633$  and  $P(-10) = 273373$ , compute  $P(1)$ .


Find the smallest positive integer  $k$  such that  $k!$  ends in at least  $43$  zeroes.


Let  $n\in \mathbb{Z}_{> 0}$ . The  set  $S$  contains all positive integers written in decimal form that simultaneously satisfy the following conditions:
[list=1][*] each element of  $S$  has exactly  $n$  digits;
[*] each element of  $S$  is divisible by  $3$ ;
[*] each element of  $S$  has all its digits from the set  $\{3,5,7,9\}$  [/list]
Find  $\mid S\mid$

Solution: \boxed{\frac{4^n + 2}{3}}


Let  $x,y\in\mathbb{R}$  be such that  $x = y(3-y)^2$  and  $y = x(3-x)^2$ . Find all possible values of  $x+y$ .

Solution: \boxed{0, 3, 4, 5, 8}


Find the largest  $K$  satisfying the following:

Given any closed intervals  $A_1,\ldots, A_N$  of length  $1$  where  $N$  is an arbitrary positive integer. If their union is  $[0,2021]$ , then we can always find  $K$  intervals from  $A_1,\ldots, A_N$  such that the intersection of any two of them is empty.

Solution: \boxed{ K = 1011 }


Let  $A = (a_1, a_2, \ldots, a_{2001})$  be a sequence of positive integers. Let  $m$  be the number of 3-element subsequences  $(a_i,a_j,a_k)$  with  $1 \leq i < j < k \leq 2001$ , such that  $a_j = a_i + 1$  and  $a_k = a_j + 1$ .  Considering all such sequences  $A$ , find the greatest value of  $m$ .


How many positive integers less that  $200$  are relatively prime to either  $15$  or  $24$ ?


Let  $P(x)=x+1$  and  $Q(x)=x^2+1.$  Consider all sequences  $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$  such that  $(x_1,y_1)=(1,3)$  and  $(x_{k+1},y_{k+1})$  is either  $(P(x_k), Q(y_k))$  or  $(Q(x_k),P(y_k))$  for each  $k.  $  We say that a positive integer  $n$  is nice if  $x_n=y_n$  holds in at least one of these sequences. Find all nice numbers.

Solution: \boxed{ n = 3 }


In triangle  $ABC$ , let  $I, O, H$  be the incenter, circumcenter and orthocenter, respectively. Suppose that  $AI = 11$  and  $AO = AH = 13$ . Find  $OH$ .

*Proposed by Kevin You*


A positive integer is called *oneic* if it consists of only  $1$ 's. For example, the smallest three oneic numbers are  $1$ ,  $11$ , and  $111$ . Find the number of  $1$ 's in the smallest oneic number that is divisible by  $63$ .


Determine all natural numbers  $n$  for which there exists a permutation  $(a_1,a_2,\ldots,a_n)$  of the numbers  $0,1,\ldots,n-1$  such that, if  $b_i$  is the remainder of  $a_1a_2\cdots a_i$  upon division by  $n$  for  $i=1,\ldots,n$ , then  $(b_1,b_2,\ldots,b_n)$  is also a permutation of  $0,1,\ldots,n-1$ .

Solution: \boxed{ n }


Find the least positive integer  $n$  such that the prime factorizations of  $n$ ,  $n + 1$ , and  $n + 2$  each have exactly two factors (as  $4$  and  $6$  do, but  $12$  does not).


Consider polynomial functions  $ax^2 -bx +c$  with integer coefficients which have two distinct zeros in the open interval  $(0,1).$  Exhibit with proof the least positive integer value of  $a$  for which such a polynomial exists.


Let  $f(x)$  be a function such that  $f(1) = 1234$ ,  $f(2)=1800$ , and  $f(x) = f(x-1) + 2f(x-2)-1$  for all integers  $x$ . Evaluate the number of divisors of
    \[\sum_{i=1}^{2022}f(i)\]

*2022 CCA Math Bonanza Tiebreaker Round #4*


Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is  $\frac{1}{29}$  of the original integer.


Determine all positive integers  $n$  with at least  $4$  factors such that  $n$  is the sum the squares of its  $4$  smallest factors.


Find all triplets  $ (x,y,z) $  of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \]

Solution: \boxed{ (6, 2, 10) }


Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?


Find all the real numbers  $k$  that have the following property: For any non-zero real numbers  $a$  and  $b$ ,  it is true that at least one of the following numbers:   $$ a, b,\frac{5}{a^2}+\frac{6}{b^3} $$  is less than or equal to  $k$ .


Positive sequences  $\{a_n\},\{b_n\}$  satisfy: $a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$ .
Find the minimum value of  $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$ ,where  $m$  is a given positive integer.


Leo the fox has a  $5$  by  $5$  checkerboard grid with alternating red and black squares. He fills in the
grid with the numbers  $1, 2, 3, \dots, 25$  such that any two consecutive numbers are in adjacent squares
(sharing a side) and each number is used exactly once. He then computes the sum of the numbers in
the  $13$  squares that are the same color as the center square. Compute the maximum possible sum Leo
can obtain.


Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let  $f^{(k)}$  denote the  $k$ th derivative of  $f$ , compute  $f^{(10)}(1)$ . For the sake of this problem, note that  $10!=3628800$ .


Find the maximal possible finite number of roots of the equation   $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where  $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$  are distinct reals.

Solution: \boxed{ 49 }


Jane and Josh wish to buy a candy. However Jane needs seven more cents to buy the candy, while John needs one more cent. They decide to buy only one candy together, but discover that they do not have enough money. How much does the candy cost?


If the rotational inertia of a sphere about an axis through the center of the sphere is  $I$ , what is the rotational inertia of another sphere that has the same density, but has twice the radius? $ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $


Let  $ABCD$  be a square of side length  $1$ , and let  $E$  and  $F$  be points on  $BC$  and  $DC$  such that  $\angle{EAF}=30^\circ$  and  $CE=CF$ . Determine the length of  $BD$ .

*2015 CCA Math Bonanza Lightning Round #4.2*


Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself).

*Proposed by Yang Liu*


Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign " $+$ " or a multiplication sign " $\times$ " between each pair of adjacent digits.  (For example, her expression could be  $4 + 3\times 3$ , with value 13.)  Each positive digit is equally likely, each arithmetic sign (" $+$ " or " $\times$ ") is equally likely, and all choices are independent.  What is the expected value (average value) of her expression?


Let   $B_n$  be the set of all sequences of length  $n$ , consisting of zeros and ones. For every two sequences   $a,b \in B_n$  (not necessarily different) we define strings   $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$  and  $\delta_0\delta_1\delta_2 \dots \delta_n$  such that  $\varepsilon_0=\delta_0=0$  and   $$  \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).  $$  . Let  $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$  . Find   $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $ .
.

Solution: \boxed{n \cdot 4^{n-1}}


(a) Find the smallest number of lines drawn on the plane so that they produce exactly 2022 points of intersection. (Note: For 1 point of intersection, the minimum is 2; for 2 points, minimum is 3; for 3 points, minimum is 3; for 4 points, minimum is 4; for 5 points, the minimum is 4, etc.)
(b) What happens if the lines produce exactly 2023 intersections?

Solution: \boxed{ k = 65 }


In the quadrilateral  $ABCD$ , we have  $\measuredangle BAD = 100^{\circ}$ ,  $\measuredangle BCD = 130^{\circ}$ , and  $AB=AD=1$  centimeter. Find the length of diagonal  $AC$ .

Solution: \boxed{1 \text{ cm}}


Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six?

Clarification: If  $A$  is a friend of  $B$  then  $B$  is a friend of  $A$ .


Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to  $2$  and the sum of its terms is  $200$ . How many progressions does Omar's list have?


Let  $ABC$  be a triangle with  $|AB|=|AC|=26$ ,  $|BC|=20$ . The altitudes of  $\triangle ABC$  from  $A$  and  $B$  cut the opposite sides at  $D$  and  $E$ , respectively. Calculate the radius of the circle passing through  $D$  and tangent to  $AC$  at  $E$ .


Alice and Bob play a game together as a team on a  $100 \times 100$  board with all unit squares initially white. Alice sets up the game by coloring exactly  $k$  of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least  $10$  red squares and color all of the remaining squares in it red. What is the
smallest  $k$  such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

Proposed by *Nikola Velov, Macedonia*


Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine  $ \angle{AQC}$ .

Solution: \boxed{ 45^\circ }


Let  $p$  be a permutation of the set  $S_n = \{1, 2, \dots, n\}$ .  An element  $j \in S_n$  is called a fixed point of  $p$  if  $p(j) = j$ .  Let  $f_n$  be the number of permutations having no fixed points, and  $g_n$  be the number with exactly one fixed point.  Show that  $|f_n - g_n| = 1$ .

Solution: \boxed{ |f_n - g_n| = 1 }


In triangle  $ABC$ , side  $AB$  has length  $10$ , and the  $A$ - and  $B$ -medians have length  $9$  and  $12$ , respectively. Compute the area of the triangle.

*Proposed by Yannick Yao*


In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals  $5$  red chips,  $3$  red tile are equal to  $ 8$  black tiles, and a black tile is worth  $15$ .
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chosen so that their scores add up to  $560$  and there are no more than five tiles of the same color.


Let  $a$  be a real number. Find the minimum value of  $\int_0^1 |ax-x^3|dx$ .

How many solutions (including University Mathematics )are there for the problem?

Any advice would be appreciated.  :)


A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length  $5$ , and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are  $12$  possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point  $(2021,2021)$ ?

Solution: \boxed{ 578 }


Let  $n \geq 4$  be an even integer. Consider an  $n \times n$  grid. Two cells ( $1 \times 1$  squares) are *neighbors* if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors.
An integer from 1 to 4 is written inside each square according to the following rules:

- If a cell has a 2 written on it, then at least two of its neighbors contain a 1.
- If a cell has a 3 written on it, then at least three of its neighbors contain a 1.
- If a cell has a 4 written on it, then all of its neighbors contain a 1.

Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?


In  $\triangle ABC$ ,  $AB = 40$ ,  $BC = 60$ , and  $CA = 50$ . The angle bisector of  $\angle A$  intersects the circumcircle of  $\triangle ABC$  at  $A$  and  $P$ . Find  $BP$ .

*Proposed by Eugene Chen*


Let  $S$  be a subset of  $\{1,2,\dots,2017\}$  such that for any two distinct elements in  $S$ , both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in  $S$ .


Find all real numbers  $x,y,z$  such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$

Solution: \boxed{x = y = z = 1}


In a country there are  $15$  cities, some pairs of which are connected by a single two-way airline of a company. There are  $3$  companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?


For  $k=1,2,\dots$ , let  $f_k$  be the number of times
\[\sin\left(\frac{k\pi x}{2}\right)\]
attains its maximum value on the interval  $x\in[0,1]$ . Compute
\[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]

Solution: \boxed{ \frac{1}{4} }


You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that none of the digits are prime, 0, or 1, and that the average value of the digits is 5.

How many combinations will you have to try?


A positive integer  $n$  is fixed. Numbers  $0$  and  $1$  are placed in all cells (exactly one number in any cell) of a  $k \times n$  table ( $k$  is a number of the rows in the table,  $n$  is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets  $R$ 1 and  $R$ 2 there exists a nonempty set  $S$  of the columns such that on the intersection of any row from  $R$ 1 with the columns from  $S$  there are even number of  $1's$  while on the intersection of any row from  $R$ 2 with the columns from  $S$  there are odd number of  $1's$ .
 Find the greatest number of  $k$  such that there exists at least one nice  $k \times n$  table.

Solution: \boxed{ n }


Point  $P$  lies on the diagonal  $AC$  of square  $ABCD$  with  $AP>CP$ . Let  $O_1$  and  $O_2$  be the circumcenters of triangles  $ABP$  and  $CDP$  respectively. Given that  $AB=12$  and  $\angle O_1 P O_2 = 120^\circ$ , then  $AP=\sqrt{a}+\sqrt{b}$  where  $a$  and  $b$  are positive integers. Find  $a+b$ .


Let  $S$  be the set of all real numbers  $x$  such that  $0 \le x \le 2016 \pi$  and  $\sin x < 3 \sin(x/3)$ .  The set  $S$  is the union of a finite number of disjoint intervals.  Compute the total length of all these intervals.


At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions left (for example, the list 2,  3, 4, 6 closes right after we add 12). Which is the maximum numbers which can be written on a list that had closed, if the list had at the beginning 10 numbers?

Solution: \boxed{ 2^{10} - 1 = 1023 }


Let  $ABCD$  be an inscribed trapezoid such that the sides  $[AB]$  and  $[CD]$  are parallel. If  $m(\widehat{AOD})=60^\circ$  and the altitude of the trapezoid is  $10$ , what is the area of the trapezoid?


For a positive integer  $n$ , define  $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer  $k$  for which  $7?9?=5?k?$ .

*Proposed by Tristan Shin*


How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?

*2019 CCA Math Bonanza Individual Round #5*


What is the least posssible number of cells that can be marked on an  $n \times n$  board such that for each  $m >\frac{ n}{2}$  both diagonals of any  $m \times m$   sub-board contain a marked cell?

Solution: \boxed{ n }


The **Collaptz function** is defined as   $$ C(n) = \begin{cases} 3n - 1 & n\textrm{~odd},  \frac{n}{2} & n\textrm{~even}.\end{cases} $$
We obtain the **Collaptz sequence** of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of  $13$  begins with  $13, 38, 19, 56, 28, \cdots$  and so on. Find the sum of the three smallest positive integers  $n$  whose Collaptz sequences do not contain  $1,$  or in other words, do not **collaptzse**.

*Proposed by Andrew Wu and Jason Wang*


Find all positive integers   $(m, n)$  that satisfy   $$ m^2 =\sqrt{n} +\sqrt{2n + 1}. $$

Solution: \boxed{ (13, 4900) }


Find the functions  $ f:\mathbb{N}\longrightarrow\mathbb{N} $  that satisfy the following relation:  $$  \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} .  $$

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


Find  $100m+n$  if  $m$  and  $n$  are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0  i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \]*Proposed by Aaron Lin*


Find all positive integers  $k$  satisfying: there is only a finite number of positive integers  $n$ , such that the positive integer solution  $x$  of  $xn+1\mid n^2+kn+1$  is not unique.

Solution: \boxed{ k \neq 2 }


Let  $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$ . For every non-empty subset of  $ M(n )$  we consider the product of its elements. How big is the sum over all these products?


 $ P(x)$  is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence  $ P(1)$ ,  $ P(2)$ ,  $ P(3)$ ,  $ \dots?$

*Proposed by A. Golovanov*


Let  $n$  be a positive integer and consider an arrangement of  $2n$  blocks in a straight line, where  $n$  of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let  $A$  be the minimum number of swaps needed to make the first  $n$  blocks all red and  $B$  be the minimum number of swaps needed to make the first  $n$  blocks all blue. Show that  $A+B$  is independent of the starting arrangement and determine its value.


A word is an ordered, non-empty sequence of letters, such as  $word$  or  $wrod$ . How many distinct  $3$ -letter words can be made from a subset of the letters  $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears?


We call the polynomial  $P (x)$  simple if the coefficient of each of its members belongs to the set  $\{-1, 0, 1\}$ .
Let  $n$  be a positive integer,  $n> 1$ . Find the smallest possible number of terms with a non-zero coefficient in a simple  $n$ -th degree polynomial with all values at integer places are divisible by  $n$ .


Define  $ a \circledast b = a + b-2ab $ . Calculate the value of  $$ A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014} $$


For  $n \geq 3$ , let  $(b_0, b_1,..., b_{n-1}) = (1, 1, 1, 0, ..., 0).$  Let  $C_n =
(c_{i, j})$  the  $n \times n$  matrix defined by  $c_{i, j} = b _{(j -i) \mod n}$ . Show
that  $\det (C_n) = 3$  if  $n$  is not a multiple of 3 and  $\det (C_n) = 0$  if  $n$
is a multiple of 3.


In parallelogram  $ABCD$ ,  $AC=10$  and  $BD=28$ . The points  $K$  and  $L$  in the plane of  $ABCD$  move in such a way that  $AK=BD$  and  $BL=AC$ . Let  $M$  and  $N$  be the midpoints of  $CK$  and  $DL$ , respectively. What is the maximum walue of  $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$  ?


We have a calculator with two buttons that displays and integer  $x$ . Pressing the first button replaces  $x$  by  $\lfloor \frac{x}{2} \rfloor$ , and pressing the second button replaces  $x$  by  $4x+1$ . Initially, the calculator displays  $0$ . How many integers less than or equal to  $2014$  can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here,  $\lfloor y \rfloor$  denotes the greatest integer less than or equal to the real number  $y$ ).


A jacket was originally priced  $\textdollar 100$ . The price was reduced by  $10\%$  three times and increased by  $10\%$  four times in some order. To the nearest cent, what was the final price?


Find all non-negative integer numbers  $n$  for which there exists integers  $a$  and  $b$  such that  $n^2=a+b$  and  $n^3=a^2+b^2.$

Solution: \boxed{0, 1, 2}


There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different
rooms will have people in them?

Solution: \boxed{ 61 }


Let  $a_1 = 2,$  and for  $n\ge 1,$  let  $a_{n+1} = 2a_n + 1.$  Find the smallest value of an  $a_n$  that is not a prime number.


Given a triangle  $ABC$  with integer side lengths, where  $BD$  is an angle bisector of  $\angle ABC$ ,  $AD=4$ ,  $DC=6$ , and  $D$  is on  $AC$ , compute the minimum possible perimeter of  $\triangle ABC$ .


For real numbers  $a$  and  $b$ , define  $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$  Find the smallest possible value of the expression   $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$

Solution: \boxed{4 \sqrt{2018}}


In isosceles right-angled triangle  $ABC$ ,  $CA = CB = 1$ .  $P$  is an arbitrary point on the sides of  $ABC$ . Find the maximum of  $PA \cdot PB \cdot PC$ .


Suppose  $X$  is a random variable that takes on only nonnegative integer values, with  $E[X]=1,$   $E[X^2]=2,$  and  $E[X^3]=5.$  (Here  $E[Y]$  denotes the expectation of the random variable  $Y.$ ) Determine the smallest possible value of the probability of the event  $X=0.$


Find all triples  $(p, q, r)$  of prime numbers for which  $4q - 1$  is a prime number and   $$ \frac{p + q}{p + r} = r - p $$  holds.

*(Walther Janous)*

Solution: \boxed{ (2, 3, 3) }


Suppose that  $a$  and  $b$  are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of  $p$ ?

Solution: \boxed{ 5 }


Tim and Allen are playing a match of *tenus*.  In a match of *tenus*, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability  $3/4$ , and in the even-numbered games, Allen wins with probability  $3/4$ .  What is the expected number of games in a match?


In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths  $a$  and  $b$ . If the right triangle has area  $T$  and is inscribed in a circle of area  $C$ , find  $ab$  in terms of  $T$  and  $C$ .

Solution: \boxed{\frac{\pi T^2}{C}}


If  $a_0$  is a positive real number, consider the sequence  $\{a_n\}$  defined by:

\[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \]

Show that there exist a real number  $a > 0$  such that:

**i.)** for all  $a_0 \geq a,$  the sequence  $\{a_n\} \rightarrow \infty,$

**ii.)** for all  $a_0 < a,$  the sequence   $\{a_n\} \rightarrow 0.$

Solution: \boxed{ a = 2 }


Find all values of the positive integer  $k$  that has the property:
There are no positive integers  $a,b$  such that the expression  $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$  is a composite positive number.

Solution: \boxed{ k = 1 }


Let  $y=f(x)$  be a function of the graph of broken line connected by points  $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$  in the  $x$  - $y$  plane.
Find the minimum value of  $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ *2010 Tohoku University entrance exam/Economics, 2nd exam*


Let  $ P_1$  be a regular  $ r$ -gon and  $ P_2$  be a regular  $ s$ -gon  $ (r\geq s\geq 3)$  such that each interior angle of  $ P_1$  is  $ \frac {59}{58}$  as large as each interior angle of  $ P_2$ . What's the largest possible value of  $ s$ ?


Determine the smallest integer  $n > 1$  with the property that  $n^2(n - 1)$  is divisible by 2009.


Let  $A$  be the set  $\{k^{19}-k: 1<k<20, k\in N\}$ . Let  $G$  be the GCD of all elements of  $A$ .
Then the value of  $G$  is?


On January  $20$ ,  $2018$ , Sally notices that her  $7$  children have ages which sum to a perfect square: their ages are  $1$ ,  $3$ ,  $5$ ,  $7$ ,  $9$ ,  $11$ , and  $13$ , with  $1+3+5+7+9+11+13=49$ . Let  $N$  be the age of the youngest child the next year the sum of the  $7$  children's ages is a perfect square on January  $20$ th, and let  $P$  be that perfect square. Find  $N+P$ .

*2018 CCA Math Bonanza Lightning Round #2.3*


Nine tiles are numbered  $1, 2, 3, \ldots, 9,$  respectively.  Each of three players randomly selects and keeps three of the tile, and sums those three values.  The probability that all three players obtain an odd sum is  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$


A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers  $n$  such that it is possible to give each of the directors a set of keys to  $n$  different locks, according to the requirements and regulations of the company.

Solution: \boxed{ n = 6 }


Circle  $o$  contains the circles  $m$  ,  $p$  and  $r$ , such that they are tangent to  $o$  internally and any two of them are tangent between themselves. The radii of the circles  $m$  and  $p$  are equal to  $x$  . The circle  $r$  has radius  $1$  and passes through the center of the circle  $o$ . Find the value of  $x$  .


Medians  $AM$  and  $BE$  of a triangle  $ABC$   intersect at  $O$ . The points  $O, M, E, C$  lie on one circle. Find the length of  $AB$  if  $BE = AM =3$ .


In a  $2 \times 8$  squared board, you want to color each square red or blue in such a way that on each  $2 \times 2$  sub-board there are at least  $3$  boxes painted blue. In how many ways can this coloring be done?

Note. A  $2 \times 2$  board is a square made up of  $4$  squares that have a common vertex.


 $ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$   satisfy the given conditions

 $ a)$   $ f(0,0)\equal{}1$  ,  $ f(0,1)\equal{}1$  ,

 $ b)$   $ \forall k \notin \left\{0,1\right\}$    $ f(0,k)\equal{}0$  and

 $ c)$   $ \forall n \geq 1$  and  $ k$  ,  $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$

find the sum  $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$


What is the largest possible value of the expression   $$ gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 ) $$   for naturals  $n$ ?

Click to expandoriginal wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n?


Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer  $N$  for which there exists a rectangular prism with integer side lengths and *bigness* $N$  and another one with integer side lengths and *bigness* $N + 1$ .