Let the definite integral  $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$ .

(1) Find  $ I_0,\ I_{\minus{}1},\ I_2$ .

(2) Find  $ I_1$ .

(3) Express  $ I_{n\plus{}2}$  in terms of  $ I_n$ .

(4) Find  $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$ .

(5) Evaluate the definite integrals  $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$  in using the avobe results.

You are not allowed to use the formula of integral for  $ \sqrt{x^2\plus{}1}$  directively here.

Solution: \boxed{\frac{2 + \pi}{8}}


Let  $k$  be a positive integer. In the coordinate plane, circle  $\omega$  has positive integer radius and is tangent to both axes. Suppose that  $\omega$  passes through  $(1,1000+k)$ . Compute the smallest possible value of  $k$ .

[i]Proposed by Luke Robitaille


a) For each  $n \ge 2$ , find the maximum constant  $c_{n}$  such that
 $\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$
for all positive reals  $a_{1},a_{2},\ldots,a_{n}$  such that  $a_{1}a_{2}\cdots a_{n}= 1$ .

b) For each  $n \ge 2$ , find the maximum constant  $d_{n}$  such that
 $\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$
for all positive reals  $a_{1},a_{2},\ldots,a_{n}$  such that  $a_{1}a_{2}\cdots a_{n}= 1$ .

Solution: \boxed{ c_n = 1 }


When the integer  $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$  is divided by 9, what is the remainder?


A sequence  $ (S_n), n \geq 1$  of sets of natural numbers with  $ S_1 = \{1\}, S_2 = \{2\}$  and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine  $ S_{1024}.$


Jimmy runs a successful pizza shop.  In the middle of a busy day, he realizes that he is running low on ingredients.  Each pizza must have 1 lb of dough,  $\frac14$  lb of cheese,  $\frac16$  lb of sauce, and  $\frac13$  lb of toppings, which include pepperonis, mushrooms, olives, and sausages.  Given that Jimmy currently has 200 lbs of dough, 20 lbs of cheese, 20 lbs of sauce, 15 lbs of pepperonis, 5 lbs of mushrooms, 5 lbs of olives, and 10 lbs of sausages, what is the maximum number of pizzas that JImmy can make?


Determine all functions  $f:\mathbb{R}\rightarrow \mathbb{R}$  such that the set
\[\left \{   \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\]
is finite, and for all  $x \in \mathbb{R}$ \[f(x-1-f(x)) = f(x) - x - 1\]

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


Let  $N=30^{2015}$ .  Find the number of ordered 4-tuples of integers  $(A,B,C,D)\in\{1,2,\ldots,N\}^4$  (not necessarily distinct) such that for every integer  $n$ ,  $An^3+Bn^2+2Cn+D$  is divisible by  $N$ .


The increasing sequence  $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$  is formed with positive integers which are powers of  $3$  or sums of different powers of  $3$ . Which number is in the  $100^{th}$  position?


There is a box containing  $100$  balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between  $0$  and  $100$ , inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form  $\tfrac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p + q$ .


Let  $ABC$  be a triangle with  $AB = 9$ ,  $BC = 10$ , and  $CA = 17$ . Let  $B'$  be the reflection of the point  $B$  over the line  $CA$ . Let  $G$  be the centroid of triangle  $ABC$ , and let  $G'$  be the centroid of triangle  $AB'C$ . Determine the length of segment  $GG'$ .


Let  $n\geq 2$  be a positive integer. The sets  $A_{1},A_{2},\ldots, A_{n}$  and  $B_{1},B_{2},\ldots, B_{n}$  of positive integers are such that  $A_{i}\cap B_{j}$  is non-empty  $\forall i,j\in\{1,2,\ldots ,n\}$  and  $A_{i}\cap A_{j}=\o$ ,  $B_{i}\cap B_{j}=\o$   $\forall i\neq j\in \{1,2,\ldots, n\}$ . We put the elements of each set in a descending order and calculate the differences between consecutive elements in this new order. Find the least possible value of the greatest of all such differences.

Solution: \boxed{n}


At Stanford in 1988, human calculator Shakuntala Devi was asked to compute  $m = \sqrt[3]{61{,}629{,}875}$  and  $n = \sqrt[7]{170{,}859{,}375}$ .  Given that  $m$  and  $n$  are both integers, compute  $100m+n$ .

*Proposed by Evan Chen*


For each positive integer  $n$ , let  $S(n)$  be the sum of the digits of  $n$ . Determines the smallest positive integer  $a$  such that there are infinite positive integers  $n$  for which you have  $S (n) -S (n + a) = 2018$ .


Define a regular  $n$ -pointed star to be the union of  $n$  line segments  $P_1P_2, P_2P_3,\ldots, P_nP_1$  such that

 $\bullet$  the points  $P_1, P_2,\ldots, P_n$  are coplanar and no three of them are collinear,
 $\bullet$  each of the  $n$  line segments intersects at least one of the other line segments at a point other than an endpoint,
 $\bullet$  all of the angles at  $P_1, P_2,\ldots, P_n$  are congruent,
 $\bullet$  all of the  $n$  line segments  $P_2P_3,\ldots, P_nP_1$  are congruent, and
 $\bullet$  the path  $P_1P_2, P_2P_3,\ldots, P_nP_1$  turns counterclockwise at an angle of less than 180 degrees at each vertex.

There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?


Cyclic quadrilateral  $ABCD$  satisfies  $\angle ABD = 70^\circ$ ,  $\angle ADB=50^\circ$ , and  $BC=CD$ .  Suppose  $AB$  intersects  $CD$  at point  $P$ , while  $AD$  intersects  $BC$  at point  $Q$ .  Compute  $\angle APQ-\angle AQP$ .


Alice the ant starts at vertex  $A$  of regular hexagon  $ABCDEF$  and moves either right or left each move with equal probability. After  $35$  moves, what is the probability that she is on either vertex  $A$  or  $C$ ?

*2015 CCA Math Bonanza Lightning Round #5.3*


Trapezoid  $ABCD$  is an isosceles trapezoid with  $AD=BC$ .  Point  $P$  is the intersection of the diagonals  $AC$  and  $BD$ .  If the area of  $\triangle ABP$  is  $50$  and the area of  $\triangle CDP$  is  $72$ , what is the area of the entire trapezoid?

[i]Proposed by David Altizio


Find the least positive integer that cannot be represented as  $\frac{2^a-2^b}{2^c-2^d}$  for some positive integers  $a, b, c, d$ .


The columns of an  $ n\times n$  board are labeled  $ 1$  to  $ n$ . The numbers  $ 1,2,...,n$  are arranged in the board so that the numbers in each row and column are pairwise different. We call a cell "good" if the number in it is greater than the label of its column. For which  $ n$  is there an arrangement in which each row contains equally many good cells?

Solution: \boxed{ n }


How many six-letter words formed from the letters of AMC do not contain the substring AMC? (For example, AMAMMC has this property, but AAMCCC does not.)


Suppose there are  $2017$  spies, each with  $\frac{1}{2017}$ th of a secret code. They communicate by telephone; when two of them talk, they share all information they know with each other. What is the minimum number of telephone calls that are needed for all 2017 people to know all parts of the code?


Let  $\triangle ABC$  have median  $CM$  ( $M\in AB$ ) and circumcenter  $O$ . The circumcircle of  $\triangle AMO$  bisects  $CM$ . Determine the least possible perimeter of  $\triangle ABC$  if it has integer side lengths.


Let  $a, b, c, d$  be the roots of the quartic polynomial  $f(x) = x^4 + 2x + 4$ . Find the value of  $$ \frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}. $$


Find all postitive integers n  such that  $$ \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot  \left\lfloor \frac{n}{4} \right\rfloor=n^2 $$  where  $\lfloor x \rfloor$  represents the largest integer less than the real number  $x$ .


A real number  $a$  is chosen randomly and uniformly from the interval  $[-20, 18]$ . The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\] are all real can be written in the form  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .


A set  $S$  of unit cells of an  $n\times n$  array,  $n\geq 2$ , is said *full* if each row and  each column of the array contain at least one element of  $S$ , but which has this property no more when any of its elements is removed. A full set having maximum cardinality is said *fat*, while a full set of minimum cardinality is said *meagre*.

i) Determine the cardinality  $m(n)$  of the meagre sets, describe all meagre sets and give their count.
ii) Determine the cardinality  $M(n)$  of the fat sets, describe all fat sets and give their count.

*(Dan Schwarz)*

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


Two regular tetrahedrons  $A$  and  $B$  are made with the 8 vertices of a unit cube. (this way is unique)

What's the volume of  $A\cup B$ ?


For a positive integer  $n$ , let  $f(n)$  be the sum of the positive integers that divide at least one of the nonzero base  $10$  digits of  $n$ . For example,  $f(96)=1+2+3+6+9=21$ . Find the largest positive integer  $n$  such that for all positive integers  $k$ , there is some positive integer  $a$  such that  $f^k(a)=n$ , where  $f^k(a)$  denotes  $f$  applied  $k$  times to  $a$ .

*2021 CCA Math Bonanza Lightning Round #4.3*


Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.


A non-negative integer  $n$  is said to be *squaredigital* if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.

Solution: \boxed{0, 1, 81}


In a regular hexagon  $ABCDEF$  of side length  $8$  and center  $K$ , points  $W$  and  $U$  are chosen on  $\overline{AB}$  and  $\overline{CD}$  respectively such that  $\overline{KW} = 7$  and  $\angle WKU = 120^{\circ}$ . Find the area of pentagon  $WBCUK$ .

*Proposed by Bradley Guo*


A circle is inscribed in a regular octagon with area  $2024$ . A second regular octagon is inscribed in the circle, and its area can be expressed as  $a + b\sqrt{c}$ , where  $a, b, c$  are integers and  $c$  is square-free. Compute  $a + b + c$ .


If  $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$  can be written in the form  $a^b$  for positive integers  $a,b$  with  $a$  squarefree, find  $a+b$ .

*Proposed by Deyuan Li and Andrew Milas*


 $5.$ Let x be a real number with  $0<x<1$  and let  $0.c_1c_2c_3...$  be the decimal expansion of x.Denote by  $B(x)$  the set of all subsequences of  $c_1c_2c_3$  that consist of 6 consecutive digits.
For instance ,  $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of  $B(x)$  as  $x$  varies among all irrational numbers with  $0<x<1$


It is known that  $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$  for some rational number  $q.$  Determine  $q.$  Here,  $0\leq\arctan(x)<\frac{\pi}{2}$  for  $0\leq x <\infty.$

Solution: \boxed{ q = \frac{3}{8} }


Find the number of  $4$ -digit numbers (in base  $10$ ) having non-zero digits and which are divisible by  $4$  but not by  $8$ .


Rhombus  $PQRS$  is inscribed in rectangle  $ABCD$  so that vertices  $P$ ,  $Q$ ,  $R$ , and  $S$  are interior points on sides  $\overline{AB}$ ,  $\overline{BC}$ ,  $\overline{CD}$ , and  $\overline{DA}$ , respectively. It is given that  $PB=15$ ,  $BQ=20$ ,  $PR=30$ , and  $QS=40$ . Let  $m/n$ , in lowest terms, denote the perimeter of  $ABCD$ . Find  $m+n$ .

Solution: \boxed{ 677 }


Suppose  $x$  is a positive real number such that  $\{x\}, [x]$  and  $x$  are in a geometric progression. Find the least positive integer  $n$  such that  $x^n > 100$ . (Here  $[x]$  denotes the integer part of  $x$  and  $\{x\} = x - [x]$ .)


Let  $L$  be the number formed by  $2022$  digits equal to  $1$ , that is,  $L=1111\dots 111$ .
Compute the sum of the digits of the number  $9L^2+2L$ .


A tree has 10 pounds of apples at dawn.  Every afternoon, a bird comes and eats x pounds of apples.  Overnight, the amount of food on the tree increases by 10%.  What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?


Square  $ABCD$  has side length  $13$ , and points  $E$  and  $F$  are exterior to the square such that  $BE=DF=5$  and  $AE=CF=12$ . Find  $EF^{2}$ .
[asy]
size(200);
defaultpen(fontsize(10));
real x=22.61986495;
pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x);
draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7));
dot(A^^B^^C^^D^^E^^F);
pair point=(13,13);
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $D$ ", D, dir(point--D));
label(" $E$ ", E, dir(point--E));
label(" $F$ ", F, dir(point--F));[/asy]


Let  $ n $   $(n\geq2)$  be an integer. Find the greatest possible value of the expression   $$ E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2} $$   if the positive real numbers  $a_1,a_2,\ldots,a_n$  satisfy  $a_1+a_2+\ldots+a_n=\frac{n}{2}.$  What are the values of  $a_1,a_2,\ldots,a_n$  when the greatest value is achieved?


Find the positive integer  $n$  such that the least common multiple of  $n$  and  $n - 30$  is  $n + 1320$ .


There are  $10001$  students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of  $k$  societies. Suppose that the following conditions hold:

*i.)* Each pair of students are in exactly one club.

*ii.)* For each student and each society, the student is in exactly one club of the society.

*iii.)* Each club has an odd number of students. In addition, a club with  ${2m+1}$  students ( $m$  is a positive integer) is
in exactly  $m$  societies.

Find all possible values of  $k$ .

*Proposed by Guihua Gong, Puerto Rico*


Find the sum of all the digits in the decimal representations of all the positive integers less than  $1000.$


The equation  $ ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0$  has three distinct solutions. How many distinct solutions does the following equation have:

 $ 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?$


Let  $u$  be a real number. On the coordinate plane, consider two parabolas  $C_1: y=-x^2+1,\ C_2: y=(x-u)^2+u$ .
The range of  $u$  such that  $C_1$  and  $C_2$  have at least one point of intersection is expressed by  $a\leq u\leq b$  for some real numbers  $a,\ b$ .

(1) Find the values of  $a,\ b$ .

(2) When  $u$  satisfies  $a\leq u\leq b$ , let  $P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$  be the point of intersections of  $C_1$  and  $C_2$ .

Note : if the point of intersection is just one, then we regard the intersection as  $P_1=P_2$ .

Express  $2|x_1y_2-x_2y_1|$  in terms of  $u$ .

(3) Let  $f(u)$  be the expression obtained in (2), evaluate  $I=\int_ a^b f(u)du.$

Solution: \boxed{I = \frac{21\pi}{8}}


Find  $$  \inf_{\substack{ n\ge 1  a_1,\ldots ,a_n >0  a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) .  $$

Solution: \boxed{-\pi}


Kevin colors three distinct squares in a  $3\times 3$  grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.


Let  $f$  be a function such that  $f (x + y) = f (x) + f (y)$  for all  $x,y \in R$  and  $f (1) = 100$ . Calculate  $\sum_{k = 1}^{10}f (k!)$ .


Given the following system of equations:  $$ \begin{cases} R I +G +SP = 50   R I +T + M = 63  G +T +SP = 25   SP + M = 13    M +R I = 48  N = 1 \end{cases} $$
Find the value of L that makes  $LMT +SPR I NG = 2023$  true.


Find the greatest real  $k$  such that, for every tetrahedron  $ABCD$  of volume  $V$ , the product of areas of faces  $ABC,ABD$  and  $ACD$  is at least  $kV^2$ .

Solution: \boxed{ \frac{9}{2} }


What is the minimum number of planes determined by  $6$  points in space which are not all coplanar, and among which no three are collinear?


Two distinct points  $A$  and  $B$  are chosen at random from 15 points equally spaced around a circle centered at  $O$  such that each pair of points  $A$  and  $B$  has the same probability of being chosen. The probability that the perpendicular bisectors of  $OA$  and  $OB$  intersect strictly inside the circle can be expressed in the form  $\frac{m}{n}$ , where  $m,n$  are relatively prime positive integers. Find  $m+n$ .

*Ray Li.*

Solution: \boxed{ 11 }


Find the smallest positive integer  $n$  such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers  $m>1$ .

Solution: \boxed{ 2^{1987} }


Let  $ p>2 $  be a prime number. For any permutation  $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $  of the set  $ S = \{ 1, 2, \cdots , p \} $ , let  $ f( \pi ) $  denote the number of multiples of  $ p $  among the following  $ p $  numbers:
\[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)  \]
Determine the average value of  $ f( \pi) $  taken over all permutations  $ \pi $  of  $ S $ .

Solution: \boxed{2 - \frac{1}{p}}


A  $3 \times 3$  grid of unit cells is given. A *snake of length  $k$* is an animal which occupies an ordered  $k$ -tuple of cells in this grid, say  $(s_1, \dots, s_k)$ . These cells must be pairwise distinct, and  $s_i$  and  $s_{i+1}$  must share a side for  $i = 1, \dots, k-1$ . After being placed in a finite  $n \times n$  grid, if the snake is currently occupying  $(s_1, \dots, s_k)$  and  $s$  is an unoccupied cell sharing a side with  $s_1$ , the snake can *move* to occupy  $(s, s_1, \dots, s_{k-1})$  instead. The snake has *turned around* if it occupied  $(s_1, s_2, \dots, s_k)$  at the beginning, but after a finite number of moves occupies  $(s_k, s_{k-1}, \dots, s_1)$  instead.

Find the largest integer  $k$  such that one can place some snake of length  $k$  in a  $3 \times 3$  grid which can turn around.


Two circumferences of radius  $1$  that do not intersect,  $c_1$  and  $c_2$ , are placed inside an angle whose vertex is  $O$ .  $c_1$  is tangent to one of the rays of the angle, while  $c_2$  is tangent to the other ray. One of the common internal tangents of  $c_1$  and  $c_2$  passes through  $O$ , and the other one intersects the rays of the angle at points  $A$  and  $B$ , with  $AO=BO$ . Find the distance of point  $A$  to the line  $OB$ .


Suppose that  $2^{2n+1}+ 2^{n}+1=x^{k}$ , where  $k\geq2$  and  $n$  are positive integers. Find all possible values of  $n$ .

Solution: \boxed{ n = 4 }


Find the number of ordered triples of positive integers  $(a, b, c)$  such that  $abc$  divides  $(ab + 1)(bc + 1)(ca + 1)$ .


Determine the smallest possible value of  $$ |2^m - 181^n|, $$  where  $m$  and  $n$  are positive integers.

Solution: \boxed{ 7 }


A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?


Find all 4-digit numbers  $\overline{abcd}$  that are multiples of  $11$ , such that the 2-digit number  $\overline{ac}$  is a multiple of  $7$  and  $a + b + c + d = d^2$ .


Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the  $\emph{last}$  forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?


Let  $n \geq 3$  be a positive integer. Find the smallest positive real  $k$ , satisfying the following condition: if  $G$  is a connected graph with  $n$  vertices and  $m$  edges, then it is always possible to delete at most  $k(m-\lfloor \frac{n} {2} \rfloor)$  edges, so that the resulting graph has a proper vertex coloring with two colors.

Solution: \boxed{ k = \frac{1}{2} }


You are given a positive integer  $n$ . What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$  so that there are no two chosen numbers  $x > y$  for which  $x - y = (x, y)$ ?

Here  $(x, y)$  denotes the greatest common divisor of  $x, y$ .

*Proposed by Anton Trygub*

Solution: \boxed{n}


A cake has a shape of triangle with sides  $19,20$  and  $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't  stick out of the plate. What is the minimal diameter of the plate?


Let  $d$  be the greatest common divisor of  $2^{30^{10}}-2$  and  $2^{30^{45}}-2$ .  Find the remainder when  $d$  is divided by  $2013$ .


Find all pairs of integers  $(a,b)$  so that each of the two cubic polynomials   $$ x^3+ax+b \, \, \text{and} \, \, x^3+bx+a $$   has all the roots to be integers.

*Proposed by Prithwijit De and Sutanay Bhattacharya*

Solution: \boxed{(0, 0)}


Let  $a$  and  $b$  be positive integers such that  $a>b$  and the difference between  $a^2+b$  and  $a+b^2$  is prime. Compute all possible pairs  $(a,b)$ .

Solution: \boxed{(2, 1)}


For positive integer  $n,$  let  $s(n)$  be the sum of the digits of n when n is expressed in base ten. For example,  $s(2022) = 2 + 0 + 2 + 2 = 6.$  Find the sum of the two solutions to the equation  $n - 3s(n) = 2022.$


In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo).
What is the maximal possible number of edges in such a graph?

Solution: \boxed{ 42550 }


David is taking a true/false exam with  $9$  questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly  $5$  of the answers are True. In accordance with this, David guesses the answers to all  $9$  questions, making sure that exactly  $5$  of his answers are True. What is the probability he answers at least  $5$  questions correctly?


Let  $p$ ,  $q$ ,  $r$ , and  $s$  be 4 distinct primes such that  $p+q+r+s$  is prime, and the numbers  $p^2+qr$  and  $p^2+qs$  are both perfect squares. What is the value of  $p+q+r+s$ ?


Square  $SEAN$  has side length  $2$  and a quarter-circle of radius  $1$  around  $E$  is cut out. Find the radius of the largest circle that can be inscribed in the remaining figure.

Solution: \boxed{5 - 3\sqrt{2}}


For arbiterary integers  $n,$  find the continuous function  $f(x)$  which satisfies the following equation. \[\lim_{h\rightarrow 0}\frac{1}{h}\int_{x-nh}^{x+nh}f(t) dt=2f(nx).\] Note that  $x$  can range all real numbers and  $f(1)=1.$

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


Define  $ n!!$  to be  $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$  for  $ n$  odd and  $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$  for  $ n$  even. When  $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$  is expressed as a fraction in lowest terms, its denominator is  $ 2^ab$  with  $ b$  odd. Find  $ \displaystyle \frac{ab}{10}$ .


Find (with proof) all natural numbers  $n$  such that, for some natural numbers  $a$  and  $b$ ,  $a\ne b$ , the digits in the decimal representations of the two numbers  $n^a+1$  and  $n^b+1$  are in reverse order.

Solution: \boxed{ n = 3 }


In  $\triangle ABC$  points  $M,O$  are midpoint of  $AB$  and circumcenter. It is true, that  $OM=R-r$ . Bisector of external  $\angle A$  intersect  $BC$  at  $D$  and bisector of external  $\angle C$  intersect  $AB$  at  $E$ .
Find possible values of  $\angle CED$ *D. Shiryaev*


Let  $S$  be the set of all real values of  $x$  with  $0 < x < \pi/2$  such that  $\sin x$ ,  $\cos x$ , and  $\tan x$  form the side lengths (in some order) of a right triangle.  Compute the sum of  $\tan^2 x$  over all  $x$  in  $S$ .


Let  $n$  be a positive integer. Positive numbers  $a$ ,  $b$ ,  $c$  satisfy  $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ . Find the greatest possible value of   $$ E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}} $$


Find the remainder when  $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$  is divided by  $43$ . Your answer should be an integer between  $0$  and  $42$ .


What is the greatest number of integers that can be selected from a set  of  $2015$  consecutive numbers so that no sum of any two selected numbers is divisible by their difference?


A frog is standing in a center of a  $3 \times 3$  grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with their current square uniformly at random, and jumps onto the lilypad on their chosen square. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected number of times the frog jumps?

*2021 CCA Math Bonanza Lightning Round #3.2*


Find all sequences of positive integers  $\{a_n\}_{n=1}^{\infty}$ , for which  $a_4=4$  and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural  $n \geq 2$ .

*Peter Boyvalenkov*

Solution: \boxed{a_n = n}


An isosceles triangle has angles of  $50^\circ,x^\circ,$  and  $y^\circ$ . Find the maximum possible value of  $x-y$ .

[i]Proposed by Nathan Ramesh


A game is played on an  ${n \times  n}$   chessboard. At the beginning there are  ${99}$   stones on each square. Two players  ${A}$   and  ${B}$   take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player  ${A}$   takes the first turn. Determine all n for which player  ${A}$   has a winning strategy.

Solution: \boxed{ n }


Let  $ABCD$  be a rhombus with angle  $\angle A = 60^\circ$ . Let  $E$  be a point, different from  $D$ , on the line  $AD$ . The lines  $CE$  and  $AB$  intersect at  $F$ . The lines  $DF$  and  $BE$  intersect at  $M$ . Determine the angle  $\angle BMD$  as a function of the position of  $E$  on  $AD.$


For each positive integer  $k$ , let  $d(k)$  be the number of positive divisors of  $k$  and  $\sigma(k)$  be the sum of positive divisors of  $k$ . Let  $\mathbb N$  be the set of all positive integers. Find all functions  $f: \mathbb{N} \to \mathbb N$  such that \begin{align*}
			f(d(n+1)) &= d(f(n)+1)\quad \text{and}
			f(\sigma(n+1)) &= \sigma(f(n)+1)
		\end{align*}
	for all positive integers  $n$ .

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


16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.


Jane is 25 years old.  Dick is older than Jane.  In  $n$  years, where  $n$  is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age.  Let  $d$  be Dick's present age.  How many ordered pairs of positive integers  $(d,n)$  are possible?


How many ordered triplets  $(a, b, c)$  of positive integers such that  $30a + 50b + 70c \leq 343$ .


Find the  $2019$ th strictly positive integer  $n$  such that  $\binom{2n}{n}$  is not divisible by  $5$ .


The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.

Find the number of possible colorings that satisfies the above conditions.


Find all ordered pairs  $(a,b)$  of positive integers that satisfy  $a>b$   and the equation  $(a-b)^{ab}=a^bb^a$ .

Solution: \boxed{(4, 2)}


Compute the number of subsets  $S$  of  $\{0,1,\dots,14\}$  with the property that for each  $n=0,1,\dots,
6$ , either  $n$  is in  $S$  or both of  $2n+1$  and  $2n+2$  are in  $S$ .

*Proposed by Evan Chen*


In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for  $b+e+h$ ?

  $ \begin{tabular}{cccc} &a&b&c  &d&e&f  + & g&h&i  \hline 1&6&6&5 \end{tabular}$ *Proposed by Eugene Chen*


Let  $f : N \to R$  be a function, satisfying the following condition:
for every integer  $n > 1$ , there exists a prime divisor  $p$  of  $n$  such that  $f(n) = f \big(\frac{n}{p}\big)-f(p)$ .
If  $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$ ,  determine the value of  $f(2007^2) + f(2008^3) + f(2009^5)$


Find the smallest value of the expression  $|253^m - 40^n|$  over all pairs of positive integers  $(m, n)$ .

*Proposed by Oleksii Masalitin*


Let *Revolution* $(x) = x^3 +Ux^2 +Sx + A$ , where  $U$ ,  $S$ , and  $A$  are all integers and  $U +S + A +1 = 1773$ . Given that *Revolution* has exactly two distinct nonzero integer roots  $G$  and  $B$ , find the minimum value of  $|GB|$ .

*Proposed by Jacob Xu*

Solution*Solution.*  $\boxed{392}$ Notice that  $U + S + A + 1$  is just *Revolution* $(1)$  so *Revolution* $(1) = 1773$ . Since  $G$  and  $B$  are integer roots we write *Revolution* $(X) = (X-G)^2(X-B)$  without loss of generality. So Revolution $(1) = (1-G)^2(1-B) = 1773$ .  $1773$  can be factored as  $32 \cdot 197$ , so to minimize  $|GB|$  we set  $1-G = 3$  and  $1-B = 197$ . We get that  $G = -2$  and  $B = -196$  so  $|GB| = \boxed{392}$ .


Determine the maximum possible real value of the number  $ k$ , such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers  $ a,b,c\ge 0$  with  $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$ .


Determine the values of the real parameter  $a$ , such that the equation
\[\sin 2x\sin 4x-\sin x\sin 3x=a\]
has a unique solution in the interval  $[0,\pi)$ .

Solution: \boxed{ a = 1 }


Suppose  $P (x)$  is a polynomial with real coefficients such that  $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$  for all real numbers  $t$ . Compute the largest possible value of  $P(P(P(P(1))))$ .


Let  $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $  be a function such that for all  $n > 1$  there is a prime divisor  $p$  of  $n$  such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that  $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $ . Determine  $ f(2014^2)+f(2015^3)+f(2016^5) $ .


For a positive integer  $n$  not divisible by  $211$ , let  $f(n)$  denote the smallest positive integer  $k$  such that  $n^k - 1$  is divisible by  $211$ . Find the remainder when   $$ \sum_{n=1}^{210} nf(n) $$   is divided by  $211$ .

*Proposed by ApraTrip*


Isosceles triangle  $\triangle{ABC}$  has  $\angle{ABC}=\angle{ACB}=72^\circ$  and  $BC=1$ . If the angle bisector of  $\angle{ABC}$  meets  $AC$  at  $D$ , what is the positive difference between the perimeters of  $\triangle{ABD}$  and  $\triangle{BCD}$ ?

*2019 CCA Math Bonanza Tiebreaker Round #2*


One hundred and one of the squares of an  $n\times n$  table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible  $n$ .


Let  $ABCD$  be a parallelogram such that  $\angle BAD = 60^{\circ}.$  Let  $K$  and  $L$  be the midpoints of  $BC$  and  $CD,$  respectively. Assuming that  $ABKL$  is a cyclic quadrilateral, find  $\angle ABD.$


Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a *single* and that between a boy and a girl was called a *mixed single*. The total number of boys differed from the total number of girls by at most 1. The total number of singles differed from the total number of mixed singles by at most 1. At most how many schools were represented by an odd number of players?

Solution: \boxed{ 3 }


Given an integer  $n\ge 2$ , compute  $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$ , where all  $n$ -element permutations are considered, and where  $\ell(\sigma)$  is the number of disjoint cycles in the standard decomposition of  $\sigma$ .

Solution: \boxed{ n! }


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

*Proposed by David Altizio*


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


Find all naturals  $k$  such that  $3^k+5^k$  is the power of a natural number with the exponent  $\ge 2$ .

Solution: \boxed{ k = 1 }


Triangle  $ABC$  has right angle at  $B$ , and contains a point  $P$  for which  $PA = 10$ ,  $PB = 6$ , and  $\angle APB = \angle BPC = \angle CPA$ .  Find  $PC$ .

[asy]
pair A=(0,5), B=origin, C=(12,0), D=rotate(-60)*C, F=rotate(60)*A, P=intersectionpoint(A--D, C--F);
draw(A--P--B--A--C--B^^C--P);
dot(A^^B^^C^^P);
pair point=P;
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $P$ ", P, NE);[/asy]


Determine all functions  $ f$  mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:
(1)  $ f(mn) \equal{} f(m)\plus{}f(n)$ ,
(2)  $ f(2008) \equal{} 0$ , and
(3)  $ f(n) \equal{} 0$  for all  $ n \equiv 39\pmod {2008}$ .

Solution: \boxed{ f(n) = 0 }


There are  $27$  cards, each has some amount of ( $1$  or  $2$  or  $3$ ) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a *match* such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a *match* be cause they have distinct amount of shapes, distinct shapes but the same color of shapes.
What is the maximum number of cards that we can choose such that non of the triples make a *match*?

*Proposed by Amin Bahjati*


Find the smallest positive integer  $n$  such that if we color in red  $n$  arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.


Let   ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are  $ b $  a's in total. That is  $ a\uparrow\uparrow b $  is given by the recurrence \[  a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of  $  3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) $  when divided by  $ 60 $ ?

Solution: \boxed{ 27 }


If the base of a rectangle is increased by  $ 10\%$  and the area is unchanged, then the altitude is decreased by:

 $ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$

Solution: \boxed{9 \frac{1}{11}\%}


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

Solution: \boxed{2\sqrt{2} - 2}


Joshua likes to play with numbers and patterns.  Joshua's favorite number is  $6$  because it is the units digit of his birth year,  $1996$ .  Part of the reason Joshua likes the number  $6$  so much is that the powers of  $6$  all have the same units digit as they grow from  $6^1$ :
\begin{align*}6^1&=6,6^2&=36,6^3&=216,6^4&=1296,6^5&=7776,6^6&=46656,\vdots\end{align*}
However, not all units digits remain constant when exponentiated in this way.  One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above.  Michael responds, "You tell me!"  Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover.  "See if you can find the units digit of  $2008^{2008}$ ," Michael challenges.  After a little while, Joshua finds an answer which Michael confirms is correct.  What is Joshua's correct answer (the units digit of  $2008^{2008}$ )?


A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list  $10,34,34,22,30,22$  the sublist  $22,30,22$  is good and  $10,34,34,22$  is not. A list is very good if all its sublists are good. Find the minimum value of  $k$  such that there exists a very good list of length  $2019$  with  $k$  different values on it.


Find all real polynomials  $ f$  with  $ x,y \in \mathbb{R}$  such that
\[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
\]

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


Let  $S = \{1, 2, \ldots, 2016\}$ , and let  $f$  be a randomly chosen bijection from  $S$  to itself. Let  $n$  be the smallest
positive integer such that  $f^{(n)}(1) = 1$ , where  $f^{(i)}(x) = f(f^{(i-1)}(x))$ . What is the expected value of  $n$ ?


The sequence  $a_1,a_2,\dots,a_{13}$  is a geometric sequence with  $a_1=a$  and common ratio  $r$ , where  $a$  and  $r$  are positive integers. Given that  $$ \log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015, $$  find the number of possible ordered pairs  $(a,r)$ .


Determine the maximal size of a set of positive integers with the following properties: $1.$  The integers consist of digits from the set  $\{ 1,2,3,4,5,6\}$ . $2.$  No digit occurs more than once in the same integer. $3.$  The digits in each integer are in increasing order. $4.$  Any two integers have at least one digit in common (possibly at different positions). $5.$  There is no digit which appears in all the integers.


If positive integers  $p,q,r$  are such that the quadratic equation  $px^2-qx+r=0$  has two distinct real roots in the open interval  $(0,1)$ , find the minimum value of  $p$ .


Consider a round table on which  $2014$  people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated  $2$  seats away from the person at the head of the table.


A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\bullet$  All its digits are different. $\bullet$  The digit on the far left is equal to the sum of the other four digits.  Example:  $91350$  is an Ecuadorian number since  $9 = 1 + 3 + 5 + 0$ , but  $54210$  is not since  $5 \ne 4 + 2 + 1 + 0$ .
Find how many Ecuadorian numbers exist.


Determine the number of positive integers  $n$  satisfying:

- $n<10^6$
- $n$  is divisible by 7
- $n$  does not contain any of the digits 2,3,4,5,6,7,8.


Find an integral solution of the equation
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \]
(Note  $\lfloor u \rfloor$  stands for the greatest integer less than or equal to  $u$ .)


A country has  $ 1998$  airports connected by some direct flights. For any three airports, some two are not connected by a direct flight. What is the maximum number of direct flights that can be offered?


Tyler rolls two  $ 4025 $  sided fair dice with sides numbered  $ 1, \dots , 4025 $ . Given that the number on the ﬁrst die is greater than or equal to the number on the second die, what is the probability that the number on the ﬁrst die is less than or equal to  $ 2012 $ ?


Determine all integers  $n \geq 2$  for which the number  $11111$  in base  $n$  is a perfect square.

Solution: \boxed{n = 3}


Let  $p\ge 3$  be a prime number. Each side of a triangle is divided into  $p$  equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.

Solution: \boxed{3p^2 - 3p + 1}


Let  $N$  be the greatest integer multiple of  $8,$  no two of whose digits are the same. What is the remainder when  $N$  is divided by  $1000?$


Given a parameterized curve  $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$ .

Find the area bounded by the curve  $ C$ , the  $ x$  axis and two lines  $ x\equal{}\pm 1$ .


Suppose that  $n$  is a positive integer and let \[d_{1}<d_{2}<d_{3}<d_{4}\] be the four smallest positive integer divisors of  $n$ . Find all integers  $n$  such that \[n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.\]

Solution: \boxed{ n = 130 }


Alexa wrote the first  $16$  numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had  $2015$  numbers in total.
What was the last number she wrote?


The number  $123454321$  is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by  $9$ . What is the fewest number of digits he could have erased?

*Ray Li*


How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?


Your math friend Steven rolls five fair icosahedral dice (each of which is labelled  $1,2, \dots,20$  on its sides). He conceals the results but tells you that at least half the rolls are  $20$ . Suspicious, you examine the first two dice and find that they show  $20$  and  $19$  in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show  $20$ ?


We call a set “sum free” if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of  $\{ 1, 2, \ldots , 2n - 1 \}$ .

Solution: \boxed{n}


Solve the system of equation for  $(x,y) \in \mathbb{R}$   $$ \left\{\begin{matrix}
\sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5
  3x^2+4xy=24
\end{matrix}\right. $$
Explain your answer

Solution: \boxed{(2, 1.5)}


Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length  $11$ .


Let  $p$  be a prime.  It is given that there exists a unique nonconstant function  $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$  such that  $\chi(1) = 1$  and  $\chi(mn) = \chi(m)\chi(n)$  for all  $m, n \not\equiv 0 \pmod p$  (here the product  $mn$  is taken mod  $p$ ). For how many positive primes  $p$  less than  $100$  is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual  $a^{-1}$  denotes multiplicative inverse.

*Proposed by David Altizio*


We select  $16$  cells on an  $8\times 8$  chessboard. What is the minimal number of pairs of selected cells in the same row or column?

Solution: \boxed{ 16 }


A snowman is built on a level plane by placing a ball radius  $6$  on top of a ball radius  $8$  on top of a ball radius  $10$  as shown. If the average height above the plane of a point in the snowman is  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers, find  $m + n$ .

[asy]
size(150);
draw(circle((0,0),24));
draw(ellipse((0,0),24,9));
draw(circle((0,-56),32));
draw(ellipse((0,-56),32,12));
draw(circle((0,-128),40));
draw(ellipse((0,-128),40,15));
[/asy]


Find, with proof, all real numbers  $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$ , such that  $ (2 \minus{} \sin 2x)\sin (x \plus{} \frac {\pi}{4}) \equal{} 1$ .

Solution: \boxed{ x = \frac{\pi}{4} }


The mayor of a city wishes to establish a transport system with at least one bus line, in which:
- each line passes exactly three stops,
- every two different lines have exactly one stop in common,
- for each two different bus stops there is exactly one line that passes through both.
Determine the number of bus stops in the city.

Solution: \boxed{ s = 7 }


In an  $m \times n$  grid, each square is either filled or not filled. For each square, its *value* is defined as  $0$  if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let  $f(m,n)$  be the largest total value of squares in the grid. Determine the minimal real constant  $C$  such that   $$ \frac{f(m,n)}{mn} \le C $$  holds for any positive integers  $m,n$ *CSJL*

Solution: \boxed{ C = 2 }


Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability  $\tfrac12$  the lowest note that is higher than the note he just played, and with probability  $\tfrac12$  the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?


We say that a polygon  $P$  is inscribed in another polygon  $Q$  when all the vertices of  $P$  belong to the perimeter of  $Q$ . We also say in this case that  $Q$  is circumscribed to  $P$ . Given a triangle  $T$ , let  $\ell$  be the largest side of a square inscribed in  $T$   and  $L$  is the shortest side of a square circumscribed to  $T$  . Find the smallest possible value of the ratio  $L/\ell$  .


For a constant  $c$ , a sequence  $a_n$  is defined by  $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find  $\lim_{n\to\infty} a_n$ .


Let n be a set of integers.  $S(n)$  is defined as the sum of the elements of n.  $T=\{1,2,3,4,5,6,7,8,9\}$  and A and B are subsets of T such that A  $\cup$   $B=T$  and A  $\cap$   $B=\varnothing$ . The probability that  $S(A)\geq4S(B)$  can be expressed as  $\frac{p}{q}$ . Compute  $p+q$ .

*2022 CCA Math Bonanza Team Round #8*


Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example,  $23$  and  $37$  are "very prime" numbers, but  $237$  and  $357$  are not. Find the largest "prime" number (with justification!).


The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the  $16$  possible colored  $2\times2$  squares.


Define  $a_k = (k^2 + 1)k!$  and  $b_k = a_1 + a_2 + a_3 + \cdots + a_k$ . Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where  $m$  and  $n$  are relatively prime natural numbers. Find  $n - m$ .


Find all four-digit natural numbers  $\overline{xyzw}$  with the property that their sum plus the sum of their digits equals  $2003$ .


Given mobile points  $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$  on the  $x$ - $y$  plane.
Denote by  $D$  the part in which line segment  $PQ$  sweeps. Find the volume  $V$  generated by a rotation of  $D$  around the  $x$ -axis.


A positive integer  $n$  is called  $\textit{un-two}$  if there does not exist an ordered triple of integers  $(a,b,c)$  such that exactly two of  $$ \dfrac{7a+b}{n},\;\dfrac{7b+c}{n},\;\dfrac{7c+a}{n} $$  are integers. Find the sum of all un-two positive integers.

*Proposed by **stayhomedomath***


Ben works quickly on his homework, but tires quickly. The first problem takes him  $1$  minute to solve, and the second problem takes him  $2$  minutes to solve. It takes him  $N$  minutes to solve problem  $N$  on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.


Consider a set  $X$  with  $|X| = n\geq 1$  elements. A family  $\mathcal{F}$  of distinct subsets of  $X$  is said to have property  $\mathcal{P}$  if there exist  $A,B \in \mathcal{F}$  so that  $A\subset B$  and  $|B\setminus A| = 1$ .

i) Determine the least value  $m$ , so that any family  $\mathcal{F}$  with  $|\mathcal{F}| > m$  has property  $\mathcal{P}$ .
ii) Describe all families  $\mathcal{F}$  with  $|\mathcal{F}| = m$ , and not having property  $\mathcal{P}$ .

(*Dan Schwarz*)


A positive integer  $m$  is perfect if the sum of all its positive divisors,  $1$  and  $m$  inclusive, is equal to  $2m$ . Determine the positive integers  $n$  such that  $n^n + 1$  is a perfect number.

Solution: \boxed{ n = 3 }


Parallelogram  $ABCD$  is given such that  $\angle ABC$  equals  $30^o$  . Let  $X$  be the foot of the perpendicular from  $A$  onto  $BC$ , and  $Y$  the foot of the perpendicular from  $C$  to  $AB$ . If  $AX = 20$  and  $CY = 22$ , find the area of the parallelogram.


Let  $z$  be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of  $z$ .

*Proposed by Michael Tang*


A permutation  $(a_1, a_2, a_3, \dots, a_{2012})$  of  $(1, 2, 3, \dots, 2012)$  is selected at random. If  $S$  is the expected value of
		\[
		\sum_{i = 1}^{2012} | a_i - i |,
		\]
		then compute the sum of the prime factors of  $S$ .

*Proposed by Aaron Lin*


A parabola has focus  $F$  and vertex  $V$  , where  $VF = 1$ 0. Let  $AB$  be a chord of length  $100$  that passes through  $F$ . Determine the area of  $\vartriangle VAB$ .


Find the least positive integer  $ n$  so that the polynomial  $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$  has at least one root of modulus  $ 1$ .


Given a square  $ABCD$ . Let  $P\in{AB},\ Q\in{BC},\ R\in{CD}\ S\in{DA}$  and  $PR\Vert BC,\ SQ\Vert AB$  and let  $Z=PR\cap SQ$ . If  $BP=7,\ BQ=6,\ DZ=5$ , then find the side length of the square.


Let  $a$  be a positive real number,  $n$  a positive integer, and define the *power tower*  $a\uparrow n$  recursively with  $a\uparrow 1=a$ , and  $a\uparrow(i+1)=a^{a\uparrow i}$  for  $i=1,2,3,\ldots$ .  For example, we have  $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has  $155$  digits.  For each positive integer  $k$ , let  $x_k$  denote the unique positive real number solution of the equation  $x\uparrow k=10\uparrow (k+1)$ .  Which is larger:  $x_{42}$  or  $x_{43}$ ?

Solution: \boxed{ x_{42} }


In rectangle  $ABCD$ , point  $M$  is the midpoint of  $AB$  and  $P$  is a point on side  $BC$ . The perpendicular bisector of  $MP$  intersects side  $DA$  at point  $X$ . Given that  $AB = 33$  and  $BC = 56$ , find the least possible value of  $MX$ .

*Proposed by Michael Tang*


A quadrilateral  $ABCD$  has a right angle at  $\angle ABC$  and satisfies  $AB = 12$ ,  $BC = 9$ ,  $CD = 20$ , and  $DA = 25$ . Determine  $BD^2$ .
.


Compute the number of positive integers  $n \leq 50$  such that there exist distinct positive integers  $a,b$  satisfying
	\[
		\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
	\]


Let  $a$  and  $b$  be positive integers that satisfy  $ab-7a-11b+13=0$ . What is the minimum possible value of  $a+b$ ?


Toph wants to tile a rectangular  $m\times n$  square grid with the  $6$  types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs  $(m,n)$  is this possible?

Solution: \boxed{ (m, n) }


On the whiteboard, the numbers are written sequentially:  $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$ . Andi has to paste a  $+$  (plus) sign or  $-$  (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.


Let  $n{}$  be a positive integer. What is the smallest sum of digits that  $5^n + 6^n + 2022^n$  can take?


A positive integer  $n$  is called *mythical* if every divisor of  $n$  is two less than a prime.  Find the unique mythical number with the largest number of divisors.

*Proposed by Evan Chen*


Solution: \boxed{(4, -6, 20, 2)}


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

Solution: \boxed{ n = 3 }


Let  $ n \geq 3$  be an odd integer. Determine the maximum value of
\[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\]
where  $ x_{i}$  are positive real numbers from the interval  $ [0,1]$ .

Solution: \boxed{n - 2 + \sqrt{2}}


Let triangle  $ABC$  with incenter  $I$  and circumcircle  $\Gamma$  satisfy  $AB = 6\sqrt{3}, BC = 14,$  and  $CA = 22$ . Construct points  $P$  and  $Q$  on rays  $BA$  and  $CA$  such that  $BP = CQ = 14$ . Lines  $PI$  and  $QI$  meet the tangents from  $B$  and  $C$  to  $\Gamma$ , respectively, at points  $X$  and  $Y$ . If  $XY$  can be expressed as  $a\sqrt{b} - c$  for positive integers  $a,b,c$  with  $c$  squarefree, find  $a + b + c$ .

*Proposed by Andrew Wu*


A subset  $B$  of  $\{1, 2, \dots, 2017\}$  is said to have property  $T$  if any three elements of  $B$  are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property  $T$  may contain.


We define a sequence of natural numbers by the initial values  $a_0 = a_1 = a_2 = 1$  and the recursion   $$  a_n = \bigg \lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \bigg \rfloor  $$
for all  $n \ge 3$ . Find the value of  $a_{2022}$ .


Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime integers. Find  $m+n$ .

*Proposed by **AOPS12142015***


Ria writes down the numbers  $1,2,\cdots, 101$  in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen?


Each number in the list  $1,2,3,\ldots,10$  is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . What is  $m+n$ ?

*2021 CCA Math Bonanza Team Round #9*


Anton and Britta play a game with the set  $M=\left \{ 1,2,\dots,n-1 \right \}$  where  $n \geq 5$  is an odd integer. In each step Anton removes a number from  $M$  and puts it in his set  $A$ , and Britta removes a number from  $M$  and puts it in her set  $B$  (both  $A$  and  $B$  are empty to begin with). When  $M$  is empty, Anton picks two distinct numbers  $x_1, x_2$  from  $A$  and shows them to Britta. Britta then picks two distinct numbers  $y_1, y_2$  from  $B$ . Britta wins if $(x_1x_2(x_1-y_1)(x_2-y_2))^{\frac{n-1}{2}}\equiv 1\mod n$ otherwise Anton wins. Find all  $n$  for which Britta has a winning strategy.

Solution: \boxed{ n }


Let  $ABC$  be a triangle whose angles measure  $A$ ,  $B$ ,  $C$ , respectively.  Suppose  $\tan A$ ,  $\tan B$ ,  $\tan C$  form a geometric sequence in that order. If  $1\le \tan A+\tan B+\tan C\le 2015$ , find the number of possible integer values for  $\tan B$ .  (The values of  $\tan A$  and  $\tan C$  need not be integers.)

*Proposed by Justin Stevens*


in a right-angled triangle  $ABC$  with  $\angle C=90$ , $a,b,c$  are the corresponding sides.Circles  $K.L$  have their centers on  $a,b$  and are tangent to  $b,c$ ; $a,c$  respectively,with radii  $r,t$ .find the greatest real number  $p$  such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.

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


Dave has a pile of fair standard six-sided dice. In round one, Dave selects eight of the dice and rolls them. He calculates the sum of the numbers face up on those dice to get  $r_1$ . In round two, Dave selects  $r_1$  dice and rolls them. He calculates the sum of the numbers face up on those dice to get  $r_2$ . In round three, Dave selects  $r_2$  dice and rolls them. He calculates the sum of the numbers face up on those dice to get  $r_3$ . Find the expected value of  $r_3$ .


On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?


Let  $p$  be a polynomial with integer coefficients such that  $p(15)=6$ ,  $p(22)=1196$ , and  $p(35)=26$ . Find an integer  $n$  such that  $p(n)=n+82$ .


How many distinct positive integers can be expressed in the form  $ABCD - DCBA$ , where  $ABCD$  and  $DCBA$  are 4-digit positive integers?  (Here  $A$ ,  $B$ ,  $C$  and  $D$  are digits, possibly equal.)
Clarification:   $A$  and  $D$  can't be zero (because otherwise  $ABCD$  or  $DCBA$  wouldn't be a true 4-digit integer).


Let  $(F_n)_{n\in{N^*}}$  be the Fibonacci sequence defined by
 $F_1=1$ ,  $F_2=1$ ,  $F_{n+1}=F_n+F_{n-1}$  for every  $n\geq{2}$ . Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]


Sequence  ${u_n}$  is defined with  $u_0=0,u_1=\frac{1}{3}$  and  $$ \frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1}) $$    $\forall n=1,2,...$ Show that  $|u_n|\leq1$   $\forall n\in\mathbb{N}.$

Solution: \boxed{ |u_n| \leq 1 }


The number  $16^4+16^2+1$  is divisible by four distinct prime numbers. Compute the sum of these four primes.

*2018 CCA Math Bonanza Lightning Round #3.1*


Marisela is putting on a juggling show! She starts with  $1$  ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.


Let  $O$  and  $I$  be the circumcenter and incenter of triangle  $ABC$ . The perpendicular from  $I$  to  $OI$  meets  $AB$  and the external bisector of angle  $C$  at points  $X$  and  $Y$  respectively. In what ratio does  $I$  divide the segment  $XY$ ?

Solution: \boxed{ 1:2 }


A four-digit number has the following properties:

(a) It is a perfect square;

(b) Its first two digits are equal

(c) Its last two digits are equal.

Find all such four-digit numbers.


Triangle  $ABC$  has sidelengths  $AB=1$ ,  $BC=\sqrt{3}$ , and  $AC=2$ . Points  $D,E$ , and  $F$  are chosen on  $AB, BC$ , and  $AC$  respectively, such that  $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of  $[DEF]^2$  can be expressed as  $\frac{a}{b}$  for positive integers  $a, b$  with  $\gcd (a, b) = 1$ , find  $a + b$ . (Here  $[DEF]$  denotes the area of triangle  $DEF$ .)

*Proposed by Vismay Sharan*


Let  $m$  be a given positive integer which has a prime divisor greater than  $\sqrt {2m} +1 $ . Find the minimal positive integer  $n$  such that there exists a finite set  $S$  of distinct positive integers satisfying the following two conditions:

**I.**  $m\leq x\leq n$  for all  $x\in S$ ;

**II.** the product of all elements in  $S$  is the square of an integer.

Solution: \boxed{n = m + p}


Solve the following system of equations for real  $x,y$  and  $z$ :
\begin{eqnarray*}
x &=& \sqrt{2y+3}
y &=& \sqrt{2z+3}
z &=& \sqrt{2x+3}.
\end{eqnarray*}

Solution: \boxed{x = y = z = 3}


Given that  $5^{2018}$  has  $1411$  digits and starts with  $3$  (the leftmost non-zero digit is  $3$ ), for how many integers  $1\leq n\leq2017$  does  $5^n$  start with  $1$ ?

*2018 CCA Math Bonanza Tiebreaker Round #3*


Let  $P$  be a square pyramid whose base consists of the four vertices  $(0, 0, 0), (3, 0, 0), (3, 3, 0)$ , and  $(0, 3, 0)$ , and whose apex is the point  $(1, 1, 3)$ . Let  $Q$  be a square pyramid whose base is the same as the base of  $P$ , and whose apex is the point  $(2, 2, 3)$ . Find the volume of the intersection of the interiors of  $P$  and  $Q$ .


Let the positive integer  $n$  have at least for positive divisors and  $0<d_1<d_2<d_3<d_4$  be its least positive divisors. Find all positive integers  $n$  such that:
\[ n=d_1^2+d_2^2+d_3^2+d_4^2. \]


Given an integer number  $n \geq 3$ , consider  $n$  distinct points on a circle, labelled  $1$  through  $n$ .
Determine the maximum number of closed chords  $[ij]$ ,  $i \neq j$ , having pairwise non-empty intersections.

*János Pach*

Solution: \boxed{ n }


A group of  $n$  people play a board game with the following rules:
1) In each round of the game exactly  $3$  people play
2) The game ends after exactly  $n$  rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of  $n$


Find all solutions to  $aabb=n^4-6n^3$ , where  $a$  and  $b$  are non-zero digits, and  $n$  is an integer.  ( $a$  and  $b$  are not necessarily distinct.)


Find the least positive integer  $n$  such that the decimal representation of the binomial coefficient  $\dbinom{2n}{n}$  ends in four zero digits.


$a,b,c,x,y,z$  are positive real numbers and  $bz+cy=a$ ,  $az+cx=b$ ,  $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$


Let there be a regular polygon of  $n$  sides with center  $O$ . Determine the highest possible number of vertices  $k$   $(k \geq 3)$ , which can be coloured in green, such that  $O$  is strictly outside of any triangle with  $3$  vertices coloured green. Determine this  $k$  for  $a)  n=2019$  ;   $b)  n=2020$ .


Let  $S$  be the set of all 3-tuples  $(a, b, c)$  of positive integers such that  $a + b + c = 2013$ . Find   $$ \sum_{(a,b,c)\in S} abc. $$


Let  $ T$  be the set of all positive integer divisors of  $ 2004^{100}$ . What is the largest possible number of elements of a subset  $ S$  of  $ T$  such that no element in  $ S$  divides any other element in  $ S$ ?


Twelve  $1$ 's and ten  $-1$ 's are written on a chalkboard.  You select 10 of the numbers and compute their product, then add up these products for every way of choosing 10 numbers from the 22 that are written on the chalkboard.  What sum do you get?


Find all integer solutions  $(p, q, r)$  of the equation  $r + p ^ 4 = q ^ 4$  with the following conditions:  $\bullet$   $r$  is a positive integer with exactly  $8$  positive divisors.  $\bullet$    $p$  and  $q$  are prime numbers.

Solution: \boxed{ (2, 5, 609) }


If positive real numbers  $x,y,z$  satisfy the following system of equations, compute  $x+y+z$ .  $$ xy+yz = 30 $$    $$ yz+zx = 36 $$    $$ zx+xy = 42 $$  *Proposed by Nathan Xiong*


Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\]


The positive integers  $a$  and  $b$  are such that the numbers  $15a+16b$  and  $16a-15b$  are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?


Suppose that  $0^\circ < A < 90^\circ$  and  $0^\circ < B < 90^\circ$  and \[\left(4+\tan^2 A\right)\left(5+\tan^2 B\right) = \sqrt{320}\tan A\tan B\] Determine all possible values of  $\cos A\sin B$ .


A number is called *purple* if it can be expressed in the form  $\frac{1}{2^a 5^b}$  for positive integers  $a > b$ . The sum of all purple numbers can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

*Proposed by Eugene Chen*


Consider the sequence of numbers defined recursively by  $t_1=1$  and for  $n>1$  by  $t_n=1+t_{(n/2)}$  when  $n$  is even and by  $t_n=\frac{1}{t_{(n-1)}}$  when  $n$  is odd. Given that  $t_n=\frac{19}{87}$ , the sum of the digits of  $n$  is $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 19 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 23$


Let  $ABC$  be an acute triangle with orthocenter  $H$ . Let  $M$ ,  $N$  and  $R$  be the midpoints of  $AB$ ,  $BC$  an  $AH$ , respectively. If  $A\hat{B}C=70^\large\circ$ , compute  $M\hat{N}R$ .


Given a positive integer  $n$ , suppose that  $P(x,y)$  is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all  $x,y\in\{0,1,2,\dots,n\}$ } \] What is the minimum degree of  $P$ ?

*Proposed by Loke Zhi Kin*

Solution: \boxed{ 2n }


Willy Wonka has  $n$  distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is  $p(n)$ , then we have
        \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
             $n$  &  $1$  &  $2$  &  $3$  &  $4$  &  $5$  &  $6$  &  $7$  &  $8$  &  $9$  &  $10$  \hline
             $p(n)$  &  $1$  &  $2$  &  $5$  &  $15$  &  $52$  &  $203$  &  $877$  &  $4140$  &  $21147$  &  $115975$  \hline
        \end{tabular}
    Define a *splitting* of the  $n$  distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has  $8$  candies, what is the sum of the number of groups over all splittings he can use?

*2020 CCA Math Bonanza Lightning Round #3.4*


A circle with diameter  $23$  is cut by a chord  $AC$ . Two different circles can be inscribed between the large circle and  $AC$ . Find the sum of the two radii.


Circle inscribed in square  $ABCD$  , is tangent to sides  $AB$  and  $CD$  at points  $M$  and  $K$  respectively. Line  $BK$  intersects this circle at the point  $L, X$  is the midpoint of  $KL$ . Find the angle  $\angle MXK $ .


For nonnegative real numbers  $x,y,z$  and  $t$  we know that  $|x-y|+|y-z|+|z-t|+|t-x|=4$ .
Find the minimum of  $x^2+y^2+z^2+t^2$ .

*proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi*


We define a sequence of positive integers  $a_1,a_2,a_3,\dots$  as follows: Let  $a_1=1$  and iteratively, for  $k =2,3,\dots$  let  $a_k$  be the largest prime factor of  $1+a_1a_2\cdots a_{k-1}$ . Show that the number  $11$  is not an element of this sequence.

Solution: \boxed{ 11 }


A nonnegative integer  $n$  is said to be  $\textit{squarish}$  is it satisfies the following conditions: $\textbf{(i)}$  it contains no digits of  $9$ ; $\textbf{(ii)}$  no matter how we increase exactly one of its digits with  $1$ , the outcome is a square.

Find all squarish nonnegative integers. $\textit{(Vlad Robu)}$

Solution: \boxed{0, 3, 8, 15}


A geometric progression of positive integers has  $n$  terms; the first term is  $10^{2015}$  and the last term is an odd positive integer. How many possible values of  $n$  are there?

*Proposed by Evan Chen*


A one-person game with two possible outcomes is played as follows:
After each play, the player receives either  $a$  or  $b$  points, where  $a$  and  $b$  are integers with  $0 < b < a < 1986$ . The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer  $x \geq 1986$  can be obtained as the total score whereas  $1985$  and  $663$  cannot. Determine  $a$  and  $b.$

Solution: \boxed{a = 332, b = 7}


**9.** Let  $w=f(x)$  be regular in  $ \left | z \right |\leq 1$ . For  $0\leq r \leq 1$ , denote by c, the image by  $f(z)$  of the circle   $\left | z \right | = r$ . Show that if the maximal length of the chords of  $c_{1}$  is  $1$ , then for every  $r$  such that  $0\leq r \leq 1$ , the maximal length of the chords of c, is not greater than  $r$ . **(F. 1)**

Solution: \boxed{r}


We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures  $75$  degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated  $75$  degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated  $75$  degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection?


Find the number of ordered pairs  $(x,y)$  of positive integers that satisfy  $x\le 2y\le 60$  and  $y\le 2x\le 60.$


Consider the sets  $A_1=\{1\}$ ,  $A_2=\{2,3,4\}$ ,  $A_3=\{5,6,7,8,9\}$ , etc. Let  $b_n$  be the arithmetic mean of the smallest and the greatest element in  $A_n$ . Show that the number  $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$  is a prime integer.

Solution: \boxed{ 1999 }


Two distinct squares of the  $8\times8$  chessboard  $C$  are said to be adjacent if they have a vertex or side in common.
Also,  $g$  is called a  $C$ -gap if for every numbering of the squares of  $C$  with all the integers  $1, 2, \ldots, 64$  there exist twoadjacent squares whose numbers differ by at least  $g$ . Determine the largest  $C$ -gap  $g$ .


Suppose  $w,x,y,z$  satisfy \begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of  $w$  can be expressed in lowest terms as  $w_1/w_2$  for some integers  $w_1,w_2>0$ .  Find  $w_1+w_2$ .

Solution: \boxed{ 25 + 2 = 27 }


Find the sum of first two integers  $n > 1$  such that  $3^n$  is divisible by  $n$  and  $3^n - 1$  is divisible by  $n - 1$ .


Consider a rectangle  $ABCD$  with  $AB = a$  and  $AD = b.$  Let  $l$  be a line through  $O,$  the center of the rectangle, that cuts  $AD$  in   $E$  such that  $AE/ED = 1/2$ . Let  $M$  be any point on  $l,$  interior to the rectangle.
Find the necessary and suﬃcient condition on  $a$  and  $b$  that the four distances from M to lines  $AD, AB, DC, BC$  in this order form an arithmetic progression.

Solution: \boxed{a = b}


Integers from 1 to 100 are placed in a row in some order. Let us call a number *large-right*, if it is greater than each number to the right of it; let us call a number *large-left*, is it is greater than each number to the left of it. It appears that in the row there are exactly  $k$  large-right numbers and exactly  $k$  large-left numbers. Find the maximal possible value of  $k$ .

Solution: \boxed{ k = 50 }


Consider the set  $E = \{5, 6, 7, 8, 9\}$ . For any partition  ${A, B}$  of  $E$ , with both  $A$  and  $B$  non-empty, consider the number obtained by adding the product of elements of  $A$  to the product of elements of  $B$ . Let  $N$  be the largest prime number amonh these numbers. Find the sum of the digits of  $N$ .


On the bisector  $AL$  of triangle  $ABC$  a point  $K$  is chosen such that  $\angle BKL=\angle KBL=30^{\circ}$ . Lines  $AB$  and  $CK$  intersect at point  $M$ , lines  $AC$  and  $BK$  intersect at point  $N$ . FInd the measure of angle  $\angle AMN$ *Proposed by D. Shiryaev, S. Berlov*


Find the number of permutations  $x_1, x_2, x_3, x_4, x_5$  of numbers  $1, 2, 3, 4, 5$  such that the sum of five products  $$ x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2 $$  is divisible by  $3$ .


When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio  $2$  to  $1$ .  How many integer values of  $k$  are there such that  $0<k\leq 2007$  and the area between the parabola  $y=k-x^2$  and the  $x$ -axis is an integer?

[asy]
import graph;
size(300);
defaultpen(linewidth(0.8)+fontsize(10));
real k=1.5;
real endp=sqrt(k);
real f(real x) {
return k-x^2;
}
path parabola=graph(f,-endp,endp)--cycle;
filldraw(parabola, lightgray);
draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0));
label("Region I", (0,2*k/5));
label("Box II", (51/64*endp,13/16*k));
label("area(I) =  $\frac23$ \,area(II)",(5/3*endp,k/2));
[/asy]


Let  $\alpha$  and  $\beta$  be positive integers such that   $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$ . Find the minimum possible value of  $\beta$ .


There are several different positive integers written on the blackboard, and the sum of any two different numbers should be should be a prime power. At this time, find the maximum possible number of integers written on the blackboard. A prime power is an integer expressed in the form  $p^n$  using a prime number  $p$  and a non-negative integer number  $n$ .


Let  $a_1, a_2, \ldots$  and  $b_1, b_2, \ldots$  be sequences such that  $a_ib_i - a_i - b_i = 0$  and  $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$  for all  $i \ge 1$ .  If  $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$ , then what is  $b_{6}$ ?

*Proposed by Andrew Wu*


Let  $ABC$  be a triangle in which ( ${BL}$ is the angle bisector of  ${\angle{ABC}}$   $\left( L\in AC \right)$ ,  ${AH}$  is an altitude of $\vartriangle ABC$   $\left( H\in BC \right)$  and  ${M}$ is the midpoint of the side  ${AB}$ . It is known  that the midpoints of the segments  ${BL}$  and  ${MH}$  coincides. Determine the internal angles of triangle  $\vartriangle ABC$ .


Let  $P$  be a  $10$ -degree monic polynomial with roots  $r_1, r_2, . . . , r_{10} \ne $  and let  $Q$  be a  $45$ -degree monic polynomial with roots  $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$  where  $i < j$  and  $i, j \in \{1, ... , 10\}$ . If  $P(0) = Q(1) = 2$ , then  $\log_2 (|P(1)|)$  can be written as  $a/b$  for relatively prime integers  $a, b$ . Find  $a + b$ .


Bob writes a random string of  $5$  letters, where each letter is either  $A, B, C,$  or  $D$ . The letter in each position is independently chosen, and each of the letters  $A, B, C, D$  is chosen with equal probability. Given that there are at least two  $A's$  in the string, find the probability that there are at least three  $A's$  in the string.


The polynomial of seven variables  $$ Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2) $$  is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:  $$ Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2. $$  Find all possible values of  $P_1(1,1,\ldots,1)$ .

*(A. Yuran)*


A sequence  $\{a_n\}$  is defined by  $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$

Find the constant number  $c$  such that  $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

Solution: \boxed{ c = 5 }


**Q13.** Determine the greatest value of the sum  $M=11xy+3x+2012yz$ , where  $x,y,z$  are non negative integers satisfying condition  $x+y+z=1000.$


Find the smallest prime number that can not be written in the form  $\left| 2^a-3^b \right|$  with non-negative integers  $a,b$ .


Find the minimum area of the part bounded by the parabola  $ y\equal{}a^3x^2\minus{}a^4x\ (a>0)$  and the line  $ y\equal{}x$ .


Determine all four-digit numbers  $\overline{abcd}$  which are perfect squares and for which the equality holds: $\overline{ab}=3 \cdot \overline{cd} + 1$ .


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


Suppose that  $p,q$  are prime numbers such that  $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$  is an integer.
Show that  $p = q$ .

Solution: \boxed{ p = q }


Find all quadruples  $(a, b, c, d)$  of non-negative integers such that  $ab =2(1 + cd)$  and there exists a non-degenerate triangle with sides of length  $a - c$ ,  $b - d$ , and  $c + d$ .

Solution: \boxed{ (1, 2, 0, 1) }


Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela will do it. If 2 or more of the dice are a multiple of 3 neither one of them will do it. How many dice should be rolled so that the risk is the same for both Claudia and Adela?

Solution: \boxed{ x = 2 }


Let  $x_1=1$  and  $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$ , for  $n=1,2,3,\ldots $  where  $x$  denotes the largest integer not greater than  $x$ . Determine  $x_{1997}$ .


Define  $ \{ a_n \}_{n\equal{}1}$  as follows:  $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$  is the sum of the digits of  $ a_{n\minus{}1}$ . What is the value of  $ a_5$ ?


Triangles  $\triangle ABC$  and  $\triangle A'B'C'$  lie in the coordinate plane with vertices  $A(0,0)$ ,  $B(0,12)$ ,  $C(16,0)$ ,  $A'(24,18)$ ,  $B'(36,18)$ , and  $C'(24,2)$ . A rotation of  $m$  degrees clockwise around the point  $(x,y)$ , where  $0<m<180$ , will transform  $\triangle ABC$  to  $\triangle A'B'C'$ . Find  $m+x+y$ .


Let  $p$  be an odd prime number. For positive integer  $k$  satisfying  $1\le k\le p-1$ , the number of divisors of  $kp+1$  between  $k$  and  $p$  exclusive is  $a_k$ . Find the value of  $a_1+a_2+\ldots + a_{p-1}$ .


The rectangle in the figure has dimensions  $16$  x  $20$  and is divided into  $10$  smaller equal rectangles. What is the perimeter of each of the  $10$  smaller rectangles?


What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?


Find (in terms of  $n \ge 1$ ) the number of terms with odd coefficients after expanding the product:
\[\prod_{1 \le i < j \le n} (x_i + x_j)\]
e.g., for  $n = 3$  the expanded product is given by  $x_1^2 x_2 + x_1^2 x_3 + x_2^2 x_3 + x_2^2 x_1 + x_3^2 x_1 + x_3^2 x_2 + 2x_1 x_2 x_3$  and so the answer would be  $6$ .

Solution: \boxed{ n! }


Define mutually externally tangent circles  $\omega_1$ ,  $\omega_2$ , and  $\omega_3$ . Let  $\omega_1$  and  $\omega_2$  be tangent at  $P$ . The common external tangents of  $\omega_1$  and  $\omega_2$  meet at  $Q$ . Let  $O$  be the center of  $\omega_3$ . If  $QP = 420$  and  $QO = 427$ , find the radius of  $\omega_3$ .

*Proposed by Tanishq Pauskar and Mahith Gottipati*


An international firm has 250 employees, each of whom speaks several languages. For each pair of employees,  $(A,B)$ , there is a language spoken by  $A$  and not  $B$ , and there is another language spoken by  $B$  but not  $A$ . At least how many languages must be spoken at the firm?


Determine the smallest positive integer  $ n$  such that there exists positive integers  $ a_1,a_2,\cdots,a_n$ , that smaller than or equal to  $ 15$  and are not necessarily distinct, such that the last four digits of the sum,

\[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\]

Is  $ 2001$ .


A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of  $\tfrac{2}{3}$  of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?


Let  $ABCD$  be a square and  $O$  is your center. Let  $E,F,G,H$  points in the segments  $AB,BC,CD,AD$  respectively, such that  $AE = BF = CG = DH$ . The line  $OA$  intersects the segment  $EH$  in the point  $X$ ,  $OB$  intersects  $EF$  in the point  $Y$ ,  $OC$  intersects  $FG$  in the point  $Z$  and  $OD$  intersects  $HG$  in the point  $W$ . If the  $(EFGH) = 1$ . Find: $(ABCD) \times (XYZW)$ Note  $(P)$  denote the area of the polygon  $P$ .


Find the smallest positive integer  $n$  for which we can find an integer  $m$  such that  $\left[\frac{10^n}{m}\right] = 1989$ .

Solution: \boxed{ n = 7 }


How many ways can the integers from  $-7$  to  $7$  inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?


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

Solution: \boxed{ 2033 }


How many integers can be expressed in the form:  $\pm 1 \pm 2 \pm 3 \pm 4 \pm \cdots \pm 2018$ ?


Suppose that  $|x_i| < 1$  for  $i = 1, 2, \dots, n$ .  Suppose further that
\[ |x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|. \]
What is the smallest possible value of  $n$ ?


Let  $K$  be a closed plane curve such that the distance between any two points of  $K$  is always less than  $1.$  Show that  $K$  lies in a circle of radius  $\frac{1}{\sqrt{3}}.$


Let  $a$  and  $b$  be integer solutions to  $17a+6b=13$ . What is the smallest possible positive value for  $a-b$ ?


Cat and Claire are having a conversation about Cat's favorite number.

Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits,  $\overline{AB}$ , such that  $A$  and  $B$  are both factors of  $\overline{AB}$ ."

Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"

Cat says, "That's cool, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!"

Claire says, "Now I know your favorite number!"

What is Cat's favorite number?

*Proposed by Andrew Wu*


Let  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.


Let  $a,b,c,d,e,f$  be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors  $x-x_i$ , with  $x_i>0$  for  $i=1,2,\ldots,8$ . Determine all possible values of  $f$ .


Find the total number of triples of integers  $(x,y,n)$  satisfying the equation  $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$ , where  $n$  is either  $2012$  or  $2013$ .


Consider an acute triangle  $ABC$  of area  $S$ . Let  $CD \perp AB$  ( $D \in AB$ ),   $DM \perp AC$  ( $M \in AC$ ) and  $DN \perp BC$  ( $N \in BC$ ). Denote by  $H_1$  and  $H_2$  the orthocentres of the triangles  $MNC$ , respectively  $MND$ . Find the area of the quadrilateral  $AH_1BH_2$  in terms of  $S$ .

Solution: \boxed{ S }


A circle with area  $40$  is tangent to a circle with area  $10$ . Let R be the smallest rectangle containing both circles. The area of  $R$  is  $\frac{n}{\pi}$ . Find  $n$ .

[asy]
defaultpen(linewidth(0.7)); size(120);
real R = sqrt(40/pi), r = sqrt(10/pi);
draw(circle((0,0), R)); draw(circle((R+r,0), r));
draw((-R,-R)--(-R,R)--(R+2*r,R)--(R+2*r,-R)--cycle);[/asy]

Solution: \boxed{ 240 }


Leticia has a  $9\times 9$  board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has  $4$  friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:

 - If the square is green, write the number of red friends plus twice the number of blue friends.

 - If the square is red, write the number of blue friends plus twice the number of green friends.

 - If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.


Find the largest integer  $n$  which equals the product of its leading digit and the sum of its digits.


A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer  $N$  such that, regardless of the students' sitting positions, at the end there is a set  $S$  consisting of  $N$  students satisfying the following property: every  two of them have never been sitting together.


In a Cartesian coordinate plane, call a rectangle  $standard$  if all of its sides are parallel to the  $x$ - and  $y$ - axes, and call a set of points  $nice$  if no two of them have the same  $x$ - or  $y$ - coordinate. First, Bert chooses a nice set  $B$  of  $2016$  points in the coordinate plane. To mess with Bert, Ernie then chooses a set  $E$  of  $n$  points in the coordinate plane such that  $B\cup E$  is a nice set with  $2016+n$  points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in  $B$  and no points in  $E$  in its interior. For a given nice set  $B$  that Bert chooses, define  $f(B)$  as the smallest positive integer  $n$  such that Ernie can find a nice set  $E$  of size  $n$  with the aforementioned properties. Help Bert determine the minimum and maximum possible values of  $f(B)$ .

*Yannick Yao*


A prime number  $p$  is a **moderate** number if for every  $2$  positive integers  $k > 1$  and  $m$ , there exists k positive integers  $n_1, n_2, ..., n_k $  such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If  $q$  is the smallest **moderate** number, then determine the smallest prime  $r$  which is not moderate and  $q < r$ .

Solution: \boxed{ 7 }


Determine the smallest natural number  $n$  for which there exist distinct nonzero naturals  $a, b, c$ , such that  $n=a+b+c$  and  $(a + b)(b + c)(c + a)$  is a perfect cube.


Determine the least  $n\in\mathbb{N}$  such that  $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$  has at least  $2010$  positive factors.


Find  $n$  such that  $n - 76$  and  $n + 76$  are both cubes of positive integers.


4. Let  $v$  and  $w$  be two randomly chosen roots of the equation  $z^{1997} -1 = 0$  (all roots are equiprobable). Find the probability that  $\sqrt{2+\sqrt{3}}\le |u+w|$


We call a number *perfect* if the sum of its positive integer divisors(including  $1$  and  $n$ ) equals  $2n$ . Determine all *perfect* numbers  $n$  for which  $n-1$  and  $n+1$  are prime numbers.


Let  $ABCD$  be a convex quadrilateral with  $\angle{DAC}= \angle{BDC}= 36^\circ$  ,  $\angle{CBD}= 18^\circ$  and  $\angle{BAC}= 72^\circ$ . The diagonals and intersect at point  $P$  . Determine the measure of  $\angle{APD}$ .


Given two positive integers  $m,n$ , we say that a function  $f : [0,m] \to \mathbb{R}$  is  $(m,n)$ -*slippery* if it has the following properties:

i)  $f$  is continuous;
ii)  $f(0) = 0$ ,  $f(m) = n$ ;
iii) If  $t_1, t_2\in [0,m]$  with  $t_1 < t_2$  are such that  $t_2-t_1\in \mathbb{Z}$  and  $f(t_2)-f(t_1)\in\mathbb{Z}$ , then  $t_2-t_1 \in \{0,m\}$ .

Find all the possible values for  $m, n$  such that there is a function  $f$  that is  $(m,n)$ -slippery.

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


For how many integers  $a$  with  $|a| \leq 2005$ , does the system
 $x^2=y+a$
 $y^2=x+a$
have integer solutions?


There are  $24$  different pencils,  $4$  different colors, and  $6$  pencils of each color. They were given to  $6$  children in such a way that each got  $4$  pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.

P.S. for 10 grade gives same problem with  $40$  pencils,  $10$  of each color and  $10$  children.

Solution: \boxed{ 5 }


Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number.  This number is exactly nine times the product Sarah should have obtained.  What is the sum of the two-digit number and the three-digit number?


Find the positive integer  $n$  such that  $10^n$  cubic centimeters is the same as 1 cubic kilometer.


Let  $A$ ,  $B$ ,  $C$ ,  $D$  be four points on a circle in that order. Also,  $AB=3$ ,  $BC=5$ ,  $CD=6$ , and  $DA=4$ . Let diagonals  $AC$  and  $BD$  intersect at  $P$ . Compute  $\frac{AP}{CP}$ .


 How many (possibly empty) sets of lattice points  $\{P_1, P_2, ... , P_M\}$ , where each point  $P_i =(x_i, y_i)$  for  $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line  $P_iP_j$  is positive for each  $1 \le i < j \le M$  ? An infinite slope, e.g.  $P_i$  is vertically above  $P_j$  , does not count as positive.


In a school there are  $1200$  students. Each student is part of exactly  $k$  clubs. For any  $23$  students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of  $k$ .


Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.


Let  $\alpha$  be a root of  $x^6-x-1$ , and call two polynomials  $p$  and  $q$  with integer coefficients  $\textit{equivalent}$  if  $p(\alpha)\equiv q(\alpha)\pmod3$ .  It is known that every such polynomial is equivalent to exactly one of  $0,1,x,x^2,\ldots,x^{727}$ .  Find the largest integer  $n<728$  for which there exists a polynomial  $p$  such that  $p^3-p-x^n$  is equivalent to  $0$ .


Let  $a$  and  $b$  be positive integers such that  $(a^3 - a^2 + 1)(b^3 - b^2 + 2) = 2020$ . Find  $10a + b$ .


There are  $5$  accents in French, each applicable to only specific letters as follows:

- The cédille: ç
- The accent aigu: é
- The accent circonflexe: â, ê, î, ô, û
- The accent grave: à, è, ù
- The accent tréma: ë, ö, ü

Cédric needs to write down a phrase in French. He knows that there are  $3$  words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If  $n$  is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of  $n$ .


Two knights placed on distinct square of an  $8\times8$  chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is  $\sqrt{5}.$  In how many ways can two identical knights be placed on distinct squares of an  $8\times8$  chessboard such that they do NOT attack each other?


**Q14.** Let be given a trinagle  $ABC$  with  $\angle A=90^o$  and the bisectrices of angles  $B$  and  $C$  meet at  $I$ . Suppose that  $IH$  is perpendicular to  $BC$  ( $H$  belongs to  $BC$ ). If  $HB=5 \text{cm}, \; HC=8 \text{cm}$ , compute the area of  $\triangle ABC$ .


In every row of a grid  $100 \times n$  is written a permutation of the numbers  $1,2 \ldots, 100$ . In one move you can choose a row and swap two non-adjacent numbers with difference  $1$ . Find the largest possible  $n$ , such that at any moment, no matter the operations made, no two rows may have the same permutations.

Solution: \boxed{ 2^{99} }


A single elimination tournament is held with  $2016$  participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner.

[i]Proposed by Nathan Ramesh


For positive integers  $m$  and  $n$ , find the smalles possible value of  $|2011^m-45^n|$ .

*(Swiss Mathematical Olympiad, Final round, problem 3)*


Let  $a$  and  $b$  be positive integers such that  $(2a+b)(2b+a)=4752$ . Find the value of  $ab$ .

*Proposed by James Lin*


Let  $A=\{1,2,3,\ldots,40\}$ . Find the least positive integer  $k$  for which it is possible to partition  $A$  into  $k$  disjoint subsets with the property that if  $a,b,c$  (not necessarily distinct) are in the same subset, then  $a\ne b+c$ .


Let  $\mathcal{P}$  be a parallelepiped with side lengths  $x$ ,  $y$ , and  $z$ .  Suppose that the four space diagonals of  $\mathcal{P}$  have lengths  $15$ ,  $17$ ,  $21$ , and  $23$ .  Compute  $x^2+y^2+z^2$ .


In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer  $n$  for which any tennis tournament with  $n$  people has a group of four tennis players that is ordered.

*Ray Li*


The first  $510$  positive integers are written on a blackboard:  $1, 2, 3, ..., 510$ . An *operation* consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.


Let  $p_1,p_2,p_3$  be primes with  $p_2\neq p_3$  such that  $4+p_1p_2$  and  $4+p_1p_3$  are perfect squares. Find all possible values of  $p_1,p_2,p_3$ .

Solution: \boxed{ (7, 11, 3) }


Determine all squarefree positive integers  $n\geq 2$  such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where  $d_1,d_2,\ldots,d_k$  are all the positive divisors of  $n$ .

Solution: \boxed{ n = 6 }


Kris is asked to compute  $\log_{10} (x^y)$ , where  $y$  is a positive integer and  $x$  is a positive real number. However, they misread this as  $(\log_{10} x)^y$  , and compute this value. Despite the reading error, Kris still got the right answer. Given that  $x > 10^{1.5}$  , determine the largest possible value of  $y$ .


Positive integers 1 to 9 are written in each square of a  $ 3 \times 3 $  table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers  $ a, b, c$  with either non-negative numbers  $ a-x, b-x, c+x $  or  $ a+x, b-x, c-x$ , where  $ x $  is a positive number and can vary in each operation.
(1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)?
\[ a) \begin{array}{ccc} 1 & 2 & 3  4 & 5 & 6  7 & 8 & 9 \end{array} )\]
\[ b) \begin{array}{ccc} 2 & 8 & 5  9 & 3 & 4  6 & 7 & 1 \end{array} )\]
(2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Solution: \boxed{ 5 }


Points  $M$ ,  $N$ ,  $P$  are selected on sides  $\overline{AB}$ ,  $\overline{AC}$ ,  $\overline{BC}$ , respectively, of triangle  $ABC$ .  Find the area of triangle  $MNP$  given that  $AM=MB=BP=15$  and  $AN=NC=CP=25$ .

*Proposed by Evan Chen*


Suppose that  $P(x)$  is a monic quadratic polynomial satisfying  $aP(a) = 20P(20) = 22P(22)$  for some integer  $a\neq 20, 22$ . Find the minimum possible positive value of  $P(0)$ .

*Proposed by Andrew Wu*

(Note: wording changed from original to specify that  $a \neq 20, 22$ .)


The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to  $ 100$ . How many cells are there that are on the distance  $ 50$  from each of the three cells?


Find the sum of all positive integers  $B$  such that  $(111)_B=(aabbcc)_6$ , where  $a,b,c$  represent distinct base  $6$  digits,  $a\neq 0$ .


**i.)** Calculate  $x$  if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})}  \]

**ii.)** For each positive number  $x,$  let \[ k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)}  \] Calculate the minimum value of  $k.$


Alice is thinking of a positive real number  $x$ , and Bob is thinking of a positive real number  $y$ . Given that  $x^{\sqrt{y}}=27$  and  $(\sqrt{x})^y=9$ , compute  $xy$ .


Suppose we list the decimal representations of the positive even numbers from left to right. Determine the  $2015^{th}$  digit in the list.


Inside the circle with center  $O$ , points  $A$  and  $B$  are marked so that  $OA = OB$ . Draw a point  $M$  on the circle from which the sum of the distances to points  $A$  and  $B$  is the smallest among all possible.

Solution: \boxed{ M }


Find the smallest prime number  $p$  that cannot be represented in the form  $|3^{a} - 2^{b}|$ , where  $a$  and  $b$  are non-negative integers.

Solution: \boxed{ 41 }


Let  $n \geq 2$  be a fixed integer. Find the least constant  $C$  such the inequality

\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]

holds for any  $x_{1}, \ldots ,x_{n} \geq 0$  (the sum on the left consists of  $\binom{n}{2}$  summands). For this constant  $C$ , characterize the instances of equality.


How many ways can you mark 8 squares of an  $8\times8$  chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)


Find the sum of the two smallest odd primes  $p$  such that for some integers  $a$  and  $b$ ,  $p$  does not divide  $b$ ,  $b$  is even, and  $p^2=a^3+b^2$ .

*2021 CCA Math Bonanza Individual Round #13*


**Q8.** Given a triangle  $ABC$  and  $2$  point  $K \in AB, \; N \in BC$  such that  $BK=2AK, \; CN=2BN$  and  $Q$  is the common point of  $AN$  and  $CK$ . Compute  $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$