An ant walks around on the coordinate plane. It moves from the origin to  $(3,4)$ , then to  $(-9, 9)$ , then back to the origin. How many units did it walk? Express your answer as a decimal rounded to the nearest tenth.


Let  $a$ ,  $b$ ,  $c$  be positive real numbers for which \[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b. \] If  $a+b+c = \frac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $m+n$ .

*Proposed by Evan Chen*


A store had  $376$  chocolate bars. Min bought some of the bars, and Max bought  $41$  more of the bars than Min bought. After that, the store still had three times as many chocolate bars as Min bought. Find the number of chocolate bars that Min bought.


(**3**) Let  $ \ell$  be the line through  $ (0,0)$  and tangent to the curve  $ y \equal{} x^3 \plus{} x \plus{} 16$ . Find the slope of  $ \ell$ .


Find the largest possible value of  $a+b$  less than or equal to  $2007$ , for which  $a$  and  $b$  are relatively prime, and such that there is some positive integer  $n$  for which \[\frac{2^3-1}{2^3+1}\cdot\frac{3^3-1}{3^3+1}\cdot\frac{4^3-1}{4^3+1}\cdots\frac{n^3-1}{n^3+1} = \frac ab.\]


Inside the rectangle  $ABCD$  is taken a point  $M$  such that  $\angle BMC + \angle AMD = 180^o$ . Determine the sum of the angles  $BCM$  and  $DAM$ .


3. Consider all 100-digit positive integers such that each decimal digit of these equals  $2,3,4,5,6$ , or 7 . How many of these integers are divisible by  $2^{100}$  ?

	Pavel Kozhevnikov


Given a prime number  $p > 2$ . Find the least  $n\in Z_+$ , for which every set of  $n$  perfect squares not divisible by  $p$  contains nonempty subset with product of all it's elements equal to  $1\ (\text{mod}\ p)$


The degrees of polynomials  $P$  and  $Q$  with real coefficients do not exceed  $n$ . These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of  $Q \left( - \frac{1}{2} \right)$ .


Find the positive integer  $n$  such that a convex polygon with  $3n + 2$  sides has  $61.5$  percent fewer diagonals than a convex polygon with  $5n - 2$  sides.


Convex pentagon  $ABCDE$  satisfies  $AB \parallel DE$ ,  $BE \parallel CD$ ,  $BC \parallel AE$ ,  $AB = 30$ ,  $BC = 18$ ,  $CD = 17$ , and  $DE = 20$ . Find its area.

*Proposed by Michael Tang*


Find all primes  $p$ , such that there exist positive integers  $x$ ,  $y$  which satisfy  $$ \begin{cases}
p + 49 = 2x^2
p^2 + 49 = 2y^2
\end{cases} $$

Solution: \boxed{ p = 23 }


Determine the real value of  $t$  that minimizes the expression
\[
  \sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}.
\]


Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$  be real strictly positive real numbers such that  $abcde = 1$ . Then is true the following inequality:  $$ \frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2} $$


Starting with an empty string, we create a string by repeatedly appending one of the letters  $H$ ,  $M$ ,  $T$  with probabilities  $\frac 14$ ,  $\frac 12$ ,  $\frac 14$ , respectively, until the letter  $M$  appears twice consecutively. What is the expected value of the length of the resulting string?


Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers  $x$ .

*2019 CCA Math Bonanza Lightning Round #3.4*


Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?


Find the smallest prime number  $p$  for which the number  $p^3+2p^2+p$  has exactly  $42$  divisors.


The only prime factors of an integer  $n$  are 2 and 3. If the sum of the divisors of  $n$  (including itself) is  $1815$ , find  $n$ .


Let  $P$  be a  $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of  $P$  a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)


Let  $ABCD$  be a rhombus of sides  $AB = BC = CD= DA = 13$ . On the side  $AB$  construct the rhombus   $BAFC$  outside  $ABCD$  and such that the side  $AF$   is parallel to the diagonal  $BD$  of  $ABCD$ . If the area of  $BAFE$  is equal to  $65$ , calculate the area of  $ABCD$ .


The function  $ f$  has the property that for each real number  $ x$  in its domain,  $ 1/x$  is also in its domain and
\[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x.
\]What is the largest set of real numbers that can be in the domain of  $ f$ ?
 $ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}
\textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$

Solution: \boxed{\{-1, 1\}}


For  $n$  measured in degrees, let  $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$ .  Evaluate   $$  4\sum^{30}_{n=1} n \cdot T(n). $$


Compute the maximum real value of  $a$  for which there is an integer  $b$  such that  $\frac{ab^2}{a+2b} = 2019$ . Compute the maximum possible value of  $a$ .


There is a complex number  $K$  such that the quadratic polynomial  $7x^2 +Kx + 12 - 5i$  has exactly one root, where  $i =\sqrt{-1}$ . Find  $|K|^2$ .


A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?


Let  $\mathbb{N}$  denote the set of positive integers, and let  $S$  be a set. There exists a function  $f :\mathbb{N} \rightarrow S$  such that if  $x$  and  $y$  are a pair of positive integers with their difference being a prime number, then  $f(x) \neq f(y)$ . Determine the minimum number of elements in  $S$ .


Two players, \(A\) (first player) and \(B\), take alternate turns in playing a game using 2016 chips as follows: *the player whose turn it is, must remove \(s\) chips from the remaining pile of chips, where \(s \in \{ 2,4,5 \}\)*. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?

Solution: \boxed{ B }


[asy]size(8cm);

real w = 2.718; // width of block
real W = 13.37; // width of the floor
real h = 1.414; // height of block
real H = 7; // height of block + string
real t = 60; // measure of theta

pair apex = (w/2, H); // point where the strings meet

path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the block
draw(shift(-W/2,0)*block); // draws white block
path arrow = (w,h/2)--(w+W/8,h/2); // path of the arrow
draw(shift(-W/2,0)*arrow, EndArrow); // draw the arrow

picture pendulum; // making a pendulum...
draw(pendulum, block); // block
fill(pendulum, block, grey); // shades block
draw(pendulum, (w/2,h)--apex); // adds in string

add(pendulum); // adds in block + string
add(rotate(t, apex) * pendulum); // adds in rotated block + string

dot(" $\theta$ ", apex, dir(-90+t/2)*3.14); // marks the apex and labels it with theta
draw((apex-(w,0))--(apex+(w,0))); // ceiling
draw((-W/2-w/2,0)--(w+W/2,0)); // floor[/asy]
A block of mass  $m=\text{4.2 kg}$  slides through a frictionless table with speed  $v$  and collides with a block of identical mass  $m$ , initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle  $\theta=12^\circ$ , as labeled in the diagram. It takes a time  $ t = \text {1.0 s} $  for the block to swing up to this peak. Find  $10v$ , in  $\text{m/s}$  and round to the nearest integer. Do not approximate  $ \theta \approx 0 $ ; however, assume  $\theta$  is small enough as to use the small-angle approximation for the period of the pendulum.

*(Ahaan Rungta, 6 points)*


In the middle of the school year,  $40\%$  of Poolesville magnet students decided to transfer to the Blair magnet, and  $5\%$  of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from  $400$  students to  $480$  students, how many students does the Poolesville magnet have after the transferring has occurred?


A number is between  $500$  and  $1000$  and has a remainder of  $6$  when divided by  $25$  and a remainder of  $7$  when divided by  $9$ . Find the only odd number to satisfy these requirements.


For all natural numbers  $n$ , consider the polynomial  $P_n(x) = x^{n+2}-2x+1$ .
(a) Show that the equation  $P_n(x)=0$  has exactly one root  $c_n$  in the open interval  $(0,1)$ .
(b) Find  $lim_{n \to \infty}c_n$ .


A coin that comes up heads with probability  $ p > 0$  and tails with probability  $ 1\minus{}p > 0$  independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to  $ \frac{1}{25}$  of the probability of five heads and three tails. Let  $ p \equal{} \frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .


Call a positive integer  $n\geq 2$  *junk* if there exist two distinct  $n$  digit binary strings  $a_1a_2\cdots a_n$  and  $b_1b_2\cdots b_n$  such that

- $a_1+a_2=b_1+b_2,$
- $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$  for all  $2\leq i\leq n-1,$  and
- $a_{n-1}+a_n=b_{n-1}+b_n$ .

Find the number of junk positive integers less than or equal to  $2016$ .

[i]Proposed by Nathan Ramesh


Compute the smallest integer  $n\geq 4$  such that  $\textstyle\binom n4$  ends in  $4$  or more zeroes (i.e. the rightmost four digits of  $\textstyle\binom n4$  are  $0000$ ).


Let  $p(n)$  denote the product of decimal digits of a positive integer  $n$ . Computer the sum  $p(1)+p(2)+\ldots+p(2001)$ .


Find all real parameters  $a$  for which the equation  $x^8 +ax^4 +1 = 0$  has four real roots forming an arithmetic progression.


The integer  $n$  is the smallest positive multiple of 15 such that every digit of  $n$  is either 8 or 0.  Compute  $\frac{n}{15}$ .


A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola  $y^2=x^2-x+1$  in the first quadrant.  This ray makes an angle of  $\theta$  with the positive  $x$ -axis.  Compute  $\cos\theta$ .


Show that if  $t_1 , t_2, t_3, t_4, t_5$  are real numbers, then  $$ \sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}. $$

Solution: \boxed{ e^{e^{e^{e}}} }


The set of quadruples  $(a,b,c,d)$  where each of  $a,b,c,d$  is either  $0$  or  $1$  is *called vertices of the four dimensional unit cube* or *4-cube* for short. Two vertices are called *adjacent*, if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from  $(0,0,0,0)$  and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to  $(0,0,0,0)$  after  $4042$  turns? Note that it is <u>NOT</u> forbidden for the robot to pass through  $(0,0,0,0)$  before the  $4042$ -nd turn.

Solution: \boxed{2^{4041} + 2^{8081}}


A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.


In the acute triangle  $ABC$  the circle through  $B$  touching the line  $AC$  at  $A$  has centre  $P$ , the circle through  $A$  touching the line  $BC$  at  $B$  has centre  $Q$ . Let  $R$  and  $O$  be the circumradius and circumcentre of triangle  $ABC$ , respectively. Show that  $R^2 = OP  \cdot OQ$ .

Solution: \boxed{ R^2 = OP \cdot OQ }


$\textbf{Problem 4.}$  The number of perfect inhabitants of a city was a perfect square, in other words, a whole number squared. with  $100$  people plus the new number of inhabitants turned out to be a perfect square plus one. Now, with another increase of  $100$  people, the number of inhabitants is again a perfect square. What was the number of inhabitants original city?


Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right triangle), cutting it along the altitude to its hypotenuse and randomly discarding one of the two pieces once again, and continues doing this forever. As the number of iterations of this process approaches infinity, the total length of the cuts made in the paper approaches a real number  $l$ . Compute  $[\mathbb{E}(l)]^2$ , that is, the square of the expected value of  $l$ .

*Proposed by Matthew Kroesche*


A plane intersects a right circular cylinder of radius  $1$  forming an ellipse.  If the major axis of the ellipse of  $50\%$  longer than the minor axis, the length of the major axis is $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$


We color some unit squares in a  $ 99\times 99 $  square grid  with one of  $ 5 $  given distinct colors, such that each color appears  the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.


Let  $H$  be a convex, equilateral heptagon whose angles measure (in degrees)  $168^\circ$ ,  $108^\circ$ ,  $108^\circ$ ,  $168^\circ$ ,  $x^\circ$ ,  $y^\circ$ , and  $z^\circ$  in clockwise order.  Compute the number  $y$ .

Solution: \boxed{ y = 132^\circ }


Let  $ABC$  be a triangle with area  $1$  and  $P$  the middle of the side   $[BC]$ .  $M$  and  $N$  are two points of  $[AB]-\left \{ A,B \right \} $  and  $[AC]-\left \{ A,C \right \}$  respectively such that  $AM=2MB$  and $CN=2AN$ .  The two lines  $(AP)$  and  $(MN)$  intersect in a point  $D$ . Find the area of the triangle  $ADN$ .


Find all ordered triples  $(a,b, c)$  of positive integers which satisfy  $5^a + 3^b - 2^c = 32$

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


suppose that  $p$  is a prime number. find that smallest  $n$  such that there exists a non-abelian group  $G$  with  $|G|=p^n$ .

SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.

the exam time was 5 hours.

Solution: \boxed{n = 3}


Find the continuous function  $f(x)$  such that the following equation holds  for any real number  $x$ .

\[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\]

*1977 Keio University entrance exam/Medicine*

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


Each of the first  $150$  positive integers is painted on a different marble, and the  $150$  marbles are placed in a bag. If  $n$  marbles are chosen (without replacement) from the bag, what is the smallest value of  $n$  such that we are guaranteed to choose three marbles with consecutive numbers?


One base of a trapezoid is 100 units longer than the other base.  The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio  $2: 3.$   Let  $x$  be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area.  Find the greatest integer that does not exceed  $x^2/100.$


Let  $ABC$  be a triangle in which  $\measuredangle{A}=135^{\circ}$ . The perpendicular to the line  $AB$  erected at  $A$  intersects the side  $BC$  at  $D$ , and the angle bisector of  $\angle B$  intersects the side  $AC$  at  $E$ .
Find the measure of  $\measuredangle{BED}$ .


Rectangle  $ABCD$  has sides  $AB = 3$ ,  $BC = 2$ . Point  $ P$  lies on side  $AB$  is such that the bisector of the angle  $CDP$  passes through the midpoint  $M$  of  $BC$ . Find  $BP$ .


Find the smallest positive integer  $n$  such that there are at least three distinct ordered pairs  $(x,y)$  of positive integers such that \[x^2-y^2=n.\]


There is a lamp on each cell of a  $2017 \times 2017$  board. Each lamp is either on or off. A lamp is called *bad* if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.)


Find the largest real number  $k$ , such that for any positive real numbers  $a,b$ ,  $$ (a+b)(ab+1)(b+1)\geq kab^2 $$


Arthur is driving to David’s house intending to arrive at a certain time. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. If he drives at  n  km/h, he will arrive exactly on time. What is the value of n?


A child lines up  $2020^2$  pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below  $250$ . How many bricks remain in the end?


A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are  $189$ ,  $320$ ,  $287$ ,  $234$ ,  $x$ , and  $y$ . Find the greatest possible value of  $x+y$ .


How many positive integers less than or equal to  $1000$  are divisible by  $2$  and  $3$  but not by  $5$ ?

*2015 CCA Math Bonanza Individual Round #6*


Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \]*Proposed by Evan Chen*


A truck is initially moving at velocity  $v$ . The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force  $F$  to the truck. The truck rolls a distance  $x$  before coming to a stop, and the time it takes to stop is  $t$ .

Which of the following expressions is equal the initial momentum of the truck (i.e. the momentum before the driver starts braking)? $\textbf{(A) } Fx
\textbf{(B) }  Ft/2
\textbf{(C) } Fxt
\textbf{(D) } 2Ft
\textbf{(E) } 2Fx/v$


Solve in the interval  $ (2,\infty ) $  the following equation:  $$  1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) .  $$

Solution: \boxed{x = 3}


Let  $a,b,c,d,e$  be positive real numbers. Find the largest possible value for the expression  $$ \frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}. $$


Write an integer between  $1$  and  $9$  in each square of the  $5 \times 7$  grid as shown in the figure below.
[asy]
import math;
size(200);

add(shift(0, 0)*grid(7, 5));
[/asy]
Find the maximum possible value of the sum of the  $35$  numbers written when writing different numbers for any two squares that share a vertex.


Let  $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$ . Find the least positive integer  $ a$  such that  $ 65$  divides  $ f(x)$  for every integer  $ x$ .


$(HUN 6)$  Find the positions of three points  $A,B,C$  on the boundary of a unit cube such that  $min\{AB,AC,BC\}$  is the greatest possible.

Solution: \boxed{ \sqrt{2} }


Let  $ABC$  be an equilateral triangle, and let  $D$  and  $F$  be points on sides  $BC$  and  $AB$ , respectively, with  $FA=5$  and  $CD=2$ .  Point  $E$  lies on side  $CA$  such that  $\angle DEF = 60^\circ$ .  The area of triangle  $DEF$  is  $14\sqrt{3}$ .  The two possible values of the length of side  $AB$  are  $p \pm q\sqrt{r}$ , where  $p$  and  $q$  are rational, and  $r$  is an integer not divisible by the square of a prime.  Find  $r$ .


Let  $n$  be a positive integer, and let  $a>0$  be a real number. Consider the equation:
\[ \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 \]
How many solutions ( $x_1, x_2 \cdots , x_n$ ) does this equation have, such that:
\[ 0 \leq x_i \leq a, i \in N^+ \]


Let  $ABCD$  be a tetrahedron. Let  $a$  be the length of  $AB$  and let  $S$  be the area of the projection of the tetrahedron onto a plane perpendicular to  $AB$ . Determine the volume of the tetrahedron in terms of  $a$  and  $S$ .

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


Let  $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form  $\frac{b}{2^a}$ , where  $a$  is a non-negative integer and  $b$  is an odd integer (not necessarily positive). Find  $a+b$ .

*2017 CCA Math Bonanza Tiebreaker Round #3*


Find the least  $n\in N$  such that among any  $n$  rays in space sharing a common origin there exist two which form an acute angle.


Let  $d_k$  be the greatest odd divisor of  $k$  for  $k = 1, 2, 3, \ldots$ . Find  $d_1 + d_2 + d_3 + \ldots + d_{1024}$ .


The numbers  $1,2,...,100$  are written in a board. We are allowed to choose any two numbers from the board  $a,b$  to delete them and replace on the board the number  $a+b-1$ .
What are the possible numbers u can get after  $99$  consecutive operations of these?


Let  $f : R \to R$  be a function satisfying the equation  $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$  for all real numbers  $x$ . What is the value of  $f(85)$ ?


Melinda has three empty boxes and  $12$  textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  Are relatively prime positive integers. Find  $m+n$ .


Let  $a$  be a positive real number such that  $\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}$ .  Then  $\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .


A wooden cube, whose edges are one centimeter long, rests on a horizontal surface.  Illuminated by a point source of light that is  $x$  centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface.  The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters.  Find the greatest integer that does not exceed  $1000x.$


On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are  $k$  arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of  $k$ ?


For a sequence  $s = (s_1, s_2, . . . , s_n)$ , define  $$ F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2. $$  Consider the sequence  $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$ . Let  $R$  be the sum of all  $F(m)$  for all non-empty subsequences  $m$  of  $S$ . Find the remainder when  $R$  is divided by  $1000$ .

Note: A subsequence is a sequence that can be obtained from another sequence by deleting some non-negative number of values without changing the order.


In convex quadrilateral  $KLMN$  side  $\overline{MN}$  is perpendicular to diagonal  $\overline{KM}$ , side  $\overline{KL}$  is perpendicular to diagonal  $\overline{LN}$ ,  $MN = 65$ , and  $KL = 28$ . The line through  $L$  perpendicular to side  $\overline{KN}$  intersects diagonal  $\overline{KM}$  at  $O$  with  $KO = 8$ . Find  $MO$ .


Let  $M$  be the intersection point of the medians  $AD$  and  $BE$  of a right triangle  $ABC$  ( $\angle C=90^\circ$ ). It is known that the circumcircles of triangles  $AEM$  and  $CDM$  are tangent. Find the angle  $\angle BMC.$


Find all natural integers  $n$  such that  $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$  is a square.

Solution: \boxed{ n = 1 }


The number  $2^{29}$  has a  $9$ -digit decimal representation that contains all but one of the  $10$  (decimal) digits. Determine which digit is missing


A building contractor needs to pay his  $108$  workers  $\$ 200 $ each. He is carrying $ 122 $ one hundred dollar bills and $ 188 $ fifty dollar bills. Only $ 45 $ workers get paid with two $ \ $100$  bills. Find the number of workers who get paid with four  $\$ 50$ bills.


George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi, and vegetable jiaozi. Each person orders exactly  $15$  jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiaozi in both orders.


Bob sends a secret message to Alice using her RSA public key  $n = 400000001.$  Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of  $n.$  Eve knows that  $n = pq,$  a product of two prime factors. Find  $p$  and  $q.$

Solution: \boxed{ p = 20201 }


We choose 100 points in the coordinate plane. Let  $N$  be the number of triples  $(A,B,C)$  of distinct chosen points such that  $A$  and  $B$  have the same  $y$ -coordinate, and  $B$  and  $C$  have the same  $x$ -coordinate. Find the greatest value that  $N$  can attain considering all possible ways to choose the points.


Consider all the real sequences  $x_0,x_1,\cdots,x_{100}$  satisfying the following two requirements:
(1) $x_0=0$ ;
(2)For any integer  $i,1\leq i\leq 100$ ,we have  $1\leq x_i-x_{i-1}\leq 2$ .
Find the greatest positive integer  $k\leq 100$ ,so that for any sequence  $x_0,x_1,\cdots,x_{100}$  like this,we have
\[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]


Let  $ABC$  be a triangle with  $\angle BAC = 90^\circ$ . Construct the square  $BDEC$  such as  $A$  and the square are at opposite sides of  $BC$ . Let the angle bisector of  $\angle BAC$  cut the sides  $[BC]$  and  $[DE]$  at  $F$  and  $G$ , respectively. If  $|AB|=24$  and  $|AC|=10$ , calculate the area of quadrilateral  $BDGF$ .


Suppose three boba drinks and four burgers cost  $28$  dollars, while two boba drinks and six burgers cost  $\$  37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.


Let be a real number  $ a\in \left[ 2+\sqrt 2,4 \right] . $  Find  $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $


Given an acute triangle  $ABC$  with altituties AD and BE. O circumcinter of  $ABC$ .If o lies on the segment DE then find the value of  $sinAsinBcosC$


For positive integers  $n$ , let  $M_n$  be the  $2n+1$  by  $2n+1$  skew-symmetric matrix for which each entry in the first  $n$  subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of  $M_n$ . (According to one definition, the rank of a matrix is the largest  $k$  such that there is a  $k \times k$  submatrix with nonzero determinant.)

One may note that
\begin{align*}
M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1  1 & 0 & -1  -1 & 1 & 0
\end{array}\right)
M_2 &= \left( \begin{array}{ccccc} 0 & -1 & -1 & 1
& 1  1 & 0 & -1 & -1 & 1  1 & 1 & 0 & -1 & -1  -1 & 1 & 1 & 0 &
-1  -1 & -1 & 1 & 1 & 0 \end{array} \right).
\end{align*}

Solution: \boxed{ 2n }


If  $p$ ,  $q$ , and  $r$  are nonzero integers satisfying \[p^2+q^2 = r^2,\] compute the smallest possible value of  $(p+q+r)^2$ .

*Proposed by David Altizio*


Let  $A$  and  $B$  be two sets such that  $A \cup B$  is the set of the positive integers, and  $A \cap B$  is the empty set. It is known that if two positive integers have a prime larger than  $2013$  as their difference, then one of them is in  $A$  and the other is in  $B$ . Find all the possibilities for the sets  $A$  and  $B$ .

Solution: \boxed{ A }


Let  $ABC$  be an isosceles triangle with  $AB = BC$ . Point  $E$  lies on the side  $AB$ , and  $ED$  is the perpendicular from  $E$  to  $BC$ . It is known that  $AE = DE$ . Find  $\angle DAC$ .


Three concentric circles have radii  $3$ ,  $4$ , and  $5$ .  An equilateral triangle with one vertex on each circle has side length  $s$ .  The largest possible area of the triangle can be written as  $a+\frac{b}{c}\sqrt{d}$ , where  $a,b,c$  and  $d$  are positive integers,  $b$  and  $c$  are relatively prime, and  $d$  is not divisible by the square of any prime.  Find  $a+b+c+d$ .

Solution: \boxed{ 41 }


$2016$  numbers written on the board:  $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016},  ..., \frac{2016}{2016}$ . In one move, it is allowed to choose any two numbers  $a$  and  $b$  written on the board, delete them, and write the number  $3ab - 2a - 2b + 2$  instead. Determine what number will remain written on the board after  $2015$  moves.


Let  $\{x\}$  denote the fractional part of  $x$ , which means the unique real  $0\leq\{x\}<1$  such that  $x-\{x\}$  is an integer. Let  $f_{a,b}(x)=\{x+a\}+2\{x+b\}$  and let its range be  $[m_{a,b},M_{a,b})$ . Find the minimum value of  $M_{a,b}$  as  $a$  and  $b$  range along all real numbers.


In how many ways can we enter numbers from the set  $\{1,2,3,4\}$  into a  $4\times 4$  array so that all of the following conditions hold?

(a) Each row contains all four numbers.

(b) Each column contains all four numbers.

(c) Each "quadrant" contains all four numbers.  (The quadrants are the four corner  $2\times 2$  squares.)


Define a function  $g :\mathbb{N} \rightarrow \mathbb{R}$ Such that  $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$ .
Find the last 2 digits in the decimal representation of  $g(2021)$ .


Triangle  $\triangle PQR$ , with  $PQ=PR=5$  and  $QR=6$ , is inscribed in circle  $\omega$ . Compute the radius of the circle with center on  $\overline{QR}$  which is tangent to both  $\omega$  and  $\overline{PQ}$ .


Consider a  $5 \times 5$  grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do  $\textbf{not}$  form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?


In the figure below, how many ways are there to select two squares which do not share an edge?

[asy]
size(3cm);
for (int t = -2; t <= 2; t=t+1) { draw( shift((t,0))*unitsquare ) ; if (t!=0) draw( shift((0,t))*unitsquare ); }
[/asy]

*Proposed by Evan Chen*


There are  $522$  people at a beach, each of whom owns a cat, a dog, both, or neither. If  $20$  percent of cat-owners also own a dog,  $70$  percent of dog-owners do not own a cat, and  $50$  percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?


Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where  $\alpha,\beta$  and  $\gamma$  are positive and  $\alpha + \beta + \gamma = 180^{\circ}$ .


Let  $p=4k+3$  be a prime number. Find the number of different residues mod p of  $(x^{2}+y^{2})^{2}$  where  $(x,p)=(y,p)=1.$

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


Determine all functions  $ f: \mathbb{N} \rightarrow \mathbb{N}$  which satisfy:

 $ f(x\plus{}f(y))\equal{}f(x)\plus{}y$  for all  $ x,y \in \mathbb{N}$ .

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


Let  $ABCDEF$  be a regular hexagon, and let  $G$ ,  $H$ ,  $I$ ,  $J$ ,  $K$ , and  $L$  be the midpoints of sides  $AB$ ,  $BC$ ,  $CD$ ,  $DE$ ,  $EF$ , and  $FA$ , respectively. The intersection of lines  $\overline{AH}$ ,  $\overline{BI}$ ,  $\overline{CJ}$ ,  $\overline{DK}$ ,  $\overline{EL}$ , and  $\overline{FG}$  bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of  $ABCDEF$ .


Let  $A$  and  $B$  be two finite sets such that there are exactly  $144$  sets which are subsets of  $A$  or subsets of  $B$ . Find the number of elements in  $A \cup B$ .


Ed and Sue bike at equal and constant rates.  Similarly, they jog at equal and constant rates, and they swim at equal and constant rates.  Ed covers  $ 74$  kilometers after biking for  $ 2$  hours, jogging for  $ 3$  hours, and swimming for  $ 4$  hours, while Sue covers  $ 91$  kilometers after jogging for  $ 2$  hours, swimming for  $ 3$  hours, and biking for  $ 4$  hours.  Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour.  Find the sum of the squares of Ed's biking, jogging, and swimming rates.


Let  $n$  be a positive integer. E. Chen and E. Chen play a game on the  $n^2$  points of an  $n \times n$  lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of  $n$  between  $1$  and  $2013$  (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.

*Ray Li*


Let  $(x_n)$  be a sequence of positive integers defined as follows:  $x_1$  is a fixed six-digit number and for any  $n \geq 1$ ,  $x_{n+1}$  is a prime divisor of  $x_n + 1$ . Find  $x_{19} + x_{20}$ .


Let f be a function that satisfies the following conditions: $(i)$  If  $x > y$  and  $f(y) - y \geq v \geq f(x) - x$ , then  $f(z) = v + z$ , for some number  $z$  between  $x$  and  $y$ . $(ii)$  The equation  $f(x) = 0$  has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$   $f(0) = 1$ . $(iv)$   $f(1987) \leq 1988$ . $(v)$   $f(x)f(y) = f(xf(y) + yf(x) - xy)$ .

Find  $f(1987)$ .

*Proposed by Australia.*


What is the least real value of the expression  $\sqrt{x^2-6x+13} + \sqrt{x^2-14x+58}$  where  $x$  is a real number? $
\textbf{(A)}\ \sqrt {39}
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ \frac {43}6
\qquad\textbf{(D)}\ 2\sqrt 2 + \sqrt {13}
\qquad\textbf{(E)}\ \text{None of the above}
$


Let  $ABCD$  be a cyclic quadrilateral with  $\angle ADC = \angle DBA$ . Furthermore, let  $E$  be the projection of  $A$  on  $BD$ . Show that  $BC = DE - BE$  .

Solution: \boxed{ BC = DE - BE }


Let  $S_n$  be the sum of reciprocal values of non-zero digits of all positive integers up to (and including)  $n$ . For instance,  $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$  .
Find the least positive integer  $k$  making the number  $k!\cdot  S_{2016}$  an integer.

Solution: \boxed{ 7 }


Let  $x\geq 5, y\geq 6, z\geq 7$  such that  $x^2+y^2+z^2\geq 125$ . Find the minimum value of  $x+y+z$ .


Define  $f(x,y)=\frac{xy}{x^2+y^2\ln(x^2)^2}$  if  $x\ne0$ , and  $f(0,y)=0$  if  $y\ne0$ . Determine whether  $\lim_{(x,y)\to(0,0)}f(x,y)$  exists, and find its value is if the limit does exist.


Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster *consumes* cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many minutes does it take Oscar the Grouch to bake one cookie?


Jackie and Phil have two fair coins and a third coin that comes up heads with probability  $ \frac47$ . Jackie flips the three coins, and then Phil flips the three coins. Let  $ \frac{m}{n}$  be the probability that Jackie gets the same number of heads as Phil, where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m \plus{} n$ .


Let  $S$  be area of the given parallelogram  $ABCD$  and the points  $E,F$  belong to  $BC$  and  $AD$ , respectively, such that  $BC = 3BE, 3AD = 4AF$ . Let  $O$  be the intersection of  $AE$  and  $BF$ . Each straightline of  $AE$  and  $BF$  meets that of  $CD$  at points  $M$  and  $N$ , respectively. Determine area of triangle  $MON$ .


There are  $100$  students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly  $20$  students. The  $100$  students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?


Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is  $\frac{C - T + S}{M}$  ?


Let  $n$  be a positive integer greater than  $2$ . Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where  $x_{n+1}=x_{1}$  and  $x_{n+2}=x_{2}$ .

Solution: \boxed{ x_i = 2 }


Bread draws a circle. He then selects four random distinct points on the circumference of the circle to form a convex quadrilateral. Kwu comes by and randomly chooses another 3 distinct points (none of which are the same as Bread's four points) on the circle to form a triangle. Find the probability that Kwu's triangle does not intersect Bread's quadrilateral, where two polygons intersect if they have at least one pair of sides intersecting.

*Proposed by Nathan Cho*


Find all functions  $f:\mathbb{N}\rightarrow \mathbb{N}$  such that the inequality   $$ f(x)+yf(f(x))\le x(1+f(y)) $$  holds for all positive integers  $x, y$ .

Proposed by Adrian Beker.

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


Let  $ABCD$  be a rectangle with side lengths  $AB = CD = 5$  and  $BC = AD = 10$ .  $W, X, Y, Z$  are points on  $AB, BC, CD$  and  $DA$  respectively chosen in such a way that  $WXYZ$  is a kite, where  $\angle ZWX$  is a right angle. Given that  $WX = WZ = \sqrt{13}$  and  $XY = ZY$ , determine the length of  $XY$ .


Find the maximum value of the real constant  $C$  such that  $x^{2}+y^{2}+1\geq C(x+y)$ , and  $ x^{2}+y^{2}+xy+1\geq  C(x+y)$  for all reals  $x,y$ .


What is the smallest prime number  $p$  such that  $p^3+4p^2+4p$  has exactly  $30$  positive divisors ?


A sequence of real numbers  $ x_0, x_1, x_2, \ldots$  is defined as follows:  $ x_0 \equal{} 1989$  and for each  $ n \geq 1$

\[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\]

Calculate the value of  $ \sum^{1989}_{n\equal{}0} 2^n x_n.$


Every member, starting from the third one, of two sequences  $\{a_n\}$  and  $\{b_n\}$  equals to the sum of two preceding ones. First members are:  $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$ . How many natural numbers are encountered in both sequences (may be on the different places)?


Let  $\alpha$  and  $\beta$  be reals. Find the least possible value of   $$ (2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2. $$


Given is an acute angled triangle  $ ABC$ . Determine all points  $ P$  inside the triangle with
\[1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2\]

Solution: \boxed{ O }


An altitude  $h$  of a triangle is increased by a length  $m$ . How much must be taken from the corresponding base  $b$  so that the area of the new triangle is one-half that of the original triangle?  ${{ \textbf{(A)}\ \frac{bm}{h+m}\qquad\textbf{(B)}\ \frac{bh}{2h+2m}\qquad\textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad\textbf{(D)}\ \frac{b(m+h)}{2m+h} }\qquad\textbf{(E)}\ \frac{b(2m+h)}{2(h+m)} } $

Solution: \boxed{\textbf{(E)}}


The domain of the function  $f(x) = \text{arcsin}(\log_{m}(nx))$  is a closed interval of length  $\frac{1}{2013}$ , where  $m$  and  $n$  are positive integers and  $m > 1$ . Find the remainder when the smallest possible sum  $m+n$  is divided by  $1000$ .


The diagonals of convex quadrilateral  $BSCT$  meet at the midpoint  $M$  of  $\overline{ST}$ .  Lines  $BT$  and  $SC$  meet at  $A$ , and  $AB = 91$ ,  $BC = 98$ ,  $CA = 105$ .  Given that  $\overline{AM} \perp \overline{BC}$ , find the positive difference between the areas of  $\triangle SMC$  and  $\triangle BMT$ .

*Proposed by Evan Chen*


Consider a sequence of eleven squares that have side lengths  $3, 6, 9, 12,\ldots, 33$ . Eleven copies of a single square each with area  $A$  have the same total area as the total area of the eleven squares of the sequence. Find  $A$ .


Let  $0\leq p,r\leq 1$  and consider the identities  $$ a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2. $$  Show that  $$  a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}. $$


Find the least real number  $m$  such that with all  $5$  equilaterial triangles with sum of areas  $m$  we can cover an equilaterial triangle with side 1.

*O. Mushkarov, N. Nikolov*


A positive integer  $k$  is  $2014$ -ambiguous if the quadratics  $x^2+kx+2014$  and  $x^2+kx-2014$  both have two integer roots. Compute the number of integers which are  $2014$ -ambiguous.


There are 51 senators in a senate. The senate needs to be divided into  $n$  committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does *not* necessarily hate senator A.) Find the smallest  $n$  such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.


How many ordered pairs of integers  $(m,n)$  are there such that  $m$  and  $n$  are the legs of a right triangle with an area equal to a prime number not exceeding  $80$ ?


Let  $ n > 1, n \in \mathbb{Z}$  and  $ B \equal{}\{1,2,\ldots, 2^n\}.$  A subset  $ A$  of  $ B$  is called weird if it contains exactly one of the distinct elements  $ x,y \in B$  such that the sum of  $ x$  and  $ y$  is a power of two. How many weird subsets does  $ B$  have?

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


Let  $T$  be a triangle with inscribed circle  $C.$  A square with sides of length  $a$  is circumscribed about the same circle  $C.$  Show that the total length of the parts of the edge of the square interior to the triangle  $T$  is at least  $2 \cdot a.$

Solution: \boxed{ 2a }


Let  $p$  be a prime number, and define a sequence by:  $x_i=i$  for  $i=,0,1,2...,p-1$  and  $x_n=x_{n-1}+x_{n-p}$  for  $n \geq p$
Find the remainder when  $x_{p^3}$  is divided by  $p$ .


For a positive integer  $n$ , let  $s(n)$  denote the sum of the binary digits of  $n$ . Find the sum  $s(1)+s(2)+s(3)+...+s(2^k)$  for each positive integer  $k$ .

Solution: \boxed{ 2^{k-1}k + 1 }


A quadruplet of distinct positive integers  $(a, b, c, d)$  is called  $k$ -good if the following conditions hold:

1. Among  $a, b, c, d$ , no three form an arithmetic progression.

2. Among  $a+b, a+c, a+d, b+c, b+d, c+d$ , there are  $k$  of them, forming an arithmetic progression. $a)$  Find a  $4$ -good quadruplet. $b)$  What is the maximal  $k$ , such that there is a  $k$ -good quadruplet?

Solution: \boxed{ k = 4 }


Let  $T$  denote the  $15$ -element set  $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let  $S$  be a subset of  $T$  in which all six digits  $1,2,\ldots ,6$  appear and in which no three elements together use all these six digits. Determine the largest possible size of  $S$ .


For a positive integer  $N$ , let  $f_N$  be the function defined by
\[ f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right). \]
Determine the smallest constant  $M$  such that  $f_N (x)\le M$  for all  $N$  and all real  $x$ .


Find all functions  $f:\mathbb Z\to \mathbb Z$  such that for all surjective functions  $g:\mathbb Z\to \mathbb Z$ ,  $f+g$  is also surjective. (A function  $g$  is surjective over  $\mathbb Z$  if for all integers  $y$ , there exists an integer  $x$  such that  $g(x)=y$ .)

*Proposed by Sean Li*

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


Let  $ a,b$  and  $ c$  real numbers such that the equation  $ ax^2\plus{}bx\plus{}c\equal{}0$  has two distinct real solutions  $ p_1,p_2$  and the equation  $ cx^2\plus{}bx\plus{}a\equal{}0$  has two distinct real solutions  $ q_1,q_2$ . We know that the numbers  $ p_1,q_1,p_2,q_2$  in that order, form an arithmetic progression. Show that  $ a\plus{}c\equal{}0$ .

Solution: \boxed{ a + c = 0 }


Five non-overlapping equilateral triangles meet at a common vertex so that the angles between adjacent triangles are all congruent. What is the degree measure of the angle between two adjacent triangles?

[asy]
size(100);
defaultpen(linewidth(0.7));
path equi=dir(300)--dir(240)--origin--cycle;
for(int i=0;i<=4;i=i+1)
draw(rotate(72*i,origin)*equi);
[/asy]


When Meena turned 16 years old, her parents gave her a cake with  $n$  candles, where  $n$  has exactly 16 different positive integer divisors.  What is the smallest possible value of  $n$ ?


Alexander has chosen a natural number  $N>1$  and has written down in a line,and in increasing order,all his positive divisors  $d_1<d_2<\ldots <d_s$  (where  $d_1=1$  and  $d_s=N$ ).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these  $s-1$  numbers (the greatest common divisors) is equal to  $N-2$ .Find all possible values of  $N$ .


Let  $a_1$ ,  $a_2$ ,  $\cdots$  be a sequence such that  $a_1=a_2=\frac 15$ , and for  $n \ge 3$ ,   $$ a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}. $$   Find the smallest integer  $n$  such that  $a_n>1-5^{-2022}$ .


We inscribe a circle  $\omega$  in equilateral triangle  $ABC$  with radius  $1$ . What is the area of the region inside the triangle but outside the circle?

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


Niki has  $15$  dollars more than twice as much money as her sister Amy. If Niki gives Amy  $30$  dollars, then Niki will have hals as much money as her sister. How many dollars does Niki have?


Find the number of positive integers less than or equal to  $2017$  that have at least one pair of adjacent digits that are both even. For example, count the numbers  $24$ ,  $1862$ , and  $2012$ , but not  $4$ ,  $58$ , or  $1276$ .


How many lattice points  $(v, w, x, y, z)$  does a  $5$ -sphere centered on the origin, with radius  $3$ , contain on its surface or in its interior?


A  $8\times 8$  board is given. Seven out of  $64$  unit squares are painted black. Suppose that there exists a positive  $k$  such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area  $k$  containing no black squares. Find the maximum value of  $k$ .


In the unit squre For the given natural number  $n \geq 2$  find the smallest number  $k$  that from each set of  $k$  unit squares of the  $n$ x $n$  chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even


Find all real parameters  $p$  for which the equation  $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$ has three distinct real roots which are sides of a right triangle.

Solution: \boxed{ p = \sqrt{2} }


Let  $ABC$  be a triangle with sides  $AB = 6$ ,  $BC = 10$ , and  $CA = 8$ . Let  $M$  and  $N$  be the midpoints of  $BA$  and  $BC$ , respectively. Choose the point  $Y$  on ray  $CM$  so that the circumcircle of triangle  $AMY$  is tangent to  $AN$ . Find the area of triangle  $NAY$ .


A convex quadrilateral is determined by the points of intersection of the curves  $x^4+y^4=100$  and  $xy=4$ ; determine its area.


If a square is partitioned into  $ n$  convex polygons, determine the maximum possible number of edges in the obtained figure.

(You may wish to use the following theorem of Euler: If a polygon is partitioned into  $ n$  polygons with  $ v$  vertices and  $ e$  edges in the resulting figure, then  $ v\minus{}e\plus{}n\equal{}1$ .)

Solution: \boxed{ 3n + 1 }


A box contains answer  $4032$  scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among
which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.


Let  $n$  be an positive integer. Find the smallest integer  $k$  with the following property; Given any real numbers  $a_1 , \cdots , a_d $  such that  $a_1 + a_2 + \cdots + a_d = n$  and  $0 \le a_i \le 1$  for  $i=1,2,\cdots ,d$ , it is possible to partition these numbers into  $k$  groups (some of which may be empty) such that the sum of the numbers in each group is at most  $1$ .

Solution: \boxed{ 2n-1 }


 The deputies in a parliament were split into  $10$  fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation.


Find the smallest natural  $ k $  such that among any  $ k $  distinct and pairwise coprime naturals smaller than  $ 2018, $  a prime can be found.

*Vlad Robu*


If the GCD of  $a$  and  $b$  is  $12$  and the LCM of  $a$  and  $b$  is  $168$ , what is the value of  $a\times b$ ?

*2016 CCA Math Bonanza L1.3*


The quadrilateral  $ABCD$  is an isosceles trapezoid with  $AB = CD = 1$ ,  $BC = 2$ , and  $DA = 1+ \sqrt{3}$ .  What is the measure of  $\angle ACD$  in degrees?


\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\]
where  $n$  is an positive integer. Find the largest  $k$  such that  $n^k$  divides  $[A]$ .


Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?

Solution: \boxed{ n = 13 }


Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity  $ x$  and inversely proportional to  $ y$ , the number of hours he slept the night before. If the price of HMMT is  $ \$ 12 $ when $  x\equal{}8 $ and $  y\equal{}4 $, how many dollars does it cost when $  x\equal{}4 $ and $  y\equal{}8$?


For each permutation  $ a_1, a_2, a_3, \ldots,a_{10}$  of the integers  $ 1,2,3,\ldots,10,$  form the sum
\[ |a_1 \minus{} a_2| \plus{} |a_3 \minus{} a_4| \plus{} |a_5 \minus{} a_6| \plus{} |a_7 \minus{} a_8| \plus{} |a_9 \minus{} a_{10}|.\]
The average value of all such sums can be written in the form  $ p/q,$  where  $ p$  and  $ q$  are relatively prime positive integers.  Find  $ p \plus{} q.$


Two circles have radii  $15$  and  $95$ . If the two external tangents to the circles intersect at  $60$  degrees, how far apart are the centers of the circles?


Five rays  $\overrightarrow{OA}$ , $\overrightarrow{OB}$ ,  $\overrightarrow{OC}$ ,  $\overrightarrow{OD}$ , and  $\overrightarrow{OE}$   radiate in a clockwise order from  $O$  forming four non-overlapping angles such that  $\angle EOD = 2\angle COB$ ,  $\angle COB = 2\angle BOA$ , while  $\angle DOC = 3\angle BOA$ . If  $E$ ,  $O$ ,  $A$  are collinear with  $O$  between  $A$  and  $E$ , what is the degree measure of  $\angle DOB?$


Determine all functions  $f: \mathbb N\to \mathbb N$  such that for every positive integer  $n$  we have: \[ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. \]

Solution: \boxed{ f(n) = n + 667 }


Positive integers  $a,b,c$  exist such that  $a+b+c+1$ ,  $a^2+b^2+c^2 +1$ ,  $a^3+b^3+c^3+1,$  and  $a^4+b^4+c^4+7459$  are all multiples of  $p$  for some prime  $p$ . Find the sum of all possible values of  $p$  less than  $1000$ .


Starting with a list of three numbers, the “*Make-My-Day*” procedure creates a new list by replacing each number by the sum of the other two. For example, from  $\{1, 3, 8\}$  “*Make-My-Day*” gives  $\{11, 9, 4\}$  and a new “*MakeMy-Day*” leads to  $\{13, 15, 20\}$ . If we begin with  $\{20, 1, 8\}$ , what is the maximum difference between two numbers on the list after  $2018$  consecutive “*Make-My-Day*”s?


Let  $\Gamma_1, \Gamma_2, \Gamma_3$  be three pairwise externally tangent circles with radii  $1,2,3,$  respectively. A circle passes through the centers of  $\Gamma_2$  and  $\Gamma_3$  and is externally tangent to  $\Gamma_1$  at a point  $P.$  Suppose  $A$  and  $B$  are the centers of  $\Gamma_2$  and  $\Gamma_3,$  respectively. What is the value of  $\frac{{PA}^2}{{PB}^2}?$ *Proposed by Kyle Lee*


Each pair  $(x, y)$  of nonnegative integers is assigned number  $f(x, y)$  according the conditions: $f(0, 0) = 0$ ; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$ , $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$  for  $x, y \ge 0$ .
Let  $n$  be a fixed nonnegative integer and let  $a$ ,  $b$  be nonnegative integers such that  $f(a, b) = n$ . Decide how many numbers satisfy the equation  $f(a, x) + f(b, x) = n$ .


Find all pairs  $ (p, q)$  of primes such that  $ {p}^{p}\plus{}{q}^{q}\plus{}1$  is divisible by  $ pq$ .

Solution: \boxed{(5, 2)}


Let  $ABC$  be an acute-angled, nonisosceles triangle. Altitudes  $AA'$  and  $BB' $ meet at point  $H$ , and the medians of triangle  $AHB$  meet at point  $M$ . Line  $CM$  bisects segment  $A'B'$ . Find angle  $C$ .

(D. Krekov)


A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:

*The  $50$  lightest frogs represented  $30\%$  of the total mass of all the frogs in the pond, while the  $44$  heaviest frogs represented  $27\%$  of the total mass.*As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?


A plane has several seats on it, each with its own price, as shown below(attachment).  $2n-2$  passengers wish to take this plane, but none of them wants to sit with any other passenger in the same column or row. The captain realize that, no matter how he arranges the passengers, the total money he can collect is the same. Proof this fact, and compute how much money the captain can collect.


Let  $R$  be the rectangle on the cartesian plane with vertices  $(0,0)$ ,  $(5,0)$ ,  $(5,7)$ , and  $(0,7)$ . Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by  $R$ .

*Proposed by Boyan Litchev*

Solution*Solution*.  $\boxed{85}$ We have  $(6-n)(8-n)$  distinct squares with side length  $n$ , so the total number of squares is  $5 \cdot 7+4 \cdot 6+3 \cdot 5+2 \cdot 4+1\cdot 3 = \boxed{85}$ .


For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}  \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of  $\sum_{k=1}^{200} f(k)$ .


**3/1/32.** The bisectors of the internal angles of parallelogram  $ABCD$  determine a quadrilateral with the same area as  $ABCD$ . Given that  $AB > BC$ , compute, with proof, the ratio  $\frac{AB}{BC}$ .

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


For which pairs of positive integers  $(m,n)$  there exists a set  $A$  such that for all positive integers  $x,y$ , if  $|x-y|=m$ , then at least one of the numbers   $x,y$  belongs to the set  $A$ , and if  $|x-y|=n$ , then at least one of the numbers  $x,y$  does not belong to the set  $A$ ?

*Adapted by Dan Schwarz from A.M.M.*

Solution: \boxed{ (m, n) }


Let  $k$  be a positive real number. In the   $X-Y$  coordinate plane, let  $S$  be the set of all points of the form  $(x,x^2+k)$  where  $x\in\mathbb{R}$ . Let  $C$  be the set of all circles whose center lies in  $S$ , and which are tangent to  $X$ -axis. Find the minimum value of  $k$  such that any two circles in  $C$  have at least one point of intersection.


For each nonnegative integer  $n$  we define  $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers  $A_0,A_1,\ldots, A_{1999}$ .

*Romania*


For any positive integer  $n$ , let  $a_n$  be the number of pairs  $(x,y)$  of integers satisfying  $|x^2-y^2| = n$ .
(a) Find  $a_{1432}$  and  $a_{1433}$ .
(b) Find  $a_n$  .

Solution: \boxed{ a_{1432} = 8 }


For positive reals  $p$  and  $q$ , define the *remainder* when  $p$  and  $q$  as the smallest nonnegative real  $r$  such that  $\tfrac{p-r}{q}$  is an integer. For an ordered pair  $(a, b)$  of positive integers, let  $r_1$  and  $r_2$  be the remainder when  $a\sqrt{2} + b\sqrt{3}$  is divided by  $\sqrt{2}$  and  $\sqrt{3}$  respectively. Find the number of pairs  $(a, b)$  such that  $a, b \le 20$  and  $r_1 + r_2 = \sqrt{2}$ .


Let  $S(N)$  be the number of 1's in the binary representation of an integer  $N$ , and let  $D(N) = S(N + 1) - S(N)$ .  Compute the sum of  $D(N)$  over all  $N$  such that  $1 \le N \le 2017$  and  $D(N) < 0$ .


Betty Lou and Peggy Sue take turns flipping switches on a  $100 \times 100$  grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.


On a blackboard , the 2016 numbers  $\frac{1}{2016} , \frac{2}{2016} ,... \frac{2016}{2016}$  are written.
One can perfurm the following operation : Choose any numbers in the blackboard, say  $a$  and $ b$  and replace them by  $2ab-a-b+1$ .
After doing 2015 operation , there will only be one number  $t$  Onthe blackboard .
Find all possible values of  $ t$ .


Let  $p$  be a fixed odd prime. A  $p$ -tuple  $(a_1,a_2,a_3,\ldots,a_p)$  of integers is said to be *good* if

- **(i)**  $0\le a_i\le p-1$  for all  $i$ , and
- **(ii)**  $a_1+a_2+a_3+\cdots+a_p$  is not divisible by  $p$ , and
- **(iii)**  $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$  is divisible by  $p$ .

Determine the number of good  $p$ -tuples.


Find the total number of solutions to the following system of equations:

 $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37}
b(a + d)\equiv b \pmod{37}
c(a + d)\equiv c \pmod{37}
bc + d^2\equiv d \pmod{37}
ad - bc\equiv 1 \pmod{37} \end{array}$


A round table has  $2 N$  chairs around it. Due to social distancing guidelines, no two people are allowed to sit next to each other. How many different ways are there to choose seats around the table on which  $N-1$  guests can be seated?


Let  $\lfloor x \rfloor$  denote the largest integer less than or equal to  $x$ , and let  $\{x\}$  denote the fractional part of  $x$ . For example,  $\lfloor \pi \rfloor=3$ , and  $\{\pi\}=0.14159\dots$ , while  $\lfloor 100 \rfloor=100$  and  $\{100\}=0$ . If  $n$  is the largest solution to the equation  $\frac{\lfloor n \rfloor}{n}=\frac{2015}{2016}$ , compute  $\{n\}$ .


[4] Let  $D$  be the set of divisors of  $100$ . Let  $Z$  be the set of integers between  $1$  and  $100$ , inclusive. Mark chooses an element  $d$  of  $D$  and an element  $z$  of  $Z$  uniformly at random. What is the probability that  $d$  divides  $z$ ?


Convex quadrilateral  $ABCD$  satisfies  $\angle{CAB} = \angle{ADB} = 30^{\circ}, \angle{ABD} = 77^{\circ}, BC = CD$  and  $\angle{BCD} =n^{\circ}$  for some positive integer  $n$ . Compute  $n$ .


Determine all positive integers  $n$  with the property that the third root of  $n$  is obtained by removing its last three decimal digits.

Solution: \boxed{ n = 32768 }


Let  $ABCD$  be a square and  $P$  be a point on the shorter arc  $AB$  of the circumcircle of the square. Which values can the expression  $\frac{AP+BP}{CP+DP}$  take?

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


Given a positive integer  $ n\geq 2$ , let  $ B_{1}$ ,  $ B_{2}$ , ...,  $ B_{n}$  denote  $ n$  subsets of a set  $ X$  such that each  $ B_{i}$  contains exactly two elements. Find the minimum value of  $ \left|X\right|$  such that for any such choice of subsets  $ B_{1}$ ,  $ B_{2}$ , ...,  $ B_{n}$ , there exists a subset  $ Y$  of  $ X$  such that:
 (1)  $ \left|Y\right| \equal{} n$ ;
 (2)  $ \left|Y \cap B_{i}\right|\leq 1$  for every  $ i\in\left\{1,2,...,n\right\}$ .

Solution: \boxed{ 2n - 1 }


Find the number of three-digit positive integers where the digits are three different prime numbers. For example, count 235 but not 553.


Find all polynomials  $P$  with integer coefficients such that  $P (0)\ne  0$  and   $$ P^n(m)\cdot P^m(n) $$   is a square of an integer for all nonnegative integers  $n, m$ .

*Remark:* For a nonnegative integer  $k$  and an integer  $n$ ,  $P^k(n)$  is defined as follows:  $P^k(n) = n$  if  $k = 0$  and  $P^k(n)=P(P(^{k-1}(n))$  if  $k >0$ .

Proposed by Adrian Beker.

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


**Evaluate**

  $\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$


What is the last two digits of the number  $(11^2 + 15^2 + 19^2 +  ...  + 2007^2)^2$ ?


A natural number  $n$  is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number  $n$ . Find all numbers  $n$  with this property.


For a positive integer  $n$  determine all  $n\times n$  real matrices  $A$  which have only real eigenvalues and such that there exists an integer  $k\geq n$  with  $A + A^k = A^T$ .

Solution: \boxed{A = 0}


Find the number of ways to color  $n \times m$  board with white and black
colors such that any  $2 \times 2$  square contains the same number of black and white cells.


Find  $A^2$ , where  $A$  is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}


Point  $P$  is in the interior of  $\triangle ABC$ . The side lengths of  $ABC$  are  $AB = 7$ ,  $BC = 8$ ,  $CA = 9$ . The three foots of perpendicular lines from  $P$  to sides  $BC$ ,  $CA$ ,  $AB$  are  $D$ ,  $E$ ,  $F$  respectively. Suppose the minimal value of  $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$  can be written as  $\frac{a}{b}\sqrt{c}$ , where  $\gcd(a,b) = 1$  and  $c$  is square free, calculate  $abc$ .
[asy]
size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black;
pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B);
D(A--B--C--cycle); D(P--D); D(P--E); D(P--F);
D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE));
[/asy]


Set  $M$  contains  $n \ge 2$  positive integers. It's known that for any two different  $a, b \in M$ ,  $a^2+1$  is divisible by  $b$ . What is the largest possible value of  $n$ ?

*Proposed by Oleksiy Masalitin*


Consider an  $n\times{n}$  grid formed by  $n^2$  unit squares. We define the centre of a unit square as the intersection of its diagonals.
Find the smallest integer  $m$  such that, choosing any  $m$  unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram.


Let  $ABCD$  be a rectangle. We consider the points  $E\in CA,F\in AB,G\in BC$  such that  $DC\perp CA,EF\perp AB$  and  $EG\perp BC$ . Solve in the set of rational numbers the equation  $AC^x=EF^x+EG^x$ .


Let  $S_0 = \varnothing$  denote the empty set, and define  $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$  for every positive integer  $n$ .  Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]

*Proposed by Evan Chen*


$\textbf{Problem 4.}$  What is the largest number such that dividing  $17$  or  $30$  by this number, the same remainder is obtained in both cases


A square sheet of paper that measures  $18$  cm on a side has corners labeled  $A$ ,  $B$ ,  $C$ , and  $D$  in clockwise order. Point  $B$  is folded over to a point  $E$  on  $\overline{AD}$  with  $DE=6$  cm and the paper is creased. When the paper is unfolded, the crease intersects side  $\overline{AB}$  at  $F$ . Find the number of centimeters in  $FB$ .


**a)** Let  $ a,b $  two non-negative integers such that  $ a^2>b. $  Show that the equation  $$  \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1  $$  has an infinite number of solutions in the non-negative integers. Here,  $ \lfloor\alpha\rfloor $  denotes the floor of  $ \alpha. $ **b)** Find the floor of  $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $  where  $ n $  is a natural number. Justify.

Solution: \boxed{ 1 }


Jay has a  $24\times 24$  grid of lights, all of which are initially off. Each of the  $48$  rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the  $48$  rows and columns exactly once, such that after each toggle he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules which has been expended by all the lights after all  $48$  toggles.

[i]Proposed by James Lin


Determine the real number  $a$  having the property that  $f(a)=a$  is a relative minimum of  $f(x)=x^4-x^3-x^2+ax+1$ .

Solution: \boxed{ a = 1 }


Let ABCD be a trapezoid with  $AB \parallel CD, AB = 5, BC = 9, CD = 10,$  and  $DA = 7$ . Lines  $BC$  and  $DA$  intersect at point  $E$ . Let  $M$  be the midpoint of  $CD$ , and let  $N$  be the intersection of the circumcircles of  $\triangle BMC$  and  $\triangle DMA$  (other than  $M$ ). If  $EN^2 = \tfrac ab$  for relatively prime positive integers  $a$  and  $b$ , compute  $100a + b$ .


Find the largest  $a$  for which there exists a polynomial  $$ P(x) =a x^4 +bx^3 +cx^2 +dx +e $$  with real coefficients which satisfies  $0\leq P(x) \leq 1$  for  $-1 \leq x \leq 1.$


If  $f(n)$  denotes the number of divisors of  $2024^{2024}$  that are either less than  $n$  or share at least one prime factor with  $n$ , find the remainder when   $$ \sum^{2024^{2024}}_{n=1} f(n) $$   is divided by  $1000$ .


Let  $A,B,C$  be the midpoints of the three sides  $B'C', C'A', A'B'$  of the triangle  $A'B'C'$  respectively. Let  $P$  be a point inside  $\Delta ABC$ , and  $AP,BP,CP$  intersect with  $BC, CA, AB$  at  $P_a,P_b,P_c$ , respectively. Lines  $P_aP_b, P_aP_c$  intersect with  $B'C'$  at  $R_b, R_c$  respectively, lines  $P_bP_c, P_bP_a$  intersect with  $C'A'$  at  $S_c, S_a$  respectively. and lines  $P_cP_a, P_cP_b$  intersect with  $A'B'$  at  $T_a, T_b$ , respectively. Given that  $S_c,S_a, T_a, T_b$  are all on a circle centered at  $O$ .

Show that  $OR_b=OR_c$ .

Solution: \boxed{OR_b = OR_c}


In convex quadrilateral  $ABCD$ ,  $\angle ADC = 90^\circ + \angle BAC$ .  Given that  $AB = BC = 17$ , and  $CD = 16$ , what is the maximum possible area of the quadrilateral?

*Proposed by Thomas Lam*


Let  $d(m)$  denote the number of positive integer divisors of a positive integer  $m$ . If  $r$  is the number of integers   $n \leqslant 2023$  for which  $\sum_{i=1}^{n} d(i)$  is odd. , find the sum of digits of  $r.$


Compute the  $\textit{number}$  of ordered quadruples  $(w,x,y,z)$  of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1
wxy^2 + wx^2z + w^2yz + xyz^2 &=2
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*}


Let  $m_n$  be the smallest value of the function  ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$
Show that  $m_n  \to \frac{1}{2}$ , as  $n \to \infty.$


Let  $ABC$  be a triangle in the plane with  $AB = 13$ ,  $BC = 14$ ,  $AC = 15$ . Let  $M_n$  denote the smallest possible value of  $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$  over all points  $P$  in the plane. Find  $\lim_{n \to \infty} M_n$ .


Let  $ d_1,d_2 \ldots, d_r$  be the positive divisors of  $ n$
 $ 1\equal{}d_1<d_2< \ldots <d_r\equal{}n$
If  $ (d_7)^2 \plus{} (d_{15})^2\equal{} (d_{16})^2$  find all posible values of  $ d_{17}$

Solution: \boxed{ d_{17} = 28 }


Determine all polynomials  $P(x)$  with degree  $n\geq 1$  and integer coefficients so that for every real number  $x$  the following condition is satisfied   $$ P(x)=(x-P(0))(x-P(1))(x-P(2))\cdots (x-P(n-1)) $$

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


A triangle has vertices  $P=(-8,5)$ ,  $Q=(-15,-19)$ , and  $R=(1,-7)$ . The equation of the bisector of  $\angle P$  can be written in the form  $ax+2y+c=0$ . Find  $a+c$ .


A semicircle with diameter length  $16$  contains a circle radius  $3$  tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written as  $\frac{n + k\sqrt{2}}{m}$  where  $m$ ,  $n$ , and  $k$  are positive integers and  $m$  and  $n$  have no factors in common. Find  $m + n + k$ .

[asy]
size(200);
pair O=(0,0);
real R=10, r=4.7;
draw(arc(O,R,0,180)--cycle);
pair P=(sqrt((R-r)^2-r^2),r),Q;
draw(circle(P,r));
real a=0,b=r,c;
for(int k=0;k<20;++k)
{
c=(a+b)/2;
Q=(-sqrt((R-c)^2-c^2),c);
if(abs(P-Q)>c+r) a=c; else b=c;
}
draw(circle(Q,c));[/asy]


Let points  $A,B,$  and  $C$  lie on a line such that  $AB=1, BC=1,$  and  $AC=2.$  Let  $C_1$  be the circle centered at  $A$  passing through  $B,$  and let  $C_2$  be the circle centered at  $A$  passing through  $C.$  Find the area of the region outside  $C_1,$  but inside  $C_2.$


Math teacher wrote in a table a polynomial  $P(x)$  with integer coefficients and he said:
"Today my daughter have a birthday.If in polynomial  $P(x)$  we have  $x=a$  where  $a$  is the age of my daughter we have  $P(a)=a$  and  $P(0)=p$  where  $p$  is a prime number such that  $p>a$ ."
How old is the daughter of math teacher?


In a table consisting of  $n$  by  $n$  small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour.
What is the largest possible value of  $n$ ?

Solution: \boxed{ n = 4 }


Find all primes  $ p$  for that there is an integer  $ n$  such that there are no integers  $ x,y$  with  $ x^3 \plus{} y^3 \equiv n \mod p$  (so not all residues are the sum of two cubes).

E.g. for  $ p \equal{} 7$ , one could set  $ n \equal{} \pm 3$  since  $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$ , thus  $ x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$  only.


Let  $ABCDEF$  be a regular hexagon with sidelength  $6$ , and construct squares  $ABGH$ ,  $BCIJ$ ,  $CDKL$ ,  $DEMN$ ,  $EFOP$ , and  $FAQR$  outside the hexagon. Find the perimeter of dodecagon  $HGJILKNMPORQ$ .

*Proposed by Andrew Wu*


For a positive integer  $n$ , let  $v(n)$  denote the largest integer  $j$  such that  $n$  is divisible by  $2^j$ .  Let  $a$  and  $b$  be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive.  What is the probability that  $v(a) > v(b)$ ?


Find all triples of pairwise distinct positive integers  $(a,b,c)$ , which satisfy the following conditions:  $2a-1$  is divisible by  $b$ ,  $2b-1$  is divisible  $c$  and  $2c-1$  is divisible by  $a$ .

*Proposed by Bohdan Rublyov*

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


In triangle  $ABC$ ,  $AB=13$ ,  $BC=14$  and  $CA=15$ .  Segment  $BC$  is split into  $n+1$  congruent segments by  $n$  points.  Among these points are the feet of the altitude, median, and angle bisector from  $A$ . Find the smallest possible value of  $n$ .

*Proposed by Evan Chen*

Solution: \boxed{ 27 }


Let  $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$ .
(a) Show that  $s(n)$  is an integer whenever  $n$  is an integer.
(b) How many integers  $n$  with  $0 < n \le 2008$  are such that  $s(n)$  is divisible by  $4$ ?


Find the greatest real number  $k$  such that, for any positive  $a,b,c$  with  $a^{2}>bc$ ,  $(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)$ .


 $m$  boys and  $n$  girls ( $m>n$ ) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen?

*Hendrata Dharmawan, Boston, USA*

Solution: \boxed{m - n}


Point  $A$  is located at  $(0,0)$ . Point  $B$  is located at  $(2,3)$  and is the midpoint of  $AC$ . Point  $D$  is located at  $(10,0)$ . What are the coordinates of the midpoint of segment  $CD$ ?

Solution: \boxed{(7, 3)}


The point  $M$  varies on the segment  $AB$  that measures  $2$  m.
a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates,  $x$ , and  $y$ , are, respectively, the areas of the squares of sides  $AM$  and  $MB$  .
b) Find out what kind of curve it is. (Suggestion: make a  $45^o$  axis rotation).
c) Find the area of the enclosure between the curve obtained and the coordinate axes.


$x,y$  are real numbers such that   $$ x^2+y^2=1    ,  20x^3-15x=3 $$  Find the value of  $|20y^3-15y|$ .(K. Tyshchuk)


Given a finite set  $S \subset \mathbb{R}^3$ , define  $f(S)$  to be the mininum integer  $k$  such that there exist  $k$  planes that divide  $\mathbb{R}^3$  into a set of regions, where no region contains more than one point in  $S$ . Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate  $M(200) \cdot m(200)$ .


Let  $[n] = \{1, 2, 3, ... ,n\}$  and for any set  $S$ , let $ P(S)$  be the set of non-empty subsets of  $S$ . What is the last digit of  $|P(P([2013]))|$ ?


Let  $x$  be a number such that  $x +\frac{1}{x}=-1$ . Determine the value of   $x^{1994} +\frac{1}{x^{1994}}$ .


Suppose  $N$  is any positive integer. Add the digits of  $N$  to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers  $N \le 1000$ , such that the final single-digit number  $n$  is equal to  $5$ .
Example:  $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$  will be counted as one such integer.


A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.


On the plane  $S$  in a space, given are unit circle  $C$  with radius 1 and the line  $L$ . Find the volume of the solid bounded by the curved surface formed by the point  $P$  satifying the following condition  $(a),\ (b)$ . $(a)$  The point of intersection  $Q$  of the line passing through  $P$  and perpendicular to  $S$  are on the perimeter or the inside of  $C$ . $(b)$  If  $A,\ B$  are the points of intersection of the line passing through  $Q$  and pararell to  $L$ , then  $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$ .

Solution: \boxed{\pi}


What is the maximal dimension of a linear subspace  $ V$  of the vector space of real  $ n \times n$  matrices such that for all  $ A$  in  $ B$  in  $ V$ , we have  $ \text{trace}\left(AB\right) \equal{} 0$  ?


Consider the real function defined by  $f(x) =\frac{1}{|x + 3| + |x + 1| + |x - 2| + |x -5|}$  for all  $x \in R$ .
a) Determine its maximum.
b) Graphic representation.


Let  $x_0=1$ , and let  $\delta$  be some constant satisfying  $0<\delta<1$ . Iteratively, for  $n=0,1,2,\dots$ , a point  $x_{n+1}$  is chosen uniformly form the interval  $[0,x_n]$ . Let  $Z$  be the smallest value of  $n$  for which  $x_n<\delta$ . Find the expected value of  $Z$ , as a function of  $\delta$ .

Solution: \boxed{1 - \ln \delta}


For any positive integer  $n$ , let  $D_n$  denote the set of all positive divisors of  $n$ , and let  $f_i(n)$  denote the size of the set   $$ F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \} $$  where  $i = 1, 2$ .
Determine the smallest positive integer  $m$  such that  $2f_1(m) - f_2(m) = 2017$ .

Solution: \boxed{ 2 \cdot 5^{2016} }


Find the largest positive integer  $k{}$  for which there exists a convex polyhedron  $\mathcal{P}$  with 2022 edges, which satisfies the following properties:

- The degrees of the vertices of  $\mathcal{P}$  don’t differ by more than one, and
- It is possible to colour the edges of  $\mathcal{P}$  with  $k{}$  colours such that for every colour  $c{}$ , and every pair of vertices  $(v_1, v_2)$  of  $\mathcal{P}$ , there is a monochromatic path between  $v_1$  and  $v_2$  in the colour  $c{}$ .

*Viktor Simjanoski, Macedonia*

Solution: \boxed{ k = 2 }


In a right triangle rectangle  $ABC$  such that  $AB = AC$ ,  $M$  is the midpoint of   $BC$ . Let  $P$  be a point on the perpendicular bisector of   $AC$ , lying in the semi-plane determined by  $BC$  that does not contain  $A$ . Lines  $CP$  and  $AM$  intersect at  $Q$ . Calculate the angles that form the lines  $AP$  and  $BQ$ .

Solution: \boxed{ 90^\circ }


Let  $n\ge3$  be a positive integer. Find the real numbers  $x_1\ge0,\ldots,x_n\ge 0$ , with  $x_1+x_2+\ldots +x_n=n$ , for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value.

Solution: \boxed{ n^2 }


Let  $T = \{ 1, 2, 3, \dots, 14, 15 \}$ .  Say that a subset  $S$  of  $T$  is *handy* if the sum of all the elements of  $S$  is a multiple of  $5$ .  For example, the empty set is handy (because its sum is 0) and  $T$  itself is handy (because its sum is 120).  Compute the number of handy subsets of  $T$ .


Ket  $f(x) = x^{2} +ax + b$ . If for all nonzero real  $x$   $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$  and the roots of  $f(x) = 0$  are integers, what is the value of  $a^{2}+b^{2}$ ?


Find  $x$  so that the arithmetic mean of  $x, 3x, 1000$ , and  $3000$  is  $2018$ .


A set  $S$  has  $7$  elements. Several  $3$ -elements subsets of  $S$  are listed, such
that any  $2$  listed subsets have exactly  $1$  common element. What is the maximum number of subsets that can be listed?


Let  $m$  and  $n$  be positive integers. Find the smallest positive integer  $s$  for which there exists an  $m \times n$  rectangular array of positive integers such that

- each row contains  $n$  distinct consecutive integers in some order,
- each column contains  $m$  distinct consecutive integers in some order, and
- each entry is less than or equal to  $s$ .

*Proposed by Ankan Bhattacharya.*

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


Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a *grade* is a nonnegative decimal number with finitely many digits after the decimal point.

Given a grade with digits  $a_1a_2 \dots a_m.b_1b_2 \dots b_n$ , Joe first rounds the number to the nearest  $10^{-n+1}$ th place. He then repeats the procedure on the new number, rounding to the nearest  $10^{-n+2}$ th, then rounding the result to the nearest  $10^{-n+3}$ th, and so on, until he obtains an integer.  For example, he rounds the number  $2014.456$  via  $2014.456 \to 2014.46 \to 2014.5 \to 2015$ .

There exists a rational number  $M$  such that a grade  $x$  gets rounded to at least  $90$  if and only if  $x \ge M$ .  If  $M = \tfrac pq$  for relatively prime integers  $p$  and  $q$ , compute  $p+q$ .

*Proposed by Yang Liu*


Yao Ming is  $7\text{ ft and }5\text{ in}$  tall. His basketball hoop is  $10$  feet from the ground. Given that there are $12$  inches in a foot, how many inches must Yao jump to touch the hoop with his head?


The ternary expansion  $x = 0.10101010\cdots$  is given. Give the binary expansion of  $x$ .
Alternatively, transform the binary expansion  $y = 0.110110110 \cdots$  into a ternary expansion.

Solution: \boxed{ 0.011_2 }


Find the number of positive integers  $n$  for which

(i)  $n \leq 1991$ ;

(ii) 6 is a factor of  $(n^2 + 3n +2)$ .


Triangle  $ABC$  has  $BC=1$  and  $AC=2$ . What is the maximum possible value of  $\hat{A}$ .


Given the parallelogram  $ABCD$ . The circle  $S_1$  passes through the vertex  $C$  and touches the sides  $BA$  and  $AD$  at points  $P_1$  and  $Q_1$ , respectively. The circle  $S_2$  passes through the vertex  $B$  and touches the side  $DC$  at points  $P_2$  and  $Q_2$ , respectively. Let  $d_1$  and  $d_2$  be the distances from  $C$  and  $B$  to the lines  $P_1Q_1$  and  $P_2Q_2$ , respectively.
Find all possible values of the ratio  $d_1:d_2$ .

*(I. Voronovich)*

Solution: \boxed{ 1 }


Let  $f (x)$  be a function mapping real numbers to real numbers. Given that  $f (f (x)) =\frac{1}{3x}$ , and  $f (2) =\frac19$ , find  $ f\left(\frac{1}{6}\right)$ .

*Proposed by Zachary Perry*


Compute the smallest positive integer  $a$  for which   $$ \sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7 $$


Find the minimum value of  $k$  such that there exists two sequence  ${a_i},{b_i}$  for  $i=1,2,\cdots ,k$  that satisfies the following conditions.

(i) For all  $i=1,2,\cdots ,k,$   $a_i,b_i$  is the element of  $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all  $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all  $i=1,2,\cdots, k, a_i\le a_{i+1}$  and  $b_i\le b_{i+1}.$ (iv)  $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$


Let  $\triangle XOY$  be a right-angled triangle with  $\angle XOY=90^\circ$ .  Let  $M$  and  $N$  be the midpoints of legs  $OX$  and  $OY$ , respectively.  Find the length  $XY$  given that  $XN=22$  and  $YM=31$ .


Druv has a  $33 \times 33$  grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides?


 Determine twice differentiable functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  which verify relation

    \[
    \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
    \]

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


Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each  $\frac{1}{3}$ . The probability that Club Truncator will finish the season with more wins than losses is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .


An equilateral  $12$ -gon has side length  $10$  and interior angle measures that alternate between  $90^\circ$ ,  $90^\circ$ , and  $270^\circ$ . Compute the area of this  $12$ -gon.

*Proposed by Connor Gordon*


Let  $P$  be a point inside a square  $ABCD$  such that  $PA:PB:PC$  is  $1:2:3$ . Determine the angle  $\angle BPA$ .


Consider  $7$  points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords.


Robin goes birdwatching one day.  he sees three types of birds:  penguins, pigeons, and robins.   $\frac23$  of the birds he sees are robins.   $\frac18$  of the birds he sees are penguins.  He sees exactly  $5$  pigeons.  How many robins does Robin see?


From vertex  $A$  of an equilateral triangle  $ABC$ , a ray  $Ax$  intersects  $BC$  at point  $D$ . Let  $E$  be a point on  $Ax$  such that  $BA =BE$ . Calculate  $\angle AEC$ .


The rectangular faces of rectangular prism  $A$  have perimeters  $12$ ,  $16$ , and  $24$ . The rectangular faces of rectangular prism  $B$  have perimeters  $12$ ,  $16$ , and  $20$ . Let  $V_A$  denote the volume of  $A$  and  $V_B$  denote the volume of  $B$ . Find  $V_A-V_B$ .


Real numbers  $a,b,c$  with  $a\neq b$  verify   $$ a^2(b+c)=b^2(c+a)=2023. $$   Find the numerical value of  $E=c^2(a+b)$ .


2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Solution: \boxed{ 11 }


Determine all non negative integers  $k$  such that there is a function  $f : \mathbb{N} \to \mathbb{N}$  that satisfies
\[ f^n(n) = n + k \]
for all  $n \in \mathbb{N}$

Solution: \boxed{ k = 0 }


A contest has six problems worth seven points each. On any given problem, a contestant can score either  $0$ ,  $1$ , or  $7$  points. How many possible total scores can a contestant achieve over all six problems?


Consider the numbers  $\{24,27,55,64,x\}$ . Given that the mean of these five numbers is prime and the median is a multiple of  $3$ , compute the sum of all possible positive integral values of  $x$ .


Given is a convex pentagon  $ABCDE$  in which  $\angle A = 60^o$ ,  $\angle  B = 100^o$ ,  $\angle  C = 140^o$ .
Show that this pentagon can be placed in a circle with a radius of  $\frac23 AD$ .

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


Find the smallest number  $a$  such that a square of side  $a$  can contain five disks of radius  $1$ , so that no two of the disks have a common interior point.

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


Let  ${(a_n)_{n\ge1}} $  be a sequence with   ${a_1 = 1} $   and   ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$  for all  ${n \ge 1}$ , where  ${\lfloor x \rfloor}$  denotes the greatest integer less than or equal to  ${x}$ . Find all  ${n \le 2013}$  such that  ${a_n}$  is a perfect square

Solution: \boxed{ n = 1 }


One plant is now  $44$  centimeters tall and will grow at a rate of  $3$  centimeters every  $2$  years. A second plant is now  $80$  centimeters tall and will grow at a rate of  $5$  centimeters every  $6$  years. In how many years will the plants be the same height?


A positive integer is called *cool* if it can be expressed in the form  $a!\cdot b!+315$  where  $a,b$  are positive integers. For example,  $1!\cdot 1!+315=316$  is a cool number. Find the sum of all cool numbers that are also prime numbers.

[i]Proposed by Evan Fang


 $ 30$  students participated in the mathematical Olympiad. Each of them was given  $ 8$  problems to solve. Jury estimated their work with the following rule:
1) Each problem was worth  $ k$  points, if it wasn't solved by exactly  $ k$  students;
2) Each student received the maximum possible points in each problem or got  $ 0$  in it;
Lasha got the least number of points. What's the maximal number of points he could have?
Remark: 1) means that if the problem was solved by exactly  $ k$  students, than each of them got  $ 30 \minus{} k$  points in it.


The sizes of the freshmen class and the sophomore class are in the ratio  $5:4$ . The sizes of the sophomore class and the junior class are in the ratio  $7:8$ . The sizes of the junior class and the senior class are in the ratio  $9:7$ . If these four classes together have a total of  $2158$  students, how many of the students are freshmen?


Let  $S$  be the set of all ordered triples  $(p,q,r)$  of prime numbers for which at least one rational number  $x$  satisfies  $px^2+qx+r=0.$  Which primes appear in seven or more elements of  $S?$


Let's say a positive integer  $ n$  is *atresvido* if the set of its divisors (including 1 and  $ n$ ) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.


Find all quadrupels  $(a, b, c, d)$  of positive real numbers that satisfy the following two equations:
\begin{align*}
ab + cd &= 8,
abcd &= 8 + a + b + c + d.
\end{align*}

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


Find the greatest common divisor of all number in the set  $( a^{41} - a | a \in \mathbb{N} and \geq 2 )$  . What is your guess if 41 is replaced by a natural number  $n$

Solution: \boxed{ 13530 }


Determine the smallest rational number  $\frac{r}{s}$  such that  $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$  whenever  $k, m,$  and  $n$  are positive integers that satisfy the inequality  $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$ .


What is the sum of all positive integers  $n$  such that  $\text{lcm}(2n, n^2) = 14n - 24$ ?


How many  $8$ -digit numbers in base  $4$  formed of the digits  $1,2, 3$  are divisible by  $3$ ?


Deﬁne the sequence of positive integers  $a_n$  recursively by  $a_1=7$  and  $a_n=7^{a_{n-1}}$  for all  $n\geq 2$ .  Determine the last two digits of  $a_{2007}$ .


Suppose that  $a$  and  $b$  are real numbers such that  $\sin(a)+\sin(b)=1$  and  $\cos(a)+\cos(b)=\frac{3}{2}$ . If the value of  $\cos(a-b)$  can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , determine  $100m+n$ .

*Proposed by Michael Ren*


In quadrilateral  $ABCD, \angle B$  is a right angle, diagonal  $\overline{AC}$  is perpendicular to  $\overline{CD},$   $AB=18, BC=21,$  and  $CD=14.$   Find the perimeter of  $ABCD$ .


Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of  $N$  (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal  $N$  they may show such a trick?
*K. Knop, O. Leontieva*

Solution: \boxed{ 101 }


The train schedule in Hummut is hopelessly unreliable. Train  $A$  will enter Intersection  $X$  from the west at a random time between  $9:00$  am and  $2:30$  pm; each moment in that interval is equally likely. Train  $B$  will enter the same intersection from the north at a random time between  $9:30$  am and  $12:30$  pm, independent of Train  $A$ ; again, each moment in the interval is equally likely. If each train takes  $45$  minutes to clear the intersection, what is the probability of a collision today?


If  $x \in R-\{-7\}$ , determine the smallest value of the expression  $$ \frac{2x^2 + 98}{(x + 7)^2} $$


One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle.  A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced.  Tony fetches a tape measure from the garage and measures a distance of almost exactly  $12$  feet between a consecutive pair of trees.  Tony realizes the need to include the width of the trees in his measurements.  Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around  $15$  feet.  He then measures  $2$  feet to either side of the first and last trees in the row before the ends of the yard.  Tony uses these measurements to estimate the width of the yard.  If there are six trees in the row of trees, what is Tony's estimate in feet?

[asy]
size(400);
	defaultpen(linewidth(0.8));
	draw((0,-3)--(0,3));
	int d=8;
	for(int i=0;i<=5;i=i+1)
	{
	draw(circle(7/2+d*i,3/2));
	}
	draw((5*d+7,-3)--(5*d+7,3));
	draw((0,0)--(2,0),Arrows(size=7));
	draw((5,0)--(2+d,0),Arrows(size=7));
	draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7));
label(" $2$ ",(1,0),S);
label(" $12$ ",((7+d)/2,0),S);
label(" $15$ ",((7+3*d)/2,0),S);
[/asy]


The cells of a  $100 \times 100$  table are colored white. In one move, it is allowed to select some  $99$  cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?

*S. Berlov*


On a circle of center  $O$ , let  $A$  and  $B$  be points on the circle such that   $\angle AOB = 120^o$ . Point  $C$  lies on  the small arc  $AB$  and point  $D$  lies on the segment  $AB$ . Let also  $AD = 2, BD = 1$  and  $CD = \sqrt2$ . Calculate the area of triangle  $ABC$ .


For a fixed integer  $n\geqslant2$  consider the sequence  $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$ . Find all  $n$  for which the sequence  $a_k$  increases starting from some number.

Solution: \boxed{ n = 2 }


Triangle  $ABC$  satisfies  $\tan A \cdot \tan B = 3$  and  $AB = 5$ . Let  $G$  and  $O$  be the centroid and circumcenter of  $ABC$  respectively. The maximum possible area of triangle  $CGO$  can be written as  $\frac{a\sqrt{b}}{c}$  for positive integers  $a$ ,  $b$ , and  $c$  with  $a$  and  $c$  relatively prime and  $b$  not divisible by the square of any prime. Find  $a + b + c$ .


Adam and Bettie are playing a game. They take turns generating a random number between  $0$  and  $127$  inclusive. The numbers they generate are scored as follows: $\bullet$  If the number is zero, it receives no points. $\bullet$  If the number is odd, it receives one more point than the number one less than it. $\bullet$  If the number is even, it receives the same score as the number with half its value.
if Adam and Bettie both generate one number, the probability that they receive the same score is  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Find  $p$ .


Find the smallest value that the expression takes  $x^4 + y^4 - x^2y - xy^2$ , for positive numbers  $x$  and  $y$  satisfying  $x + y \le 1$ .


Let  $a$ ,  $b$ ,  $c$ , and  $d$  be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}} c@{\hspace{3pt}} c@{\hspace{3pt}} c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008, ac&=&bd&=&1000.\end{array}\]If  $S=a+b+c+d$ , compute the value of  $\lfloor S\rfloor$ .


Let  $k>1$  be a positive integer and  $n>2018$  an odd positive integer. The non-zero rational numbers  $x_1,x_2,\ldots,x_n$  are not all equal and:  $$ x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1} $$  Find the minimum value of  $k$ , such that the above relations hold.


Beto plays the following game with his computer: initially the computer randomly picks  $30$  integers from  $1$  to  $2015$ , and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer  $k$  and some if the numbers written on the chalkboard, and subtracts  $k$  from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all  $30$  numbers to  $0$ , in which case the game ends. Find the minimal number  $n$  such that, regardless of which numbers the computer chooses, Beto can end the game in at most  $n$  turns.


Two points  $A, B$  are randomly chosen on a circle with radius  $100.$  For a positive integer  $x$ , denote  $P(x)$  as the probability that the length of  $AB$  is less than  $x$ . Find the minimum possible integer value of  $x$  such that  $\text{P}(x) > \frac{2}{3}$ .


Triangle  $ABC$  lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes  $63$ ,  $73$ ,  $97$ . Suppose the curve  $\mathcal V$  with equation  $y=(x+3)(x^2+3)$  passes through the vertices of  $ABC$ . Find the sum of the slopes of the three tangents to  $\mathcal V$  at each of  $A$ ,  $B$ ,  $C$ .

*Proposed by Akshaj*


Given a digits { $0,1,2,...,9$ } . Find the number of numbers of 6 digits which cantain  $7$  or  $7$ 's digit and they is permulated(For example 137456 and 314756 is one numbers).


The  $25$  member states of the European Union set up a committee with the following rules:
1) the committee should meet daily;
2) at each meeting, at least one member should be represented;
3) at any two different meetings, a different set of member states should be represented;
4) at  $n^{th}$  meeting, for every  $k<n$ , the set of states represented should include at least one state that was represented at the  $k^{th}$  meeting.

For how many days can the committee have its meetings?

Solution: \boxed{ 2^{24} }


Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard?

*Author: Alexander Golovanov*


Let  $ABCD$  be a parallelogram. Let  $E$  be the midpoint of  $AB$  and  $F$  be the midpoint of  $CD$ . Points  $P$  and  $Q$  are on segments  $EF$  and  $CF$ , respectively, such that  $A, P$ , and  $Q$  are collinear. Given that  $EP = 5$ ,  $P F = 3$ , and  $QF = 12$ , find  $CQ$ .


Let  $\mathbb{Z^+}$  denote the set of positive integers. Find all functions  $f:\mathbb{Z^+} \to \mathbb{Z^+}$  satisfying the condition  $$  f(a) + f(b) \mid (a + b)^2 $$  for all  $a,b \in \mathbb{Z^+}$

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


For a natural number  $n$ , let  $n'$  denote the number obtained by deleting zero digits, if any. (For example, if  $n = 260$ ,  $n' = 26$ , if  $n = 2020$ ,  $n' = 22$ .),Find the number of  $3$ -digit numbers  $n$  for which  $n'$  is a divisor of  $n$ , different from  $n$ .


Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.

Solution: \boxed{160 \text{ seconds}}


Initially five variables are defined:  $a_1=1, a_2=0, a_3=0, a_4=0, a_5=0.$  On a turn, Evan can choose an integer  $2 \le i \le 5.$  Then, the integer  $a_{i-1}$  will be added to  $a_i$ . For example, if Evan initially chooses  $i = 2,$  then now  $a_1=1, a_2=0+1=1, a_3=0, a_4=0, a_5=0.$  Find the minimum number of turns Evan needs to make  $a_5$  exceed  $1,000,000.$


Let  $\operatorname{cif}(x)$  denote the sum of the digits of the number  $x$  in the decimal system. Put  $a_1=1997^{1996^{1997}}$ , and  $a_{n+1}=\operatorname{cif}(a_n)$  for every  $n>0$ . Find  $\lim_{n\to\infty}a_n$ .