Let  $ n$  be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of  $ (n \plus{} 1)^{3} \minus{} 1$  points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains  $ S$  but does not include  $ (0,0,0)$ .

*Author: Gerhard Wöginger, Netherlands*

Solution: \boxed{ 3n }


Given that for reals  $a_1,\cdots, a_{2004},$  equation  $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$  has  $2006$  positive real solution, find the maximum possible value of  $a_1.$


For each real number  $x$ , let  $f(x)$  be the minimum of the numbers  $4x+1$ ,  $x+2$ , and  $-2x+4$ . Then the maximum value of  $f(x)$  is
 $\text{(A)} \ \frac 13 \qquad \text{(B)} \ \frac 12 \qquad \text{(C)} \ \frac 23 \qquad \text{(D)} \ \frac 52 \qquad \text{(E)} \ \frac 83$


Let  $ABC$  be a triangle.  The incircle  $\omega$  of  $\triangle ABC$ , which has radius  $3$ , is tangent to  $\overline{BC}$  at  $D$ .  Suppose the length of the altitude from  $A$  to  $\overline{BC}$  is  $15$  and  $BD^2 + CD^2 = 33$ .  What is  $BC$ ?


Given a pair  $(a_0, b_0)$  of real numbers, we define two sequences  $a_0, a_1, a_2,...$  and  $b_0, b_1, b_2, ...$  of real numbers by  $a_{n+1}= a_n + b_n$  and  $b_{n+1}=a_nb_n$  for all  $n = 0, 1, 2,...$ . Find all pairs  $(a_0, b_0)$  of real numbers such that  $a_{2022}= a_0$  and  $b_{2022}= b_0$ .

Solution: \boxed{ (a, 0) }


A soccer coach named  $C$  does a header drill with two players  $A$  and  $B$ , but they all forgot to put sunscreen on their foreheads.  They solve this issue by dunking the ball into a vat of sunscreen before starting the drill.  Coach  $C$  heads the ball to  $A$ , who heads the ball back to  $C$ , who then heads the ball to  $B$ , who heads the ball back to  $C$ ;
this pattern  $CACBCACB\ldots\,$  repeats ad infinitum.  Each time a person heads the ball,  $1/10$  of the sunscreen left on the ball ends up on the person's forehead.  In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?


Square  $ABCD$  has side length  $4$ . Points  $E$  and  $F$  are the midpoints of sides  $AB$  and  $CD$ , respectively. Eight  $1$  by  $2$  rectangles are placed inside the square so that no two of the eight rectangles overlap (see diagram). If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers  $m$  and  $n$  so that  $\tfrac{m}{n}$  is the probability that none of the rectangles crosses the line segment  $EF$  (as in the arrangement on the right). Find  $m + n$ .

[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(10pt));
real r = 7;
path square=origin--(4,0)--(4,4)--(0,4)--cycle;
draw(square^^shift((r,0))*square,linewidth(1));
draw((1,4)--(1,0)^^(3,4)--(3,0)^^(0,2)--(1,2)^^(1,3)--(3,3)^^(1,1)--(3,1)^^(2,3)--(2,1)^^(3,2)--(4,2));
draw(shift((r,0))*((2,4)--(2,0)^^(0,2)--(4,2)^^(0,1)--(4,1)^^(0,3)--(2,3)^^(3,4)--(3,2)));
label("A",(4,4),NE);
label("A",(4+r,4),NE);
label("B",(0,4),NW);
label("B",(r,4),NW);
label("C",(0,0),SW);
label("C",(r,0),SW);
label("D",(4,0),SE);
label("D",(4+r,0),SE);
label("E",(2,4),N);
label("E",(2+r,4),N);
label("F",(2,0),S);
label("F",(2+r,0),S);
[/asy]


In triangle  $\triangle ABC$ , the angle  $\angle BAC$  is less than  $90^o$ . The perpendiculars from  $C$  on  $AB$  and from  $B$  on  $AC$  intersect the circumcircle of  $\triangle ABC$  again at  $D$  and  $E$  respectively. If  $|DE| =|BC|$ , find the measure of the angle  $\angle BAC$ .


Let  $E(n)$  denote the largest integer  $k$  such that  $5^k$  divides  $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$  Calculate  $$ \lim_{n\to \infty} \frac{E(n)}{n^2 }. $$


A convex  $n$ -gon  $P$ , where  $n > 3$ , is dissected into equal triangles by diagonals non-intersecting inside it. Which values of  $n$  are possible, if  $P$  is circumscribed?

Solution: \boxed{ n = 4 }


There are  $169$  lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly  $19$  switches at once. (Every time you use this remote control, you can choose which  $19$  switches are to be changed.)

(a) Given that at the beginning some lamps are on, can you turn all the lamps off, using the remote control?
(b) Given that at the beginning all lamps are on, how many times do you need to use the remote control to turn all lamps off?


We have a five-digit positive integer  $N$ . We select every pair of digits of  $N$  (and keep them in order) to obtain the  $\tbinom52 = 10$  numbers  $33$ ,  $37$ ,  $37$ ,  $37$ ,  $38$ ,  $73$ ,  $77$ ,  $78$ ,  $83$ ,  $87$ .  Find  $N$ .

*Proposed by Lewis Chen*


Let  $A$  and  $B$  be two sets of non-negative integers, define  $A+B$  as the set of the values obtained when we sum any (one) element of the set  $A$  with any (one) element of the set  $B$ . For instance, if  $A=\{2,3\}$  and  $B=\{0,1,2,5\}$  so  $A+B=\{2,3,4,5,7,8\}$ .
Determine the least integer  $k$  such that there is a pair of sets  $A$  and  $B$  of non-negative integers with  $k$  and  $2k$  elements, respectively, and $A+B=\{0,1,2,\dots, 2019,2020\}$


Let  $x$  and  $y$  be distinct real numbers such that
\[ \sqrt{x^2+1}+\sqrt{y^2+1}=2021x+2021y. \]
Find, with proof, the value of
\[ \left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right). \]


For each positive integer  $n$ , consider the highest common factor  $h_n$  of the two numbers  $n!+1$  and  $(n+1)!$ . For  $n<100$ , find the largest value of  $h_n$ .


How many coins can be placed on a  $10 \times 10$  board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?


What is the sum of real numbers satisfying the equation  $\left \lfloor \frac{6x+5}{8}  \right \rfloor = \frac{15x-7}{5}$ ? $
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ \frac{81}{90}
\qquad\textbf{(C)}\ \frac{7}{15}
\qquad\textbf{(D)}\ \frac{4}{5}
\qquad\textbf{(E)}\ \frac{19}{15}
$


Find the least positive integer  $m$  such that  $lcm(15,m) = lcm(42,m)$ . Here  $lcm(a, b)$  is the least common multiple of  $a$  and  $b$ .


The set of positive real numbers  $x$  that satisfy  $2 | x^2 - 9 | \le 9 | x | $  is an interval  $[m, M]$ . Find  $10m + M$ .

Solution: \boxed{ 21 }


Compute the  $100^{\text{th}}$  smallest positive integer  $n$  that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4},  \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4},  \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] Here  $\lfloor \cdot \rfloor$  denotes the greatest integer function.

*Proposed by Michael Tang*


Let  $ABCD$  be a square such that  $A=(0,0)$  and  $B=(1,1)$ .  $P(\frac{2}{7},\frac{1}{4})$  is a point inside the square. An ant starts walking from  $P$ , touches  $3$  sides of the square and comes back to the point  $P$ . The least possible distance traveled by the ant can be expressed as  $\frac{\sqrt{a}}{b}$ , where  $a$  and  $b$  are integers and  $a$  not divisible by any square number other than  $1$ . What is the value of  $(a+b)$ ?


One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.


Alex launches his boat into a river and heads upstream at a constant speed. At the same time at a point  $8$  miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at  $6$  miles per hour in still water, but the river is  owing downstream at  $2\frac{3}{10}$  miles per hour. Alex and Alice will meet at a point that is  $\frac{m}{n}$  miles from Alex's starting point, where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


Let  $ABCD$  be a square and  $X$  a point such that  $A$  and  $X$  are on opposite sides of  $CD$ . The lines  $AX$  and  $BX$  intersect  $CD$  in  $Y$  and  $Z$  respectively. If the area of  $ABCD$  is  $1$  and the area of  $XYZ$  is  $\frac{2}{3}$ , determine the length of  $YZ$


Let  $r,s,t$  be the roots of the equation  $x(x-2)(3x-7)=2$ . Show that  $r,s,t$  are real and positive and determine  $\arctan r+\arctan s +\arctan t$ .


A positive integer  $n$  is called*bad*if it cannot be expressed as the product of two distinct positive integers greater than  $1$ . Find the number of bad positive integers less than  $100. $ *Proposed by Michael Ren*


Alex has an  $20 \times 16$  grid of lightbulbs, initially all off. He has  $36$  switches, one for each row and column. Flipping the switch for the  $i$ th row will toggle the state of each lightbulb in the  $i$ th row (so that if it were on before, it would be off, and  vice versa). Similarly, the switch for the  $j$ th column will toggle the state of each bulb in the  $j$ th column. Alex makes some (possibly empty) sequence of switch flips, resulting in some configuration of the lightbulbs and their states. How many distinct possible configurations of lightbulbs can Alex achieve with such a sequence? Two configurations are distinct if there exists a lightbulb that is on in one configuration and off in another.


99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?


In  $ \vartriangle ABC$ ,  $\angle B = 46^o$  and  $\angle C = 48^o$  . A circle is inscribed in   $ \vartriangle ABC$  and the points of tangency are connected to form  $PQR$ . What is the measure of the largest angle in  $\vartriangle P QR$ ?


The roots of the polynomial  $P(x) = x^3 + 5x + 4$  are  $r$ ,  $s$ , and  $t$ . Evaluate  $(r+s)^4 (s+t)^4 (t+r)^4$ .

*Proposed by Eugene Chen*


Determine all pairs  $(a, b)$  of integers having the following property:
there is an integer  $d \ge  2$  such that  $a^n + b^n + 1$  is divisible by  $d$  for all positive integers  $n$ .

Solution: \boxed{ (a, b) }


Answer the following questions.

(1) For  $x\geq 0$ , show that  $x-\frac{x^3}{6}\leq \sin x\leq x.$ (2) For  $x\geq 0$ , show that  $\frac{x^3}{3}-\frac{x^5}{30}\leq \int_0^x t\sin t\ dt\leq \frac{x^3}{3}.$ (3) Find the limit

\[\lim_{x\rightarrow 0} \frac{\sin x-x\cos x}{x^3}.\]

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


In  $ xy$  plane, find the minimum volume of the solid by rotating the region boubded by the parabola  $ y \equal{} x^2 \plus{} ax \plus{} b$  passing through the point  $ (1,\ \minus{} 1)$  and the  $ x$  axis about the  $ x$  axis


Two circles  $ G_1$  and  $ G_2$  intersect at two points  $ M$  and  $ N$ . Let  $ AB$  be the line tangent to these circles at  $ A$  and  $ B$ , respectively, so that  $ M$  lies closer to  $ AB$  than  $ N$ . Let  $ CD$  be the line parallel to  $ AB$  and passing through the point  $ M$ , with  $ C$  on  $ G_1$  and  $ D$  on  $ G_2$ . Lines  $ AC$  and  $ BD$  meet at  $ E$ ; lines  $ AN$  and  $ CD$  meet at  $ P$ ; lines  $ BN$  and  $ CD$  meet at  $ Q$ . Show that  $ EP \equal{} EQ$ .

Solution: \boxed{ EP = EQ }


A *quadratic* number is a real root of the equations  $ax^2 + bx + c = 0$  where  $|a|,|b|,|c|\in\{1,2,\ldots,10\}$ . Find the smallest positive integer  $n$  for which at least one of the intervals  $$ \left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right) $$  does not contain any quadratic number.


Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim.


How many positive integers  $a$  with  $a\le 154$  are there such that the coefficient of  $x^a$  in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?

*Author: Ray Li*


A classroom has  $30$  seats arranged into  $5$  rows of  $6$  seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?


Eddy draws  $6$  cards from a standard  $52$ -card deck. What is the probability that four of the cards that he draws have the same value?


Let  $\mathcal{F}$  be the set of all functions  $f : (0,\infty)\to (0,\infty)$  such that  $f(3x) \geq f( f(2x) )+x$  for all  $x$ . Find the largest  $A$  such that  $f(x) \geq A x$  for all  $f\in\mathcal{F}$  and all  $x$ .


Suppose  $ A $  is a subset of   $ n $ -elements taken from  $ 1,2,3,4,...,2009 $  such that the difference of any two numbers in  $ A $  is not a prime number. Find the largest value of   $ n $  and the set   $ A $  with this number of elements.

Solution: \boxed{ n = 503 }


The quadrilateral  $ABCD$  with perpendicular diagonals is inscribed in the circle with center  $O$ , the points  $M,N$  are the midpoints of  $[BC]$  and  $[CD]$  respectively. Find the ratio of areas of the figures  $OMCN$  and  $ABCD$


For each positive integer  $m$  and  $n$  define function  $f(m, n)$  by  $f(1, 1) = 1$ ,  $f(m+ 1, n) = f(m, n) +m$  and  $f(m, n + 1) = f(m, n) - n$ . Find the sum of all the values of  $p$  such that  $f(p, q) = 2004$  for some  $q$ .


[b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are  $189, 320, 287, 264, x$ , and y. Find the greatest possible value of:  $x + y$ .

NOTE: There is a high chance that this problems was copied.


For some positive integers  $m>n$ , the quantities  $a=\text{lcm}(m,n)$  and  $b=\gcd(m,n)$  satisfy  $a=30b$ .  If  $m-n$  divides  $a$ , then what is the value of  $\frac{m+n}{b}$ ?

*Proposed by Andrew Wu*


An ant lies on each corner of a  $20 \times 23$  rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After  $10$  seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.


For  $n$  people, if it is known that

(a) there exist two people knowing each other among any three people, and

(b) there exist two people not knowing each other among any four people.

Find the maximum of  $n$ .

Here, we assume that if  $A$  knows  $B$ , then  $B$  knows  $A$ .


Find the sum of   $$ \frac{\sigma(n) \cdot d(n)}{ \phi (n)} $$   over all positive  $n$  that divide  $ 60$ .

Note: The function  $d(i)$  outputs the number of divisors of  $i$ ,   $\sigma (i)$  outputs the sum of the factors of  $i$ , and  $\phi  (i)$  outputs the number of positive integers less than or equal to  $i$  that are relatively prime to  $i$ .


Two players,  $B$  and  $R$ , play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with  $B$ . On  $B$ 's turn,  $B$  selects one white unit square and colors it blue. On  $R$ 's turn,  $R$  selects two white unit squares and colors them red. The players alternate until  $B$  decides to end the game. At this point,  $B$  gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score  $B$  can guarantee?

(A *simple polygon* is a polygon (not necessarily convex) that does not intersect itself and has no holes.)

*Proposed by David Torres*


Find all functions  $f:\mathbb{N} \to \mathbb{N}$  so that for any distinct positive integers  $x,y,z$  the value of  $x+y+z$  is a perfect square if and only if  $f(x)+f(y)+f(z)$  is a perfect square.

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


Let  $f : \mathbb R \to \mathbb R$  be a continuous function. Suppose that the restriction of  $f$  to the set of irrational numbers is injective. What can we say about  $f$ ? Answer the analogous question if  $f$  is restricted to rationals.

Solution: \boxed{ f }


Let  $d$  and  $n$  be positive integers such that  $d$  divides  $n$ ,  $n > 1000$ , and  $n$  is not a perfect square.  The minimum possible value of  $\left\lvert d - \sqrt{n} \right\rvert$  can be written in the form  $a\sqrt{b} + c$ , where  $b$  is a positive integer not divisible by the square of any prime, and  $a$  and  $c$  are nonzero integers (not necessarily positive).  Compute  $a+b+c$ .

*Proposed by Matthew Lerner-Brecher*


Let  $ABC$  be an equilateral triangle with side length  $1$ . This triangle is rotated by some angle about its center to form triangle  $DEF.$  The intersection of  $ABC$  and  $DEF$  is an equilateral hexagon with an area that is  $\frac{4} {5}$  the area of  $ABC.$  The side length of this hexagon can be expressed in the form  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

*Author: Ray Li*


Twelve congruent disks are placed on a circle  $C$  of radius 1 in such a way that the twelve disks cover  $C$ , no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below.  The sum of the areas of the twelve disks can be written in the from  $\pi(a-b\sqrt{c})$ , where  $a,b,c$  are positive integers and  $c$  is not divisible by the square of any prime. Find  $a+b+c$ .

[asy]
real r=2-sqrt(3);
draw(Circle(origin, 1));
int i;
for(i=0; i<12; i=i+1) {
draw(Circle(dir(30*i), r));
dot(dir(30*i));
}
draw(origin--(1,0)--dir(30)--cycle);
label("1", (0.5,0), S);[/asy]


Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?


A number  $x_n$  of the form 10101...1 has  $n$  ones. Find all  $n$  such that  $x_n$  is prime.

Solution: \boxed{ n = 2 }


Patricia has a rectangular painting that she wishes to frame. The frame must also be rectangular and will extend  $3\text{ cm}$  outward from each of the four sides of the painting. When the painting is framed, the area of the frame not covered by the painting is  $108\text{ cm}^2$ . What is the perimeter of the painting alone (without the frame)?

Solution: \boxed{24 \text{ cm}}


Find the least possible value for the fraction  $$ \frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)} $$  over all distinct positive integers  $a, b, c$ .
By  $lcm(x, y)$  we mean the least common multiple of  $x, y$  and by  $gcd(x, y)$  we mean the greatest common divisor of  $x, y$ .


At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?


Let  $\vartriangle ABC$  be a triangle such that the area $ [ABC] = 10$  and  $\tan (\angle ABC) = 5$ . If the smallest possible value of  $(\overline{AC})^2$  can be expressed as  $-a + b\sqrt{c}$  for positive integers  $a, b, c$ , what is  $a + b + c$ ?


Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.


Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100  $\text{kg/m}^3$ , let  $f(r)$  denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find  $\lim_{r\to0} \left(f(r)\right) $ . Express your answer is meters and round to the nearest integer. Assume the density of water is 1000  $\text{kg/m}^3$ .

*(B. Dejean, 6 points)*


The sum of two integers is  $8$ . The sum of the squares of those two integers is  $34$ . What is the product of the two
integers?


Let  $f: \mathbb{R}^2 \to \mathbb{R}$  be a function satisfying the following property: If  $A, B, C \in \mathbb{R}^2$  are
 the vertices of an equilateral triangle with sides of length  $1$ , then
   $$ f(A) + f(B) + f(C)  = 0. $$   Show that  $f(x) = 0$  for all  $x \in \mathbb{R}^2$ .

*Proposed by Ilir Snopce*

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


In a triangle, the ratio of the interior angles is  $1 : 5 : 6$ , and the longest
side has length  $12$ . What is the length of the altitude (height) of the triangle that
is perpendicular to the longest side?


A dart board looks like three concentric circles with radii of 4, 6, and 8.  Three darts are thrown at the board so that they stick at three random locations on then board.  The probability that one dart sticks in each of the three regions of the dart board is  $\dfrac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers.   Find  $m + n$ .


Numbers  $1, 2, \dots , n$  are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?

Solution: \boxed{ 2n - 2 }


$ f(x)$  is a given polynomial whose degree at least 2. Define the following polynomial-sequence:  $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all  $ n \in N$ . Let  $ r_n$  be the average of  $ g_n(x)$ 's roots. If  $ r_{19}\equal{}99$ , find  $ r_{99}$ .


Let  $a$ ,  $b$  and  $c$  be complex numbers such that  $abc = 1$ . Find the value of the cubic root of

\begin{tabular}{|ccc|} $b + n^3c$  &   $n(c - b)$  &  $n^2(b - c)$  $n^2(c - a)$  &  $c + n^3a$  &  $n(a - c)$  $n(b - a)$   &  $n^2(a - b)$  &  $a + n^3b$ \end{tabular}

Solution: \boxed{n^3 + 1}


Find the minimum value of  $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$ .

Please solve the problem without using partial differentiation for those who don't learn it.

1961 Waseda University entrance exam/Science and Technology


Let  $n$  be an integer greater than 1. If all digits of  $97n$  are odd, find the smallest possible value of  $n$ .


Solve in positive integers the following equation   $$ \left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125, $$   where  $[a]$  is the integer part of the real number  $a$ .


How many  $3$ -inch-by- $5$ -inch photos will it take to completely cover the surface of a  $3$ -foot-by- $5$ -foot poster? $\text{(A) }24\qquad\text{(B) }114\qquad\text{(C) }160\qquad\text{(D) }172\qquad\text{(E) }225$


The triangle with side lengths  $3, 5$ , and  $k$  has area  $6$  for two distinct values of  $k$ :  $x$  and  $y$ . Compute  $|x^2 -y^2|$ .


What are the last  $2$  digits of the number  $2018^{2018}$  when written in base  $7$ ?


In the convex quadrilateral  $ABCD$  angle  $\angle{BAD}=90$ , $\angle{BAC}=2\cdot\angle{BDC}$  and  $\angle{DBA}+\angle{DCB}=180$ . Then find the angle  $\angle{DBA}$


Let  $e > 0$  be a given real number. Find the least value of  $f(e)$  (in terms of  $e$  only) such that the inequality  $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$  holds for all real numbers  $a, b, c, d$ .


A uniform ladder of mass  $m$  and length  $\mathcal{L}$  is resting on a wall. A man of mass  $m$  climbs up the ladder and is in perfect equilibrium with the ladder when he is  $\frac{2}{3}\mathcal{L}$  the way up the ladder. The ladder makes an angle of  $ \theta = 30^\circ $  with the horizontal floor. If the coefficient of static friction between the ladder and the wall is the same as that between the ladder and the floor, which is  $\mu$ , what is  $\mu$ , expressed to the nearest thousandth?

*(Proposed by Ahaan Rungta)*

Solution: \boxed{ \mu \approx 0.678 }


A straight line connects City A at  $(0, 0)$  to City B, 300 meters away at  $(300, 0)$ . At time  $t=0$ , a bullet train instantaneously sets out from City A to City B while another bullet train simultaneously leaves from City B to City A going on the same train track. Both trains are traveling at a constant speed of  $50$  meters/second.

Also, at  $t=0$ , a super
y stationed at  $(150, 0)$  and restricted to move only on the train tracks travels towards City B. The
y always travels at 60 meters/second, and any time it hits a train, it instantaneously reverses its direction and travels at the same speed. At the moment the
trains collide, what is the total distance that the y will have traveled? Assume each train is a point
and that the trains travel at their same respective velocities before and after collisions with the y


A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer buys both bread and milk is .30. What is the probability that the customer would buy either bread or milk or both?


Find the largest  $r$  such that  $4$  balls each of radius  $r$  can be packed into a regular tetrahedron with side length  $1$ . In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If  $r$  can be expressed in the form  $\frac{\sqrt{a}+b}{c}$  where  $a, b, c$  are integers such that  $\gcd(b, c) = 1$ , what is  $a + b + c$ ?

Solution: \boxed{ 15 }


For a sequence  $x_1,x_2,\ldots,x_n$  of real numbers, we define its  $\textit{price}$  as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given  $n$  real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price  $D$ . Greedy George, on the other hand, chooses  $x_1$  such that  $|x_1 |$  is as small as possible; among the remaining numbers, he chooses  $x_2$  such that  $|x_1 + x_2 |$  is as small as possible, and so on. Thus, in the  $i$ -th step he chooses  $x_i$  among the remaining numbers so as to minimise the value of  $|x_1 + x_2 + \cdots  x_i |$ . In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price  $G$ .

Find the least possible constant  $c$  such that for every positive integer  $n$ , for every collection of  $n$  real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality  $G\le cD$ .

*Proposed by Georgia*

Solution: \boxed{ c = 2 }


There are  $n{}$  wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed.

The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all  $2^n$  layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

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


Call a pair of integers  $a$  and  $b$  square makers , if  $ab+1$  is a perfect square.
Determine for which  $n$  is it possible to divide the set  $\{1,2, \dots , 2n\}$  into  $n$  pairs of square makers.

Solution: \boxed{ n }


Let  $N$  be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to  $N$ , independently and uniformly at random. Let  $p_N$  denote the probability that the product of these two integers has a units digit of  $0$ . The maximum possible value of  $p_N$  over all possible choices of  $N$  can be written as  $\tfrac ab,$  where  $a$  and  $b$  are relatively prime positive integers. Compute  $100a + b$ .


In a school, there are  $n$  students and some of them are friends each other. (Friendship is mutual.) Define  $ a, b $  the minimum value which satisfies the following conditions:
(1) We can divide students into  $ a $  teams such that two students in the same team are always friends.
(2) We can divide students into  $ b $  teams such that two students in the same team are never friends.
Find the maximum value of  $ N = a+b $  in terms of  $n$ .

Solution: \boxed{ n + 1 }


Consider the set  $\mathcal{T}$  of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let  $\triangle \in \mathcal{T}$  be the triangle with least perimeter. If  $a^{\circ}$  is the largest angle of  $\triangle$  and  $L$  is its perimeter, determine the value of  $\frac{a}{L}$ .


Determine all positive integers  $n$  with the property that   $n = (d(n))^2$ . Here  $d(n)$  denotes the number of positive divisors of  $n$ .

Solution: \boxed{1 \text{ and } 9}


Sam Wang decides to evaluate an expression of the form  $x +2 \cdot 2+ y$ . However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find  $x + y$ .

*Proposed by Edwin Zhao*

Solution*Solution.*  $\boxed{4}$ We have  $x+2*2+y=x \cdot 2+2 \cdot y$ . When simplifying, we have  $x+y+4=2x+2y$ , and  $x+y=4$ .


A *festive* number is a four-digit integer containing one of each of the digits  $0, 1, 2$ , and  $4$  in its decimal representation. How many festive numbers are there?


Let  $f: \mathbb{R} \rightarrow \mathbb{R}$  be a function such that   $$ \displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,} $$   for all  $x \in  \mathbb{R}.$  Find all possible values of  $f(2).$


The adjoining figure shows two intersecting chords in a circle, with  $B$  on minor arc  $AD$ .  Suppose that the radius of the circle is 5, that  $BC = 6$ , and that  $AD$  is bisected by  $BC$ .  Suppose further that  $AD$  is the only chord starting at  $A$  which is bisected by  $BC$ .  It follows that the sine of the minor arc  $AB$  is a rational number.  If this fraction is expressed as a fraction  $m/n$  in lowest terms, what is the product  $mn$ ?
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1));
draw(Circle(origin, 1)^^A--D^^B--C);
real r=0.05;
pair M1=midpoint(M--D), M2=midpoint(M--A);
draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D)));
draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D)));
pair point=origin;
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $D$ ", D, dir(point--D));[/asy]


In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a  $12 \times 12$  board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.


Ben is throwing darts at a circular target with diameter 10.  Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target.  Ben gets  $\lceil 5-x \rceil$  points for having the dart land  $x$  units away from the center of the target.  What is the expected number of points that Ben can earn from throwing a single dart?  (Note that  $\lceil y \rceil$  denotes the smallest integer greater than or equal to  $y$ .)


Find the number of ordered triples  $(a, b)$  of positive integers with  $a < b$  and  $100 \leq a, b \leq 1000$  satisfy  $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$ ?


A quadrilateral  $ABCD$  is inscribed in a circle. Suppose that  $|DA| =|BC|= 2$  and $ |AB|  = 4$ . Let  $E $  be the intersection  point of lines  $BC$  and  $DA$ . Suppose that  $\angle AEB = 60^o$  and that  $|CD| <|AB|$ . Calculate the radius of the circle.


Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics  $B$  test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of the questions correctly, what is the average possible value of your friend’s score? Recall that each question is worth the point value shown at the beginning of each question.


A set  $ X$  of positive integers is called *nice* if for each pair  $ a$ ,  $ b\in X$  exactly one of the numbers  $ a \plus{} b$  and  $ |a \minus{} b|$  belongs to  $ X$  (the numbers  $ a$  and  $ b$  may be equal). Determine the number of nice sets containing the number 2008.

*Author: Fedor Petrov*


Determine the maximum integer  $ n $  such that for each positive integer  $ k \le \frac{n}{2} $  there are two positive divisors of  $ n $  with difference  $ k $ .


In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most  $100$  flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest  $d$  for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding  $d$ ?


A Mathlon is a competition in which there are  $M$  athletic events.  Such a competition was held in which only  $A$ ,  $B$ , and  $C$  participated.  In each event  $p_1$  points were awarded for first place,  $p_2$  for second and  $p_3$  for third, where  $p_1 > p_2 > p_3 > 0$  and  $p_1$ ,  $p_2$ ,  $p_3$  are integers.  The final scores for  $A$  was 22, for  $B$  was 9 and for  $C$  was also 9.   $B$  won the 100 metres.  What is the value of  $M$  and who was second in the high jump?

Solution: \boxed{ M = 5 }


Cyclic quadrilateral  $ABCD$  has  $AC\perp BD$ ,  $AB+CD=12$ , and  $BC+AD=13$ . FInd the greatest possible area of  $ABCD$ .


What is the third digit after the decimal point of the decimal representation of  $\sqrt[3]{2+\sqrt 5} + \sqrt[3]{2-\sqrt 5}$ ? $
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of the above}
$


$162$  pluses and  $144$  minuses are placed in a  $30\times 30$  table in such a way that each row and each column contains at most  $17$  signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.


The pages of a book are numbered 1 through  $n$ . When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice?


Find the largest positive integer  $n$  such that  $\sigma(n) = 28$ , where  $\sigma(n)$  is the sum of the divisors of  $n$ , including  $n$ .


Find the remainder when \[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\] is divided by  $ 1000$ .


A cube has eight vertices (corners) and twelve edges.  A segment, such as  $x$ , which joins two vertices not joined by an edge is called a diagonal.  Segment  $y$  is also a diagonal.  How many diagonals does a cube have?

[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label(" $x$ ",(2.75,3.5),NNE);
label(" $y$ ",(4.125,1.5),NNE);
[/asy] $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$


Let  $A_o =\{1, 2\}$  and for  $n> 0, A_n$  results from  $A_{n-1}$  by adding the natural numbers to  $A_{n-1}$  which can be represented as the sum of two different numbers from  $A_{n-1}$ . Let   $a_n = |A_n |$  be the number of numbers in  $A_n$ . Determine  $a_n$  as a function of   $n$ .

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


Let   $T  =  TNFTPP$ .  As   $n$   ranges  over  the  integers,  the  expression   $n^4  -  898n^2  +  T  -  2160$   evaluates  to just  one  prime  number.  Find  this  prime.

[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$T=2161$.


Some of the  $n + 1$  cities in a country (including the capital city) are connected by one-way or two-way airlines. No two cities are connected by both a one-way airline and a two-way airline, but there may be more than one two-way airline between two cities. If  $d_A$  denotes the number of airlines from a city  $A$ , then  $d_A \le n$  for any city  $A$  other than the capital city and  $d_A + d_B \le n$  for any two cities  $A$  and  $B$  other than the capital city which are not connected by a two-way airline. Every airline has a return, possibly consisting of several connected flights. Find the largest possible number of two-way airlines and all configurations of airlines for which this largest number is attained.

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


Let  $A_1B_1C_1$ ,  $A_2B_2C_2$ , and  $A_3B_3C_3$  be three triangles in the plane. For  $1 \le i \le3$ , let  $D_i $ ,  $E_i$ , and  $F_i$  be the midpoints of  $B_iC_i$ ,  $A_iC_i$ , and  $A_iB_i$ , respectively. Furthermore, for  $1 \le i \le 3$  let  $G_i$  be the centroid of  $A_iB_iC_i$ .
Suppose that the areas of the triangles  $A_1A_2A_3$ ,  $B_1B_2B_3$ ,  $C_1C_2C_3$ ,  $D_1D_2D_3$ ,  $E_1E_2E_3$ , and  $F_1F_2F_3$  are  $2$ ,  $3$ ,  $4$ ,  $20$ ,  $21$ , and  $2020$ , respectively. Compute the largest possible area of  $G_1G_2G_3$ .


Let the rest energy of a particle be  $E$ . Let the work done to increase the speed of this particle from rest to  $v$  be  $W$ . If  $ W = \frac {13}{40} E $ , then  $ v = kc $ , where  $ k $  is a constant. Find  $10000k$  and round to the nearest whole number.

*(Proposed by Ahaan Rungta)*


**Q12.** Find all positive integers  $P$  such that the sum and product of all its divisors are  $2P$  and  $P^2$ , respectively.


There are a family of  $5$  siblings. They have a pile of at least  $2$  candies and are trying to split them up
amongst themselves. If the  $2$  oldest siblings share the candy equally, they will have  $1$  piece of candy left over.
If the  $3$  oldest siblings share the candy equally, they will also have  $1$  piece of candy left over. If all  $5$  siblings
share the candy equally, they will also have  $1$  piece left over. What is the minimum amount of candy required
for this to be true?


Find the greatest positive integer  $N$  with the following property: there exist integers  $x_1, . . . , x_N$  such that  $x^2_i - x_ix_j$  is not divisible by  $1111$  for any  $i\ne j.$

Solution: \boxed{ 1000 }


Given a triangle with acute angle  $ \angle BEC,$  let  $ E$  be the midpoint of  $ AB.$  Point  $ M$  is chosen on the opposite ray of  $ EC$  such that  $ \angle BME \equal{} \angle ECA.$  Denote by  $ \theta$  the measure of angle  $ \angle BEC.$  Evaluate  $ \frac{MC}{AB}$  in terms of  $ \theta.$

Solution: \boxed{\cos \theta}


How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.


Let  $ABCD$  be a square, and let  $E$  be a point external to  $ABCD$  such that  $AE = CE = 9$  and  $BE = 8$ . Compute the side length of  $ABCD$ .

Solution: \boxed{7 - 4\sqrt{2}}


A positive integer  $N$  is given. Panda builds a tree on  $N$  vertices, and writes a real number on each vertex, so that  $1$  plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be  $M$  and the minimal number written  $m$ . Mink then gives Panda  $M-m$  kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

Solution: \boxed{ (N-1)^2 }


Let  $\phi(n)$  denote the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ . Over all integers  $1\le n \le 100$ , find the maximum value of  $\phi(n^2+2n)-\phi(n^2)$ .

*Proposed by Vincent Huang*


Let  $f(x)=x^2+x$  for all real  $x$ . There exist positive integers  $m$  and  $n$ , and distinct nonzero real numbers  $y$  and  $z$ , such that  $f(y)=f(z)=m+\sqrt{n}$  and  $f(\frac{1}{y})+f(\frac{1}{z})=\frac{1}{10}$ . Compute  $100m+n$ .

*Proposed by Luke Robitaille*


The trinomial  $f(x)$  is such that  $(f(x))^3-f(x)=0$  has three real roots. Find the y-coordinate of the vertex of  $f(x)$ .


What is the largest  $n$  such that there exists a non-degenerate convex  $n$ -gon such that each of its angles are an integer number of degrees, and are all distinct?


Let  $ f(x)$  be a function such that  $ 1\minus{}f(x)\equal{}f(1\minus{}x)$  for  $ 0\leq x\leq 1$ .

Evaluate  $ \int_0^1 f(x)\ dx$ .


Let  $ABC$  be a triangle and let  $P$  be a point not lying on any of the three lines  $AB$ ,  $BC$ , or  $CA$ . Distinct points  $D$ ,  $E$ , and  $F$  lie on lines  $BC$ ,  $AC$ , and  $AB$ , respectively, such that  $\overline{DE}\parallel \overline{CP}$  and  $\overline{DF}\parallel \overline{BP}$ . Show that there exists a point  $Q$  on the circumcircle of  $\triangle AEF$  such that  $\triangle BAQ$  is similar to  $\triangle PAC$ .

*Andrew Gu*

Solution: \boxed{Q}


Let  $a_1,a_2,a_3,a_4,a_5$  be distinct real numbers. Consider all sums of the form  $a_i + a_j$  where  $i,j \in \{1,2,3,4,5\}$  and  $i \neq j$ . Let  $m$  be the number of distinct numbers among these sums. What is the smallest possible value of  $m$ ?


Determine the maximum value of  $m$ , such that the inequality

\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]

holds for every  $a,b,c \in \mathbb{R} \setminus \{0\}$  with  $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$ .
When does equality occur?


Define a sequence  $\{a_n\}_{n \geq 1}$  recursively by  $a_1=1$ ,  $a_2=2$ , and for all integers  $n \geq 2$ ,  $a_{n+1}=(n+1)^{a_n}$ . Determine the number of integers  $k$  between  $2$  and  $2020$ , inclusive, such that  $k+1$  divides  $a_k - 1$ .

*Proposed by Taiki Aiba*


A  $4\times4\times4$  cube is composed of  $64$  unit cubes. The faces of  $16$  unit cubes are to be coloured red. A colouring is called interesting if there is exactly  $1$  red unit cube in every  $1\times1\times 4$  rectangular box composed of  $4$  unit cubes. Determine the number of interesting colourings.


Find the smallest real number  $p$  such that the inequality  $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$  holds for all natural numbers  $n$ .

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


The lengths of the two legs of a right triangle are the two distinct roots of the quadratic  $x^2 - 36x + 70$ . What is the length of the triangle’s hypotenuse?


Suppose  $X, Y, Z$  are collinear points in that order such that  $XY = 1$  and  $YZ = 3$ . Let  $W$  be a point such that  $YW = 5$ , and define  $O_1$  and  $O_2$  as the circumcenters of triangles  $\triangle WXY$  and  $\triangle WYZ$ , respectively. What is the minimum possible length of segment  $\overline{O_1O_2}$ ?


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


There are five people, and the age differences (absolute value) for each twosome are all different. Find the smallest possible difference in age between the oldest and the youngest.


Triangle  $ABC$  has positive integer side lengths with  $AB=AC$ . Let  $I$  be the intersection of the bisectors of  $\angle B$  and  $\angle C$ . Suppose  $BI=8$ . Find the smallest possible perimeter of  $\triangle ABC$ .


Let  $ n$  be a natural number. A cube of edge  $ n$  may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for  $ n$ .

Solution: \boxed{ n = 13 }


Let  $ABC$  be a scalene triangle with inradius  $1$  and exradii  $r_A$ ,  $r_B$ , and  $r_C$  such that \[20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2.\] If \[\tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,\] then the area of  $\triangle{ABC}$  can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

*Proposed by Tristan Shin*

Solution: \boxed{ 100m + n = 201925 }


Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes.  When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital  $\textit{every}$  month, and having half the women work while only  $2$  of the  $7$  men work on some months means that some of the women risk getting burned out.  After all, nearly all the members of the volunteer group have other jobs.

Hannah comes up with a plan that the committee likes.  Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month.  Under this new plan, what is the least number of months that  $\textit{must}$  go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee  $\textit{must have}$  taught together twice (all five members are the same during two different months)?


Consider the multiplicative group  $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$  of all the roots of unity of degree  $2006^k$  for all positive integers  $k$ .

Find the number of homomorphisms  $f:A\to A$  that satisfy  $f(f(x))=f(x)$  for all elements  $x\in A$ .


Let  $A_1$ ,  $A_2$ ,  $A_3$  be three points in the plane, and for convenience, let  $A_4= A_1$ ,  $A_5 = A_2$ .  For  $n = 1$ ,  $2$ , and  $3$ , suppose that  $B_n$  is the midpoint of  $A_n A_{n+1}$ , and suppose that  $C_n$  is the midpoint of  $A_n B_n$ . Suppose that  $A_n C_{n+1}$  and  $B_n A_{n+2}$  meet at  $D_n$ , and that  $A_n B_{n+1}$  and  $C_n A_{n+2}$  meet at  $E_n$ .
Calculate the ratio of the area of triangle  $D_1 D_2 D_3$  to the area of triangle  $E_1 E_2 E_3$ .


Points  $A, B$  and  $C$  lie on the same line so that  $CA = AB$ . Square  $ABDE$  and the equilateral triangle  $CFA$ , are constructed  on the same side of line  $CB$ . Find the acute angle between straight lines  $CE$  and  $BF$ .


Assuming that the roots of  $x^3 + p \cdot x^2 + q \cdot x + r = 0$  are real and positive, find a relation between  $p,q$  and  $r$  which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.

Solution: \boxed{ p^2 = 2q + 2r + 1 }


A natural number  $k > 1$  is called *good* if there exist natural numbers  $$ a_1 < a_2 < \cdots < a_k $$  such that  $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots +  \dfrac{1}{\sqrt{a_k}} = 1 $$  .

Let  $f(n)$  be the sum of the first  $n$  *[good* numbers,  $n \geq$  1. Find the sum of all values of  $n$  for which  $f(n+5)/f(n)$  is an integer.


Consider (3-variable) polynomials
\[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\]
and
\[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\]
Determine all positive integers  $n$  such that the quotient  $Q_n(x,y,z)/P_n(x,y,z)$  is a (3-variable) polynomial with rational coefficients.

Solution: \boxed{n = 1}


An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at  $(0,0)$ , what is the probability that it will be at either  $(2,1)$  or  $(1,2)$  after  $6$  moves?

*2020 CCA Math Bonanza Individual Round #1*


Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.

(Two cells of the grid are adjacent if they have a common vertex)


Let  $A$  be a given set with  $n$  elements. Let  $k<n$  be a given positive integer. Find the maximum value of  $m$  for which it is possible to choose sets  $B_i$  and  $C_i$  for  $i=1,2,\ldots,m$  satisfying the following conditions:
[list=1]
[*] $B_i\subset A,$   $|B_i|=k,$ [*] $C_i\subset B_i$  (there is no additional condition for the number of elements in  $C_i$ ), and
[*] $B_i\cap C_j\neq B_j\cap C_i$  for all  $i\neq j.$ [/list]

Solution: \boxed{ 2^k }


A coin is flipped  $20$  times. Let  $p$  be the probability that each of the following sequences of flips occur exactly twice:

- one head, two tails, one head
- one head, one tails, two heads.

    Given that  $p$  can be expressed as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $\gcd (m,n)$ .

*2021 CCA Math Bonanza Lightning Round #1.3*


Find a positive integer  $n$  such that there is a polygon with  $n$  sides where each of its interior angles measures  $177^o$


The figure below shows two parallel lines,  $ \ell$  and  $ m$ , that are distance  $ 12$  apart:
[asy]unitsize(7);
  draw((-7, 0) -- (12, 0));
  draw((-7, 12) -- (12, 12));
  real r = 169 / 48;
  draw(circle((0, r), r));
  draw(circle((5, 12 - r), r));
  pair A = (0, 0);
  pair B = (5, 12);
  dot(A);
  dot(B);
  label(" $A$ ", A, plain.S);
  label(" $B$ ", B, plain.N);
  label(" $\ell$ ", (12, 0), plain.E);
  label(" $m$ ", (12, 12), plain.E);[/asy]
A circle is tangent to line  $ \ell$  at point  $ A$ .  Another circle is tangent to line  $ m$  at point  $ B$ .  The two circles are congruent and tangent to each other as shown.  The distance between  $ A$  and  $ B$  is  $ 13$ .  What is the radius of each circle?


How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?


Let  $M$  and  $m$  be the largest and the smallest values of  $x$ , respectively, which satisfy  $4x(x - 5) \le 375$ . Find  $M - m$ .


Note that there are exactly three ways to write the integer  $4$  as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let  $f(n)$  be the number of ways to write a natural number  $n$  as a sum of positive odd integers where the order of the summands matters.  Find the remainder when  $f(2008)$  is divided by  $100$ .


We say that a number  $n \ge 2$  has the property  $(P)$  if, in its prime factorization, at least one of the factors has an exponent   $3$ .

a) Determine the smallest number  $N$  with the property that, no matter how we choose  $N$  consecutive natural numbers, at least one of them has the property  $(P).$ b) Determine the smallest  $15$  consecutive numbers  $a_1, a_2, \ldots, a_{15}$  that do not have the property  $(P),$  such that the sum of the numbers  $5 a_1, 5 a_2, \ldots, 5 a_{15}$  is a number with the property  $(P).$

Solution: \boxed{ 16 }


Given a regular 2007-gon. Find the minimal number  $k$  such that: Among every  $k$  vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.


Find the sum of the two smallest possible values of  $x^\circ$  (in degrees) that satisfy the following equation if  $x$  is greater than  $2017^\circ$ :   $$ \cos^59x+\cos^5x=32\cos^55x\cos^54x+5\cos^29x\cos^2x\left(\cos9x+\cos x\right). $$  *2017 CCA Math Bonanza Individual Round #10*


There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?


Find all positive integers  $n$  such that for any integer  $k$  there exists an integer  $a$  for which  $a^3+a-k$  is divisible by  $n$ .

*Warut Suksompong, Thailand*

Solution: \boxed{n = 3^k}


Find the smallest integer  $k > 1$  for which  $n^k-n$  is a multiple of  $2010$  for every integer positive  $n$ .


In how many ways can the integers from  $1$  to  $n$  be ordered subject to the condition that, except for the first integer on the left, every integer differs by  $1$  from some integer to the left of it?


Point  $A$ ,  $B$ ,  $C$ , and  $D$  form a rectangle in that order. Point  $X$  lies on  $CD$ , and segments  $\overline{BX}$  and  $\overline{AC}$  intersect at  $P$ . If the area of triangle  $BCP$  is 3 and the area of triangle  $PXC$  is 2, what is the area of the entire rectangle?


Find the number of rationals  $\frac{m}{n}$  such that

(i)  $0 < \frac{m}{n} < 1$ ;

(ii)  $m$  and  $n$  are relatively prime;

(iii)  $mn = 25!$ .


A positive integer  $n>1$  is called beautiful if  $n$  can be written in one and only one way as  $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$  for some positive integers  $a_1, a_2, \ldots, a_k$ , where  $k>1$  and  $a_1 \geq a_2 \geq \cdots \geq a_k$ . (For example 6 is beautiful since  $6=3 \cdot 2 \cdot 1=3+2+1$ , and this is unique. But 8 is not beautiful since  $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$  as well as  $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$ , so uniqueness is lost.) Find the largest beautiful number less than 100.


Let  $ABC$  be an isosceles right-angled triangle. Point  $D$  is chosen on the prolongation of the hypothenuse  $AB$  beyond point  $A$  so that  $AB = 2AD$ . Points  $M$  and  $N$  on side  $AC$  satisfy the relation  $AM = NC$ . Point  $K$  is chosen on the prolongation of  $CB$  beyond point  $B$  so that  $CN = BK$ . Determine the angle between lines  $NK$  and  $DM$ .

(M.Kungozhin)


How many positive integers less than  $1998$  are relatively prime to  $1547$ ? (Two integers are relatively prime if they have no common factors besides 1.)


How many ways are there to choose distinct positive integers  $a, b, c, d$  dividing  $15^6$  such that none of  $a, b, c,$  or  $d$  divide each other? (Order does not matter.)

*Proposed by Miles Yamner and Andrew Wu*

(Note: wording changed from original to clarify)


Call a positive integer  $k$   $\textit{pretty}$  if for every positive integer  $a$ , there exists an integer  $n$  such that  $n^2+n+k$  is divisible by  $2^a$  but not  $2^{a+1}$ . Find the remainder when the  $2021$ st pretty number is divided by  $1000$ .

*Proposed by i3435*


The rhombus  $ABCD$  is given. Let  $E$  be one of the points of intersection of the circles  $\Gamma_B$  and  $\Gamma_C$ , where  $\Gamma_B$  is the circle centered at  $B$  and passing through  $C$ , and  $\Gamma_C$  is the circle centered at  $C$  and passing through  $B$ . The line  $ED$  intersects  $\Gamma_B$  at point  $F$ .
Find the value of angle  $\angle AFB$ .

*(S. Mazanik)*


Consider a rectangle  $ABCD$  with  $BC = 2 \cdot AB$ . Let  $\omega$  be the circle that touches the sides  $AB$ ,  $BC$ , and  $AD$ . A tangent drawn from point  $C$  to the circle  $\omega$  intersects the segment  $AD$  at point  $K$ . Determine the ratio  $\frac{AK}{KD}$ .

*Proposed by Giorgi Arabidze, Georgia*

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


A line with negative slope passing through the point  $(18,8)$  intersects the  $x$  and  $y$  axes at  $(a,0)$  and  $(0,b)$ , respectively.  What is the smallest possible value of  $a+b$ ?


Sean is a biologist, and is looking at a strng of length  $66$  composed of the letters  $A$ ,  $T$ ,  $C$ ,  $G$ . A *substring* of a string is a contiguous sequence of letters in the string. For example, the string  $AGTC$  has  $10$  substrings:  $A$ ,  $G$ ,  $T$ ,  $C$ ,  $AG$ ,  $GT$ ,  $TC$ ,  $AGT$ ,  $GTC$ ,  $AGTC$ . What is the maximum number of distinct substrings of the string Sean is looking at?


Let  $p=2^{16}+1$  be a prime. A sequence of  $2^{16}$  positive integers  $\{a_n\}$  is *monotonically bounded* if  $1\leq a_i\leq i$  for all  $1\leq i\leq 2^{16}$ . We say that a term  $a_k$  in the sequence with  $2\leq k\leq 2^{16}-1$  is a *mountain* if  $a_k$  is greater than both  $a_{k-1}$  and  $a_{k+1}$ . Evan writes out all possible monotonically bounded sequences. Let  $N$  be the total number of mountain terms over all such sequences he writes. Find the remainder when  $N$  is divided by  $p$ .

*Proposed by Michael Ren*


Let  $ABCD$  be a parallelogram with area 160. Let diagonals  $AC$  and  $BD$  intersect at  $E$ . Point  $P$  is on  $\overline{AE}$  such that  $EC = 4EP$ . If line  $DP$  intersects  $AB$  at  $F$ , find the area of  $BFPC$ .

*Proposed by Andy Xu*


A finite sequence of positive integers  $ m_i$  for  $ i\equal{}1,2,...,2006$  are defined so that  $ m_1\equal{}1$  and  $ m_i\equal{}10m_{i\minus{}1} \plus{}1$  for  $ i>1$ . How many of these integers are divisible by  $ 37$ ?


Find all sets  $(a, b, c)$  of different positive integers  $a$ ,  $b$ ,  $c$ , for which: *****  $2a - 1$  is a multiple of  $b$ ;  *****  $2b - 1$  is a multiple of  $c$ ; *****  $2c - 1$  is a multiple of  $a$ .

Solution: \boxed{ (13, 25, 7) }


Find the maximal constant  $ M$ , such that for arbitrary integer  $ n\geq 3,$  there exist two sequences of positive real number  $ a_{1},a_{2},\cdots,a_{n},$  and  $ b_{1},b_{2},\cdots,b_{n},$  satisfying
(1): $ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2): $ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$ .

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


Find the number ot 6-tuples  $(x_1, x_2,...,x_6)$ , where  $x_i=0,1 or 2$  and  $x_1+x_2+...+x_6$  is even


Below the standard representation of a positive integer  $n$  is the representation understood by  $n$  in the decimal system, where the first digit is different from  $0$ . Everyone positive integer n is now assigned a number  $f(n)$  by using the standard representation of  $n$  last digit is placed before the first.
Examples:  $f(1992) = 2199$ ,   $f(2000) = 200$ .
Determine the smallest positive integer  $n$  for which  $f(n) = 2n$  holds.


The sequence  $\{c_{n}\}$  is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let  $\lambda$  be the limit value  $\lim_{n\to\infty}c_{n}.$   Find  $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$


$n$  is a positive integer. Let  $a(n)$  be the smallest number for which  $n\mid a(n)!$ Find all solutions of:  $$ \frac{a(n)}{n}=\frac{2}{3} $$

Solution: \boxed{n = 9}


Find the smallest natural number  $n$  which has the following properties:

a) Its decimal representation has a 6 as the last digit.

b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number  $n$ .


A nine-digit number has the form  $\overline{6ABCDEFG3}$ , where every three consecutive digits sum to  $13$ . Find  $D$ .

*Proposed by Levi Iszler*


Inscribe a rectangle of base  $b$  and height  $h$  in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of  $h$  do the rectangle and triangle have the same area?

Solution: \boxed{ h = \frac{2}{5} }


The sequence  $ (a_n)$  satisfies  $ a_0 \equal{} 0$  and  $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$  for  $ n\ge0$ . Find the greatest integer less than or equal to  $ a_{10}$ .


Let  $s(a)$  denote the sum of digits of a given positive integer a. The sequence   $a_1, a_2,..., a_n, ...$  of positive integers is such that  $a_{n+1} = a_n+s(a_n)$  for each positive integer  $n$ . Find the greatest possible n for which it is possible to have  $a_n = 2008$ .


Basil needs to solve an exercise on summing two fractions  $\dfrac{a}{b}$  and  $\dfrac{c}{d}$ , where   $a$ ,  $b$ ,  $c$ ,  $d$  are some non-zero real numbers. But instead of summing he performed multiplication  (correctly). It appears that Basil's answer coincides with the correct answer to given exercise. Find the value of   $\dfrac{b}{a} + \dfrac{d}{c}$ .


Equilateral triangles  $ACB'$  and  $BDC'$  are drawn on the diagonals of a convex quadrilateral  $ABCD$  so that  $B$  and  $B'$  are on the same side of  $AC$ , and  $C$  and  $C'$  are on the same sides of  $BD$ . Find  $\angle BAD + \angle CDA$  if  $B'C' = AB+CD$ .

Solution: \boxed{ 120^\circ }


Find the number of pairs  $(a, b)$  of positive integers with the property that the greatest common divisor of  $a$  and  $ b$  is equal to  $1\cdot 2 \cdot 3\cdot ... \cdot50$ , and the least common multiple of  $a$  and  $ b$  is  $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$ .


Let  $P$  be a given quadratic polynomial. Find all functions  $f : \mathbb{R}\to\mathbb{R}$  such that   $$ f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}. $$

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


Let  $n$  be a positive integer  $\geq 2$  . Consider a  $n$  by  $n$  grid with all entries  $1$ . Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all  $n$  such that it is possible after a finite sequence of operations to reach a  $n$  by  $n$  grid with all entries  $-1$

Solution: \boxed{ n }


Arnold has plates weighing  $5$ ,  $15$ ,  $25$ ,  $35$ , or  $45$  pounds. He lifts a barbell, which consists of a  $45$ -pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press  $600$  pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size heavier, and Arnold was actually lifting  $470$  pounds. How many plates did Arnold have on the bar?

*2018 CCA Math Bonanza Team Round #2*

Solution: \boxed{ 13 }


Suppose  $x,y$  and  $z$  are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate  $x^2+y^2+z^2.$


Suppose that  $n$  persons meet in a meeting, and that each of the persons is acquainted to exactly  $8$  others. Any two acquainted persons have exactly  $4$  common acquaintances, and any two non-acquainted persons have exactly  $2$  common acquaintances. Find all possible values of  $n$ .

Solution: \boxed{ n = 21 }


Triangle  $ ABC$  has fixed vertices  $ B$  and  $ C$ , so that  $ BC \equal{} 2$  and  $ A$  is variable. Denote by  $ H$  and  $ G$  the orthocenter and the centroid, respectively, of triangle  $ ABC$ . Let  $ F\in(HG)$  so that  $ \frac {HF}{FG} \equal{} 3$ . Find the locus of the point  $ A$  so that  $ F\in BC$ .

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


An integer consists of 7 different digits, and is a multiple of each of its digits.

What digits are in this nubmer?

Solution: \boxed{1, 2, 3, 6, 7, 8, 9}


Let  $a, b, c$  be positive integers such that  $a + 2b +3c = 100$ .
Find the greatest value of  $M = abc$


 $\textbf{Problem 1.}$  ****There are less than  $400$  marbles.** If they are distributed among  $3$  childrens, there is one left over if they are distributed among  $7$  children, there are 2 left over. Finally if they are distributed among  $5$  children, there are none left over.

What is the largest number of the marbels you have?


A prison has  $2004$  cells, numbered  $1$  through  $2004$ . A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on  $n^{\text{th}}$  trip, turning the key in every  $n^{\text{th}}$  cell, and he finished his mission after  $2004$  trips. How many prisoners were released?


Let  $u$  be a positive rational number and  $m$  be a positive integer. Define a sequence  $q_1,q_2,q_3,\dotsc$  such that  $q_1=u$  and for  $n\geqslant 2$ :  $$ \text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}. $$
Determine all positive integers  $m$  such that the sequence  $q_1,q_2,q_3,\dotsc$  is eventually periodic for any positive rational number  $u$ .

*Remark:* A sequence  $x_1,x_2,x_3,\dotsc $  is *eventually periodic* if there are positive integers  $c$  and  $t$  such that  $x_n=x_{n+t}$  for all  $n\geqslant c$ .

*Proposed by Petar Nizié-Nikolac*

Solution: \boxed{ m }


On a  $(4n + 2)\times (4n + 2)$  square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of  $n$ , what is the largest positive integer  $k$  such that there must be a row or column that the turtle has entered at least  $k$  distinct times?

Solution: \boxed{ 2n + 2 }


Let  $S$  be the sum of all positive integers  $n$  such that  $\frac{3}{5}$  of the positive divisors of  $n$  are multiples of  $6$  and  $n$  has no prime divisors greater than  $3$ . Compute  $\frac{S}{36}$ .


Let  $ \mathcal{R}$  be the region consisting of the set of points in the coordinate plane that satisfy both  $ |8 \minus{} x| \plus{} y \le 10$  and  $ 3y \minus{} x \ge 15$ . When  $ \mathcal{R}$  is revolved around the line whose equation is  $ 3y \minus{} x \equal{} 15$ , the volume of the resulting solid is  $ \frac {m\pi}{n\sqrt {p}}$ , where  $ m$ ,  $ n$ , and  $ p$  are positive integers,  $ m$  and  $ n$  are relatively prime, and  $ p$  is not divisible by the square of any prime. Find  $ m \plus{} n \plus{} p$ .


In triangle  $ABC$  such that  $\angle ACB=90^{\circ}$ , let point  $H$  be foot of perpendicular from point  $C$  to side  $AB$ . Show that sum of radiuses of incircles of  $ABC$ ,  $BCH$  and  $ACH$  is  $CH$

Solution: \boxed{ CH }


Find the largest positive integer  $n$  such that for each prime  $p$  with  $2<p<n$  the difference  $n-p$  is also prime.


The fourth-degree equation  $x^4-x-504=0$  has  $4$  roots  $r_1$ ,  $r_2$ ,  $r_3$ ,  $r_4$ . If  $S_x$  denotes the value of  ${r_1}^4+{r_2}^4+{r_3}^4+{r_4}^4$ , compute  $S_4$ .

*2015 CCA Math Bonanza Individual Round #10*


There is a  $40\%$  chance of rain on Saturday and a  $30\%$  of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is  $\frac{a}{b}$ , where  $a$  and  $b$  are relatively prime positive integers. Find  $a+b$ .


A cube with sides of length 3cm is painted red
and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm.
If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that
are not painted at all, b the number painted on one sides, c the
number painted on two sides, and d the number painted on three
sides, determine the value a-b-c+d.


Find the number of permutations  $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$  of  $1, 2 ,3,4,5,6$  such that for any  $k, 1 \leq k \leq 5$ ,  $(p_1, \ldots, p_k)$  does not form a permutation of  $1 , 2, \ldots, k$ .


Find the smallest positive number  $\lambda $  , such that for any complex numbers  ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$  ,if   $z_1+z_2+z_3=0$ , then   $$ \left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda . $$

Solution: \boxed{ \lambda = 1 }


Given a parallelogram  $ABCD$ , let  $\mathcal{P}$  be a plane such that the distance from vertex  $A$  to  $\mathcal{P}$  is  $49$ , the distance from vertex  $B$  to  $\mathcal{P}$  is  $25$ , and the distance from vertex  $C$  to  $\mathcal{P}$  is  $36$ . Find the sum of all possible distances from vertex  $D$  to  $\mathcal{P}$ .

*Proposed by **HrishiP***


Natural number  $M$  has  $6$  divisors, such that sum of them are equal to  $3500$ .Find the all values of  $M$ .


Equilateral triangle  $ T$  is inscribed in circle  $ A$ , which has radius  $ 10$ . Circle  $ B$  with radius  $ 3$  is internally tangent to circle  $ A$  at one vertex of  $ T$ . Circles  $ C$  and  $ D$ , both with radius  $ 2$ , are internally tangent to circle  $ A$  at the other two vertices of  $ T$ . Circles  $ B$ ,  $ C$ , and  $ D$  are all externally tangent to circle  $ E$ , which has radius  $ \frac {m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m \plus{} n$ .
[asy]unitsize(2.2mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90);
pair Ep=(0,4-27/5);
pair[] dotted={A,B,C,D,Ep};

draw(Circle(A,10));
draw(Circle(B,3));
draw(Circle(C,2));
draw(Circle(D,2));
draw(Circle(Ep,27/5));

dot(dotted);
label(" $E$ ",Ep,E);
label(" $A$ ",A,W);
label(" $B$ ",B,W);
label(" $C$ ",C,W);
label(" $D$ ",D,E);[/asy]


Find the strictly increasing functions  $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$  such that  $x+y$  divides  $x f(x) + y f(y)$  for all  $x,y \in \{ 1,2,\ldots,10 \}$ .

*Cristinel Mortici*

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


A positive integer  $N$  is divided in  $n$  parts inversely proportional to the numbers  $2, 6, 12, 20, \ldots$ The smallest part is equal to  $\frac{1}{400} N$ . Find the value of  $n$ .


Find all real numbers  $a$  for which the following equation has a unique real solution:  $$ |x-1|+|x-2|+\ldots+|x-99|=a. $$


Let  $A$  be the greatest possible value of a product of positive integers that sums to  $2014$ .  Compute the sum of all bases and exponents in the prime factorization of  $A$ .  For example, if  $A=7\cdot 11^5$ , the answer would be  $7+11+5=23$ .


Let  $ABC$  be an acute triangle. Let $H$  and  $D$  be points on  $[AC]$  and  $[BC]$ , respectively, such that  $BH \perp AC$  and  $HD \perp BC$ . Let  $O_1$  be the circumcenter of  $\triangle ABH$ , and  $O_2$  be the circumcenter of  $\triangle BHD$ , and  $O_3$  be the circumcenter of  $\triangle HDC$ . Find the ratio of area of  $\triangle O_1O_2O_3$  and  $\triangle ABH$ .


A square and equilateral triangle have the same perimeter. If the triangle has area  $16\sqrt3$ , what is the area of the square?

*Proposed by Evan Chen*


For positive integers  $m$  and  $n$ , the decimal representation for the fraction  $\tfrac{m}{n}$  begins  $0.711$  followed by other digits. Find the least possible value for  $n$ .


In triangle  $ABC$ ,  $AB = 13$ ,  $BC = 14$ , and  $CA = 15$ . Let  $M$  be the midpoint of side  $AB$ ,  $G$  be the centroid of  $\triangle ABC$ , and  $E$  be the foot of the altitude from  $A$  to  $BC$ . Compute the area of quadrilateral  $GAME$ .

*Proposed by Evin Liang*

Solution*Solution*.  $\boxed{23}$ Use coordinates with  $A = (0,12)$ ,  $B = (5,0)$ , and  $C = (-9,0)$ . Then  $M = \left(\dfrac{5}{2},6\right)$  and  $E = (0,0)$ . By shoelace, the area of  $GAME$  is  $\boxed{23}$ .


Triangle  $\triangle ABC$  has circumcenter  $O$  and incircle  $\gamma$ . Suppose that  $\angle BAC =60^\circ$  and  $O$  lies on  $\gamma$ . If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers  $a$  and  $b$ , compute  $100a+b$ .

*Proposed by Kaan Dokmeci*


Determine the minimum value of prime  $p> 3$  for which there is no natural number  $n> 0$  such that  $2^n+3^n\equiv 0\pmod{p} $ .


How many ways are there to remove an  $11\times11$  square from a  $2011\times2011$  square so that the remaining part can be tiled with dominoes ( $1\times 2$  rectangles)?


Let  $N \ge 5$  be given. Consider all sequences  $(e_1,e_2,...,e_N)$  with each  $e_i$  equal to  $1$  or  $-1$ . Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length  $N$ .


Let  $S$  be the sum of all the real coefficients of the expansion of  $(1+ix)^{2009}$ .  What is  $\log_2(S)$ ?


Suppose  $z^{3}=2+2i$ , where  $i=\sqrt{-1}$ . The product of all possible values of the real part of  $z$  can be written in the form  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .


Let  $ABCDEF$  be a regular hexagon with side length  $2$ . Calculate the area of  $ABDE$ .

*2015 CCA Math Bonanza Lightning Round #1.2*


There are  $N{}$  points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum  $N{}$  for which this is possible?

*Proposed by E. Bakaev*


The graph of the function  $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$  (where  $n>1$ ) intersects the line  $y=b$  at the points  $B_1,B_2,\ldots ,B_n$  (from left to right), and the line  $y=c\ (c\not= b)$  at the points  $C_1,C_2,\ldots ,C_n$  (from left to right). Let  $P$  be a point on the line  $y=c$ , to the right to the point  $C_n$ . Find the sum
\[\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP) \]

Solution: \boxed{ 0 }


Mark's teacher is randomly pairing his class of  $16$  students into groups of  $2$  for a project. What is the probability that Mark is paired up with his best friend, Mike? (There is only one Mike in the class)

*2015 CCA Math Bonanza Individual Round #3*


**a)** Show that if two non-negative integers  $ p,q $  satisfy the property that both  $ \sqrt{2p-q} $  and  $ \sqrt{2p+q} $  are non-negative integers, then  $ q $  is even.**b)** Determine how many natural numbers  $ m $  are there such that  $ \sqrt{2m-4030} $  and  $ \sqrt{2m+4030} $  are both natural.


Let  $x, y$  be positive real numbers. If \[129-x^2=195-y^2=xy,\] then  $x = \frac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $100m+n$ .

[i]Proposed by Michael Tang


For each positive integer  $n$ , let  $s(n)$  be the sum of the digits of  $n$ . Find the smallest positive integer  $k$  such that
\[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]


Let  $n \leq 100$  be an integer. Hare puts real numbers in the cells of a  $100 \times 100$  table. By asking Hare one question, Wolf can find out the sum of all numbers of a square  $n \times n$ , or the sum of all numbers of a rectangle  $1 \times (n - 1)$  (or  $(n - 1) \times 1$ ). Find the greatest  $n{}$  such that, after several questions, Wolf can find the numbers in all cells, with guarantee.


Square  $ABCD$  has center  $O$ ,  $AB=900$ ,  $E$  and  $F$  are on  $AB$  with  $AE<BF$  and  $E$  between  $A$  and  $F$ ,  $m\angle EOF =45^\circ$ , and  $EF=400$ . Given that  $BF=p+q\sqrt{r}$ , wherer  $p,q,$  and  $r$  are positive integers and  $r$  is not divisible by the square of any prime, find  $p+q+r$ .


Triangles  $OAB,OBC,OCD$  are isoceles triangles with  $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$ . Find the area of the triangle  $OAB$  if the area of the triangle  $OCD$  is 12.


Elza draws  $2013$  cities on the map and connects some of them with  $N$  roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest  $N$  for which Elza has a winning strategy.


Let  $ABC$  be an equilateral triangle with side length  $1$ .  Say that a point  $X$  on side  $\overline{BC}$  is *balanced* if there exists a point  $Y$  on side  $\overline{AC}$  and a point  $Z$  on side  $\overline{AB}$  such that the triangle  $XYZ$  is a right isosceles triangle with  $XY = XZ$ .  Find with proof the length of the set of all balanced points on side  $\overline{BC}$ .

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


Let  $P(x), Q(x), $  and  $R(x)$  be three monic quadratic polynomials with only real roots, satisfying   $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$    $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$      for all real numbers  $x.$  What is  $P(0)+Q(0)+R(0)?$ *Proposed by Kyle Lee*


Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$  be positive reals satisfying \begin{align*} a + b &= c  a + b + c &= d  a + b + c + d &= e.\end{align*} If  $c=5$ , compute  $a+b+c+d+e$ .

*Proposed by Evan Chen*


In an acute triangle  $ABC$ , the bisector  $AL$ , the altitude  $BH$ , and the perpendicular bisector of the side  $AB$  intersect at one point. Find the value of the angle  $BAC$ .


The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament:  $2021\div 43 = 47$ . How many times more will the humanity witness such a wonderful event?

*Alexey Zaslavsky*

Solution: \boxed{ 4 }


Let  $\mathcal{T}$  be the set of ordered triples  $(x,y,z)$  of nonnegative real numbers that lie in the plane  $x+y+z=1.$   Let us say that  $(x,y,z)$  supports  $(a,b,c)$  when exactly two of the following are true:  $x\ge a, y\ge b, z\ge c.$   Let  $\mathcal{S}$  consist of those triples in  $\mathcal{T}$  that support  $\left(\frac 12,\frac 13,\frac 16\right).$   The area of  $\mathcal{S}$  divided by the area of  $\mathcal{T}$  is  $m/n,$   where  $m$  and  $n$  are relatively prime positive integers, find  $m+n.$


The positive integer equal to the expression
    \[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\]
    is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.

*Team #7*


Determine the greatest positive integer  $ n$  such that in three-dimensional space, there exist n points  $ P_{1},P_{2},\cdots,P_{n},$  among  $ n$  points no three points are collinear, and for arbitary  $ 1\leq i < j < k\leq n$ ,  $ P_{i}P_{j}P_{k}$  isn't obtuse triangle.


An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on the board have?

*Ray Li.*


Given the condition that there exist exactly  $1990$  triangles  $ABC$  with integral side-lengths satisfying the following conditions:
(i)  $\angle ABC =\frac 12 \angle BAC;$ (ii)  $AC = b.$ Find the minimal value of  $b.$


In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than  $12$  employees have a more difficult work;
(ii) At least  $30$  employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.


Solution: \boxed{ \frac{1}{2^{24}} }


Let  $E$  denote the set of all natural numbers  $n$  such that  $3 < n < 100$  and the set  $\{ 1, 2, 3, \ldots , n\}$  can be partitioned in to  $3$  subsets with equal sums. Find the number of elements of  $E$ .


In a given triangle  $ABC$ ,  $O$  is its circumcenter,  $D$  is the midpoint of  $AB$  and  $E$  is the centroid of the triangle  $ACD$ . Show that the lines  $CD$  and  $OE$  are perpendicular if and only if  $AB=AC$ .

Solution: \boxed{ AB = AC }


Let  $n>3$  be a positive integer. Equilateral triangle ABC is divided into  $n^2$  smaller congruent equilateral triangles (with sides parallel to its sides). Let  $m$  be the number of rhombuses that contain two small equilateral triangles and  $d$  the number of rhombuses that contain eight small equilateral triangles. Find the difference  $m-d$  in terms of  $n$ .

Solution: \boxed{6n - 9}


How many ways are there to fill in a  $2\times 2$  square grid with the numbers  $1,2,3,$  and  $4$  such that the numbers in any two grid squares that share an edge have an absolute difference of at most  $2$ ?

*Proposed by Andrew Wu*


A point  $M$  is chosen inside the square  $ABCD$  in such a way that  $\angle MAC = \angle MCD = x$  . Find  $\angle ABM$ .


Let  $ABCD$  be a unit square. Points  $E, F, G, H$  are chosen outside  $ABCD$  so that  $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$  . Let  $O_1, O_2, O_3, O_4$ , respectively, be the incenters of  $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$ . Show that the area of  $O_1O_2O_3O_4$  is at most  $1$ .

Solution: \boxed{ 1 }


The four points  $A(-1,2), B(3,-4), C(5,-6),$  and  $D(-2,8)$  lie in the coordinate plane. Compute the minimum possible value of  $PA + PB + PC + PD$  over all points P .


Triangle  $ABC$  lies in the Cartesian Plane and has an area of 70. The coordinates of  $B$  and  $C$  are  $(12,19)$  and  $(23,20)$ , respectively, and the coordinates of  $A$  are  $(p,q)$ . The line containing the median to side  $BC$  has slope  $-5$ . Find the largest possible value of  $p+q$ .


Bangladesh National Mathematical Olympiad 2016 Higher Secondary

<u>**Problem 2:**</u>
(a) How many positive integer factors does  $6000$  have?

(b) How many positive integer factors of  $6000$  are not perfect squares?


At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges?

*Ray Li*


When a right triangle is rotated about one leg, the volume of the cone produced is  $800 \pi$   $\text{cm}^3$ .  When the triangle is rotated about the other leg, the volume of the cone produced is  $1920 \pi$   $\text{cm}^3$ .  What is the length (in cm) of the hypotenuse of the triangle?


Given is a number with  $1998$  digits which is divisible by  $9$ . Let  $x$  be the sum of its digits, let  $y$  be the sum of the digits of  $x$ , and  $z$  the sum of the digits of  $y$ . Find  $z$ .


Find the sum of all  $x$  from  $2$  to  $1000$  inclusive such that   $$ \prod_{n=2}^x \log_{n^n}(n+1)^{n+2} $$   is an integer.

*Proposed by Deyuan Li and Andrew Milas*


A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) =  x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients  $a_0\ldots,a_7$ .  Determine the largest real number that occurs as a root of some Mediterranean polynomial.
*(Proposed by Gerhard Woeginger, Austria)*


There are  $10$  girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

Solution: \boxed{ 512 }


Assume  $A=\{a_{1},a_{2},...,a_{12}\}$  is a set of positive integers such that for each positive integer  $n \leq 2500$   there is a subset  $S$  of  $A$  whose sum of elements is  $n$ . If  $a_{1}<a_{2}<...<a_{12}$  , what is the smallest possible value of  $a_{1}$ ?


Let  $a,b$  be real numbers such that  $|a| \neq |b|$  and  $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$  Find the value of the expression  $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$


A circle of length  $10$  is inscribed in a convex polygon with perimeter  $15$ . What part of the area of this polygon is occupied by the resulting circle?


Pam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer  $N$  by the first prime, the remainder is  $1$ . When she divides  $N$  by the second prime, the remainder is  $2$ . When she divides  $N$  by the third prime, the remainder is  $3$ . When she divides  $N$  by the fourth prime, the remainder is  $4$ . Find the least possible value for  $N$ .


There is a regular  $17$ -gon  $\mathcal{P}$  and its circumcircle  $\mathcal{Y}$  on the plane.
The vertices of  $\mathcal{P}$  are coloured in such a way that  $A,B \in \mathcal{P}$  are of different colour, if the shorter arc connecting  $A$  and  $B$  on  $\mathcal{Y}$  has  $2^k+1$  vertices, for some  $k \in \mathbb{N},$  including  $A$  and  $B.$
What is the least number of colours which suffices?


Compute the number of ordered quintuples of nonnegative integers  $(a_1,a_2,a_3,a_4,a_5)$  such that  $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$  and  $5$  divides  $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$ .


Let  $N$  be the number of ways of choosing a subset of  $5$  distinct numbers from the set  $$ {10a+b:1\leq a\leq 5, 1\leq b\leq 5} $$  where  $a,b$  are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when  $N$  is divided by  $73$ ?


Find the minimum positive integer  $k$  such that there exists a function  $f$  from the set  $\Bbb{Z}$  of all integers to  $\{1, 2, \ldots k\}$  with the property that  $f(x) \neq f(y)$  whenever  $|x-y| \in \{5, 7, 12\}$ .


Consider the set of all triangles  $ OPQ$  where  $ O$  is the origin and  $ P$  and  $ Q$  are distinct points in the plane with nonnegative integer coordinates  $ (x,y)$  such that  $ 41x\plus{}y \equal{} 2009$ . Find the number of such distinct triangles whose area is a positive integer.


Suppose that  $x$  is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for  $0\le x\le \frac{\pi}{16}$


For integers a, b, c, and d the polynomial  $p(x) =$   $ax^3 + bx^2 + cx + d$  satisﬁes  $p(5) + p(25) = 1906$ . Find the minimum possible value for  $|p(15)|$ .


Since guts has 36 questions, they will be combined into posts.

1.**[5]** Farmer Yang has a  $2015$  ×  $2015$  square grid of corn plants. One day, the plant in the very center
of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased.
After how many days will all of Yang's corn plants be diseased?

2. **[5]** The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of
the hypotenuse to the length of the shorter leg.

3. **[5]** A parallelogram has  $2$  sides of length  $20$  and  $15$ . Given that its area is a positive integer, ﬁnd the
minimum possible area of the parallelogram.

4. **[6]** Eric is taking a biology class. His problem sets are worth  $100$  points in total, his three midterms are
worth  $100$  points each, and his ﬁnal is worth  $300$  points. If he gets a perfect score on his problem sets
and scores  $60\%$ , $70\%$ , and  $80\%$  on his midterms respectively, what is the minimum possible percentage
he can get on his ﬁnal to ensure a passing grade? (Eric passes if and only if his overall percentage is
at least  $70\%$ ).

5. **[6]** James writes down three integers. Alex picks some two of those integers, takes the average of them,
and adds the result to the third integer. If the possible ﬁnal results Alex could get are  $42$ ,  $13$ , and  $37$ ,
what are the three integers James originally chose?

6. **[6]** Let  $AB$  be a segment of length  $2$  with midpoint  $M$ . Consider the circle with center  $O$  and radius $r$  that is externally tangent to the circles with diameters  $AM$  and  $BM$  and internally tangent to the
circle with diameter  $AB$ . Determine the value of  $r$ .

7. **[7]** Let n be the smallest positive integer with exactly  $2015$  positive factors. What is the sum of
the (not necessarily distinct) prime factors of n? For example, the sum of the prime factors of  $72$  is $2 + 2 + 2 + 3 + 3 = 14$ .

8. **[7]** For how many pairs of nonzero integers  $(c, d)$  with  $-2015 \le c,d \le 2015$  do the equations  $cx = d$ and  $dx = c$  both have an integer solution?

9. **[7]** Find the smallest positive integer n such that there exists a complex number z, with positive real
and imaginary part, satisfying  $z^n = (\overline{z})^n$ .


Consider the set  $A = \{0, 1, 2, \dots , 9 \}$  and let  $(B_1,B_2, \dots , B_k)$  be a collection of nonempty subsets of  $A$  such that  $B_i \cap B_j$  has at most two elements for  $i \neq j$ . What is the maximal value of  $k \ ?$


Find all triples  $m$ ;  $n$ ;  $l$  of natural numbers such that  $m + n = gcd(m; n)^2$ ;  $m + l = gcd(m; l)^2$ ;  $n + l = gcd(n; l)^2$ :
*S. Tokarev*

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


Find the total number of times the digit ‘ $2$ ’ appears in the set of integers  $\{1,2,..,1000\}$ . For example, the digit ’ $2$ ’ appears twice in the integer  $229$ .


We know that  $201$  and  $9$  give the same remainder when divided by  $24$ . What is the smallest positive integer  $k$  such that  $201+k$  and  $9+k$  give the same remainder when divided by  $24+k$ ?

*2020 CCA Math Bonanza Lightning Round #2.1*


Let  $k$  be an integer and  $k > 1$ . Define a sequence  $\{a_n\}$  as follows: $a_0 = 0$ , $a_1 = 1$ , and $a_{n+1} = ka_n + a_{n-1}$  for  $n = 1,2,...$ .

Determine, with proof, all possible  $k$  for which there exist non-negative integers  $l,m (l \not= m)$  and positive integers  $p,q$  such that  $a_l + ka_p = a_m + ka_q$ .

Solution: \boxed{ k = 2 }


Find all functions  $f:\mathbb Z\rightarrow \mathbb Z$  such that, for all integers  $a,b,c$  that satisfy  $a+b+c=0$ , the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here  $\mathbb{Z}$  denotes the set of integers.)

*Proposed by Liam Baker, South Africa*

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


Let  $ABC$  be a triangle, and let  $P$  be a point on side  $BC$  such that  $\frac{BP}{PC}=\frac{1}{2}$ . If  $\measuredangle ABC$   $=$   $45^{\circ}$  and  $\measuredangle APC$   $=$   $60^{\circ}$ , determine  $\measuredangle ACB$  without trigonometry.


Let  $AD$  and  $BC$  be the parallel sides of a trapezium  $ABCD$ . Let  $P$  and  $Q$  be the midpoints of the diagonals  $AC$  and  $BD$ . If  $AD = 16$  and  $BC = 20$ , what is the length of  $PQ$ ?


Ben starts with an integer greater than  $9$  and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive  $1$ -digit integer. Find all possible  $1$ -digit integers Ben can end with from this process.


Positive real numbers  $x$  and  $y$  satisfy  $$ \Biggl|\biggl|\cdots\Bigl|\bigl||x|-y\bigr|-x\Bigr|\cdots -y\biggr|-x\Biggr|\ =\ \Biggl|\biggl|\cdots\Bigl|\bigl||y|-x\bigr|-y\Bigr|\cdots -x\biggr|-y\Biggr| $$  where there are  $2019$  absolute value signs  $|\cdot|$  on each side. Determine, with proof, all possible values of  $\frac{x}{y}$ .

*Proposed by Krit Boonsiriseth.*


The vertices of the convex quadrilateral  $ABCD$  lie on the parabola  $y=x^2$ . It is known that  $ABCD$  is cyclic and  $AC$  is a diameter of its circumcircle. Let  $M$  and  $N$  be the midpoints of the diagonals of  $AC$  and  $BD$  respectively. Find the length of the projection of the segment  $MN$  on the axis  $Oy$ .


Let  $S$  be a subset of  $\{1, 2, 3, \cdots, 1989 \}$  in which no two members differ by exactly  $4$  or by exactly  $7$ . What is the largest number of elements  $S$  can have?


The elliptic curve  $y^2=x^3+1$  is tangent to a circle centered at  $(4,0)$  at the point  $(x_0,y_0)$ .  Determine the sum of all possible values of  $x_0$ .


Let  $S = \{1, 22, 333, \dots , 999999999\}$ . For how many pairs of integers  $(a, b)$  where  $a, b \in S$  and  $a < b$  is it the case that  $a$  divides  $b$ ?


For how many nonempty subsets  $S \subseteq \{1, 2, \ldots , 10\}$  is the sum of all elements in  $S$  even?

*Proposed by Andrew Wen*


What is the smallest positive multiple of  $225$  that can be written using
digits  $0$  and  $1$  only?


Find the number of ordered  $64$ -tuples  $\{x_0,x_1,\dots,x_{63}\}$  such that  $x_0,x_1,\dots,x_{63}$  are distinct elements of  $\{1,2,\dots,2017\}$  and
\[x_0+x_1+2x_2+3x_3+\cdots+63x_{63}\]
is divisible by  $2017.$


Let  $ n, m$  be positive integers of different parity, and  $ n > m$ .  Find all integers  $ x$  such that  $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$  is a perfect square.

Solution: \boxed{ x = 0 }


If  $a$  and  $b$  are positive integers such that  $a^2-b^4= 2009$ , find  $a+b$ .


The altitudes in triangle  $ABC$  have lengths 10, 12, and 15. The area of  $ABC$  can be expressed as  $\tfrac{m\sqrt n}p$ , where  $m$  and  $p$  are relatively prime positive integers and  $n$  is a positive integer not divisible by the square of any prime. Find  $m + n + p$ .

[asy]
import olympiad;
size(200);
defaultpen(linewidth(0.7)+fontsize(11pt));
pair A = (9,8.5), B = origin, C = (15,0);
draw(A--B--C--cycle);
pair D = foot(A,B,C), E = foot(B,C,A), F = foot(C,A,B);
draw(A--D^^B--E^^C--F);
label(" $A$ ",A,N);
label(" $B$ ",B,SW);
label(" $C$ ",C,SE);
[/asy]

Solution: \boxed{ 254 }


Let  $x,y,z$  be complex numbers such that $\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$  $\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$  $\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$ If  $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$  where  $m,n$  are positive integers with  $GCD(m,n)=1$ , find  $m+n$ .


Solution: \boxed{ T_n = n! + 2^n }


For each integer from 1 through 2019, Tala calculated the product of its digits.  Compute the sum of all 2019 of Tala's products.


A regular  $ n\minus{}$ gon  $ A_1A_2A_3 \cdots A_k \cdots A_n$  inscribed in a circle of radius  $ R$  is given. If  $ S$  is a point on the circle, calculate \[ T \equal{} \sum^n_{k\equal{}1} SA^2_k.\]


If we divide number  $19250$  with one number, we get remainder  $11$ . If we divide number  $20302$  with the same number, we get the reamainder  $3$ . Which number is that?


Let  $p_1 = 2, p_2 = 3, p_3 = 5 ...$  be the sequence of prime numbers. Find the least positive even integer  $n$  so that  $p_1 + p_2 + p_3 + ... + p_n$  is not prime.


A set  $D$  of positive integers is called *indifferent* if there are at least two integers in the set, and for any two distinct elements  $x,y\in D$ , their positive difference  $|x-y|$  is also in  $D$ . Let  $M(x)$  be the smallest size of an indifferent set whose largest element is  $x$ . Compute the sum  $M(2)+M(3)+\dots+M(100)$ .

*Proposed by Yannick Yao*


Let  $x,y,z>0$  such that  $$ (x+y+z)\left(\frac1x+\frac1y+\frac1z\right)=\frac{91}{10} $$  Compute  $$ \left[(x^3+y^3+z^3)\left(\frac1{x^3}+\frac1{y^3}+\frac1{z^3}\right)\right] $$  where  $[.]$  represents the integer part.

*Proposed by Marian Cucoanoeş and Marius Drăgan*


A two-digit integer  $\underline{a}\,\, \underline{b}$  is multiplied by  $9$ . The resulting three-digit integer is of the form  $\underline{a} \,\,\underline{c} \,\,\underline{b}$  for some digit  $c$ . Evaluate the sum of all possible  $\underline{a} \,\, \underline{b}$ .


Three real numbers  $x, y, z$  are such that  $x^2 + 6y = -17, y^2 + 4z = 1$  and  $z^2 + 2x = 2$ . What is the value of  $x^2 + y^2 + z^2$ ?


Let  $p,q$  be prime numbers such that  $n^{3pq}-n$  is a multiple of  $3pq$  for **all** positive integers  $n$ . Find the least possible value of  $p+q$ .


$ABCD$  is a rectangle.  $E$  is a point on  $AB$  between  $A$  and  $B$ , and  $F$  is a point on  $AD$  between  $A$  and  $D$ . The area of the triangle  $EBC$  is  $16$ , the area of the triangle  $EAF$  is  $12$  and the area of the triangle  $FDC$  is 30. Find the area of the triangle  $EFC$ .


Given a positive integer  $N$  (written in base  $10$ ), define its *integer substrings* to be integers that are equal to strings of one or more consecutive digits from  $N$ , including  $N$  itself. For example, the integer substrings of  $3208$  are  $3$ ,  $2$ ,  $0$ ,  $8$ ,  $32$ ,  $20$ ,  $320$ ,  $208$ ,  $3208$ . (The substring  $08$  is omitted from this list because it is the same integer as the substring  $8$ , which is already listed.)

What is the greatest integer  $N$  such that no integer substring of  $N$  is a multiple of  $9$ ? (Note:  $0$  is a multiple of  $9$ .)


The sequence of positive integers  $a_0, a_1, a_2, . . .$  is defined by  $a_0 = 3$  and   $$ a_{n+1} - a_n = n(a_n - 1) $$   for all  $n \ge 0$ . Determine all integers  $m \ge 2$  for which  $gcd (m, a_n) = 1$  for all  $n \ge 0$ .

Solution: \boxed{ m = 2^k }


Let  $A=\{1,2,\ldots, 2006\}$ . Find the maximal number of subsets of  $A$  that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements.


In the parallelogram  $ABCD$ , a line through  $C$  intersects the diagonal  $BD$  at  $E$  and  $AB$  at  $F$ . If  $F$  is the midpoint of  $AB$  and the area of  $\vartriangle BEC$  is  $100$ , find the area of the quadrilateral  $AFED$ .


Let  $S(n)$  denote the sum of digits of a natural number  $n$ . Find all  $n$  for which  $n+S(n)=2004$ .

Solution: \boxed{ n = 2001 }


Find the number of arithmetic sequences  $a_1,a_2,a_3$  of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.

*Proposed by Sammy Charney*


A rectangular storage bin measures  $10$  feet by  $12$  feet, is  $3$  feet tall, and sits on a flat plane.  A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground  $4$  feet away from the base of the storage bin as shown.  The number of cubic feet of dirt needed to form the pile can be written as   $m + n \pi$  where   $m$  and  $n$  are positive integers.  Find  $m + n.$


Find all pairs  $(p, n)$  with  $n>p$ , consisting of a positive integer  $n$  and a prime  $p$ , such that  $n^{n-p}$  is an  $n$ -th power of a positive integer.

Solution: \boxed{ (4, 2) }


Let  $F_k(a,b)=(a+b)^k-a^k-b^k$  and let  $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs  $(a,b)$  with  $a,b\in S$  and  $a\leq b$  is  $\frac{F_5(a,b)}{F_3(a,b)}$  an integer?


Let  $X={1,2,3,4,5,6,7,8,9,10}$  and  $A={1,2,3,4}$ . Find the number of  $4$ -element subsets  $Y$  of  $X$  such that  $10\in Y$  and the intersection of  $Y$  and  $A$  is not empty.


Let  $c_1,c_2,\ldots,c_{6030}$  be 6030 real numbers. Suppose that for any 6030 real numbers  $a_1,a_2,\ldots,a_{6030}$ , there exist 6030 real numbers  $\{b_1,b_2,\ldots,b_{6030}\}$  such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for  $n=1,2,\ldots,6030$ . Find  $c_{6030}$ .

*Victor Wang.*


In triangle  $ABC$  we have  $|AB| \ne |AC|$ . The bisectors of  $\angle ABC$  and  $\angle  ACB$  meet  $AC$  and  $AB$  at  $E$  and  $F$ , respectively, and intersect at I. If  $|EI| = |FI|$  find the measure of  $\angle  BAC$ .


How many rational solutions for  $x$  are there to the equation  $x^4+(2-p)x^3+(2-2p)x^2+(1-2p)x-p=0$  if  $p$  is a prime number?


$A$  and  $B$  are on a circle of radius  $20$  centered at  $C$ , and  $\angle ACB = 60^\circ$ .  $D$  is chosen so that  $D$  is also on the circle,  $\angle ACD = 160^\circ$ , and  $\angle DCB = 100^\circ$ . Let  $E$  be the intersection of lines  $AC$  and  $BD$ . What is  $DE$ ?


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