Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will **not** happen is given by  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .


The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius  $12$  and circle B has radius  $42,$  find the radius of circle C.


Determine the number of  $2021$ -tuples of positive integers such that the number  $3$  is an element of the tuple and consecutive elements of the tuple differ by at most  $1$ .

*Walther Janous (Austria)*

Solution: \boxed{3^{2021} - 2^{2021}}


The sets  $A$  and  $B$  are subsets of the positive integers. The sum of any two distinct elements of  $A$  is an element of  $B$ . The quotient of any two distinct elements of  $B$  (where we divide the largest by the smallest of the two) is an element of  $A$ . Determine the maximum number of elements in  $A\cup B$ .


Let  $x_1$ ,  $x_2$ , and  $x_3$  be the roots of the polynomial  $x^3+3x+1$ .  There are relatively prime positive integers  $m$  and  $n$  such that  $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ .  Find  $m+n$ .


Find the largest positive integer  $n$  such that  $n\varphi(n)$  is a perfect square.  ( $\varphi(n)$  is the number of integers  $k$ ,  $1 \leq k \leq n$  that are relatively prime to  $n$ )


Let  $d \geq 2$  be a positive integer. Define the sequence  $a_1,a_2,\dots$  by  $$ a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1. $$  Determine all pairs of positive integers  $(p, q)$  such that  $a_p$  divides  $a_q$ .

Solution: \boxed{ (p, q) }


Find distinct positive integers  $n_1<n_2<\dots<n_7$  with the least possible sum, such that their product  $n_1 \times n_2 \times \dots \times n_7$  is divisible by  $2016$ .


There is a fence that consists of  $n$  planks arranged in a line. Each plank is painted with one of the available  $100$  colors. Suppose that for any two distinct colors  $i$  and  $j$ , there is a plank with color  $i$  located to the left of a (not necessarily adjacent) plank with color  $j$ . Determine the minimum possible value of  $n$ .


How many ten-digit whole numbers satisfy the following property: they have only  $2$  and  $5$  as digits, and there are no consecutive  $2$ 's in the number (i.e. any two  $2$ 's are separated by at least one  $5$ )?


You are given a rectangular playing field of size  $13 \times 2$  and any number of dominoes of sizes  $2\times 1$  and  $3\times 1$ . The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?

(Walther Janous)


In this problem, the symbol  $0$  represents the number zero, the symbol  $1$  represents the number seven, the symbol  $2$  represents the number five, the symbol  $3$  represents the number three, the symbol  $4$  represents the number four, the symbol  $5$  represents the number two, the symbol  $6$  represents the number nine, the symbol  $7$  represents the number one, the symbol  $8$  represents an arbitrarily large positive integer, the symbol  $9$  represents the number six, and the symbol  $\infty$  represents the number eight. Compute the value of  $\left|0-1+2-3^4-5+6-7^8\times9-\infty\right|$ .

*2016 CCA Math Bonanza Lightning #5.2*


The integers  $1, 2, \dots, 64$  are written in the squares of a  $8 \times 8$  chess board, such that for each  $1 \le i < 64$ , the numbers  $i$  and  $i+1$  are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?


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

Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!"

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! My favorite number is not among those four numbers, though."

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

What is Cat's favorite number?

*Proposed by Andrew Wu and Andrew Milas*


A triangular array of numbers has a first row consisting of the odd integers  $ 1,3,5,\ldots,99$  in increasing order.  Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry.  Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it.  How many entries in the array are multiples of  $ 67$ ?
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label("3", (2,0));
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label("99", (10,0));

label("4", (1,-1));
label("8", (3,-1));
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label("196", (9,-1));
label(rotate(90)*" $\cdots$ ", (6,-2));[/asy]


Find the smallest value of the expression  $|3 \cdot 5^m - 11 \cdot 13^n|$  for all  $m,n \in N$ .

(Folklore)


Of a rhombus  $ABCD$  we know the circumradius  $R$  of  $\Delta ABC$  and  $r$  of  $\Delta BCD$ . Construct the rhombus.

Solution: \boxed{ ABCD }


A set of positive integers is said to be *pilak* if it can be partitioned into 2 disjoint subsets  $F$  and  $T$ , each with at least  $2$  elements, such that the elements of  $F$  are consecutive Fibonacci numbers, and the elements of  $T$  are consecutive triangular numbers. Find all positive integers  $n$  such that the set containing all the positive divisors of  $n$  except  $n$  itself is pilak.


Determine all pairs of positive integers  $(m,n)$  for which an  $m\times n$  rectangle can be tiled with (possibly rotated) L-shaped trominos.

Solution: \boxed{ (m, n) = (3a, 2b) }


Denote by  $\mathbb{Q}^+$  the set of positive rational numbers. A function  $f : \mathbb{Q}^+ \to \mathbb{Q}$  satisfies

•  $f(p) = 1$  for all primes  $p$ , and

•  $f(ab) = af(b) + bf(a)$  for all  $ a,b \in \mathbb{Q}^+ $ .

For which positive integers  $n$  does the equation  $nf(c) = c$  have at least one solution  $c$  in  $\mathbb{Q}^+$ ?

Solution: \boxed{ n }


Andrew flips a fair coin  $5$  times, and counts the number of heads that appear. Beth flips a fair coin  $6$  times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.


Real numbers  $a$  and  $b$  satisfy the system of equations   $$ \begin{cases} a^3-a^2+a-5=0  b^3-2b^2+2b+4=0  \end{cases} $$    Find the numerical value of the sum  $a+ b$ .


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

Solution: \boxed{ a = -39.0625 }


Given an arbitrary angle  $\alpha$ , compute $cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$  and  $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$  .
Generalize this result and justify your answer.

Solution: \boxed{ 0 }


Consider  $n \geq 2$  distinct points in the plane  $A_1,A_2,...,A_n$  . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?


Define a sequence  $(a_n)$  by  $a_0 =0$  and  $a_n = 1 +\sin(a_{n-1}-1)$  for  $n\geq 1$ . Evaluate  $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k. $$


Determine the smallest positive integer  $n$  whose prime factors are all greater than  $18$ , and that can be expressed as  $n = a^3 + b^3$  with positive integers  $a$  and  $b$ .


Let  $S$  be the sum of all real  $x$  such that  $4^x = x^4$ .  Find the nearest integer to  $S$ .


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


Sristan Thin is walking around the Cartesian plane. From any point  $\left(x,y\right)$ , Sristan can move to  $\left(x+1,y\right)$  or  $\left(x+1,y+3\right)$ . How many paths can Sristan take from  $\left(0,0\right)$  to  $\left(9,9\right)$ ?

*2019 CCA Math Bonanza Individual Round #3*


A positive integer  $n$  is said to be  $m$ -free if  $n \leq m!$  and  $\gcd(i,n)=1$  for each  $i=1,2,...,m$ . Let  $\mathcal{S}_k$  denote the sum of the squares of all the  $k$ -free integers. Find the remainder when  $\mathcal{S}_7-\mathcal{S}_6$  is divided by  $1000$ .

*Proposed by **FedeX333X***


In how many ways can  $100$  be written as the sum of three positive integers  $x, y$ , and  $z$  satisfying  $x < y < z$  ?


A 2 meter long bookshelf is filled end-to-end with 46 books. Some of the books are 3 centimeters thick while all the others are 5 centimeters thick. Find the number of books on the shelf that are 3 centimeters thick.


**(a)** What is the maximal number of acute angles in a convex polygon?**(b)** Consider  $m$  points in the interior of a convex  $n$ -gon. The  $n$ -gon is partitioned into triangles whose vertices are among the  $n + m$  given points (the vertices of the  $n$ -gon and the given points). Each of the  $m$  points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.

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


The NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each worth a set number of points. However, the NEMO is weighted using Fibonacci numbers; that is, the  $n^{\text{th}}$  problem is worth  $F_n$  points, where  $F_1 = F_2 = 1$  and  $F_n = F_{n-1} + F_{n-2}$  for  $n \ge 3$ . The two problem writers are fair people, so they make sure that each of them is responsible for problems worth an equal number of total points. Compute the number of ways problem writing assignments can be distributed between the two writers.

*Proposed by Lewis Chen*


In base- $2$  notation, digits are  $0$  and  $1$  only and the places go up in powers of  $-2$ . For example,  $11011$  stands for  $(-2)^4+(-2)^3+(-2)^1+(-2)^0$  and equals number  $7$  in base  $10$ . If the decimal number  $2019$  is expressed in base  $-2$  how many non-zero digits does it contain ?


All the prime numbers are written in order,  $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers  $a$  and  $b$  with  $a - b \geq 2$ , such that  $p_a - p_b$  divides  $2(a-b)$ .

Solution: \boxed{ (4, 2) }


A semicircle has diameter  $AB$  with  $AB = 100$ . Points  $C$  and  $D$  lie on the semicircle such that  $AC = 28$  and  $BD = 60$ . Find  $CD$ .


Let  $a>0$  and  $\mathcal{F} = \{f:[0,1] \to \mathbb{R} : f \text{ is concave and } f(0)=1 \}.$  Determine   $$ \min_{f \in \mathcal{F}} \bigg\{ \left( \int_0^1 f(x)dx\right)^2 - (a+1) \int_0^1 x^{2a}f(x)dx \bigg\}. $$


What is the number of ways in which one can choose  $60$  unit squares from a  $11 \times 11$  chessboard such that no two chosen squares have a side in common?


Let  $k>1$  be a given positive integer. A set  $S$  of positive integers is called *good* if we can colour the set of positive integers in  $k$  colours such that each integer of  $S$  cannot be represented as sum of two positive integers of the same colour. Find the greatest  $t$  such that the set  $S=\{a+1,a+2,\ldots ,a+t\}$  is *good* for all positive integers  $a$ .

*A. Ivanov, E. Kolev*


If  $\alpha$ ,  $\beta$ , and  $\gamma$  are the roots of  $x^3 - x - 1 = 0$ , compute  $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$ .


Let  $ABC$  be triangle with incenter  $I$ . A point  $P$  in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that  $AP \geq AI$ , and that equality holds if and only if  $P=I$ .

Solution: \boxed{ AP \geq AI }


In parallelogram  $ABCD$ ,  $AB = 10$ , and  $AB = 2BC$ . Let  $M$  be the midpoint of  $CD$ , and suppose that  $BM = 2AM$ . Compute  $AM$ .


$N\geq9$  distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than  $1$ . It happens that for very  $8$  distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values  $N$  for which this is possible. *(F. Nilov)*

Solution: \boxed{ N = 9 }


3. Let n > 1 be an integer and  $a_1, a_2, . . . , a_n$  be positive reals with sum 1.
a) Show that there exists a constant c ≥ 1/2 so that $\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$ ,
where  $a_0 = 0$ .
b) Show that ’the best’ value of c is at least  $\frac{\pi}{4}$ .

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


Suppose  $a$ ,  $b$ ,  $c$ , and  $d$  are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762,  ab+cd &= 260,  ac+bd &= 365,  ad+bc &= 244. \end{aligned}\] Compute  $abcd.$ *Proposed by Michael Tang*


Kevin is in first grade, so his teacher asks him to calculate  $20+1\cdot 6+k$ , where  $k$  is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates  $(20+1)\cdot (6+k)$ . Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his answer?

*Proposed by James Lin*


Solve in the set  $R$  the equation  $$ 2 \cdot [x] \cdot  \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16} $$  where  $[x]$  and  $\{x\}$  represent the integer part and the fractional part of the real number  $x$ , respectively.


Let  $a, b, c$  be the roots of the equation  $x^3-9x^2+11x-1 = 0$ , and define  $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$ .
Compute  $s^4 -18s^2 - 8s$  .


Given that  $x$ ,  $y$  and  $z$  are positive integers such that  $x^3y^5z^6$  is a perfect 7th power of a positive integer, show that also  $x^5y^6z^3$  is a perfect 7th power.

Solution: \boxed{ x^5 y^6 z^3 }


For a positive real number  $  [x] $  be its integer part. For example,   $[2.711] = 2, [7] = 7, [6.9] = 6$ .  $z$  is the maximum real number such that [ $\frac{5}{z}$ ] + [ $\frac{6}{z}$ ] = 7. Find the value of $ 20z$ .


Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let  $m$  be the smallest number of students who could study both languages, and let  $M$  be the largest number of students who could study both languages. Find  $M-m$ .


How many positive integers which are less or equal with  $2013$  such that  $3$  or  $5$  divide the number.


There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists  $A$  and  $B$  are taking turns ( $A$  is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?

Solution: \boxed{ B }


For every point on the plane, one of  $ n$  colors are colored to it such that:

 $ (1)$  Every color is used infinitely many times.

 $ (2)$  There exists one line such that all points on this lines are colored exactly by one of two colors.

Find the least value of  $ n$  such that there exist four concyclic points with pairwise distinct colors.

Solution: \boxed{ n = 5 }


A sequence of positive integers is defined by  $a_0=1$  and  $a_{n+1}=a_n^2+1$  for each  $n\ge0$ . Find  $\text{gcd}(a_{999},a_{2004})$ .


Find all functions  $f : \mathbb{N} \to \mathbb{N}$  such that for any positive integers  $m, n$  the number  $$ (f(m))^2+ 2mf(n) + f(n^2) $$  is the square of an integer.

*Proposed by Fedir Yudin*

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


A robot is standing on the bottom left vertex  $(0,0)$  of a  $5\times5$  grid, and wants to go to  $(5,5)$ , only moving to the right  $(a,b)\mapsto(a+1,b)$  or upward  $(a,b)\mapsto(a,b+1)$ . However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path  $(a,b)\mapsto(a-1,b+1)$ . As the grid is surrounded by walls, the robot cannot go outside the region  $0\leq a,b\leq5$ . Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.


Let  $n$  be a positive integer. A pair of  $n$ -tuples  $(a_1,\cdots{}, a_n)$  and  $(b_1,\cdots{}, b_n)$  with integer entries is called an *exquisite pair* if  $$ |a_1b_1+\cdots{}+a_nb_n|\le 1. $$  Determine the maximum number of distinct  $n$ -tuples with integer entries such that any two of them form an exquisite pair.

*Pakawut Jiradilok and Warut Suksompong, Thailand*

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


It takes  $3$  rabbits  $5$  hours to dig  $9$  holes. It takes  $5$  beavers  $36$  minutes to build  $2$  dams. At this rate, how many more minutes does it take  $1$  rabbit to dig  $1$  hole than it takes  $1$  beaver to build  $1$  dam?

*2016 CCA Math Bonanza Team #1*


What is the least positive integer by which  $2^5 \cdot  3^6 \cdot  4^3 \cdot  5^3 \cdot 6^7$  should be multiplied so that, the product is a perfect square?


During a break,  $n$  children at school sit in a circle around their teacher to play a game.  The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on.  Determine the values of  $n$  for which eventually, perhaps after many rounds, all children will have at least one candy each.

Solution: \boxed{ n = 2^k }


On January 15 in the stormy town of Stormville, there is a  $50\%$  chance of rain. Every day, the probability of it raining has a  $50\%$  chance of being  $\frac{2017}{2016}$  times that of the previous day (or  $100\%$  if this new quantity is over  $100\%$ ) and a  $50\%$  chance of being  $\frac{1007}{2016}$  times that of the previous day. What is the probability that it rains on January 20?

*2018 CCA Math Bonanza Lightning Round #3.3*


A circumference was divided in  $n$  equal parts. On each of these parts one number from  $1$  to  $n$  was placed such that the distance between consecutive numbers is always the same. Numbers  $11$ ,  $4$  and  $17$  were in consecutive positions. In how many parts was the circumference divided?


Find the sum of the squares of the values  $x$  that satisfy  $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$ .


For each integer  $k\ge2$ , the decimal expansions of the numbers  $1024,1024^2,\dots,1024^k$  are concatenated, in that order, to obtain a number  $X_k$ .  (For example,  $X_2 = 10241048576$ .)  If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of  $n$ , where  $n\ge2$  is an integer.

*Proposed by Evan Chen*


Find the last three digits in the product  $1 \cdot 3\cdot  5\cdot 7 \cdot  . . . \cdot  2009 \cdot  2011$ .


We are given an infinite deck of cards, each with a real number on it. For every real number  $x$ , there is exactly one card in the deck that has  $x$  written on it. Now two players draw disjoint sets  $A$  and  $B$  of  $100$  cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
   1. The winner only depends on the relative order of the  $200$  cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
   2. If we write the elements of both sets in increasing order as  $A =\{ a_1 , a_2 , \ldots, a_{100} \}$  and  $B= \{ b_1 , b_2 , \ldots , b_{100} \}$ , and  $a_i > b_i$  for all  $i$ , then  $A$  beats  $B$ .
   3. If three players draw three disjoint sets  $A, B, C$  from the deck,  $A$  beats  $B$  and  $B$  beats  $C$   then  $A$  also beats  $C$ .
How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets  $A$  and  $B$  such that  $A$  beats  $B$  according to one rule, but  $B$  beats  $A$  according to the other.

*Proposed by Ilya Bogdanov, Russia*


There are two points  $A$  and  $B$  in the plane.
a) Determine the set  $M$  of all points  $C$  in the plane for which  $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point  $C\in M$  such that   $\angle ACB$  is maximal and if so, determine this angle.

Solution: \boxed{ 60^\circ }


A triangle  $ABC$  with area  $1$  is given. Anja and Bernd are playing the following game: Anja chooses a point  $X$  on side  $BC$ . Then Bernd chooses a point  $Y$  on side  $CA$  und at last Anja chooses a point  $Z$  on side  $AB$ . Also,  $X,Y$  and  $Z$  cannot be a vertex of triangle  $ABC$ . Anja wants to maximize the area of triangle  $XYZ$  and Bernd wants to minimize that area.
What is the area of triangle  $XYZ$  at the end of the game, if both play optimally?


Consider digits  $\underline{A}, \underline{B}, \underline{C}, \underline{D}$ , with  $\underline{A} \neq 0,$  such that  $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$  Compute the sum of all distinct possible values of  $\underline{A} + \underline{B} + \underline{C} + \underline{D}$ .

*Proposed by Kyle Lee*


The walls of a room are in the shape of a triangle  $ABC$  with  $\angle ABC = 90^\circ$ ,  $\angle BAC = 60^\circ$ , and  $AB=6$ .  Chong stands at the midpoint of  $BC$  and rolls a ball toward  $AB$ .  Suppose that the ball bounces off  $AB$ , then  $AC$ , then returns exactly to Chong.  Find the length of the path of the ball.


**N**ew this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from  $\{0, 1, 2, 3, 4\}$ . Then, the following occurs:
• If  $n>0$ , then the player and everyone else currently on the field moves (counterclockwise) around
the square by n bases. However, if in doing so a player returns to or moves past the home base,
he/she leaves the field immediately and the team scores one point.
• If  $n=0$  (a strikeout), then the game ends immediately; the team does not score any more points.
What is the expected number of points that a given team will score in this game?


Jeffrey jogs  $3$  miles at  $4$  miles per hour. Then, he walks  $4$  miles at  $3$  miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth.


There are  $63$  houses at the distance of  $1, 2, 3, . . . , 63 \text{ km}$  from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants,  $63$  elfs named  $E_1, E_2, . . . , E_{63}$  , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf  $E_n$   takes  $n$  minutes to travel  $1 \text{ km}$  for each  $n = 1,2,...,63$  , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?


Moor has  $3$  different shirts, labeled  $T, E,$  and  $A$ . Across  $5$  days, the only days Moor can wear shirt  $T$  are days  $2$  and  $5$ . How many different sequences of shirts can Moor wear across these  $5$  days?


There is a standard deck of  $52$  cards without jokers. The deck consists of four suits(diamond, club, heart, spade) which include thirteen cards in each. For each suit, all thirteen cards are ranked from “ $2$ ” to “ $A$ ” (i.e.  $2, 3,\ldots , Q, K, A$ ). A pair of cards is called a “*straight flush*” if these two cards belong to the same suit and their ranks are adjacent. Additionally, " $A$ " and " $2$ " are considered to be adjacent (i.e. "A" is also considered as " $1$ "). For example, spade  $A$  and spade  $2$  form a “*straight flush*”; diamond  $10$  and diamond  $Q$  are not a “*straight flush*” pair. Determine how many ways of picking thirteen cards out of the deck such that all ranks are included but no “*straight flush*” exists in them.

Solution: \boxed{ 3^{13} - 3 }


Jeffrey stands on a straight horizontal bridge that measures  $20000$  meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is  $20001$  meters. To the nearest meter, how long of a pole does Jeffrey need?


Let  $q(n)$  denote the sum of the digits of a natural number  $n$ . Determine  $q(q(q(2000^{2000})))$ .


For  $a>0$ , denote by  $S(a)$  the area of the part bounded by the parabolas  $y=\frac 12x^2-3a$  and  $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of  $S(a)$ .


In the  Mathematical  Competition  of  HMS (Hellenic  Mathematical  Society)  take part  boys and  girls  who are divided  into  two  groups : *Juniors*  and  *seniors.*The number of the  boys  taking  part  of this year  competition  is  55%  of the  number  of all participants. The  ratio  of the number  of juniors  boys  to  the number of senior  boys  is equal  to the  ratio of the  number of juniors  to the number of seniors. Find  the ratio  of the number of junior  boys  to the number  of  junior  girls.

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


Let  $ABC$  be a triangle with  $\angle B=120^{o}$ . Let  $D$  be point on the  $B$ -angle bisector, such that  $\angle ADB=2\angle ACB$ . Point  $E$  lies on the segment  $AB$ , so that  $AE=AD$ . Show that  $EC=ED$ .

Solution: \boxed{ EC = ED }


In a qualification group with  $15$  volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding  $N$ . If there are at least  $7$  teams qualified, find the possible least value of  $N$ .


$\vartriangle ABC$  has side lengths  $AB = 4$  and  $AC = 9$ . Angle bisector  $AD$  bisects angle  $A$  and intersects  $BC$  at  $D$ . Let  $k$  be the ratio  $\frac{BD}{AB}$  . Given that the length  $AD$  is an integer, find the sum of all possible  $k^2$ .


Find the least positive integer  $n$  such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]


Let  $ABCD$  be a square with the point of intersection  $O$  of the diagonals and let  $P,\ Q,\ R,\ S$  be the points which are on the segments  $OA,\ OB,\ OC,\ OD$ , respectively such that  $OP=3,\ OQ=5,\ OR=4$ . If the points of intersection of the lines  $AB$  and  $PQ$ , the lines  $BC$  and  $QR$ , the lines  $CD$  and  $RS$  are collinear, then find the length of the segment  $OS$ .


Find the value of  $ a$  for which  $ \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx$  is minimized.

Solution: \boxed{\frac{315}{4 \pi^6}}


Find all possible  $\{ x_1,x_2,...x_n \}$  permutations of  $ \{1,2,...,n \}$  so that  when  $1\le i \le n-2 $  then we have   $x_i < x_{i+2}$  and when  $1 \le i \le n-3$  then we have   $x_i < x_{i+3}$  . Here  $n \ge 4$ .

Solution: \boxed{ p_n = F_{n+1} }


A box contains  $900$  cards, labeled from  $100$  to  $999$ . Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?


On the sides  $ AB $  and  $ AC $  of the triangle  $ ABC $  consider the points  $ D, $  respectively,  $ E, $  such that  $$  \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} .  $$
If  $ T $  is the intersection of  $ DC $  and  $ BE, $  determine the real number  $ \alpha $  so that:  $$  \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} .  $$


Let triangle  $ABC$  have side lengths  $AB = 13$ ,  $BC = 14$ ,  $AC = 15$ . Let  $I$  be the incenter of  $ABC$ . The circle centered at  $A$  of radius  $AI$  intersects the circumcircle of  $ABC$  at  $H$  and  $J$ . Let  $L$  be a point that lies on both the incircle of  $ABC$  and line  $HJ$ . If the minimal possible value of  $AL$  is  $\sqrt{n}$ , where  $n \in \mathbb{Z}$ , find  $n$ .


Find all pairs of natural numbers  $ (a, b)$  such that  $ 7^a \minus{} 3^b$  divides  $ a^4 \plus{} b^2$ .

*Author: Stephan Wagner, Austria*

Solution: \boxed{ (a, b) = (2, 4) }


Shaq sees the numbers  $1$  through  $2017$  written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased  $-2, -1, 0$  he would replace them with  $0$ .) If  $M$  is the maximum possible final value remaining on the board, and if m is the minimum, compute  $M - m$ .


How many ways to fill the board  $ 4\times 4$  by nonnegative integers, such that sum of the numbers of each row and each column is 3?


Find the positive integer  $n$  so that  $n^2$  is the perfect square closest to  $8 + 16 + 24 + \cdots + 8040.$


In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a  $\tfrac{2}{3}$  probability of voting for Alice and a  $\tfrac{1}{3}$  probability of voting for Celia.

What is the probability that Alice wins the election (by having more votes than Celia)?


Let  $ABC$  be a triangle such that  $AB=2$ ,  $CA=3$ , and  $BC=4$ .  A semicircle with its diameter on  $BC$  is tangent to  $AB$  and  $AC$ .  Compute the area of the semicircle.


(a) Determine all pairs  $(x, y)$  of (real) numbers with  $0 < x < 1$  and  $0 <y < 1$  for which  $x + 3y$  and  $3x + y$  are both integer. An example is  $(x,y) =( \frac{8}{3}, \frac{7}{8}) $ , because  $  x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} =  3$  and  $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$ .

(b) Determine the integer  $m > 2$  for which there are exactly  $119$  pairs  $(x,y)$  with  $0 < x < 1$  and  $0 < y < 1$  such that  $x + my$  and   $mx + y$  are integers.

Remark: if  $u \ne v,$  the pairs  $(u, v)$  and  $(v, u)$  are different.

Solution: \boxed{ m = 11 }


In acute triangle  $ABC$ , points  $D$  and  $E$  are the feet of the angle bisector and altitude from  $A$  respectively.  Suppose that  $AC - AB = 36$  and  $DC - DB = 24$ .  Compute  $EC - EB$ .


Given a natural number  $n \geq 2$ , consider all the fractions of the form  $\frac{1}{ab}$ , where  $a$  and  $b$  are natural numbers, relative primes and such that: $a < b \leq n$ ,  $a+b>n$ .
Show that for each  $n$ , the sum of all this fractions are  $\frac12$ .

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


Let  $t$  and  $n$  be fixed integers each at least  $2$ . Find the largest positive integer  $m$  for which there exists a polynomial  $P$ , of degree  $n$  and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each  $k = 0,1, ..., m$ .

*Proposed by Michael Kural*

Solution: \boxed{ n }


In  $\vartriangle ABC, AB=AC=14 \sqrt2 , D$  is the midpoint of  $CA$  and  $E$  is the midpoint of  $BD$ . Suppose  $\vartriangle CDE$  is similar to  $\vartriangle ABC$ . Find the length of  $BD$ .


Let the rational number  $p/q$  be closest to but not equal to  $22/7$  among all rational numbers with denominator  $< 100$ . What is the value of  $p - 3q$ ?


A particle is located on the coordinate plane at  $ (5,0)$ . Define a *move* for the particle as a counterclockwise rotation of  $ \pi/4$  radians about the origin followed by a translation of  $ 10$  units in the positive  $ x$ -direction. Given that the particle's position after  $ 150$  moves is  $ (p,q)$ , find the greatest integer less than or equal to  $ |p|\plus{}|q|$ .


Suppose  $m$  and  $n$  are positive integers for which $\bullet$  the sum of the first  $m$  multiples of  $n$  is  $120$ , and $\bullet$  the sum of the first  $m^3$  multiples of $ n^3$  is  $4032000$ .

Determine the sum of the first  $m^2$  multiples of  $n^2$


Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer  $n$  such that  $2^n < S^{2007} < 2^{n+1}$ .


In equilateral  $\triangle ABC$  let points  $D$  and  $E$  trisect  $\overline{BC}$ .  Then  $\sin \left( \angle DAE \right)$  can be expressed in the form  $\tfrac{a\sqrt{b}}{c}$ , where  $a$  and  $c$  are relatively prime positive integers, and  $b$  is an integer that is not divisible by the square of any prime.  Find  $a+b+c$ .


A *normal magic square* of order  $n$  is an arrangement of the integers from  $1$  to  $n^2$  in a square such that the  $n$  numbers in each row, each column, and each of the two diagonals sum to a constant, called the *magic sum* of the magic square. Compute the magic sum of a normal magic square of order  $8$ .


A block of mass  $m$  on a frictionless inclined plane of angle  $\theta$  is connected by a cord over a small frictionless, massless pulley to a second block of mass  $M$  hanging vertically, as shown. If  $M=1.5m$ , and the acceleration of the system is  $\frac{g}{3}$ , where  $g$  is the acceleration of gravity, what is  $\theta$ , in degrees, rounded to the nearest integer?
[asy]size(12cm);
pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75);

pair B = (-1,-1);
pair C = (-1,-7);
pair A = (-13,-7);
path inclined_plane = A--B--C--cycle;
draw(inclined_plane, p);
real r = 1; // for marking angles
draw(arc(A, r, 0, degrees(B-A))); // mark angle
label(" $\theta$ ", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta
draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle

real h = 1.2; // height of box
real w = 1.9; // width of box
path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box

// box on slope with label
picture box_on_slope;
filldraw(box_on_slope, box, light_grey, black);
label(box_on_slope, " $m$ ", (w/2,h/2));

pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB
pair T1 = dir(125); // point of tangency with pulley
pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box
draw(T1--X1); // string
add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope);

// picture for the hanging box
picture hanging_box;
filldraw(hanging_box, box, light_grey, black);
label(hanging_box, " $M$ ", (w/2,h/2));

pair T2 = (1,0);
pair X2 = (1,-3);
draw(T2--X2); // string
add(shift(X2-(w/2,h)) * hanging_box);

// Draws the actual pulley
filldraw(unitcircle, grey, p); // outer boundary of pulley wheel
filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel
path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley
filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm
filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy]*(Proposed by Ahaan Rungta)*


Let  $S$  be the set of numbers of the form  $n^5 - 5n^3 + 4n$ , where  $n$  is an integer that is not a multiple of  $3$ .  What is the largest integer that is a divisor of every number in  $S$ ?


Twenty-five of the numbers  $1, 2, \cdots , 50$  are chosen. Twenty-five of the numbers $ 51, 52, \cdots, 100$  are also chosen. No two chosen numbers differ by  $0$  or  $50$ . Find the sum of all  $50$  chosen numbers.


Consider the sum
\[
  S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine  $\lfloor S_{4901} \rfloor$ .  Recall that if  $x$  is a real number, then  $\lfloor x \rfloor$  (the *floor* of  $x$ ) is the greatest integer that is less than or equal to  $x$ .


What is the least integer a greater than  $14$  so that the triangle with side lengths  $a - 1$ ,  $a$ , and  $a + 1$  has integer area?


Function  $f:\mathbb{R}\rightarrow\mathbb{R}$  satisfies the following equation for all real  $x$ :   $$ f(f(x))=x^2f(x)-x+1 $$  . Find  $f(1)$


The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Find m.


Let  $ n\ge 3$  be a natural number. Find all nonconstant polynomials with real coeficcietns  $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$ , for which
\[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad  1\le k\le n,\]
for every real  $ x$  (with  $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$  and  $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$ ).

Solution: \boxed{ f_k(x) = x^2 }


Let  $c$  denote the largest possible real number such that there exists a nonconstant polynomial  $P$  with \[P(z^2)=P(z-c)P(z+c)\] for all  $z$ .  Compute the sum of all values of  $P(\tfrac13)$  over all nonconstant polynomials  $P$  satisfying the above constraint for this  $c$ .


What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping?

(A)  $Mg$ (B)  $\mu_k Mg$ (C)  $\mu_k Mg/\mu_s$ (D)  $(\mu_s + \mu_k)Mg$ (E)  $Mg/\mu_s$

Solution: \boxed{ \frac{Mg}{\mu_s} }


At Ignus School there are  $425$  students. Of these students  $351$  study mathematics,  $71$  study Latin, and  $203$  study chemistry. There are  $199$  students who study more than one of these subjects, and  $8$  students who do not study any of these subjects. Find the number of students who study all three of these subjects.


Find all positive integer pairs  $(a, b)$  such that  $\frac{2^{a+b}+1}{2^a+2^b+1}$  is an integer.**General**
Find all positive integer triples  $(t, a, b)$  such that  $\frac{t^{a+b}+1}{t^a+t^b+1}$  is an integer.

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


For how many ordered pairs of positive integers  $(a,b)$  with  $a,b<1000$  is it true that  $a$  times  $b$  is equal to  $b^2$  divided by  $a$ ? For example,  $3$  times  $9$  is equal to  $9^2$  divided by  $3$ .

*Ray Li*


A regular tetrahedron  $SABC$  of volume  $V$  is given. The midpoints  $D$  and  $E$  are taken on  $SA$  and  $SB$  respectively and the point  $F$  is taken on the edge  $SC$  such that  $SF: FC = 1: 3$ . Find the volume of the pentahedron  $FDEABC$ .


The first triangle number is  $1$ ; the second is  $1 + 2 = 3$ ; the third is  $1 + 2 + 3 = 6$ ; and so on. Find the sum of the first  $100$  triangle numbers.


Determine all real solutions  $x, y, z$  of the following system of equations:  $\begin{cases}
x^3 - 3x = 4 - y
2y^3 - 6y = 6 - z
3z^3 - 9z = 8 - x\end{cases}$

Solution: \boxed{ x = y = z = 2 }


What is the largest factor of  $130000$  that does not contain the digit  $0$  or  $5$ ?


A pair of positive integers  $(m,n)$  is called ***'steakmaker'*** if they maintain the equation 1 + 2 $^m$  = n $^2$ . For which values of m and n, the pair  $(m,n)$  are steakmaker, find the sum of  $mn$


Determine the largest positive integer  $n$  for which there exist pairwise different sets  $\mathbb{S}_1 , ..., \mathbb{S}_n$  with the following properties:   $1$ )  $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$  for any two indices  $1 \leq i, j\leq n$ , and  $2$ )  $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$  for any  $1 \leq i < j < k \leq n$ *Proposed by Ivan Matic*


The product of several distinct positive integers is divisible by  ${2006}^{2}$ . Determine the minimum value the sum of such numbers can take.


$2018$  people (call them  $A, B, C, \ldots$ ) stand in a line with each permutation equally likely. Given that  $A$  stands before  $B$ , what is the probability that  $C$  stands after  $B$ ?


When the binomial coefficient  $\binom{125}{64}$  is written out in base 10, how many zeros are at the rightmost end?


Let  $O$  and  $A$  be two points in the plane with  $OA = 30$ , and let  $\Gamma$  be a circle with center  $O$  and radius  $r$ . Suppose that there exist two points  $B$  and  $C$  on  $\Gamma$  with  $\angle ABC = 90^{\circ}$  and  $AB = BC$ . Compute the minimum possible value of  $\lfloor r \rfloor.$


Let  $a,b,c$  be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them:  $ax^2+bx+c, bx^2+cx+a,$  and  $cx^2+ax+b $ .


Define a  $good~word$  as a sequence of letters that consists only of the letters  $A,$   $B,$  and  $C$   $-$  some of these letters may not appear in the sequence  $-$  and in which  $A$  is never immediately followed by  $B,$   $B$  is never immediately followed by  $C,$  and  $C$  is never immediately followed by  $A.$  How many seven-letter good words are there?


Let  $c$  be a fixed real number. Show that a root of the equation
\[x(x+1)(x+2)\cdots(x+2009)=c\]
can have multiplicity at most  $2$ . Determine the number of values of  $c$  for which the equation has a root of multiplicity  $2$ .


Let  $f(x) = ax^2+bx+c$  be a quadratic trinomial with  $a$ , $b$ , $c$  reals such that any quadratic trinomial obtained by a permutation of  $f$ 's coefficients has an integer root (including  $f$  itself).
Show that  $f(1)=0$ .

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


Let  $a$  and  $b$  be real numbers such that  $a^5b^8=12$  and  $a^8b^{13}=18$ . Find  $ab$ .


The point  $P$  is inside of an equilateral triangle with side length  $10$  so that the distance from  $P$  to two of the sides are  $1$  and  $3$ . Find the distance from  $P$  to the third side.

Solution: \boxed{5\sqrt{3} - 4}


Find the smallest possible side of a square in which five circles of radius  $1$  can be placed, so that no two of them have a common interior point.

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


A company that sells keychains has to pay  $\mathdollar500$  in maintenance fees each day and then it pays each work  $\mathdollar15$  an hour. Each worker makes  $5$  keychains per hour, which are sold at  $\mathdollar3.10$  each. What is the least number of workers the company has to hire in order to make a profit in an  $8$ -hour workday?


Darryl has a six-sided die with faces  $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability  $1/2$  and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die  $5$  times and gets a  $1, 2, 3, 4$  and  $5$  in some unspecified
order. Compute the probability that his next roll is a  $6$ .


Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}2342\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}3636\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}4223\hline 65\end{tabular}\,.\]


Find the number of positive integers  $n$  not greater than 2017 such that  $n$  divides  $20^n + 17k$  for some positive integer  $k$ .


Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence.  Compute the greatest possible value of the common ratio of this geometric sequence.


Let  $ a_{1}, a_{2}...a_{n}$  be non-negative reals, not all zero. Show that that
(a) The polynomial  $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$  has preceisely 1 positive real root  $ R$ .
(b) let  $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$  and  $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$ . Show that  $ A^{A} \leq R^{B}$ .

Solution: \boxed{ A^A \leq R^B }


Let  $n$  be an integer with  $n\geq 2.$  Over all real polynomials  $p(x)$  of degree  $n,$  what is the largest possible number of negative coefficients of  $p(x)^2?$

Solution: \boxed{ 2n-2 }


Find all triples  $(m,p,q)$  where  $ m $  is a positive integer and  $ p , q $  are primes.
\[ 2^m p^2 + 1 = q^5 \]

Solution: \boxed{ (1, 11, 3) }


What is the largest integer  $n$  such that  $n$  is divisible by every integer less than  $\sqrt[3]{n}$ ?


A massless string is wrapped around a frictionless pulley of mass  $M$ . The string is pulled down with a force of 50 N, so that the pulley rotates due to the pull. Consider a point  $P$  on the rim of the pulley, which is a solid cylinder. The point has a constant linear (tangential) acceleration component equal to the acceleration of gravity on Earth, which is where this experiment is being held. What is the weight of the cylindrical pulley, in Newtons?

*(Proposed by Ahaan Rungta)*

NoteThis problem was not fully correct. Within friction, the pulley cannot rotate. So we responded:

Excellent observation! This is very true. To submit, I'd say just submit as if it were rotating and ignore friction. In some effects such as these, I'm pretty sure it turns out that friction doesn't change the answer much anyway, but, yes, just submit as if it were rotating and you are just ignoring friction.
So do this problem imagining that the pulley does rotate somehow.

Solution: \boxed{100 \, \text{N}}


Determine all pairs  $(p, q)$  of prime numbers such that  $p^p + q^q + 1$  is divisible by  $pq.$

Solution: \boxed{ (2, 5) }


Let  $p$  be a prime and  $n$  be a positive integer such that  $p^2$  divides  $\prod_{k=1}^n (k^2+1)$ . Show that  $p<2n$ .

Solution: \boxed{ p < 2n }


Palmer and James work at a dice factory, placing dots on dice. Palmer builds his dice correctly, placing the dots so that  $1$ ,  $2$ ,  $3$ ,  $4$ ,  $5$ , and  $6$  dots are on separate faces. In a fit of mischief, James places his  $21$  dots on a die in a peculiar order, putting some nonnegative integer number of dots on each face, but not necessarily in the correct configuration. Regardless of the configuration of dots, both dice are unweighted and have equal probability of showing each face after being rolled.

Then Palmer and James play a game. Palmer rolls one of his normal dice and James rolls his peculiar die. If they tie, they roll again. Otherwise the person with the larger roll is the winner. What is the maximum probability that James wins? Give one example of a peculiar die that attains this maximum probability.

Solution: \boxed{ \frac{17}{32} }


We are given some three element subsets of  $\{1,2, \dots ,n\}$  for which any two of them have at most one common element. We call a subset of  $\{1,2, \dots ,n\}$  *nice* if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element *nice* subset while keeping it nice, find the minimum value of  $n$ .


Three points  $X, Y,Z$  are on a striaght line such that  $XY = 10$  and  $XZ = 3$ . What is the product of all possible values of  $YZ$ ?


Let  $a,b,c,x,y,$  and  $z$  be complex numbers such that \[a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.\] If  $xy+yz+xz=67$  and  $x+y+z=2010$ , find the value of  $xyz$ .


Larry can swim from Harvard to MIT (with the current of the Charles River) in  $40$  minutes, or back (against the current) in  $45$  minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in  $15$  minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.


Find the real number  $a$  such that the integral   $$ \int_a^{a+8}e^{-x}e^{-x^2}dx $$   attain its maximum.

Solution: \boxed{a = -4.5}


The Fibonacci sequence is defined as  $f_1=f_2=1$ ,  $f_{n+2}=f_{n+1}+f_n$  ( $n\in\mathbb{N}$ ). Suppose that  $a$  and  $b$  are positive integers such that  $\frac ab$  lies between the two fractions  $\frac{f_n}{f_{n-1}}$  and  $\frac{f_{n+1}}{f_{n}}$ . Show that  $b\ge f_{n+1}$ .

Solution: \boxed{b \geq f_{n+1}}


Let  $f(n)$  be the number of ones that occur in the decimal representations of all the numbers from 1 to  $n$ .  For example, this gives  $f(8)=1$ ,  $f(9)=1$ ,  $f(10)=2$ ,  $f(11)=4$ , and  $f(12)=5$ .  Determine the value of  $f(10^{100})$ .

Solution: \boxed{10^{101} + 1}


Three distinct integers are chosen uniformly at random from the set  $$ \{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}. $$  Compute the probability that their arithmetic mean is an integer.


What is the  $22\text{nd}$  positive integer  $n$  such that  $22^n$  ends in a  $2$ ? (when written in base  $10$ ).


Five points  $A_1,A_2,A_3,A_4,A_5$  lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles  $\angle A_iA_jA_k$  can take where  $i, j, k$  are distinct integers between  $1$  and  $5$ .


Parallelogram  $ABCD$  is such that angle  $B < 90$  and  $AB<BC$ . Points E and F are on the circumference of  $\omega$  inscribing triangle ABC, such that tangents to  $\omega$  in those points pass through D. If  $\angle EDA= \angle{FDC}$ , find  $\angle{ABC}$ .


Let natural  $n \ge 2$  be given. Let Laura be a student in a class of more than  $n+2$  students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:

- for every pair of students there is exactly one problem that was solved by both students;
- for every pair of problems there is exactly one student who solved both of them;
- one specific problem was solved by Laura and exactly  $n$  other students.

Determine the number of students in Laura's class.

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


The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order).  The common difference of this arithmetic sequence can be any real number in the open interval  $(-D, D)$ .  Compute the greatest possible value of  $D$ .


Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form  $ x^3\plus{}2y^2$ , with  $ x, y$  being integers.


Let  $a$  and  $b$  be real numbers greater than 1 such that  $ab=100$ . The maximum possible value of  $a^{(\log_{10}b)^2}$  can be written in the form  $10^x$  for some real number  $x$ . Find  $x$ .


Find all functions  $ f : Z\rightarrow Z$  for which we have  $ f (0) \equal{} 1$  and  $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$ , for every natural number  $ n$ .

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


Sets  $A, B$ , and  $C$  satisfy  $|A| = 92$ ,  $|B| = 35$ ,  $|C| = 63$ ,  $|A\cap B| = 16$ ,  $|A\cap C| = 51$ ,  $|B\cap C| = 19$ . Compute the number of possible values of $ |A \cap  B \cap C|$ .


Let  $a_0 = \frac{1}{2}$  and  $a_n$  be defined inductively by
\[a_n = \sqrt{\frac{1+a_{n-1}}{2}} \text{,  $n \ge 1$ .} \]

[list=a]
[*] Show that for  $n = 0,1,2, \ldots,$
\[a_n = \cos(\theta_n) \text{ for some  $0 < \theta_n < \frac{\pi}{2}$ , }\]
and determine  $\theta_n$ .

[*] Using (a) or otherwise, calculate
\[ \lim_{n \to \infty} 4^n (1 - a_n).\]
[/list]


Find the area of the region bounded by a function  $y=-x^4+16x^3-78x^2+50x-2$  and the tangent line which is tangent to the curve at exactly two distinct points.

Proposed by Kunihiko Chikaya


Find all of the sequences  $a_1, a_2, a_3, . . .$  of real numbers that satisfy the following property: given any sequence  $b_1, b_2, b_3, . . .$  of positive integers such that for all  $n \ge 1$  we have  $b_n \ne b_{n+1}$  and  $b_n | b_{n+1}$ , then the sub-sequence  $a_{b_1}, a_{b_2}, a_{b_3}, . . .$  is an arithmetic progression.

Solution: \boxed{ a_i = c }


Let  $ABC$  be a right triangle with a right angle at  $C.$  Two lines, one parallel to  $AC$  and the other parallel to  $BC,$  intersect on the hypotenuse  $AB.$  The lines split the triangle into two triangles and a rectangle. The two triangles have areas  $512$  and  $32.$  What is the area of the rectangle?

*Author: Ray Li*


For each positive integer  $n$ , let  $s(n)$  be the sum of the squares of the digits of  $n$ . For example,  $s(15)=1^2+5^2=26$ . Determine all integers  $n\geq 1$  such that  $s(n)=n$ .


How many ordered triples  $(a, b, c)$  of odd positive integers satisfy  $a + b + c = 25?$


The graph of the equation  $x^y =y^x$  in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.

Solution: \boxed{(e, e)}


Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is  $75\%$ . If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is  $25\%$ . He decides not to make it rain today. Find the smallest positive integer  $n$  such that the probability that Lil Wayne *makes it rain*  $n$  days from today is greater than  $49.9\%$ .

Solution: \boxed{ n = 9 }


There are  $n$  cities,  $2$  airline companies in a country. Between any two cities, there is exactly one  $2$ -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of  $n$ .


What is the smallest positive integer  $n$  such that  $2^n - 1$  is a multiple of  $2015$ ?


Let  $a$ ,  $b$ ,  $c$  be positive integers and  $p$  be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b)\equiv 8\pmod{p} \] for each nonnegative integer  $n$ . Let  $m$  be the remainder when  $a^p+b^p+c^p$  is divided by  $p$ , and  $k$  the remainder when  $m^p$  is divided by  $p^4$ . Find the maximum possible value of  $k$ .

*Proposed by Justin Stevens and Evan Chen*


Find all functions  $f:\mathbb{R}\to\mathbb{R}$  such that   $$ f(x+y)\leq f(x^2+y) $$   for all  $x,y$ .

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


Let  $ x_1, x_2, \ldots, x_n$  be positive real numbers with sum  $ 1$ . Find the integer part of:
 $ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$


A square of side  $1$  is covered with  $m^2$  rectangles.
Show that there is a rectangle with perimeter at least  $\frac{4}{m}$ .

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


Given  $1962$  -digit number. It is divisible by  $9$ . Let  $x$  be the sum of its digits. Let the sum of the digits of  $x$  be  $y$ . Let the sum of the digits of  $y$  be  $z$ . Find  $z$ .


Consider  $S=\{1, 2, 3, \cdots, 6n\}$ ,  $n>1$ . Find the largest  $k$  such that the following statement is true: every subset  $A$  of  $S$  with  $4n$  elements has at least  $k$  pairs  $(a,b)$ ,  $a<b$  and  $b$  is divisible by  $a$ .

Solution: \boxed{ k = n }


The expression  $\circ \ 1\  \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$  is written on a blackboard. Catherine places a  $+$  sign or a  $-$  sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?

*Proposed by Aaron Lin*


Find all integers  $n$  such that  $n^4 + 8n + 11$  is a product of two or more consecutive integers.

Solution: \boxed{ n = 1 }


Suppose the function  $f(x)-f(2x)$  has derivative  $5$  at  $x=1$  and derivative  $7$  at  $x=2$ . Find the derivative of  $f(x)-f(4x)$  at  $x=1$ .


Let  $n$  be some positive integer and  $a_1 , a_2 , \dots , a_n$  be real numbers. Denote  $$ S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2}, $$  where  $a_{n+1} = a_1$  and  $a_{n+2} = a_2.$ 1. Show that  $S_0 - S_1 \geq 0$ .
2. Show that  $3$  is the minimum value of  $C$  such that for any  $n$  and  $a_1 , a_2 , \dots , a_n,$  there holds  $C(S_0 - S_1) \geq S_1 - S_2$ .

Solution: \boxed{ C = 3 }


Let  $S$  be a subset of the natural numbers such that  $0\in S$ , and for all  $n\in\mathbb N$ , if  $n$  is in  $S$ , then both  $2n+1$  and  $3n+2$  are in  $S$ .  What is the smallest number of elements  $S$  can have in the range  $\{0,1,\ldots, 2019\}$ ?


In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.


Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle  $C$  with radius  $30$ . Let  $K$  be the area of the region inside circle  $C$  and outside of the six circles in the ring. Find  $\lfloor K \rfloor$ .


Given seven points in the plane, some of them are connected by segments such that:

**(i)** among any three of the given points, two are connected by a segment;
**(ii)** the number of segments is minimal.

How many segments does a figure satisfying **(i)** and **(ii)** have? Give an example of such a figure.


Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.

Solution: \boxed{ n = 4 }


Let  $n$  be a positive integer with the following property:  $2^n-1$  divides a number of the form  $m^2+81$ , where  $m$  is a positive integer. Find all possible  $n$ .

Solution: \boxed{ n = 2^k }


Jae and Yoon are playing SunCraft. The probability that Jae wins the  $n$ -th game is  $\frac{1}{n+2}.$  What is the probability that Yoon wins the first six games, assuming there are no ties?


Real numbers  $x$  and  $y$  satisfy
\begin{align*}
x^2 + y^2 &= 2023
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of  $|x-y|$ .

*Proposed by Howard Halim*

Solution: \boxed{ 13\sqrt{13} }


The random variables  $X, Y$  can each take a finite number of integer values. They are not necessarily independent. Express  $P(\min(X,Y)=k)$  in terms of  $p_1=P(X=k)$ ,  $p_2=P(Y=k)$  and  $p_3=P(\max(X,Y)=k)$ .

Solution: \boxed{p_1 + p_2 - p_3}


**8.** Find all integers  $a>1$  for which the least (integer) solution  $n$  of the congruence  $a^{n} \equiv 1 \pmod{p}$  differs from 6 (p is any prime number). **(N. 9)**

Solution: \boxed{a = 2}


Let  $ABCD$  be a convex quadrilateral with  $\angle DAB =\angle B DC = 90^o$ . Let the incircles of triangles  $ABD$  and  $BCD$  touch  $BD$  at  $P$  and  $Q$ , respectively, with  $P$  lying in between  $B$  and  $Q$ . If  $AD = 999$  and  $PQ = 200$  then what is the sum of the radii of the incircles of triangles  $ABD$  and  $BDC$  ?


Find the smallest positive integer  $n$  for which  $315^2-n^2$  evenly divides  $315^3-n^3$ .

*Proposed by Kyle Lee*


Lines  $l_1$  and  $l_2$  both pass through the origin and make first-quadrant angles of  $\frac{\pi}{70}$  and  $\frac{\pi}{54}$  radians, respectively, with the positive x-axis. For any line  $l$ , the transformation  $R(l)$  produces another line as follows:  $l$  is reflected in  $l_1$ , and the resulting line is reflected in  $l_2$ . Let  $R^{(1)}(l)=R(l)$  and  $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$ . Given that  $l$  is the line  $y=\frac{19}{92}x$ , find the smallest positive integer  $m$  for which  $R^{(m)}(l)=l$ .


Let  $N$  be a given positive integer. Consider a permutation of  $1,2,3,\cdots,N$ , denoted as  $p_1,p_2,\cdots,p_N$ . For a section  $p_l, p_{l+1},\cdots, p_r$ , we call it "extreme" if  $p_l$  and  $p_r$  are the maximum and minimum value of that section. We say a permutation  $p_1,p_2,\cdots,p_N$  is "super balanced" if there isn't an "extreme" section with a length at least  $3$ . For example,  $1,4,2,3$  is "super balanced", but  $3,1,2,4$  isn't. Please answer the following questions:
1. How many "super balanced" permutations are there?
2. For each integer  $M\leq N$ . How many "super balanced" permutations are there such that  $p_1=M$ ?

*Proposed by ltf0501*


Let  $\alpha,\beta$  be the distinct positive roots of the equation of  $2x=\tan x$ .
Evaluate

\[\int_0^1 \sin \alpha x\sin \beta x\ dx\]


A right cylinder is given with a height of  $20$  and a circular base of radius  $5$ . A vertical planar cut is made into this base of radius  $5$ . A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the pieces is  $100\sqrt2$ . Then the volume of the larger piece can be written as  $a + b\pi$ , where  $a, b$  are positive integers. Find  $a + b$ .


Consider  $\triangle \natural\flat\sharp$ .  Let  $\flat\sharp$ ,  $\sharp\natural$  and  $\natural\flat$  be the answers to problems  $4$ ,  $5$ , and  $6$ , respectively.  If the incircle of  $\triangle \natural\flat\sharp$  touches  $\natural\flat$  at  $\odot$ , find  $\flat\odot$ .

*Proposed by Evan Chen*


Let  $\{a_n\}_{n\geq 1}$  be a sequence defined by  $a_n=\int_0^1 x^2(1-x)^ndx$ .
Find the real value of  $c$  such that  $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$


The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.

[i]Proposed by Clive Chan


A *9-cube* is a nine-dimensional hypercube (and hence has  $2^9$  vertices, for example). How many five-dimensional faces does it have?

(An  $n$  dimensional hypercube is defined to have vertices at each of the points  $(a_1,a_2,\cdots ,a_n)$  with  $a_i\in \{0,1\}$  for  $1\le i\le n$ )

*Proposed by Evan Chen*


A triangle has sides of lengths 5, 6, 7. What is 60 times the square of the radius of the inscribed circle?


Let  $x$ ,  $y$ ,  $z$  be positive integers satisfying  $x<y<z$  and  $x+xy+xyz=37$ . Find the greatest possible value of  $x+y+z$ .


Let  $\triangle ABC$  be a triangle with circumcenter  $O$  satisfying  $AB=13$ ,  $BC = 15$ , and  $AC = 14$ .  Suppose there is a point  $P$  such that  $PB \perp BC$  and  $PA \perp AB$ .  Let  $X$  be a point on  $AC$  such that  $BX \perp OP$ . What is the ratio  $AX/XC$ ?

*Proposed by Thomas Lam*


For reals  $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$ , let  $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$  for each  $k$ . If  $S_2 = 777$ , compute the least possible value of  $S_3$ .

*Proposed by Evan Chen*


Find all integer sequences of the form  $ x_i, 1 \le i \le 1997$ , that satisfy  $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$ .

Solution: \boxed{ x_i = 0 }


Compute the remainder when the largest integer below  $\frac{3^{123}}{5}$  is divided by  $16$ .

*2020 CCA Math Bonanza Individual Round #8*


Let  $OABC$  be a tetrahedron such that  $\angle AOB = \angle BOC = \angle COA = 90^\circ$  and its faces have integral surface areas. If  $[OAB] = 20$  and  $[OBC] = 14$ , find the sum of all possible values of  $[OCA][ABC]$ . (Here  $[\triangle]$  denotes the area of  $\triangle$ .)

*Proposed by Robin Park*


Let the circles  $k_1$  and  $k_2$  intersect at two points  $A$  and  $B$ , and let  $t$  be a common tangent of  $k_1$  and  $k_2$  that touches  $k_1$  and  $k_2$  at  $M$  and  $N$  respectively. If  $t\perp AM$  and  $MN=2AM$ , evaluate the angle  $NMB$ .


In how many different orders can the characters  $P  \ U  \ M  \ \alpha \ C$  be arranged such that the  $M$  is to the left of the  $\alpha$  and the  $\alpha$  is to the left of the  $C?$


If there is only  $1$  complex solution to the equation  $8x^3 + 12x^2 + kx + 1 = 0$ , what is  $k$ ?


Three palaces, each rotating on a duck leg, make a full round in  $30$ ,  $50$ , and  $70$  days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?


If  $a$ ,  $6$ , and  $b$ , in that order, form an arithmetic sequence, compute  $a+b$ .


What is the sum of real roots of the equation  $x^4-7x^3+14x^2-14x+4=0$ ? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$


The measures, in degrees, of the angles ,  $\alpha, \beta$  and  $\theta$  are greater than  $0$  less than  $60$ . Find the value of  $\theta$  knowing, also, that  $\alpha + \beta = 2\theta$  and that   $$ \sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ). $$


A point  $X$  exactly  $\sqrt{2}-\frac{\sqrt{6}}{3}$  away from the origin is chosen randomly. A point  $Y$  less than  $4$  away from the origin is chosen randomly. The probability that a point  $Z$  less than  $2$  away from the origin exists such that  $\triangle XYZ$  is an equilateral triangle can be expressed as  $\frac{a\pi + b}{c \pi}$  for some positive integers  $a, b, c$  with  $a$  and  $c$  relatively prime. Find  $a+b+c$ .


We color each number in the set  $S = \{1, 2, ..., 61\}$  with one of  $25$  given colors, where it is not necessary that every color gets used. Let  $m$  be the number of non-empty subsets of  $S$  such that every number in the subset has the same color. What is the minimum possible value of  $m$ ?


Let  $f$  be a function defined for the non-negative integers, such that:

a)  $f(n)=0$  if  $n=2^{j}-1$  for some  $j \geq 0$ .
b)  $f(n+1)=f(n)-1$  otherwise.

i) Show that for every  $n \geq 0$  there exists  $k \geq 0$  such that  $f(n)+n=2^{k}-1$ .
ii) Find  $f(2^{1990})$ .

Solution: \boxed{ f(2^{1990}) = -1 }


Determine all ordered pairs of positive real numbers  $(a, b)$  such that every sequence  $(x_{n})$  satisfying  $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$  must have  $\lim_{n \rightarrow \infty} x_n = 0$ .

Solution: \boxed{ (a, b) }


Find all  $t\in \mathbb Z$  such that: exists a function  $f:\mathbb Z^+\to \mathbb Z$  such that: $f(1997)=1998$  $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$

Solution: \boxed{ t = -1 }


Source: 2017 Canadian Open Math Challenge, Problem A2
-----
An equilateral triangle has sides of length  $4$ cm. At each vertex, a circle with radius  $2$ cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is  $a\cdot \pi \text{cm}^2$ . Determine  $a$ .

[center][asy]
size(2.5cm);
draw(circle((0,2sqrt(3)/3),1));
draw(circle((1,-sqrt(3)/3),1));
draw(circle((-1,-sqrt(3)/3),1));
draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle);
fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray);
draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle);
fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray);
draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle);
fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray);
[/asy][/center]


Suppose  $x$  is a random real number between  $1$  and  $4$ , and  $y$  is a random real number between  $1$  and  $9$ . If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as  $\frac mn$  where  $m$  and  $n$  are relatively prime positive integers, compute  $100m + n$ .

*Proposed by Lewis Chen*


For any non-empty subset  $A$  of  $\{1, 2, \ldots , n\}$  define  $f(A)$  as the largest element of  $A$  minus the smallest element of  $A$ . Find  $\sum f(A)$  where the sum is taken over all non-empty subsets of  $\{1, 2, \ldots , n\}$ .

Solution: \boxed{ (n-3)2^n + n + 3 }


Let  $n$  be a positive integer. All numbers  $m$  which are coprime to  $n$  all satisfy  $m^6\equiv 1\pmod n$ . Find the maximum possible value of  $n$ .


Find the greatest positive integer  $k$  such that the following inequality holds for all  $a,b,c\in\mathbb{R}^+$  satisfying  $abc=1$  \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]


The positive integers  $v, w, x, y$ , and  $z$  satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute  $10^4 v + 10^3 w + 10^2 x + 10 y + z$ .


A positive integer  $n$  is *funny* if for all positive divisors  $d$  of  $n$ ,  $d+2$  is a prime number. Find all funny numbers with the largest possible number of divisors.

Solution: \boxed{n = 135}


For real numbers  $x$ ,  $y$  and  $z$ , solve the system of equations:  $$ x^3+y^3=3y+3z+4 $$     $$ y^3+z^3=3z+3x+4 $$     $$ x^3+z^3=3x+3y+4 $$

Solution: \boxed{ x = y = z = 2 }


Find the sum of all values of  $a + b$ , where  $(a, b)$  is an ordered pair of positive integers and  $a^2+\sqrt{2017-b^2}$  is a perfect square.


$(x_{n})_{-\infty<n<\infty}$  is a sequence of real numbers which satisfies  $x_{n+1}=\frac{x_{n}^2+10}{7}$  for every  $n \in \mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values  $x_{0}$  can take.

Solution: \boxed{ [2, 5] }


Determine all functions  $f : N_0 \to R$  satisfying  $f (x+y)+ f (x-y)= f (3x)$  for all  $x,y$ .

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


One evening a theater sold 300 tickets for a concert. Each ticket sold for \ $40, and all tickets were purchased using \$ 5, \ $10, and \$ 20 bills. At the end of the evening the theater had received twice as many \ $10 bills as \$ 20 bills, and 20 more \ $5 bills than \$ 10 bills. How many bills did the theater receive altogether?


An  $m \times n$  chessboard where  $m \le n$  has several black squares such that no two rows have the same pattern. Determine the largest integer  $k$  such that we can always color  $k$  columns red while still no two rows have the same pattern.

Solution: \boxed{ n - m + 1 }


Let  $P$  be a point inside the equilateral triangle  $ABC$  such that  $6\angle  PBC = 3\angle  PAC = 2\angle  PCA$ . Find the measure of the angle  $\angle PBC$  .


Let  $n$  be a fixed positive integer.
- Show that there exist real polynomials  $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$  such that
        \[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number  $k$ , depending on  $n$ , such that the above polynomials  $p_1, p_2, \cdots, p_k$  exist.

Solution: \boxed{ k = n-1 }


For a positive integer number  $n$  we denote  $d(n)$  as the greatest common divisor of the binomial coefficients  $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$ .
Find all possible values of  $d(n)$

Solution: \boxed{ d(n) = 1 }


Alice's favorite number has the following properties:

- It has 8 distinct digits.
- The digits are decreasing when read from left to right.
- It is divisible by 180.

What is Alice's favorite number?

*Author: Anderson Wang*


The positive integer  $n$  is given and for all positive integers  $k$ ,  $1\leq k\leq n$ , denote by  $a_{kn}$  the number of all ordered sequences  $(i_1,i_2,\ldots,i_k)$  of positive integers which verify the following two conditions:

a)  $1\leq i_1<i_2< \cdots i_k \leq n$ ;

b)  $i_{r+1}-i_r \equiv 1 \pmod 2$ , for all  $r \in\{1,2,\ldots,k-1\}$ .

Compute the number  $a(n) = \sum\limits_{k=1}^n a_{kn}$ .

*Ioan Tomescu*

Solution: \boxed{F_{n+3} - 2}


Find all positive integers  $a, b,c$  greater than  $1$ , such that  $ab + 1$  is divisible by  $c, bc + 1$  is divisible by  $a$  and  $ca + 1$  is divisible by  $b$ .

Solution: \boxed{ (2, 3, 7) }


What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?


Find all real solutions to  $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$

Solution: \boxed{ x = 3 }


Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$ ,  $a, b \in \mathbb{Z}$ .


Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of  $\text A$ ,  $\text B$  or  $\text C$ , possibly repeated. The lock has three buttons, labeled  $\text A$ ,  $\text B$  and  $\text C$ . When the most recent  $3$  button-presses form the code, the lock opens. What is the minimum number of total button presses Alex needs to guarantee opening the lock?


Let  $u$  and  $v$  be integers satisfying  $0<v<u.$  Let  $A=(u,v),$  let  $B$  be the reflection of  $A$  across the line  $y=x,$  let  $C$  be the reflection of  $B$  across the y-axis, let  $D$  be the reflection of  $C$  across the x-axis, and let  $E$  be the reflection of  $D$  across the y-axis. The area of pentagon  $ABCDE$  is 451. Find  $u+v.$


If \[2011^{2011^{2012}} = x^x\] for some positive integer  $x$ , how many positive integer factors does  $x$  have?

*Author: Alex Zhu*


Find the number of the subsets  $B$  of the set  $\{1,2,\cdots, 2005 \}$  such that the sum of the elements of  $B$  is congruent to  $2006$  modulo  $2048$


The natural numbers  $m$  and  $k$   satisfy the equality   $$ 1001 \cdot 1002 \cdot  ... \cdot 2010 \cdot 2011 = 2^m (2k + 1) $$  . Find the number  $m$ .


Alice and Bob take turns alternatively on a  $2020\times2020$  board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?

*Proposed by Seyed Reza Hosseini*


A circle intersects the  $y$ -axis at two points  $(0, a)$  and  $(0, b)$  and is tangent to the line  $x+100y = 100$  at  $(100, 0)$ . Compute the sum of all possible values of  $ab - a - b$ .


Let  $n$  be a positive integer. There are  $n$  ants walking along a line at constant nonzero speeds. Different ants need not walk at the same speed or walk in the same direction. Whenever two or more ants collide, all the ants involved in this collision instantly change directions. (Different ants need not be moving in opposite directions when they collide, since a faster ant may catch up with a slower one that is moving in the same direction.) The ants keep walking indefinitely.
Assuming that the total number of collisions is finite, determine the largest possible number of collisions in terms of  $n$ .

Solution: \boxed{ \frac{n(n-1)}{2} }


For how many positive integers  $n$  less than or equal to  $1000$  is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real  $t$ ?


For an integer  $n\ge 3$  and a permutation  $\sigma=(p_{1},p_{2},\cdots ,p_{n})$  of  $\{1,2,\cdots , n\}$ , we say  $p_{l}$  is a  $landmark$  point if  $2\le l\le n-1$  and  $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$ . For example , for  $n=7$ ,
the permutation  $(2,7,6,4,5,1,3)$  has four landmark points:  $p_{2}=7$ ,  $p_{4}=4$ ,  $p_{5}=5$  and  $p_{6}=1$ . For a given  $n\ge 3$  , let  $L(n)$  denote the number of permutations of  $\{1,2,\cdots ,n\}$  with exactly one landmark point. Find the maximum  $n\ge 3$  for which  $L(n)$  is a perfect square.


Let  $A_1A_2A_3$  be an acute-angled triangle inscribed into a unit circle centered at  $O$ . The cevians from  $A_i$  passing through  $O$  meet the opposite sides at points  $B_i$   $(i = 1, 2, 3)$  respectively.
[list=a]
[*] Find the minimal possible length of the longest of three segments  $B_iO$ .
[*] Find the maximal possible length of the shortest of three segments  $B_iO$ .
[/list]


Determine the maximal value of  $ k $ , such that for positive reals  $ a,b $  and  $ c $  from inequality  $ kabc >a^3+b^3+c^3 $  it follows that  $ a,b $  and  $ c $  are sides of a triangle.

Solution: \boxed{ k = 5 }


Find the smallest positive value taken by  $a^3 + b^3 + c^3 - 3abc$  for positive integers  $a$ ,  $b$ ,  $c$  .
 Find all  $a$ ,  $b$ ,  $c$  which give the smallest value


Let  $O$  be the origin.  $y = c$  intersects the curve  $y = 2x - 3x^3$  at  $P$  and  $Q$  in the first quadrant and cuts the y-axis at  $R$ . Find  $c$  so that the region  $OPR$  bounded by the y-axis, the line  $y = c$  and the curve has the same area as the region between  $P$  and  $Q$  under the curve and above the line  $y = c$ .


For which  $ p$  prime numbers, there is an integer root of the polynominal  $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$  such that  $ Q(x)$  is a polynominal with integer coefficients?

Solution: \boxed{ p = 2 }


If  $f$  is a polynomial, and  $f(-2)=3$ ,  $f(-1)=-3=f(1)$ ,  $f(2)=6$ , and  $f(3)=5$ , then what is the minimum possible degree of  $f$ ?


A polygon on a coordinate grid is built of  $ 2005$  dominoes  $ 1 \times 2$ . What is the smallest number of sides of an even length such a polygon can have?


The  $ 52$  cards in a deck are numbered  $ 1, 2, \ldots, 52$ . Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let  $ p(a)$  be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards  $ a$  and  $ a\plus{}9$ , and Dylan picks the other of these two cards. The minimum value of  $ p(a)$  for which  $ p(a)\ge\frac12$  can be written as  $ \frac{m}{n}$ . where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .

Solution: \boxed{ 263 }


The dollar is now worth  $\frac{1}{980}$  ounce of gold. After the  $n^{th}$  7001 billion dollars bailout package passed by congress, the dollar gains  $\frac{1}{2{}^2{}^{n-1}}$  of its  $(n-1)^{th}$  value in gold. After four bank bailouts, the dollar is worth  $\frac{1}{b}(1-\frac{1}{2^c})$  in gold, where  $b, c$  are positive integers. Find  $b + c$ .


For each positive integer  $n$ , let  $g(n)$  be the sum of the digits when  $n$  is written in binary.  For how many positive integers  $n$ , where  $1\leq n\leq 2007$ , is  $g(n)\geq 3$ ?


Find all pairs of natural numbers  $(n, k)$  with the following property:
Given a  $k\times k$  array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of  $n$ .

Note: A path from the left to the right edge is a sequence of cells of the array  $a_1, a_2, ... , a_m$  so that  $a_1$  is a cell of the leftmost column,  $a_m$  is the cell of the rightmost column, and  $a_{i}$ ,  $a_{i+1}$  share an edge for all  $i = 1, 2, ... , m -1$ .

Solution: \boxed{ (n, k) }


If  $a$  and  $b$  are complex numbers such that  $a^2 + b^2 = 5$  and  $a^3 + b^3 = 7$ , then their sum,  $a + b$ , is real. The greatest possible value for the sum  $a + b$  is  $\tfrac{m+\sqrt{n}}{2}$  where  $m$  and  $n$  are integers. Find  $n.$


Let  $S$  be a set. We say  $S$  is  $D^\ast$ *-finite* if there exists a function  $f : S \to S$  such that for every nonempty proper subset  $Y \subsetneq S$ , there exists a  $y \in Y$  such that  $f(y) \notin Y$ .  The function  $f$  is called a *witness* of  $S$ .  How many witnesses does  $\{0,1,\cdots,5\}$  have?

*Proposed by Evan Chen*


Find all positive integers  $k$  such that there exists a positive integer  $n$ , for which  $2^n + 11$  is divisible by  $2^k - 1$ .

Solution: \boxed{ k = 4 }


A  $\emph{planar}$  graph is a connected graph that can be drawn on a sphere without edge crossings. Such a drawing will divide the sphere into a number of faces. Let  $G$  be a planar graph with  $11$  vertices of degree  $2$ ,  $5$  vertices of degree  $3$ , and  $1$  vertex of degree  $7$ . Find the number of faces into which  $G$  divides the sphere.


Find the smallest real constant  $p$  for which the inequality holds  $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$  with any positive real numbers  $a, b$ .

Solution: \boxed{ p = 1 }


Let  $f:\mathbb{R} \to \mathbb{R}$  be a continuous function . Suppose
 \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$  and all  $t>0$  . Then show that there exists a constant  $c$  such that  $f(x)=cx\ \forall x$

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


Find all primes that can be written both as a sum and as a difference of two primes (note that  $ 1$  is not a prime).


Richard and Shreyas are arm wrestling against each other. They will play  $10$  rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than  $3$  streaks? For example, if we denote Richard winning by  $R$  and Shreyas winning by  $S,$   $SSRSSRRRRR$  is one such outcome, with  $4$  streaks.


Let  $n, m$  be positive integers. A set  $S$  of positive integers is called  $(n, m)$ -good, if:

(1)  $m \in S$ ;
(2) for all  $a\in S$ , all divisors of  $a$  are also in  $S$ ;
(3) for all distinct  $a, b \in S$ ,  $a^n+b^n \in S$ .

For which  $(n, m)$ , the only  $(n, m)$ -good set is  $\mathbb{N}$ ?

Solution: \boxed{ (n, m) }


Let  $ABC$  be at triangle with incircle  $\Gamma$ . Let  $\Gamma_1$ ,  $\Gamma_2$ ,  $\Gamma_3$  be three circles inside  $\triangle ABC$  each of which is tangent to  $\Gamma$  and two sides of the triangle and their radii are  $1,4,9$ . Find the radius of  $\Gamma$ .


A cylindrical log has diameter  $ 12$  inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a  $ 45^\circ$  angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as  $ n\pi,$  where  $ n$  is a positive integer. Find  $ n.$


Let  $0=x_0<x_1<\cdots<x_n=1$  .Find the largest real number $ C$  such  that for any positive integer  $ n $  ,  we have   $$ \sum_{k=1}^n  x^2_k (x_k - x_{k-1})>C $$


Let  $n$  be a positive integer with  $d$  digits, all different from zero. For  $k = 0,. . . , d - 1$ , we define  $n_k$  as the number obtained by moving the last  $k$  digits of  $n$  to the beginning. For example, if  $n = 2184$  then  $n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842$ . For  $m$  a positive integer, define  $s_m(n)$  as the number of values  $k$  such that  $n_k$  is a multiple of  $m.$  Finally,  define  $a_d$  as the number of integers  $n$  with  $d$  digits all nonzero, for which  $s_2 (n) + s_3 (n) + s_5 (n) = 2d.$
Find  \[\lim_{d \to \infty} \frac{a_d}{5^d}.\]


For the all  $(m,n,k)$  positive integer triples such that  $|m^k-n!| \le n$  find the maximum value of  $\frac{n}{m}$ *Proposed by Melih Üçer*


The angles  $A$  and  $B$  of base of the isosceles triangle  $ABC$  are equal to  $40^o$ . Inside  $\vartriangle ABC$ ,  $P$  is such that  $\angle PAB = 30^o$  and  $\angle PBA = 20^o$ . Calculate, without table,  $\angle PCA$ .


The positive integers  $a, b, c$  are such that  $$ gcd \,\,\, (a, b, c) = 1, $$    $$ gcd \,\,\,(a, b + c) > 1, $$    $$ gcd \,\,\,(b, c + a) > 1, $$    $$ gcd \,\,\,(c, a + b) > 1. $$  Determine the smallest possible value of  $a + b + c$ .

Clarification: gcd stands for greatest common divisor.


Find all natural numbers  $n$ , such that there exist relatively prime integers  $x$  and  $y$  and an integer  $k > 1$  satisfying the equation  $3^n =x^k + y^k$ .

*A. Kovaldji, V. Senderov*

Solution: \boxed{ n = 2 }


Find all pairs  $(a, b)$  of coprime positive integers, such that  $a<b$  and   $$ b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000} $$   for all positive integers  $n$ .

Solution: \boxed{ (3, 5) }


Consider two circles of radius one, and let  $O$  and  $O'$  denote their centers.  Point  $M$  is selected on either circle.  If  $OO' = 2014$ , what is the largest possible area of triangle  $OMO'$ ?

*Proposed by Evan Chen*


The cheering team of Ubon Ratchathani University sits on the amphitheater that has  $441$  seats arranged into a  $21\times 21$  grid. Every seat is occupied by exactly one person, and each person has a blue sign and a yellow sign.

Count the number of ways for each person to raise one sign so that each row and column has an odd number of people raising a blue sign.


There are rhombus  $ABCD$  and circle  $\Gamma_B$ , which is centred at  $B$  and has radius  $BC$ , and circle  $\Gamma_C$ , which is centred at  $C$  and has radius  $BC$ . Circles  $\Gamma_B$  and  $\Gamma_C$  intersect at point  $E$ . The line  $ED$  intersects  $\Gamma_B$  at point  $F$ . Find all possible values of  $\angle AFB$ .


nic $\kappa$ y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers  $d>0$  such that for every positive integer  $n,$  nic $\kappa$ y can label at least  $dn^2$  cells of an  $n\times n$  square.

*Proposed by Mihir Singhal and Michael Kural*

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


Determine the smallest positive integer  $a$  for which there exist a prime number  $p$  and a positive integer  $b \ge 2$  such that   $$ \frac{a^p -a}{p}=b^2. $$


For real numbers  $x, y$  and  $z$  it is known that  $x + y = 2$  and  $xy = z^2 + 1$ .
Find the value of the expression  $x^2 + y^2+ z^2$ .


Find the minimal positive integer  $m$ , so that there exist positive integers  $n>k>1$ , which satisfy $11...1=11...1.m$ , where the first number has  $n$  digits  $1$ , and the second has  $k$  digits  $1$ .


In the right parallelopiped  $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ , with  $AB=12\sqrt{3}$  cm and  $AA^{\prime}=18$  cm, we consider the points  $P\in AA^{\prime}$  and  $N\in A^{\prime}B^{\prime}$  such that  $A^{\prime}N=3B^{\prime}N$ . Determine the length of the line segment  $AP$  such that for any position of the point  $M\in BC$ , the triangle  $MNP$  is right angled at  $N$ .


Let  $a$  and  $b$  be positive integers such that  $\text{gcd}(a, b) + \text{lcm}(a, b)$  is a multiple of  $a+1$ . If  $b \le a$ , show that  $b$  is a perfect square.

Solution: \boxed{ b }


Determine, with proof, the smallest positive integer  $n$  with the following property: For every choice of  $n$  integers, there exist at least two whose sum or difference is divisible by  $2009$ .

Solution: \boxed{ n = 1006 }


Evaluate  $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$

You may not use  $ \tan ^{ \minus{} 1} x$  or Complex Integral here.


Given the equilateral triangle  $ABC$ . It is known that the radius of the inscribed circle is in this triangle is equal to  $1$ . The rectangle  $ABDE$  is such that point  $C$  belongs to its side  $DE$ . Find the radius of the circle circumscribed around the rectangle  $ABDE$ .


Luke and Carissa are finding the sum of the first  $20$  positive integers by adding them one at a time. Luke forgets to add a number and gets an answer of  $207$ . Carissa adds a number twice by mistake and gets an answer of  $225$ . What is the sum of the number that Luke forgot and the number that Carissa added twice?


In triangles ABC and DEF, DE = 4AB, EF = 4BC, and F D = 4CA. The area of
DEF is 360 units more than the area of ABC. Compute the area of ABC.


Let  $p\geq 5$  be a prime number, and set  $M=\{1,2,\cdots,p-1\}.$  Define   $$ T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}. $$   If  $\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p},$  with  $0\leq k\leq p-1,$  where  $\left[\alpha\right]$  denotes the largest integer that does not exceed  $\alpha,$  determine the value of  $k.$


Hey,

This problem is from the VTRMC 2006.

3. Recall that the Fibonacci numbers  $ F(n)$  are defined by  $ F(0) \equal{} 0$ ,  $ F(1) \equal{} 1$  and  $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$  for  $ n \geq 2$ . Determine the last digit of  $ F(2006)$  (e.g. the last digit of 2006 is 6).

As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation.

Consider the following,

01 = 1 mod 10
01 = 1 mod 10
02 = 2 mod 10
03 = 3 mod 10
05 = 5 mod 10
08 = 6 mod 10
13 = 3 mod 10
21 = 1 mod 10
34 = 4 mod 10
55 = 5 mod 10
89 = 9 mod 10

Now, consider that between the first appearance and second apperance of  $ 5 mod 10$ , there is a difference of five terms. Following from this we see that the third appearance of  $ 5 mod 10$  occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships.

 $ F(55) \equal{} F(05) \plus{} 5({2}^{2})$

This is pretty much as far as we got, any ideas?


Let  $k$  be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where  $a$ ,  $b$ ,  $c$  are non-negative reals such that  $a+b+c=3k$ .

Solution: \boxed{ (3k-1)^{3k-1} }


Five persons wearing badges with numbers  $1, 2, 3, 4, 5$  are seated on  $5$  chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)


Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.


Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this?

*Proposed by Nathan Xiong*


A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)

Solution: \boxed{45 \text{ km}}


Let  $x$  be a positive real number. Define
\[
  A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
  B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
  C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that  $A^3+B^3+C^3 + 8ABC = 2014$ , compute  $ABC$ .

*Proposed by Evan Chen*


There are two equilateral triangles with a vertex at  $(0, 1)$ , with another vertex on the line  $y = x + 1$  and with the final vertex on the parabola  $y = x^2 + 1$ . Find the area of the larger of the two triangles.

Solution: \boxed{26\sqrt{3} + 45}


There are  $ n$   students; each student knows exactly  $d $   girl students and  $d $  boy students ("knowing" is a symmetric relation). Find all pairs  $ (n,d) $   of integers .

Solution: \boxed{ (n, d) }


Determine all positive integers  $n{}$  which can be expressed as  $d_1+d_2+d_3$  where  $d_1,d_2,d_3$  are distinct positive divisors of  $n{}$ .

Solution: \boxed{ n = 6k }


Define  $\phi_n(x)$  to be the number of integers  $y$  less than or equal to  $n$  such that  $\gcd(x,y)=1$ . Also, define  $m=\text{lcm}(2016,6102)$ . Compute  $$ \frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}. $$


A set  $A$  is endowed with a binary operation  $*$  satisfying the following four conditions:
(1) If  $a, b, c$  are elements of  $A$ , then  $a * (b * c) = (a * b) * c$  ,
(2) If  $a, b, c$  are elements of  $A$  such that  $a * c = b *c$ , then  $a = b$  ,
(3) There exists an element  $e$  of  $A$  such that  $a * e = a$  for all  $a$  in  $A$ , and
(4) If a and b are distinct elements of  $A-\{e\}$ , then  $a^3 * b = b^3 * a^2$ , where  $x^k = x * x^{k-1}$  for all integers  $k \ge 2$  and all  $x$  in  $A$ .
Determine the largest cardinality  $A$  may have.

proposed by Bojan Basic, Serbia


Let  $ABC$  be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let  $N = 2017$ , and define the  $A$ -*ntipodes* to be the points  $A_1,\dots, A_N$  to be the points on segment  $BC$  such that  $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$ , and similarly define the  $B$ ,  $C$ -ntipodes. A line  $\ell_A$  through  $A$  is called a *qevian* if it passes through an  $A$ -ntipode, and similarly we define qevians through  $B$  and  $C$ . Compute the number of ordered triples  $(\ell_A, \ell_B, \ell_C)$  of concurrent qevians through  $A$ ,  $B$ ,  $C$ , respectively.

*Proposed by Brandon Wang*


Find all subsets  $A$  of  $\left\{ 1, 2, 3, 4, \ldots \right\}$ , with  $|A| \geq 2$ , such that for all  $x,y \in A, \, x \neq y$ , we have that  $\frac{x+y}{\gcd (x,y)}\in A$ .

*Dan Schwarz*

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


Suppose that  $\{a_n\}$  is a sequence such that  $a_{n+1}=(1+\frac{k}{n})a_{n}+1$  with  $a_{1}=1$ .Find all positive integers  $k$  such that  any  $a_n$  be integer.


Consider a polynomial  $P(x,y,z)$  in three variables with integer coefficients such that for any real numbers  $a,b,c,$    $$ P(a,b,c)=0 \Leftrightarrow a=b=c. $$  Find the largest integer  $r$  such that for all such polynomials  $P(x,y,z)$  and integers  $m,n,$    $$ m^r\mid P(n,n+m,n+2m). $$  [i]Proposed by Ma Zhao Yu

Solution: \boxed{ r = 2 }


Two years ago Tom was  $25\%$  shorter than Mary. Since then Tom has grown  $20\%$  taller, and Mary has grown  $4$  inches taller. Now Mary is  $20\%$  taller than Tom. How many inches tall is Tom now?


The geometric series  $ a \plus{} ar \plus{} ar^{2} \plus{} ...$  has a sum of  $ 7$ , and the terms involving odd powers of  $ r$  have a sum of  $ 3$ .  What is  $ a \plus{} r$ ?

 $ \textbf{(A)}\ \frac {4}{3}\qquad \textbf{(B)}\ \frac {12}{7}\qquad \textbf{(C)}\ \frac {3}{2}\qquad \textbf{(D)}\ \frac {7}{3}\qquad \textbf{(E)}\ \frac {5}{2}$


Determine the smallest real constant  $c$  such that
\[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\]
for all positive integers  $n$  and all positive real numbers  $x_1,\cdots ,x_n$ .


Let  $\triangle ABC$  be equilateral with integer side length. Point  $X$  lies on  $\overline{BC}$  strictly between  $B$  and  $C$  such that  $BX<CX$ . Let  $C'$  denote the reflection of  $C$  over the midpoint of  $\overline{AX}$ . If  $BC'=30$ , find the sum of all possible side lengths of  $\triangle ABC$ .

*Proposed by Connor Gordon*


At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adults and 28 children who attended both shows. How many people does the theater seat?


In a  $k$ -player tournament for  $k > 1$ , every player plays every other player exactly once. Find with proof the smallest value of  $k$  such that it is possible that for any two players, there was a third player who beat both of them.

Solution: \boxed{ k = 7 }


Find all pairs of real numbers  $k, l$  such that inequality  $ka^2 + lb^2> c^2$  applies to the lengths of sides  $a, b, c$  of any triangle.

Solution: \boxed{ (k, l) }


For any real number  $x$ , we let  $\lfloor x \rfloor$  be the unique integer  $n$  such that  $n \leq x < n+1$ . For example.  $\lfloor 31.415 \rfloor = 31$ . Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right  \rfloor (2021).\]

*2021 CCA Math Bonanza Team Round #3*


Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]

*Proposed by Karthik Vedula*


Pedrito's lucky number is  $34117$ . His friend Ramanujan points out that  $34117 = 166^2 + 81^2 = 159^2 + 94^2$  and  $166-159 = 7$ ,  $94- 81 = 13$ . Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of positive integers, and the difference of the first integers that occur in that sum is  $7$  and in the difference between the seconds it gives  $13$ . Which is the least lucky number that Pedrito can find? Find a way to generate all the positive integers with the properties mentioned above.


Let  $ABC$  be a triangle with circumcenter  $O$ . Point  $D, E, F$  are chosen on sides  $AB, BC$  and  $AC$ , respectively, such that  $ADEF$  is a rhombus. The circumcircles of  $BDE$  and  $CFE$  intersect  $AE$  at  $P$  and  $Q$  respectively. Show that  $OP=OQ$ .

*Proposed by Ariel García*

Solution: \boxed{OP = OQ}


Let the isosceles right triangle  $ABC$  with   $\angle A= 90^o$  . The points   $E$  and  $F$  are taken on the ray  $AC$  so that  $\angle ABE   = 15^o$  and  $CE = CF$ . Determine the measure of the angle  $CBF$ .