Let  $m$  and  $n$  be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial  $\prod_{k=1}^m\, (f+k)$  , when  $f$  covers the set of  $n^{\text{th}}$  - degree polynomials with complex coefficients.

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


Let  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $E$ , and  $F$  be  $6$  points around a circle, listed in clockwise order.  We have  $AB = 3\sqrt{2}$ ,  $BC = 3\sqrt{3}$ ,  $CD = 6\sqrt{6}$ ,  $DE = 4\sqrt{2}$ , and  $EF = 5\sqrt{2}$ .  Given that  $\overline{AD}$ ,  $\overline{BE}$ , and  $\overline{CF}$  are concurrent, determine the square of  $AF$ .


Let  $S$  be a subset of  $\{1, 2, . . . , 500\}$  such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in  $S$ .


$A, B, C, D$  are points along a circle, in that order.  $AC$  intersects  $BD$  at  $X$ . If  $BC=6$ ,  $BX=4$ ,  $XD=5$ , and  $AC=11$ , find  $AB$


In an acute triangle  $ABC$ , the points  $H$ ,  $G$ , and  $M$  are located on  $BC$  in such a way that  $AH$ ,  $AG$ , and  $AM$  are the height, angle bisector, and median of the triangle, respectively. It is known that  $HG=GM$ ,  $AB=10$ , and  $AC=14$ . Find the area of triangle  $ABC$ .


Determine all prime numbers of the form  $1 + 2^p + 3^p +...+ p^p$  where  $p$  is a prime number.

Solution: \boxed{ p = 2 }


Two math students play a game with  $k$  sticks. Alternating turns, each one chooses a number from the set  $\{1,3,4\}$  and removes exactly that number of sticks from the pile (so if the pile only has  $2$  sticks remaining the next player must take  $1$ ). The winner is the player who takes the last stick. For  $1\leq k\leq100$ , determine the number of cases in which the first player can guarantee that he will win.


Given regular hexagon  $ABCDEF$ , compute the probability that a randomly chosen point inside the hexagon is inside triangle  $PQR$ , where  $P$  is the midpoint of  $AB$ ,  $Q$  is the midpoint of  $CD$ , and  $R$  is the midpoint of  $EF$ .


Let  $P_1,P_2,\dots,P_{720}$  denote the integers whose digits are a permutation of  $123456$ , arranged in ascending order (so  $P_1=123456$ ,  $P_2=123465$ , and  $P_{720}=654321$ ). What is  $P_{144}$ ?


Determine the number of angles  $\theta$  between  $0$  and  $2 \pi$ , other than integer multiples of  $\pi /2$ , such that the quantities  $\sin \theta, \cos \theta, $  and  $\tan \theta$  form a geometric sequence in some order.


Let  $ M \ge 3$  be an integer and let  $ S \equal{} \{3,4,5,\ldots,m\}$ . Find the smallest value of  $ m$  such that for every partition of  $ S$  into two subsets, at least one of the subsets contains integers  $ a$ ,  $ b$ , and  $ c$  (not necessarily distinct) such that  $ ab \equal{} c$ .**Note**: a partition of  $ S$  is a pair of sets  $ A$ ,  $ B$  such that  $ A \cap B \equal{} \emptyset$ ,  $ A \cup B \equal{} S$ .


Consider a square of side length  $12$  centimeters. Irina draws another square that has  $8$  centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?


A rectangle is inscribed in a circle of area  $32\pi$  and the area of the rectangle is  $34$ . Find its perimeter.

*2017 CCA Math Bonanza Individual Round #2*


Find all functions  $f : [0,\infty) \to [0,\infty)$  such that  $f(f(x)) +f(x) = 12x$ , for all  $x \ge 0$ .

Solution: \boxed{ f(x) = 3x }


Let  $a_1\in (0,1)$  and  $(a_n)_{n\ge 1}$  a sequence of real numbers defined by  $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$ . Evaluate  $\lim_{n\to \infty} a_n\sqrt{n}$ .


Let  $\mathcal{S}$  be the set of real numbers that can be represented as repeating decimals of the form  $0.\overline{abc}$  where  $a, b, c$  are distinct digits.  Find the sum of the elements of  $\mathcal{S}$ .


Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.


A natural number  $n$  is said to be  $good$  if  $n$  is the sum or  $r$  consecutive positive integers, for some  $r \geq 2 $ . Find the number of good numbers in the set  $\{1,2 \dots , 100\}$ .


Let the sequence  $(a_{n})$  be deﬁned by  $a_{1} = t$  and  $a_{n+1} = 4a_{n}(1 - a_{n})$  for  $n \geq 1$ . How many possible values of t are there, if  $a_{1998} = 0$ ?

Solution: \boxed{2^{1996} + 1}


Let  $f(x)=(x-a)^3$ . If the sum of all  $x$  satisfying  $f(x)=f(x-a)$  is  $42$ , find  $a$ .


The squares in a  $7\times7$  grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.


The decimal digits of a natural number  $A$  form an increasing sequence (from left to right).  Find the sum of the digits of  $9A$ .


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


Let  $a_0, a_1,\dots, a_{19} \in \mathbb{R}$  and   $$ P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}. $$   If  $P(x)=P(-x)$  for all  $x \in \mathbb{R}$ , and   $$ P(k)=k^2, $$   for  $k=0, 1, 2, \dots, 9$  then find   $$ \lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}. $$

Solution: \boxed{-(9!)^2 + 1}


Let  $n>1$  be a natural number. Find the real values of the parameter  $a$ , for which the equation  $\sqrt[n]{1+x}+\sqrt[n]{1-x}=a$  has a single real root.

Solution: \boxed{ a = 2 }


Let  $ a$ ,  $ b$ ,  $ c$  are sides of a triangle. Find the least possible value  $ k$  such that the following inequality always holds:
 $ \left|\frac{a\minus{}b}{a\plus{}b}\plus{}\frac{b\minus{}c}{b\plus{}c}\plus{}\frac{c\minus{}a}{c\plus{}a}\right|<k$
*(Vitaly Lishunov)*

Solution: \boxed{ k = 1 }


Find the maximum integral value of  $k$  such that  $0 \le k \le 2019$  and  $|e^{2\pi i  \frac{k}{2019}} - 1|$  is maximal.


A triangle has side lengths of  $x,75,100$  where  $x<75$  and altitudes of lengths  $y,28,60$  where  $y<28$ . What is the value of  $x+y$ ?

*2019 CCA Math Bonanza Team Round #2*


There is number  $1$  on the blackboard initially. The first step is to erase  $1$  and write two nonnegative reals whose sum is  $1$ . Call the smaller number of the two  $L_2$ .
For integer  $k \ge 2$ , the  ${k}$  the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the  $k+1$  on the blackboard  $L_{k+1}$ .
Find the maximum of  $L_2+L_3+\cdots+L_{2024}$ .


In triangle  $ABC$ , the angle at vertex  $B$  is  $120^o$ . Let  $A_1, B_1, C_1$  be points on the sides  $BC, CA, AB$  respectively  such that  $AA_1, BB_1, CC_1$  are bisectors of the angles of triangle  $ABC$ . Determine the angle  $\angle A_1B_1C_1$ .


Angela, Bill, and Charles each independently and randomly choose a subset of  $\{ 1,2,3,4,5,6,7,8 \}$  that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by **Th3Numb3rThr33***


You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of  $10$  m/s throughout the race. Meanwhile, Al starts running at a constant speed of  $ 1$  m/s. Thereafter, when exactly  $ 1$  second has passed from when Al last changed its speed, Al’s speed instantaneously becomes  $1$  m/s faster, so that Al runs at a constant speed of  $k$  m/s in the kth second of the race. (Start counting seconds at  $ 1$ ). Suppose Al beats you by exactly  $1$  second. How many meters was the race?


Determine all pairs of positive integers  $(m, n)$  for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors  $f(\mathbf{v}) + \mathbf{v}$ , as  $\mathbf{v}$  runs through all of  $\mathbb{Z}_m \times \mathbb{Z}_n$ , are pairwise distinct.

(For any integers  $a$  and  $b$ , the vectors  $[a, b], [a + m, b]$  and  $[a, b + n]$  are treated as equal.)

*Poland, Wojciech Nadara*

Solution: \boxed{ (m, n) }


Let  $p$  be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails.  Given that  $p$  can be written in the form  $m/n$  where  $m$  and  $n$  are relatively prime positive integers, find  $m+n$ .


Consider a  $ n \times n$  checkerboard with  $ n > 1, n \in \mathbb{N}.$  How many possibilities are there to put  $ 2n \minus{} 2$  identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the  $ n \times n$  square.

Solution: \boxed{ 2^n }


Found the smaller multiple of  $2019$  of the form  $abcabc\dots abc$ , where  $a,b$  and  $c$  are digits.


Call a day a *perfect* day if the sum of the digits of the month plus sum of the digits of the day equals the sum of digits of the year. For example, February  $28$ th,  $2028$  is a perfect day because  $2+2+8=2+0+2+8$ . Find the number of perfect days in  $2018$ .

*2018 CCA Math Bonanza Team Round #5*


Find the least

a. positive real number

b. positive integer $t$  such that the equation  $(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2$  has a solution where  $x,y$  are positive integers.

Solution: \boxed{ t = 25 }


Eve randomly chooses two  $\textbf{distinct}$  points on the coordinate plane from the set of all  $11^2$  lattice points  $(x, y)$  with  $0 \le x \le 10$ ,  $0 \le y \le 10$ . Then, Anne the ant walks from the point  $(0,0)$  to the point  $(10, 10)$  using a sequence of one-unit right steps and one-unit up steps. Let  $P$  be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of  $P$  is  $\dbinom{20}{10} \cdot \dfrac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ . Compute  $100a+b$ .

*Proposed by Michael Tang*


In isosceles  $\vartriangle ABC, AB = AC, \angle BAC$  is obtuse, and points  $E$  and  $F$  lie on sides  $AB$  and  $AC$ , respectively, so that  $AE = 10, AF = 15$ . The area of  $\vartriangle AEF$  is  $60$ , and the area of quadrilateral  $BEFC$  is  $102$ . Find  $BC$ .


Let  $0^{\circ}\leq\alpha,\beta,\gamma\leq90^{\circ}$  be angles such that \[\sin\alpha-\cos\beta=\tan\gamma\] \[\sin\beta-\cos\alpha=\cot\gamma\]
Compute the sum of all possible values of  $\gamma$  in degrees.

*Proposed by Michael Ren*


The straight line  $y=ax+16$  intersects the graph of  $y=x^3$  at  $2$  distinct points. What is the value of  $a$ ?


Let  $n,k$  be given natural numbers. Find the smallest possible cardinality of a set  $A$  with the following property: There exist subsets  $A_1,A_2,\ldots,A_n$  of  $A$  such that the union of any  $k$  of them is  $A$ , but the union of any  $k-1$  of them is never  $A$ .


You are given a square  $n \times n$ . The centers of some of some  $m$  of its  $1\times 1$  cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer  $n \geq 3$ , find the largest value of  $m$  for which it is possible.

*Proposed by Oleksiy Masalitin, Fedir Yudin*

Solution: \boxed{ 2n-1 }


Find the number of digit of  $\sum_{n=0}^{99} 3^n$ .
You may use  $\log_{10} 3=0.4771$ .

2012 Tokyo Institute of Technology entrance exam, problem 2-A


Let  $p = 101$  and let  $S$  be the set of  $p$ -tuples  $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$  of integers. Let  $N$  denote the number of functions  $f: S \to \{0, 1, \dots, p-1\}$  such that

- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$  for all  $a, b \in S$ , and
- $f(a) = f(b)$  whenever all components of  $a-b$  are divisible by  $p$ .

Compute the number of positive integer divisors of  $N$ . (Here addition and subtraction in  $\mathbb{Z}^p$  are done component-wise.)

*Proposed by Ankan Bhattacharya*


Find all natural numbers  $n$  for which both  $n^n + 1$  and  $(2n)^{2n} + 1$  are prime numbers.

Solution: \boxed{ n = 2 }


Let  $ABC$  be an acute triangle. Let  $D$  be a point on the line parallel to  $AC$  that passes through  $B$ , such that  $\angle BDC = 2\angle BAC$  as well as such that  $ABDC$  is a convex quadrilateral. Show that  $BD + DC = AC$ .

Solution: \boxed{ BD + DC = AC }


Find the largest positive real number  $k$  such that the inequality  $$ a^3+b^3+c^3-3\ge k(3-ab-bc-ca) $$  holds for all positive real triples  $(a;b;c)$  satisfying  $a+b+c=3.$

Solution: \boxed{ k = 5 }


Let  $ABCD$  be a square of side length  $1$ , and let  $P$  be a variable point on  $\overline{CD}$ .  Denote by  $Q$  the intersection point of the angle bisector of  $\angle APB$  with  $\overline{AB}$ .  The set of possible locations for  $Q$  as  $P$  varies along  $\overline{CD}$  is a line segment; what is the length of this segment?

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


On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.


A convex polyhedron  $P$  has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does  $P$  have?


Let  $n \ge 3$  be an integer. What is the largest possible number of interior angles greater than  $180^\circ$  in an  $n$ -gon in the plane, given that the  $n$ -gon does not intersect itself and all its sides have the same length?

Solution: \boxed{ n-3 }


A number is *interesting*if it is a  $6$ -digit integer that contains no zeros, its first  $3$  digits are strictly increasing, and its last  $3$  digits are non-increasing. What is the average of all interesting numbers?


For a natural number  $n>1$  , consider the  $n-1$  points on the unit circle  $e^{\frac{2\pi ik}{n}}\ (k=1,2,...,n-1) $  . Show that the product of the distances of these points from  $1$  is  $n$ .

Solution: \boxed{n}


Define a sequence of functions recursively by  $f_1(x) = |x-1|$  and  $f_n(x)=f_{n-1}(|x-n|)$  for integers  $n > 1$ . Find the least value of  $n$  such that the sum of the zeros of  $f_n$  exceeds  $500{,}000$ .


Let  $n$  be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer.  She is permitted to make either of the following two moves:

(1)  select a row and multiply each entry in this row by  $n$ ;

(2)  select a column and subtract  $n$  from each entry in this column.

Find all possible values of  $n$  for which the following statement is true:

Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is  $0$ .

Solution: \boxed{ n = 2 }


Let  $n$  be a given positive integer. Determine all positive divisors  $d$  of  $3n^2$  such that  $n^2 + d$  is the square of an integer.

Solution: \boxed{ 3n^2 }


Compute the number of non-empty subsets  $S$  of  $\{-3, -2, -1, 0, 1, 2, 3\}$  with the following property: for any  $k \ge 1$  distinct elements  $a_1, \dots, a_k \in S$  we have  $a_1 + \dots + a_k \neq 0$ .

*Proposed by Evan Chen*


Find all positive integers  $b$  with the following property: there exists positive integers  $a,k,l$  such that  $a^k + b^l$  and  $a^l + b^k$  are divisible by  $b^{k+l}$  where  $k \neq l$ .


Let  $A_1A_2 \dots A_{4000}$  be a regular  $4000$ -gon. Let  $X$  be the foot of the altitude from  $A_{1986}$  onto diagonal  $A_{1000}A_{3000}$ , and let  $Y$  be the foot of the altitude from  $A_{2014}$  onto  $A_{2000}A_{4000}$ . If  $XY = 1$ , what is the area of square  $A_{500}A_{1500}A_{2500}A_{3500}$ ?

*Proposed by Evan Chen*


When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?


Right triangle  $ABC$  is inscribed in circle  $W$ .  $\angle{CAB}=65$  degrees, and  $\angle{CBA}=25$  degrees. The median from  $C$  to  $AB$  intersects  $W$  and line  $D$ . Line  $l_1$  is drawn tangent to  $W$  at  $A$ . Line  $l_2$  is drawn tangent to  $W$  at  $D$ . The lines  $l_1$  and  $l_2$  intersect at  $P$ Determine  $\angle{APD}$


Princeton has an endowment of  $5$  million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest  $a$  million dollars in the dorms, the students will spend an additional  $5a$  hours per week studying. If the university invests  $b$  million dollars in better food, the students will spend an additional  $3b$  hours per week studying. Finally, if the  $c$  million dollars are invested in parties, students will be more relaxed and spend  $11c - c^2$  more hours per week studying. The university wants to invest its  $5$  million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study?


How many complex numbers  $z$  such that  $\left| z \right| < 30$  satisfy the equation
\[
  e^z = \frac{z - 1}{z + 1} \, ?
\]


Let  $N$  denote the number of all natural numbers  $n$  such that  $n$  is divisible by a prime  $p> \sqrt{n}$  and  $p<20$ . What is the value of  $N$  ?


The first number in the following sequence is  $1$ . It is followed by two  $1$ 's and two  $2$ 's. This is followed by three  $1$ 's, three  $2$ 's, and three  $3$ 's. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the  $2014$ th number in this sequence.


In the trapezoid  $ABCD$  with bases  $AB$  and  $CD$ , let  $M$  be the midpoint of side  $DA$ . If  $BC=a$ ,  $MC=b$  and  $\angle MCB=150^\circ$ , what is the area of trapezoid  $ABCD$  as a function of  $a$  and  $b$ ?


Let  $x$  be a real number between 0 and  $\tfrac{\pi}{2}$  for which the function   $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$  obtains its maximum value,  $M$ . Find the value of  $M + 100\cos^2x$ .


Four men are each given a unique number from  $1$  to  $4$ , and four women are each given a unique number from  $1$  to  $4$ . How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.


Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?


Marko chose two prime numbers  $a$  and  $b$  with the same number of digits and wrote them down one after another, thus obtaining a number  $c$ . When he decreased  $c$  by the product of  $a$  and  $b$ , he got the result  $154$ . Determine the number  $c$ .


The altitude from one vertex of a triangle, the bisector from the another one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.)

(A. Zaslavsky)

Solution: \boxed{ \triangle ABC }


Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place  $1,875$  feet from the prison and was spotted by a guard with a crossbow.

The guard fired an arrow with an initial velocity of  $100 \dfrac{\text{ft}}{\text{s}}$ .  At the same time, Edward started running away with an acceleration of  $1 \dfrac{\text{ft}}{\text{s}^2}$ .  Assuming that air resistance causes the arrow to decelerate at  $1 \dfrac{\text{ft}}{\text{s}^2}$ , and that it does hit Edward, how fast was the arrow moving at the moment of impact (in  $\dfrac{\text{ft}}{\text{s}}$ )?

Solution: \boxed{75 \, \text{ft/s}}


You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with  $3$  buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially  $3$  doors are closed and  $3$  mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?


Adam and Mada are playing a game of one-on-one basketball, in which participants may take  $2$ -point shots (worth  $2$  points) or  $3$ -point shots (worth  $3$  points). Adam makes  $10$  shots of either value while Mada makes  $11$  shots of either value. Furthermore, Adam made the same number of  $2$ -point shots as Mada made  $3$ -point shots. At the end of the game, the two basketball players realize that they have the exact same number of points! How many total points were scored in the game?

*2017 CCA Math Bonanza Lightning Round #2.1*


Let  $x_1$ ,  $x_2$ , …,  $x_{10}$  be 10 numbers.  Suppose that  $x_i + 2 x_{i + 1} = 1$  for each  $i$  from 1 through 9.  What is the value of  $x_1 + 512 x_{10}$ ?


Six members of the team of Fatalia for the International Mathematical Olympiad are selected from  $13$  candidates. At the TST the candidates got  $a_1,a_2, \ldots, a_{13}$  points with  $a_i \neq a_j$  if  $i \neq j$ .

The team leader has already  $6$  candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial  $P(x)$  and finds the creative potential of each candidate by the formula  $c_i = P(a_i)$ .

For what minimum  $n$  can he always find a polynomial  $P(x)$  of degree not exceeding  $n$  such that the creative potential of all  $6$  candidates is strictly more than that of the  $7$  others?

*Proposed by F. Petrov, K. Sukhov*


Find all the functions  $f(x),$  continuous on the whole real axis, such that for every real  $x$  \[f(3x-2)\leq f(x)\leq f(2x-1).\]

*Proposed by A. Golovanov*

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


Define the sequence  $ (a_p)_{p\ge0}$  as follows:  $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$ .
Find  $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$ .


Brian starts at the point  $\left(1,0\right)$  in the plane. Every second, he performs one of two moves: he can move from  $\left(a,b\right)$  to  $\left(a-b,a+b\right)$  or from  $\left(a,b\right)$  to  $\left(2a-b,a+2b\right)$ . How many different paths can he take to end up at  $\left(28,-96\right)$ ?

*2018 CCA Math Bonanza Individual Round #14*


How many alphabetic sequences (that is, sequences containing only letters from  $a\cdots z$ ) of length  $2013$  have letters in alphabetic order?


Find the smallest integer  $n$  such that each subset of  $\{1,2,\ldots, 2004\}$  with  $n$  elements has two distinct elements  $a$  and  $b$  for which  $a^2-b^2$  is a multiple of  $2004$ .


For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.


Given is a  $n \times n$  chessboard. With the same probability, we put six pawns on its six cells. Let  $p_n$  denotes the probability that there exists a row or a column containing at least two pawns. Find  $\lim_{n \to \infty} np_n$ .


Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sits there and if he finds his seat occupied he sits somewhere else randomly. Find the probability that last two students sit on their seats.

Solution: \boxed{ \frac{1}{3} }


There is infinite sequence of composite numbers  $a_1,a_2,...,$  where  $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$  ;  $p_n$  is smallest prime divisor of  $a_n$ . It is known, that  $37|a_n$  for every  $n$ .
Find possible values of  $a_1$


Find the smallest positive integer  $n \geq 10$  such that  $n + 6$  is a prime and  $9n + 7$  is a perfect square.


**Problem 5.**Denote with  $d(a,b)$  the numbers of the divisors of natural  $a$ , which are greater or equal to  $b$ . Find all natural  $n$ , for which
 $d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$

*Ivan Landgev*


The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.


There are 2008 red cards and 2008 white cards. 2008  players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.

 $ (*)$  If you have more than one red card, then you will pass one red card to the left-neighbouring player.

If you have no red card, then you will pass one white card to the left -neighbouring player.
Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first.


Determine all triples  $(x, y, z)$  of positive real numbers which satisfies the following system of equations
\[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

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


How many  $5$ -digit numbers  $N$  (in base  $10$ ) contain no digits greater than  $3$  and satisfy the equality  $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of  $N$  cannot be zero.)

*Based on a proposal by Yannick Yao*


Let  $n \ge 2$  be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that  $n = a_1 + a_2 + \cdots$ . Determine all possible values of  $n$ .


find all  $k$  distinct integers  $a_1,a_2,...,a_k$  such that there exists an injective function  $f$  from reals to themselves such that for each positive integer  $n$  we have  $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$  .

Solution: \boxed{ \{0\} }


A fair coin is to be tossed  $10$  times. Let  $i/j$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find  $i+j$ .


A finite set  $S$  of points in the coordinate plane is called *overdetermined* if  $|S|\ge 2$  and there exists a nonzero polynomial  $P(t)$ , with real coefficients and of degree at most  $|S|-2$ , satisfying  $P(x)=y$  for every point  $(x,y)\in S$ .

For each integer  $n\ge 2$ , find the largest integer  $k$  (in terms of  $n$ ) such that there exists a set of  $n$  distinct points that is *not* overdetermined, but has  $k$  overdetermined subsets.

*Proposed by Carl Schildkraut*

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


The school store is running out of supplies, but it still has five items: one pencil (costing  $\$ 1 $), one pen (costing $ \ $1$ ), one folder (costing  $\$ 2 $), one pack of paper (costing $ \ $3$ ), and one binder (costing  $\$ 4 $). If you have $ \ $10$ , in how many ways can you spend your money? (You don't have to spend all of your money, or any of it.)


Richard has a four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars.  (One dollar is 100 cents.) What is the expected value of this final sum of money, in cents?

*Proposed by Lewis Chen*


In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes

Solution: \boxed{265 - 132\sqrt{3}}


Solution: \boxed{ 508032 }


The decimal number  $13^{101}$  is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

Solution: \boxed{ 21 }


Dave's Amazing Hotel has  $3$  floors. If you press the up button on the elevator from the  $3$ rd floor, you are immediately transported to the  $1$ st floor. Similarly, if you press the down button from the  $1$ st floor, you are immediately transported to the  $3$ rd floor. Dave gets in the elevator at the  $1$ st floor and randomly presses up or down at each floor. After doing this  $482$  times, the probability that Dave is on the first floor can be expressed as  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. What is the remainder when  $m+n$  is divided by  $1000$ ? $\text{(A) }136\qquad\text{(B) }294\qquad\text{(C) }508\qquad\text{(D) }692\qquad\text{(E) }803$


A real number  $ to $  is randomly and uniformly chosen from the  $ [- 3,4] $  interval. What is the probability that all roots of the polynomial  $ x ^ 3 + ax ^ 2 + ax + 1 $  are real?


Let  $ABC$  be an acute-angled triangle and let  $AN$ ,  $BM$  and  $CP$  the altitudes with respect to the sides  $BC$ ,  $CA$  and  $AB$ , respectively. Let  $R$ ,  $S$  be the pojections of  $N$  on the sides  $AB$ ,  $CA$ , respectively, and let  $Q$ ,  $W$  be the projections of  $N$  on the altitudes  $BM$  and  $CP$ , respectively.

(a) Show that  $R$ ,  $Q$ ,  $W$ ,  $S$  are collinear.
(b) Show that  $MP=RS-QW$ .

Solution: \boxed{ MP = RS - QW }


For an integer  $n$  let  $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$ . Furthermore, be  $S (n)$  sum of squares and  $P (n)$  the product of the squares of the elements of  $M (n)$ . For which integers  $n$  is  $S (n)$  a divisor of  $P (n)$  ?

Solution: \boxed{ n = 3 }


Find the largest  $2$ -digit number  $N$  which is divisible by  $4$ , such that all integral powers of  $N$  end with  $N$ .


The increasing sequence  $1,3,4,9,10,12,13\cdots$  consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the  $100^{\text{th}}$  term of this sequence.


Let  $n$  be a fixed positive integer. There are  $n \geq 1$  lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer  $i$  ( $1 \leq i \leq n$ ) and switch the state of the first  $i$  lamps from the left. Determine the smallest number  $k$  with the property that we can make all of the lamps be switched on using at most  $k$  moves, no matter what the initial configuration was.

*Proposed by Viktor Simjanoski and Nikola Velov*

Solution: \boxed{ k = n }


Let  $ABC$  be a triangle in which  $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio  $\frac{AB}{BC}$ .


Let  $p,q,r$  be distinct prime numbers and let
\[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \]
Find the least  $n\in\mathbb{N}$  such that for any  $B\subset A$  where  $|B|=n$ , has elements  $x$  and  $y$  such that  $x$  divides  $y$ .

*Ioan Tomescu*

Solution: \boxed{ 28 }


On  $\triangle ABC$  let points  $D,E$  on sides  $AB,BC$  respectivily such that  $AD=DE=EC$  and  $AE \ne DC$ . Let  $P$  the intersection of lines  $AE, DC$ , show that  $\angle ABC=60$  if  $AP=CP$ .


Let  $ABCD$  be a convex quadrilateral with  $\angle ABD = \angle BCD$ ,  $AD = 1000$ ,  $BD = 2000$ ,  $BC = 2001$ , and  $DC = 1999$ . Point  $E$  is chosen on segment  $DB$  such that  $\angle ABD = \angle ECD$ . Find  $AE$ .


Let  $f$  be a quadratic function which satisfies the following condition. Find the value of  $\frac{f(8)-f(2)}{f(2)-f(1)}$ .

For two distinct real numbers  $a,b$ , if  $f(a)=f(b)$ , then  $f(a^2-6b-1)=f(b^2+8)$ .


Turbo the snail sits on a point on a circle with circumference  $1$ . Given an infinite sequence of positive real numbers  $c_1, c_2, c_3, \dots$ , Turbo successively crawls distances  $c_1, c_2, c_3, \dots$  around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant  $C > 0$  with the following property: for every sequence of positive real numbers  $c_1, c_2, c_3, \dots$  with  $c_i < C$  for all  $i$ , Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.

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


A checkerboard is  $91$  squares long and  $28$  squares wide. A line connecting two opposite vertices of the checkerboard is drawn. How many squares does the line pass through?


We have a complete graph with  $n$  vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

Solution: \boxed{n}


Let  $M$  be on segment $ BC$  of  $\vartriangle ABC$  so that  $AM = 3$ ,  $BM = 4$ , and  $CM = 5$ . Find the largest possible area of  $\vartriangle ABC$ .


The area of the triangle whose altitudes have lengths  $36.4$ ,  $39$ , and  $42$  can be written as  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say  $n$ , of postage which is unattainable while all amounts larger than  $n$  are attainable? (Justify your answer.)


Let  $ABC$  be an acute triangle.  Let  $DAC,EAB$ , and  $FBC$  be isosceles triangles exterior to  $ABC$ , with  $DA=DC, EA=EB$ , and  $FB=FC$ , such that
\[
\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad
\angle CFB = 2 \angle ACB.
\]
Let  $D'$   be the intersection of lines  $DB$  and  $EF$ , let  $E'$  be the intersection of  $EC$  and  $DF$ , and let  $F'$    be the intersection of  $FA$  and  $DE$ . Find, with proof, the value of the sum
\[
\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.
\]


Let  $\triangle ABC$  satisfies  $\cos A:\cos B:\cos C=1:1:2$ , then  $\sin A=\sqrt[s]{t}$ ( $s\in\mathbb{N},t\in\mathbb{Q^+}$  and  $t$  is an irreducible fraction). Find  $s+t$ .


Find the number of ordered pairs of integers  $(a, b)$  that satisfy the inequality

\[
1 < a < b+2 < 10.
\]

*Proposed by Lewis Chen*


$S$  is a circle with  $AB$  a diameter and  $t$  is the tangent line to  $S$  at  $B$ . Consider the two points  $C$  and  $D$  on  $t$  such that  $B$  is between  $C$  and  $D$ . Suppose  $E$  and  $F$  are the intersections of  $S$  with  $AC$  and  $AD$  and  $G$  and  $H$  are the intersections of  $S$  with  $CF$  and  $DE$ . Show that  $AH=AG$ .

Solution: \boxed{AG = AH}


$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$  is a function that satisfies   $$ g(x)+g(y)=g(x+y)-xy. $$   If  $g(23)=0$ , what is the sum of all possible values of  $g(35)$ ?


Find the maximum number of natural numbers  $x_1,x_2, ... , x_m$   satisfying the conditions:
a) No  $x_i - x_j , 1 \le i < j \le m$  is divisible by  $11$ ,  and
b) The sum  $x_2x_3 ...x_m + x_1x_3  ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$  is divisible by  $11$ .


Find the smallest positive integer  $k$  such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers  $(a,b,c)$ .

Solution: \boxed{ k = 1297 }


Consider the sequence  $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ...$  Find  $n$  such that the first  $n$  terms sum up to  $2010.$


At the beginning of school year in one of the first grade classes: $i)$  every student had exatly  $20$  acquaintances $ii)$  every two students knowing each other had exactly  $13$  mutual acquaintances $iii)$  every two students not knowing each other had exactly  $12$  mutual acquaintances
Find number of students in this class


Determine all sequences  $p_1, p_2, \dots $  of prime numbers for which there exists an integer  $k$  such that the recurrence relation
\[ p_{n+2} = p_{n+1} + p_n + k \]
holds for all positive integers  $n$ .

Solution: \boxed{ p_n = p }


What is the largest integer that must divide  $n^5-5n^3+4n$  for all integers  $n$ ?

*2016 CCA Math Bonanza Lightning #2.4*


There are  $17$  students in Marek's class, and all of them took a test. Marek's score was  $17$  points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer.


Given that  $(1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ}) = 2^n$ , find  $n$ .


The base- $7$  representation of number  $n$  is  $\overline{abc}_{(7)}$ , and the base- $9$  representation of number  $n$  is  $\overline{cba}_{(9)}$ . What is the decimal (base- $10$ ) representation of  $n$ ?


For positive integer  $n$  let  $a_n$  be the integer consisting of  $n$  digits of  $9$  followed by the digits  $488$ . For example,  $a_3 = 999,488$  and  $a_7 = 9,999,999,488$ . Find the value of  $n$  so that an is divisible by the highest power of  $2$ .


Denote  $\mathbb{Z}_{>0}=\{1,2,3,...\}$  the set of all positive integers. Determine all functions  $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$  such that, for each positive integer  $n$ , $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$  is a perfect square, and $\vspace{0.1cm}$  $\hspace{1cm}ii) f(n)$  divides  $n^3$ .

*Proposed by Dorlir Ahmeti, Albania*

Solution: \boxed{ f(n) = n^3 }


Evaluate  $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$ . (Here  $\sec''$  means the second derivative of  $\sec$ ).


Let  $ABC$  be a triangle with side lengths  $5$ ,  $4\sqrt 2$ , and  $7$ .  What is the area of the triangle with side lengths  $\sin A$ ,  $\sin B$ , and  $\sin C$ ?

Solution: \boxed{ \frac{7}{25} }


Tadeo draws the rectangle with the largest perimeter that can be divided into  $2015$  squares of sidelength  $1$   $cm$  and the rectangle with the smallest perimeter that can be divided into  $2015$  squares of sidelength  $1$   $cm$ . What is the difference between the perimeters of the rectangles Tadeo drew?


Compute the largest possible number of distinct real solutions for  $x$  to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where  $a$ ,  $b$ , and  $c$  are real numbers.

[i]Proposed by Tristan Shin


The MathMatters competition consists of 10 players  $P_1$ ,  $P_2$ ,  $\dots$ ,  $P_{10}$  competing in a ladder-style tournament.  Player  $P_{10}$  plays a game with  $P_9$ : the loser is ranked 10th, while the winner plays  $P_8$ .  The loser of that game is ranked 9th, while the winner plays  $P_7$ .  They keep repeating this process until someone plays  $P_1$ : the loser of that final game is ranked 2nd, while the winner is ranked 1st.  How many different rankings of the players are possible?


The number  $734{,}851{,}474{,}594{,}578{,}436{,}096$  is equal to  $n^6$  for some positive integer  $n$ .  What is the value of  $n$ ?


A subset  $M$  of  $\{1, 2, . . . , 2006\}$  has the property that for any three elements  $x, y, z$  of  $M$  with  $x < y < z$ ,  $x+ y$  does not divide  $z$ . Determine the largest possible size of  $M$ .

Solution: \boxed{ 1004 }


On a circumference, points  $A$  and  $B$  are on opposite arcs of diameter  $CD$ . Line segments  $CE$  and  $DF$  are perpendicular to  $AB$  such that  $A-E-F-B$  (i.e.,  $A$ ,  $E$ ,  $F$  and  $B$  are collinear on this order). Knowing  $AE=1$ , find the length of  $BF$ .


Suppose you are given that for some positive integer  $n$ ,  $1! + 2! + \ldots + n!$  is a perfect square. Find the sum of all possible values of  $n$ .


Find a number with  $3$  digits, knowing that the sum of its digits is  $9$ , their product is  $24$  and also the number read from right to left is  $\frac{27}{38}$  of the original.


Find the biggest real number  $C$ , such that for every different positive real numbers  $a_1,a_2...a_{2019}$  that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$


Let  $n$  be a positive integer.  When the leftmost digit of (the standard base 10 representation of)  $n$  is shifted to the rightmost position (the units position), the result is  $n/3$ .  Find the smallest possible value of the sum of the digits of  $n$ .


Let  $T = TNFTPP$ .   $x$  and  $y$  are nonzero real numbers such that \[18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0.\] The smallest possible value of  $\tfrac{y}{x}$  is equal to  $\tfrac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, here's the value of T$T=6$.

Solution: \boxed{ 7 }


Given a line segment  $ PQ$  moving on the parabola  $ y \equal{} x^2$  with end points on the parabola. The area of the figure surrounded by  $ PQ$  and the parabola is always equal to  $ \frac {4}{3}$ . Find the equation of the locus of the mid point  $ M$  of  $ PQ$ .

Solution: \boxed{y = x^2 + 1}


Let  $a,b,c$  be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum  $a + b + c$ . Find this sum.


$2019$  students are voting on the distribution of  $N$  items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible number of items one student can receive, over all possible values of  $N$  and all possible ways of voting.


The sequence  ${a_{n}}$   $n\in \mathbb{N}$  is given in a recursive way with  $a_{1}=1$ , $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$ , for all  $n\geq 2$ .
Determine the least number  $M$ , such that  $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$  for all  $m\in \mathbb{N}$


Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Nathan Xiong*


A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at  $1$  meter per second. Suddenly someone turns on the turntable; it spins at  $30$  rpm. Consider the set  $S$  of points the mouse can reach in his car within  $1$  second after the turntable is set in motion. What is the area of  $S$ , in square meters?


Given are positive reals  $x_1, x_2,..., x_n$  such that  $\sum\frac {1}{1+x_i^2}=1$ . Find the minimal value of the expression  $\frac{\sum x_i}{\sum \frac{1}{x_i}}$  and find when it is achieved.


A wishing well is located at the point  $(11,11)$  in the  $xy$ -plane. Rachelle randomly selects an integer  $y$  from the set  $\left\{ 0, 1, \dots, 10 \right\}$ .  Then she randomly selects, with replacement, two integers  $a,b$  from the set  $\left\{ 1,2,\dots,10 \right\}$ .  The probability the line through  $(0,y)$  and  $(a,b)$  passes through the well can be expressed as  $\frac mn$ , where  $m$  and  $n$  are relatively prime positive integers.  Compute  $m+n$ .

*Proposed by Evan Chen*


We are given  $2021$  points on a plane, no three of which are collinear. Among any  $5$  of these points, at least  $4$  lie on the same circle. Is it necessarily true that at least  $2020$  of the points lie on the same circle?


Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?

[i]Proposed by Matthew Weiss


In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets  $ 1$  point and the loser gets  $ 0$  points. If the game is tied, each player gets  $ 0.5$  points. Given a positive integer  $ m$ , a tournament is said to have property  $ P(m)$  if the following holds: among every set  $ S$  of  $ m$  players, there is one player who won all her games against the other  $ m\minus{}1$  players in  $ S$  and one player who lost all her games against the other  $ m \minus{} 1$  players in  $ S$ . For a given integer  $ m \ge 4$ , determine the minimum value of  $ n$  (as a function of  $ m$ ) such that the following holds: in every  $ n$ -player round robin chess tournament with property  $ P(m)$ , the final scores of the  $ n$  players are all distinct.

Solution: \boxed{ 2m-3 }


There are three bins: one with 30 apples, one with 30 oranges, and one with 15 of each. Each is labeled "apples," "oranges," or "mixed." Given that all three labels are wrong, how many pieces of fruit must you look at to determine the correct labels?


Find the greatest possible value of  $pq + r$ , where p, q, and r are (not necessarily distinct) prime numbers satisfying  $pq + qr + rp = 2016$ .


Let  $a, b, c, d$  be a permutation of the numbers  $1, 9, 8,4$  and let  $n = (10a + b)^{10c+d}$ . Find the probability that  $1984!$  is divisible by  $n.$


Find the largest real number  $\lambda$  with the following property: for any positive real numbers  $p,q,r,s$  there exists a complex number  $z=a+bi$ ( $a,b\in \mathbb{R})$   such that   $$  |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0. $$


Yesterday (=April 22, 2003) was Gittes birthday. She notices that her age equals the sum of the 4 digits of the year she was born in.
How old is she?


Points  $A$ ,  $B$ ,  $C$ ,  $D$ , and  $E$  are on the same plane such that  $A,E,C$  lie on a line in that order,  $B,E,D$  lie on a line in that order,  $AE = 1$ ,  $BE = 4$ ,  $CE = 3$ ,  $DE = 2$ , and  $\angle AEB = 60^\circ$ . Let  $AB$  and  $CD$  intersect at  $P$ . The square of the area of quadrilateral  $PAED$  can be expressed as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

*2021 CCA Math Bonanza Individual Round #9*


The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country.


Find all polynomials  $P(x)$  with real coefficients satisfying:  $P(2017) = 2016$  and   $$ (P(x)+1)^2=P(x^2+1). $$

Solution: \boxed{P(x) = x - 1}


In how many ways can we form three teams of four players each from a group of  $12$  participants?


A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is  $m-n\sqrt[3]{p},$  where  $m,$   $n,$  and  $p$  are positive integers and  $p$  is not divisible by the cube of any prime number. Find  $m+n+p.$


Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET.


Five bricklayers working together finish a job in  $3$  hours. Working alone, each bricklayer takes at most  $36$  hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?

*Author: Ray Li*


Let  $A,B$  and  $C$  be three points on a line with  $B$  between  $A$  and  $C$ . Let  $\Gamma_1,\Gamma_2, \Gamma_3$  be semicircles, all on the same side of  $AC$  and with  $AC,AB,BC$  as diameters, respectively. Let  $l$  be the line perpendicular to  $AC$  through  $B$ . Let  $\Gamma$  be the circle which is tangent to the line  $l$ , tangent to  $\Gamma_1$  internally, and tangent to  $\Gamma_3$  externally. Let  $D$  be the point of contact of  $\Gamma$  and  $\Gamma_3$ . The diameter of  $\Gamma$  through  $D$  meets  $l$  in  $E$ . Show that  $AB=DE$ .

Solution: \boxed{AB = DE}


Find the least positive integer  $n$  such that for every prime number  $p, p^2 + n$  is never prime.


Find the number of ordered quadruples  $(a, b, c, d)$  where each of  $a, b, c,$  and  $d$  are (not necessarily distinct) elements of  $\{1, 2, 3, 4, 5, 6, 7\}$  and  $3abc + 4abd + 5bcd$  is even. For example,  $(2, 2, 5, 1)$  and  $(3, 1, 4, 6)$  satisfy the conditions.


What is the largest positive integer  $n$  less than  $10, 000$  such that in base  $4$ ,  $n$  and  $3n$  have the same number of digits; in base  $8$ ,  $n$  and  $7n$  have the same number of digits; and in base  $16$ ,  $n$  and  $15n$  have the same number of digits? Express your answer in base  $10$ .


Given a complete bipartite graph on  $n,n$  vertices (call this  $K_{n,n}$ ), we colour all its edges with  $2$  colours , red and blue . What is the least value of  $n$  such that for any colouring of the edges of the graph , there will exist at least one monochromatic  $4$  cycle ?

Solution: \boxed{ n = 5 }


P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?

Solution: \boxed{ P(x) = Q(x) }


Compute the number of rearrangements  $a_1, a_2, \dots, a_{2018}$  of the sequence  $1, 2, \dots, 2018$  such that  $a_k > k$  for  $\textit{exactly}$  one value of  $k$ .

Solution: \boxed{2^{2018} - 2019}


On a Cartesian coordinate plane, points  $(1, 2)$  and  $(7, 4)$  are opposite vertices of a square. What is the area of the square?


How many ordered triples  $(a, b, c)$  of positive integers are there such that none of  $a, b, c$  exceeds  $2010$  and each of  $a, b, c$  divides  $a + b + c$ ?


In the following addition, different letters represent different non-zero digits. What is the 5-digit number  $ABCDE$ ? $
\begin{array}{ccccccc}
A&B&C&D&E&D&B
&B&C&D&E&D&B
&&C&D&E&D&B
&&&D&E&D&B
&&&&E&D&B
&&&&&D&B
+&&&&&&B \hline
A&A&A&A&A&A&A
\end{array}
$


Let set  $\mathcal{A}$  be a 90-element subset of  $\{1,2,3,\ldots,100\},$  and let  $S$  be the sum of the elements of  $\mathcal{A}$ .  Find the number of possible values of  $S$ .


Let  $n\ge 2$  be a given integer. Find the greatest value of  $N$ , for which the following is true: there are infinitely many ways to find  $N$  consecutive integers such that none of them has a divisor greater than  $1$  that is a perfect  $n^{\mathrm{th}}$  power.

*Proposed by Péter Pál Pach, Budapest*

Solution: \boxed{ 2^n - 1 }


Let  $n = 2188 = 3^7+1$  and let  $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$  be the vertices of a regular  $n$ -gon (in that order) with center  $O$  . For  $i = 1, 2, \dots, 7$  and  $j=0,1,\dots,n-1$ , let  $A_j^{(i)}$  denote  the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo  $n$ . If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers  $p$  and  $q$ , find  $p+q$ .

*Proposed by Yang Liu*


Given are  $n$   $(n > 2)$  points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?


A cube with side length  $2$  is inscribed in a sphere. A second cube, with faces parallel to the ﬁrst, is inscribed between the sphere and one face of the ﬁrst cube. What is the length of a side of the smaller cube?


Given a positive integer  $N$ . There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly  $N$ . Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make?

Proposed by USJL, ST.

Solution: \boxed{N}


Find the smallest positive interger number  $n$  such that there exists  $n$  real numbers  $a_1,a_2,\ldots,a_n$  satisfied three conditions as follow:
a.  $a_1+a_2+\cdots+a_n>0$ ;
b.  $a_1^3+a_2^3+\cdots+a_n^3<0$ ;
c.  $a_1^5+a_2^5+\cdots+a_n^5>0$ .


Q. A light source at the point  $(0, 16)$  in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius  $2$  with center at  $(6, 10)$  casts a shadow on the X-axis. The length of the shadow can be written in the form  $m\sqrt{n}$  where  $m, n$  are positive integers and  $n$  is squarefree. Find  $m + n$ .


Points  $D$  and  $E$  are chosen on side  $BC$  of triangle  $ABC$  such that  $E$  is between  $B$  and  $D$  and  $BE=1$ ,  $ED=DC=3$ .  If  $\angle BAD=\angle EAC=90^\circ$ , the area of  $ABC$  can be expressed as  $\tfrac{p\sqrt q}r$ , where  $p$  and  $r$  are relatively prime positive integers and  $q$  is a positive integer not divisible by the square of any prime.  Compute  $p+q+r$ .

[asy]
import olympiad;
size(200);
defaultpen(linewidth(0.7)+fontsize(11pt));
pair D = origin, E = (3,0), C = (-3,0), B = (4,0);
path circ1 = arc(D,3,0,180), circ2 = arc(B/2,2,0,180);
pair A = intersectionpoint(circ1, circ2);
draw(E--A--C--B--A--D);
label(" $A$ ",A,N);
label(" $B$ ",B,SE);
label(" $C$ ",C,SW);
label(" $D$ ",D,S);
label(" $E$ ",E,S);
[/asy]


Let  $x,y,z$  be positive real numbers such that  $x^2 + y^2 = 49, y^2 + yz + z^2 = 36$  and  $x^2 + \sqrt{3}xz + z^2 = 25$ . If the value of  $2xy + \sqrt{3}yz + zx$  can be written as  $p \sqrt{q}$  where  $p,q \in \mathbb{Z}$  and  $q$  is squarefree, find  $p+q$ .


Find all functions  $ f: \mathbb{R} \mapsto \mathbb{R}$  such that  $ \forall x,y,z \in \mathbb{R}$  we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]

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


Triangle  $ABC$  has circumcircle  $\Omega$ . Chord  $XY$  of  $\Omega$  intersects segment  $AC$  at point  $E$  and segment  $AB$  at point  $F$  such that  $E$  lies between  $X$  and  $F$ . Suppose that  $A$  bisects arc  $\widehat{XY}$ . Given that  $EC = 7, FB = 10, AF = 8$ , and  $YF - XE = 2$ , find the perimeter of triangle  $ABC$ .


An island has  $10$  cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.


In a  $12\times 12$  grid, colour each unit square with either black or white, such that there is at least one black unit square in any  $3\times 4$  and  $4\times 3$  rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.


Consider the sequence of integers  $ \left( a_n\right)_{n\ge 0} $  defined as  $$  a_n=\left\{\begin{matrix}n^6-2017, & 7|n \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. .  $$  Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than  $
1 $  may have.


Suppose  $r \ge 2$  is an integer, and let  $m_1, n_1, m_2, n_2, \dots, m_r, n_r$  be  $2r$  integers such that   $$ \left|m_in_j-m_jn_i\right|=1 $$   for any two integers  $i$  and  $j$  satisfying  $1 \le i<j \le r$ . Determine the maximum possible value of  $r$ .

*Proposed by B Sury*

Solution: \boxed{ 3 }


Let  $ABC$  be a right triangle, right at  $B$ , and let  $M$  be the midpoint of the side  $BC$ . Let  $P$  be the point in
bisector of the angle  $ \angle BAC$  such that  $PM$  is perpendicular to  $BC (P$  is outside the triangle  $ABC$ ). Determine the triangle area  $ABC$  if  $PM = 1$  and  $MC = 5$ .


The convex quadrilateral  $ABCD$  has area  $1$ , and  $AB$  is produced to  $E$ ,  $BC$  to  $F$ ,  $CD$  to  $G$  and  $DA$  to  $H$ , such that  $AB=BE$ ,  $BC=CF$ ,  $CD=DG$  and  $DA=AH$ . Find the area of the quadrilateral  $EFGH$ .


Let  $P$  be a point inside a tetrahedron  $ABCD$  and let  $S_A,S_B,S_C,S_D$  be the centroids (i.e. centers of gravity) of the tetrahedra  $PBCD,PCDA,PDAB,PABC$ . Show that the volume of the tetrahedron  $S_AS_BS_CS_D$  equals  $1/64$  the volume of  $ABCD$ .


Find the smallest positive real number  $\lambda$  such that for every numbers  $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$  and  $b_1,b_2,b_3 \in (0, \infty)$  with  $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$  we have   $$ b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3). $$


Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number  $n$  greater than 10. Then, she multiplies all the digits of  $n$  and obtains an odd number. Find all possible values of the units digit of  $n$ . $\textit{Proposed by Pablo Serrano, Ecuador}$


Let  $ A$  be the region :  $ \left\{(x,\ y)|x\geq 0,\ y\geq 0, \left(\frac{x}{a}\right)^{\frac{1}{2}}\plus{}\left(\frac{y}{1\minus{}a}\right)^{\frac{1}{2}}\leq 1\right\}$  for  $ 0<a<1.$

(1) Find the area of  $ A.$

(2) Find the maximum volume of the solid obtained by revolving  $ A$  around the  $ x$  axis.

Solution: \boxed{ \frac{4\pi}{405} }


A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.

A rectangle with sides  $m\ge3$  and  $n\ge3$  consists of  $mn$  unit squares. If in each of those unit squares exactly one number is written, such that any square with side  $3$  is a magic square, then find the number of most different numbers that can be written in that rectangle.

Solution: \boxed{ 9 }


Find all real  $x$  such that  $0 < x < \pi $  and  $\frac{8}{3 sin x - sin 3x} + 3 sin^2x \le 5$ .


Define the minimum real  $C$  where for any reals  $0 = a_0 < a_{1} < \dots < a_{1000}$  then   $$ \min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000}) $$   holds. Find  $\lfloor 100C \rfloor.$


Let  $N$  be the number of ways of distributing  $8$  chocolates of different brands among  $3$  children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of  $N$ .


In the  $ x \minus{} y$  plane, find the area of the region bounded by the parameterized curve as follows.

 $ \left\{\begin{array}{ll} x \equal{} \cos 2t & \quad
y \equal{} t\sin t & \quad \end{array} \right.\ (0\leq t\leq 2\pi)$


Find all  $f:\mathbb{R}\rightarrow \mathbb{R}$  continuous functions such that  $\lim_{x\rightarrow \infty} f(x) =\infty$  and  $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$  such that   $$ f^n(x)+f^n(y)=x+y $$

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


Consider a paper punch that can be centered at any point
of the plane and that, when operated, removes from the
plane precisely those points whose distance from the
center is irrational. How many punches are needed to
remove every point?


Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .

(You may use the fact that   $\sin{n^\circ}$  is irrational for positive integers  $n$  not divisible by  $30$ .)

*Ray Li*


One writes, initially, the numbers  $1,2,3,\dots,10$  in a board. An operation is to delete the numbers  $a, b$  and write the number  $a+b+\frac{ab}{f(a,b)}$ , where  $f(a, b)$  is the sum of all numbers in the board excluding  $a$  and  $b$ , one will make this until remain two numbers  $x, y$  with  $x\geq y$ . Find the maximum value of  $x$ .


Let  $P$  be a point outside a circle  $\Gamma$  centered at point  $O$ , and let  $PA$  and  $PB$  be tangent lines to circle  $\Gamma$ . Let segment  $PO$  intersect circle  $\Gamma$  at  $C$ . A tangent to circle  $\Gamma$  through  $C$  intersects  $PA$  and  $PB$  at points  $E$  and  $F$ , respectively. Given that  $EF=8$  and  $\angle{APB}=60^\circ$ , compute the area of  $\triangle{AOC}$ .

*2020 CCA Math Bonanza Individual Round #6*

Solution: \boxed{12 \sqrt{3}}


Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in
the bottle?


There are  $10$  birds on the ground. For any  $5$  of them, there are at least  $4$  birds on a circle. Determine the least possible number of birds on the circle with the most birds.


Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4.  At each step, she rolls the die and records which number is on the bottom face.  She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.)  What is the expected number (average number) of steps that she takes?


For how many positive integers  $n$ ,  $1\leq n\leq 2008$ , can the set \[\{1,2,3,\ldots,4n\}\] be divided into  $n$  disjoint  $4$ -element subsets such that every one of the  $n$  subsets contains the element which is the arithmetic mean of all the elements in that subset?


Find all functions  $f(x)$  from nonnegative reals to nonnegative reals such that  $f(f(x))=x^4$  and  $f(x)\leq Cx^2$  for some constant  $C$ .

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


For what values of  $ n$  does there exist an  $ n \times n$  array of entries -1, 0 or 1 such that the  $ 2 \cdot n$  sums obtained by summing the elements of the rows and the columns are all different?

Solution: \boxed{ n }


In a triangle  $ABC$  the area is  $18$ , the length  $AB$  is  $5$ , and the medians from  $A$  and  $B$  are orthogonal. Find the lengths of the sides  $BC,AC$ .

Solution: \boxed{ BC = 2\sqrt{13} }


Let  $k$  be a positive integer. Find all functions  $f:\mathbb{N}\to \mathbb{N}$  satisfying the following two conditions:

• For infinitely many prime numbers  $p$  there exists a positve integer  $c$  such that  $f(c)=p^k$ .

• For all positive integers  $m$  and  $n$ ,  $f(m)+f(n)$  divides  $f(m+n)$ .

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


Let  $n$  be a positive integer. Every square in a  $n \times n$ -square grid is either white or black.
How many such colourings exist, if every  $2 \times 2$ -square consists of exactly two white and two black squares?
The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.


Find the minimum real  $x$  that satisfies  $$ \lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots $$


Each square of a  $(2^n-1) \times (2^n-1)$  board contains either  $1$  or  $-1$ .  Such an arrangement is called *successful* if each number is the product of its neighbors.  Find the number of successful arrangements.


Rectangle  $HOMF$  has  $HO=11$  and  $OM=5$ . Triangle  $ABC$  has orthocenter  $H$  and circumcenter  $O$ .  $M$  is the midpoint of  $BC$  and altitude  $AF$  meets  $BC$  at  $F$ . Find the length of  $BC$ .


How many ways are there to line up  $19$  girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

Solution: \boxed{ 262144 }


A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points  $1$  unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between  $\frac{1}{2017}$  and  $\frac{1}{2016}$  units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?

Solution: \boxed{ 17 }


	Determine the last (rightmost) three decimal digits of  $n$  where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\]


Let  $a_0=1,a_1=2,$  and  $a_n=4a_{n-1}-a_{n-2}$  for  $n\ge 2.$ Find an odd prime factor of  $a_{2015}.$


A  $6$ -inch-wide rectangle is rotated  $90$  degrees about one of its corners, sweeping out an area of  $45\pi$  square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.


A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integers where gcd
Of m,n is 1.
Take this answer and
Add the numerator and
Denominator.

*Proposed by Jeff Lin*


A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term?


How many integers  $n$  (with  $1 \le n \le 2021$ ) have the property that  $8n + 1$  is a perfect square?


A permutation  $\sigma$  of the numbers  $1,2,\ldots , 10$  is called  $\textit{bad}$  if there exist integers  $i, j, k$  which satisfy
\[1 \leq i < j < k \leq 10 \quad \text{ and }\quad \sigma(j) < \sigma(k) < \sigma(i)\]
and  $\textit{good}$  otherwise. Find the number of  $\textit{good}$  permutations.


For which positive integers  $n$  is there an  $n\times n$  matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

Solution: \boxed{ n }


The figure below depicts two congruent triangles with angle measures  $40^\circ$ ,  $50^\circ$ , and  $90^\circ$ .  What is the measure of the obtuse angle  $\alpha$  formed by the hypotenuses of these two triangles?

[asy]
import olympiad;
	size(80);
defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10));
pair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0));
draw(anglemark((4.25,0),P,(0,4.25),10));
label(" $\alpha$ ",P,2 * NE);
[/asy]


Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system: $\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}
x+y+z+w= 48
\end{cases}$

Solution: \boxed{x = y = z = w = 12}


In a triangle  $ABC$ , the median  $AD$  divides  $\angle{BAC}$  in the ratio  $1:2$ . Extend  $AD$  to  $E$  such that  $EB$  is perpendicular  $AB$ . Given that  $BE=3,BA=4$ , find the integer nearest to  $BC^2$ .


On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?


 $2020$  positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?

A. Gribalko


Suppose  $n$  is an integer  $\geq 2$ . Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\]

Solution: \boxed{ 6 }


We say that some positive integer  $m$  covers the number  $1998$ , if  $1,9,9,8$  appear in this order as digits of  $m$ . (For instance  $1998$  is covered by  $2\textbf{1}59\textbf{9}36\textbf{98}$  but not by  $213326798$ .) Let  $k(n)$  be the number of positive integers that cover  $1998$  and have exactly  $n$  digits ( $n\ge 5$ ), all different from  $0$ . What is the remainder of  $k(n)$  on division by  $8$ ?


Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:

- No two numbers are equal.
- If a number  $m$  is placed on square  $C$ , then at least  $k$  of the squares orthogonally adjacent to  $C$  have a multiple of  $m$  written on them.

What is the greatest value of  $k$  for which this is possible?

Solution: \boxed{ k = 2 }


Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs  $3$  RM ,each mango costs  $4$  RM , and each papaya costs  $5$  RM . He bought at least one of each fruit, and paid exactly  $50$  RM. What is the maximum number of fruits that he could have bought?


For which  $n\ge 2$  is it possible to find  $n$  pairwise non-similar triangles  $A_1, A_2,\ldots , A_n$  such that each of them can be divided into  $n$  pairwise non-similar triangles, each of them similar to one of  $A_1,A_2 ,\ldots ,A_n$ ?

Solution: \boxed{ n \geq 2 }


Let  $n$  be a positive integer. For a permutation  $a_1, a_2, \dots, a_n$  of the numbers  $1, 2, \dots, n$  we define  $$ b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j $$  We say that the permutation  $a_1, a_2, \dots, a_n$  is *guadiana* if the sequence  $b_1, b_2, \dots, b_n$  does not contain two consecutive equal terms. How many guadiana permutations exist?


Let  $S$  be the set of  $2\times2$ -matrices over  $\mathbb{F}_{p}$  with trace  $1$  and determinant  $0$ . Determine  $|S|$ .

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


A circle  $\omega$  has center  $O$  and radius  $r$ . A chord  $BC$  of  $\omega$  also has length  $r$ , and the tangents to  $\omega$  at  $B$  and  $C$  meet at  $A$ . Ray  $AO$  meets  $\omega$  at  $D$  past  $O$ , and ray  $OA$  meets the circle centered at  $A$  with radius  $AB$  at  $E$  past  $A$ . Compute the degree measure of  $\angle DBE$ .

*Author: Ray Li*


Find the largest  $k$  such that for every positive integer  $n$  we can find at least  $k$  numbers in the set  $\{n+1, n+2, ... , n+16\}$  which are coprime with  $n(n+17)$ .

Solution: \boxed{ k = 1 }


Let  $BCDK$  be a convex quadrilateral such that  $BC=BK$  and  $DC=DK$ .  $A$  and  $E$  are points such that  $ABCDE$  is a convex pentagon such that  $AB=BC$  and  $DE=DC$  and  $K$  lies in the interior of the pentagon  $ABCDE$ . If  $\angle ABC=120^{\circ}$  and  $\angle CDE=60^{\circ}$  and  $BD=2$  then determine area of the pentagon  $ABCDE$ .


Point  $A$  lies on the circumference of a circle  $\Omega$  with radius  $78$ .  Point  $B$  is placed such that  $AB$  is tangent to the circle and  $AB=65$ , while point  $C$  is located on  $\Omega$  such that  $BC=25$ .  Compute the length of  $\overline{AC}$ .


Let  $D$  be the foot of the altitude from  $A$  in a triangle  $ABC$ . The angle bisector at  $C$  intersects  $AB$  at a point  $E$ . Given that  $\angle CEA=\frac\pi4$ , compute  $\angle EDB$ .

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


All the numbers  $1,2,...,9$  are written in the cells of a  $3\times 3$  table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary  $2\times2$  square of the table and either decrease by  $1$  or increase by  $1$  all four numbers of the square. After some number of such moves all numbers of the table become equal to some number  $a$ . Find all possible values of  $a$ .

I.Voronovich


Let  $ABC$  be a triangle with incenter  $I$  and incircle  $\omega$ .  It is given that there exist points  $X$  and  $Y$  on the circumference of  $\omega$  such that  $\angle BXC=\angle BYC=90^\circ$ .  Suppose further that  $X$ ,  $I$ , and  $Y$  are collinear.  If  $AB=80$  and  $AC=97$ , compute the length of  $BC$ .


Consider  $7$ -gons inscribed in a circle such that all sides of the  $7$ -gon are of different length. Determine the maximal number of  $120^\circ$  angles in this kind of a  $7$ -gon.


Solution: \boxed{ a - 1 }


Fix integers  $n\ge k\ge 2$ . We call a collection of integral valued coins  $n-diverse$  if no value occurs in it more than  $n$  times. Given such a collection, a number  $S$  is  $n-reachable$  if that collection contains  $n$  coins whose sum of values equals  $S$ . Find the least positive integer  $D$  such that for any  $n$ -diverse collection of  $D$  coins there are at least  $k$  numbers that are  $n$ -reachable.

*Proposed by Alexandar Ivanov, Bulgaria.*


Find all positive integers  $n$  for which there exist real numbers  $x_1, x_2,. . . , x_n$  satisfying all of the following conditions:
(i)  $-1 <x_i <1,$  for all  $1\leq  i \leq  n.$ (ii)  $ x_1 + x_2 + ... + x_n = 0.$ (iii)  $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$

Solution: \boxed{ n = 1 }


Let  $p$  be a prime number. All natural numbers from  $1$  to  $p$  are written in a row in ascending order. Find all  $p$  such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.

*A. Khrabov*

Solution: \boxed{ p = 3 }


Let  $ABCD$  be a convex quadrilateral with  $\angle A=60^o$ . Let  $E$  and  $Z$  be the symmetric points of  $A$  wrt  $BC$  and  $CD$  respectively. If the points  $B,D,E$  and  $Z$  are collinear, then calculate the angle  $\angle BCD$ .


Let  $p>13$  be a prime of the form  $2q+1$ , where  $q$  is prime. Find the number of ordered pairs of integers  $(m,n)$  such that  $0\le m<n<p-1$  and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]

*Alex Zhu.*

NoteThe original version asked for the number of solutions to  $2^m+3^n\equiv 2^n+3^n\pmod{p}$  (still  $0\le m<n<p-1$ ), where  $p$  is a Fermat prime.

Solution: \boxed{ q-1 }


Let  $c > 1$  be a real number. A function  $f: [0 ,1 ] \to R$  is called c-friendly if  $f(0) = 0, f(1) = 1$  and  $|f(x) -f(y)| \le c|x - y|$  for all the numbers  $x ,y \in [0,1]$ . Find the maximum of the expression  $|f(x) - f(y)|$  for all *c-friendly* functions   $f$  and for all the numbers  $x,y \in [0,1]$ .

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


Let  $k$  and  $n$  be positive integers. Consider an array of  $2\left(2^n-1\right)$  rows by  $k$  columns. A  $2$ -coloring of the elements of the array is said to be *acceptable* if any two columns agree on less than  $2^n-1$  entries on the same row.

Given  $n$ , determine the maximum value of  $k$  for an acceptable  $2$ -coloring to exist.

Solution: \boxed{ k = 2^n }


Let  $x$ - minimal root of equation  $x^2-4x+2=0$ .
Find two first digits of number  $ \{x+x^2+....+x^{20} \}$  after  $0$ , where  $\{a\}$ - fractional part of  $a$ .


How many unordered triples  $A,B,C$  of distinct lattice points in  $0\leq x,y\leq4$  have the property that  $2[ABC]$  is an integer divisible by  $5$ ?

*2020 CCA Math Bonanza Tiebreaker Round #3*


A *short-sighted* rook is a rook that beats all squares in the same column and in the same row for which he can not go more than  $60$ -steps.
What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a  $100\times 100$  chessboard.


Let  $I$  be the center of the incircle of non-isosceles triangle  $ABC,A_{1}=AI\cap BC$  and  $B_{1}=BI\cap AC.$  Let  $l_{a}$  be the line through  $A_{1}$  which is parallel to  $AC$  and  $l_{b}$  be the line through  $B_{1}$  parallel to  $BC.$  Let  $l_{a}\cap CI=A_{2}$  and  $l_{b}\cap CI=B_{2}.$  Also  $N=AA_{2}\cap BB_{2}$  and  $M$  is the midpoint of  $AB.$  If  $CN\parallel IM$  find  $\frac{CN}{IM}$ .


**p22.** Find the unique ordered pair  $(m, n)$  of positive integers such that  $x = \sqrt[3]{m} -\sqrt[3]{n}$  satisfies  $x^6 + 4x^3 - 36x^2 + 4 = 0$ .**p23.** Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with  $1$  pip is face up. How many ways are there to roll the dice for  $6$  steps and end with the  $1$  face up again?**p24.** There exists a unique positive five-digit integer with all odd digits that is divisible by  $5^5$ . Find this integer.

PS. You should use hide for answers.


Let  $S=\{1,2,3,\cdots,100\}$ . Find the maximum value of integer  $k$ , such that there exist  $k$  different nonempty subsets of  $S$  satisfying the condition: for any two of the  $k$  subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Solution: \boxed{ 2^{99} - 1 }


Given is a parallelogram  $ABCD$ , with  $AB <AC <BC$ . Points  $E$  and  $F$  are selected on the circumcircle   $\omega$  of  $ABC$  so that the tangenst to  $\omega$  at these points pass through point  $D$  and the segments  $AD$  and  $CE$  intersect.
It turned out that  $\angle ABF = \angle DCE$ . Find the angle  $\angle{ABC}$ .
A. Yakubov, S. Berlov


A positive integer  $n$  is called *Olympic*, if there exists a quadratic trinomial with integer coeffecients  $f(x)$  satisfying  $f(f(\sqrt{n}))=0$ . Determine, with proof, the largest Olympic number not exceeding  $2015$ .

*A. Khrabrov*


[4] Let  $x_1,x_2,\ldots,x_{100}$  be defined so that for each  $i$ ,  $x_i$  is a (uniformly) random integer between  $1$  and  $6$  inclusive. Find the expected number of integers in the set  $\{x_1,x_1+x_2,\ldots,x_1+x_2+\cdots+x_{100}\}$  that are multiples of  $6$ .


Let  $\Omega$  be a circle with radius  $18$  and let  $\mathcal{S}$  be the region inside  $\Omega$  that the centroid of  $\triangle XYZ$  sweeps through as  $X$  varies along all possible points lying outside of  $\Omega$ ,  $Y$  varies along all possible points lying on  $\Omega$  and  $XZ$  is tangent to the circle. Compute the greatest integer less than or equal to the area of  $\mathcal{S}$ .

*Proposed by **FedeX333X***


Consider rectangle  $ABCD$  with  $AB = 6$  and  $BC = 8$ .  Pick points  $E, F, G, H$  such that the angles  $\angle AEB, \angle BFC, \angle CGD, \angle AHD$  are all right.  What is the largest possible area of quadrilateral  $EFGH$ ?

*Proposed by Akshar Yeccherla (TopNotchMath)*


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Movie-Genre, Nationality, Transport.
The attributes have the following possible values:
- Job: doctor, mechanic, musician, security-guard
- Movie-Genre: fantasy, martial-arts, spy, zombie
- Nationality: american, brazilian, dutch, spanish
- Transport: car, scooter, snowmobile, train

Given the following premises about the line of people:
- Nationality:spanish is somewhere to the right of Movie-Genre:zombie
- Transport:snowmobile is on the right of Nationality:brazilian
- Nationality:dutch is somewhere to the right of Nationality:spanish
- Job:doctor is on the left or right of Nationality:spanish
- Job:security-guard is on the far left
- Movie-Genre:martial-arts is in an even position
- Transport:car == Job:mechanic
- Nationality:spanish and Job:security-guard have different parity positions
- Movie-Genre:fantasy == Transport:car
- Transport:train is on the left or right of Nationality:american

Answer the following question:

Question: Question: What is the Nationality of the person who has/is spy??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{dutch}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Food, Nationality, Pet.
The attributes have the following possible values:
- Job: electrician, firefighter, project-manager, scientist
- Food: banana, potato, pumpkin, radish
- Nationality: nigerian, pakistani, russian, spanish
- Pet: bird, cat, hamster, mouse

Given the following premises about the line of people:
- Job:project-manager is on the far left or far right
- Job:electrician is on the far right
- Food:radish is somewhere between Pet:mouse and Nationality:pakistani
- Pet:cat is somewhere to the right of Job:firefighter
- Job:electrician == Food:pumpkin
- Food:banana is somewhere to the right of Food:potato
- Nationality:russian is on the right of Nationality:nigerian
- Nationality:nigerian is somewhere to the left of Food:radish
- Nationality:pakistani == Pet:hamster
- Nationality:pakistani is on the far left

Answer the following question:

Question: Question: What is the Job of the person who has/is radish??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{scientist}


There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Music-Genre, Hobby, Transport.
The attributes have the following possible values:
- Nationality: australian, brazilian, indian, italian, russian
- Music-Genre: ambient, hip-hop, house, rock, soul
- Hobby: baking, board-games, fishing, gardening, hiking
- Transport: bus, car, ship, skateboard, snowmobile

Given the following premises about the line of people:
- Transport:snowmobile is on the right of Nationality:italian
- Hobby:baking is on the left or right of Transport:snowmobile
- Hobby:gardening is somewhere to the right of Hobby:board-games
- Transport:ship is on the right of Nationality:australian
- Hobby:fishing == Transport:ship
- Music-Genre:soul is on the left or right of Music-Genre:house
- Nationality:brazilian is somewhere to the right of Nationality:indian
- Music-Genre:soul is somewhere to the left of Music-Genre:ambient
- Nationality:australian == Transport:car
- Transport:bus is on the right of Music-Genre:hip-hop
- Hobby:fishing is on the left or right of Nationality:indian
- Music-Genre:house == Hobby:hiking
- Nationality:italian == Transport:skateboard

Answer the following question:

Question: Question: What is the Nationality of the person who has/is house??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{indian}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Food, Music-Genre, Pet.
The attributes have the following possible values:
- Job: accountant, chef, entrepreneur, security-guard
- Food: grapefruit, lime, mango, raspberry
- Music-Genre: ambient, electronic, gospel, trance
- Pet: cat, guinea-pig, lizard, mouse

Given the following premises about the line of people:
- Pet:lizard is somewhere to the right of Job:chef
- Pet:cat is not to the left of Food:raspberry
- Music-Genre:electronic is on the left of Food:grapefruit
- Music-Genre:gospel is between Pet:cat and Food:mango
- Job:accountant is not to the left of Music-Genre:gospel
- Pet:mouse is on the left of Music-Genre:ambient
- Music-Genre:ambient is on the left of Music-Genre:gospel
- Food:grapefruit is not to the right of Job:security-guard
- Job:chef is somewhere between Food:grapefruit and Food:mango

Answer the following question:

Question: Question: What is the Music-Genre of the person who has/is lime??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{gospel}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Food, Hobby, Music-Genre.
The attributes have the following possible values:
- Beverage: 7up, cola, milk, water
- Food: cauliflower, cranberry, eggplant, pumpkin
- Hobby: board-games, filmmaking, rock-climbing, video-games
- Music-Genre: classical, indie, r&b, trance

Given the following premises about the line of people:
- Hobby:board-games is somewhere to the left of Music-Genre:indie
- Music-Genre:r&b is in an even position
- Beverage:milk is on the far left
- Music-Genre:trance is somewhere to the right of Food:cranberry
- Beverage:7up is on the left of Hobby:video-games
- Beverage:7up == Food:cauliflower
- Hobby:video-games is on the left or right of Beverage:cola
- Food:eggplant != Hobby:rock-climbing
- Music-Genre:trance is on the left or right of Hobby:filmmaking
- Music-Genre:r&b is on the far left or far right
- Food:eggplant is in an even position

Answer the following question:

Question: Question: What is the Food of the person who has/is video-games??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{pumpkin}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Beverage, Music-Genre, Pet.
The attributes have the following possible values:
- Job: analyst, coach, doctor, police-officer
- Beverage: 7up, almond-milk, cola, water
- Music-Genre: classical, funk, r&b, salsa
- Pet: chinchilla, goat, hedgehog, horse

Given the following premises about the line of people:
- Music-Genre:r&b is somewhere to the left of Beverage:cola
- Beverage:7up is in an odd position
- Pet:chinchilla and Job:police-officer have the same parity positions
- Beverage:almond-milk is in an odd position
- Music-Genre:classical is somewhere to the right of Music-Genre:funk
- Pet:goat is on the far left or far right
- Beverage:7up is on the left or right of Pet:hedgehog
- Beverage:cola is somewhere to the left of Beverage:7up
- Music-Genre:classical is somewhere to the left of Job:doctor
- Job:police-officer is on the left or right of Job:doctor
- Pet:hedgehog is not to the left of Music-Genre:classical
- Job:analyst is on the left or right of Pet:horse

Answer the following question:

Question: Question: What is the Music-Genre of the person who is police-officer??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{classical}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Food, Hobby, Movie-Genre, Music-Genre, Pet.
The attributes have the following possible values:
- Food: apple, corn, cranberry, raspberry
- Hobby: camping, dancing, photography, reading
- Movie-Genre: animation, disaster, scientific, spy
- Music-Genre: dubstep, pop, soul, techno
- Pet: lizard, mouse, pony, turtle

Given the following premises about the line of people:
- Hobby:dancing == Music-Genre:dubstep
- Movie-Genre:disaster is between Hobby:photography and Pet:lizard
- Food:corn is on the left of Movie-Genre:scientific
- Food:cranberry is somewhere to the right of Movie-Genre:disaster
- Movie-Genre:spy == Food:apple
- Pet:pony is on the right of Pet:turtle
- Music-Genre:pop is somewhere to the left of Pet:pony
- Pet:turtle is between Hobby:photography and Hobby:camping
- Music-Genre:soul is on the left or right of Hobby:camping
- Movie-Genre:animation == Music-Genre:soul

Answer the following question:

Question: Question: What is the Hobby of the person who has/is disaster??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{dancing}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Beverage, Nationality.
The attributes have the following possible values:
- Job: accountant, chef, dancer, freelancer
- Hobby: baking, hiking, rock-climbing, woodworking
- Beverage: iced-tea, lemonade, mirinda, soy-milk
- Nationality: dutch, nigerian, russian, spanish

Given the following premises about the line of people:
- Job:chef is somewhere to the right of Beverage:soy-milk
- Nationality:nigerian is somewhere between Job:accountant and Hobby:baking
- Nationality:russian is somewhere to the left of Nationality:dutch
- Hobby:woodworking is on the right of Hobby:baking
- Job:dancer is on the far left or far right
- Beverage:iced-tea is on the far left or far right
- Beverage:lemonade == Nationality:russian
- Hobby:hiking != Nationality:nigerian
- Beverage:iced-tea is in an odd position
- Beverage:soy-milk is on the right of Job:accountant

Answer the following question:

Question: Question: What is the Hobby of the person who is accountant??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{hiking}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Food, Pet.
The attributes have the following possible values:
- Job: designer, firefighter, manager, teacher
- Hobby: camping, cooking, drawing, video-games
- Food: garlic, kale, papaya, radish
- Pet: dog, ferret, hamster, horse

Given the following premises about the line of people:
- Food:garlic is somewhere to the left of Job:manager
- Job:teacher == Hobby:video-games
- Hobby:camping is somewhere to the right of Food:radish
- Pet:hamster is somewhere to the left of Pet:dog
- Pet:dog is somewhere to the left of Hobby:camping
- Job:manager is on the left of Hobby:camping
- Pet:ferret is between Job:firefighter and Job:manager
- Job:firefighter is somewhere to the left of Hobby:camping
- Hobby:drawing is on the right of Food:kale

Answer the following question:

Question: Question: What is the Job of the person who has/is video-games??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{teacher}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Music-Genre, Food, Pet.
The attributes have the following possible values:
- Nationality: australian, egyptian, japanese, pakistani
- Music-Genre: country, metal, reggae, rock
- Food: grapefruit, kale, pomegranate, watermelon
- Pet: dog, horse, mouse, rabbit

Given the following premises about the line of people:
- Pet:horse is on the left or right of Nationality:egyptian
- Music-Genre:metal is on the left or right of Music-Genre:reggae
- Food:grapefruit is not to the right of Nationality:australian
- Pet:horse is on the right of Pet:dog
- Music-Genre:country and Pet:rabbit have the same parity positions
- Food:kale is on the right of Food:watermelon
- Music-Genre:metal is on the far right
- Nationality:pakistani is somewhere to the right of Nationality:australian
- Food:watermelon and Pet:mouse have the same parity positions
- Music-Genre:rock is somewhere to the right of Music-Genre:country
- Food:pomegranate is not to the right of Nationality:japanese
- Pet:dog is on the left or right of Pet:rabbit

Answer the following question:

Question: Question: What is the Food of the person who is pakistani??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{kale}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Beverage, Movie-Genre, Pet.
The attributes have the following possible values:
- Job: bartender, dancer, manager, security-guard
- Hobby: baking, video-games, woodworking, writing
- Beverage: cola, iced-tea, milk, soy-milk
- Movie-Genre: family, fantasy, spy, superhero
- Pet: chinchilla, ferret, goldfish, rat

Given the following premises about the line of people:
- Beverage:soy-milk is somewhere to the left of Beverage:milk
- Job:bartender is on the far right
- Movie-Genre:fantasy is on the right of Movie-Genre:spy
- Beverage:cola is on the right of Pet:ferret
- Pet:chinchilla is somewhere to the left of Hobby:writing
- Movie-Genre:spy is in an even position
- Job:manager is in an even position
- Hobby:writing is somewhere to the right of Beverage:cola
- Movie-Genre:spy is somewhere to the right of Job:security-guard
- Pet:rat is between Hobby:video-games and Beverage:soy-milk
- Hobby:woodworking is on the left of Movie-Genre:superhero

Answer the following question:

Question: Question: What is the Movie-Genre of the person who has/is video-games??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{family}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Nationality, Hobby, Food, Pet.
The attributes have the following possible values:
- Nationality: argentine, indian, malaysian, thai
- Hobby: camping, collecting, reading, singing
- Food: apple, avocado, eggplant, pear
- Pet: dog, ferret, fish, guinea-pig

Given the following premises about the line of people:
- Food:pear is somewhere to the left of Food:eggplant
- Hobby:camping == Nationality:thai or Hobby:camping == Food:pear or both
- Hobby:singing is somewhere to the left of Nationality:malaysian
- Hobby:collecting and Pet:fish have the same parity positions
- Pet:fish is somewhere to the right of Pet:ferret
- Food:apple is somewhere to the left of Hobby:singing
- Nationality:argentine and Pet:dog have different parity positions
- Pet:dog is not to the left of Pet:guinea-pig
- Pet:ferret is not to the left of Nationality:malaysian
- Nationality:indian is somewhere to the left of Hobby:collecting

Answer the following question:

Question: Question: What is the Food of the person who is thai??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{eggplant}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Nationality, Transport.
The attributes have the following possible values:
- Job: firefighter, journalist, project-manager, teacher
- Hobby: baking, camping, collecting, sudoku
- Nationality: american, british, malaysian, pakistani
- Transport: boat, helicopter, roller, van

Given the following premises about the line of people:
- Nationality:malaysian is on the far right
- Job:journalist is somewhere to the right of Job:teacher
- Job:teacher is on the left or right of Nationality:american
- Transport:helicopter is somewhere to the left of Nationality:british
- Job:teacher is not to the left of Transport:roller
- Hobby:baking and Transport:van have the same parity positions
- Job:project-manager is somewhere to the left of Transport:boat
- Nationality:pakistani is not to the left of Hobby:camping
- Job:teacher is somewhere to the right of Hobby:sudoku
- Nationality:british is somewhere to the right of Transport:boat

Answer the following question:

Question: Question: What is the Job of the person who has/is camping??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{project-manager}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Beverage, Movie-Genre, Nationality.
The attributes have the following possible values:
- Job: freelancer, musician, police-officer, scientist
- Beverage: fanta, hot-chocolate, mirinda, tea
- Movie-Genre: comedy, family, fantasy, superhero
- Nationality: australian, french, russian, spanish

Given the following premises about the line of people:
- Beverage:tea == Job:musician
- Job:police-officer == Movie-Genre:fantasy
- Beverage:mirinda is between Movie-Genre:family and Nationality:australian
- Beverage:mirinda is on the left or right of Nationality:russian
- Beverage:tea is somewhere to the left of Beverage:hot-chocolate
- Movie-Genre:superhero is on the left or right of Job:police-officer
- Nationality:french is on the left of Beverage:tea
- Job:police-officer is on the far right
- Job:scientist is on the right of Job:freelancer

Answer the following question:

Question: Question: What is the Beverage of the person who is musician??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{tea}


There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Beverage, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Beverage: 7up, juice, sprite, water
- Music-Genre: jazz, metal, pop, r&b
- Pet: cat, ferret, mouse, turtle
- Sport: ice-hockey, sailing, skiing, snowboarding

Given the following premises about the line of people:
- Beverage:sprite is somewhere to the right of Pet:mouse
- Pet:mouse is somewhere between Beverage:water and Music-Genre:jazz
- Pet:ferret is on the left or right of Music-Genre:r&b
- Pet:cat is on the left or right of Sport:sailing
- Sport:snowboarding is on the left of Music-Genre:jazz
- Sport:snowboarding is somewhere to the left of Beverage:juice
- Beverage:sprite is not to the right of Music-Genre:pop
- Beverage:7up and Sport:skiing have the same parity positions
- Beverage:sprite is on the left of Sport:sailing

Answer the following question:

Question: Question: What is the Music-Genre of the person who has/is water??
 Please provide your answer in the format: \boxed{X}, where X is the answer.

Solution: \boxed{metal}