Let  $ABC$  be a triangle with  $\angle ABC=90^{\circ}$ . The square  $BDEF$  is inscribed in  $\triangle ABC$ , such that  $D,E,F$  are in the sides  $AB,CA,BC$  respectively. The inradius of  $\triangle EFC$  and  $\triangle EDA$  are  $c$  and  $b$ , respectively. Four circles  $\omega_1,\omega_2,\omega_3,\omega_4$  are drawn inside the square  $BDEF$ , such that the radius of  $\omega_1$  and  $\omega_3$  are both equal to  $b$  and the radius of  $\omega_2$  and  $\omega_4$  are both equal to  $a$ . The circle  $\omega_1$  is tangent to  $ED$ , the circle  $\omega_3$  is tangent to  $BF$ ,  $\omega_2$  is tangent to  $EF$  and  $\omega_4$  is tangent to  $BD$ , each one of these circles are tangent to the two closest circles and the circles  $\omega_1$  and  $\omega_3$  are tangents. Determine the ratio  $\frac{c}{a}$ .


Find the maximal number of edges a connected graph  $G$  with  $n$  vertices may have, so that after deleting an arbitrary cycle,  $G$  is not connected anymore.


The cost of five water bottles is \ $13, rounded to the nearest dollar, and the cost of six water bottles is \$ 16, also rounded to the nearest dollar. If all water bottles cost the same integer number of cents, compute the number of possible values for the cost of a water bottle.

*Proposed by Eugene Chen*


Show that no integer of the form  $ xyxy$  in base  $ 10$  can be a perfect cube. Find the smallest base  $ b>1$  for which there is a perfect cube of the form  $ xyxy$  in base  $ b$ .


Let  $P(x) = x^2 - 20x - 11$ . If  $a$  and  $b$  are natural numbers such that  $a$  is composite,  $\gcd(a, b) = 1$ , and  $P(a) = P(b)$ , compute  $ab$ .

Note:  $\gcd(m, n)$  denotes the greatest common divisor of  $m$  and  $n$ .

*Proposed by Aaron Lin*


Find the largest  $n$  such that the last nonzero digit of  $n!$  is  $1$ .


Let  $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$ .  When  $a$ ,  $b$ ,  $c$  run over all the real numbers, find the smallest possible value of  $F$ .


Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?

*(A. Golovanov)*


Let  $p$  be an odd prime number. Let  $g$  be a primitive root of unity modulo  $p$ . Find all the values of  $p$  such that the sets  $A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}$  and  $B=\left\{g^m:1\le m\le\frac{p-1}2\right\}$  are equal modulo  $p$ .

Solution: \boxed{ p = 3 }


Patrick tosses four four-sided dice, each numbered  $1$  through  $4$ . What's the probability their product is a multiple of four?


Kevin has  $255$  cookies, each labeled with a unique nonempty subset of  $\{1,2,3,4,5,6,7,8\}$ . Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with  $\{1,2\}$ , he eats that cookie as well as the cookies with  $\{1\}$  and  $\{2\}$ ). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

*Proposed by Ray Li*


Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of  $\$  1, \ $ 2,\$  5 $, and $ \ $ 10$  bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly  $\$ 1{,}000{,}000 $, and whoever makes the last move wins the $ \ $1{,}000{,}000$  in the pile (if the pile is worth more than  $\$ 1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?

*Proposed by Yannick Yao*


Anna and Boris move simultaneously towards each other, from points A and B respectively. Their speeds are constant, but not necessarily equal. Had Anna started 30 minutes earlier, they would have met 2 kilometers nearer to B. Had Boris started 30 minutes earlier instead, they would have met some distance nearer to A. Can this distance be uniquely determined?

*(3 points)*

Solution: \boxed{2 \text{ km}}


Ranu starts with one standard die on a table.  At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step.  After 2013 such steps, let  $D$  be the number of dice on the table.  What is the expected value (average value) of  $6^D$ ?


Let the functions  $f(\alpha,x)$  and  $g(\alpha)$  be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then  $g(\alpha)$  is a polynomial is  $\alpha$ .  Find the leading coefficient of  $g(\alpha)$ .


Let  $n$  be a fixed positive integer and consider an  $n\times n$  grid of real numbers. Determine the greatest possible number of cells  $c$  in the grid such that the entry in  $c$  is both strictly greater than the average of  $c$ 's column and strictly less than the average of  $c$ 's row.

*Proposed by Holden Mui*


In a triangle  $ABC, L$  and  $K$  are the points of intersections of the angle bisectors of  $\angle ABC$  and  $\angle BAC$  with the segments  $AC$  and  $BC$ , respectively. The segment  $KL$  is angle bisector of  $\angle AKC$ , determine  $\angle BAC$ .


A marble is placed on each  $33$  unit square of a  $10*10$  chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

Solution: \boxed{ 438 }


What is the sum of all distinct values of  $x$  that satisfy  $x^4-x^3-7x^2+13x-6=0$ ?

*2018 CCA Math Bonanza Lightning Round #1.4*


Let  $n$  be a positive integer. Mariano divides a rectangle into  $n^2$  smaller rectangles by drawing  $n-1$  vertical lines and  $n-1$  horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer  $k$  for which it is possible that Emilio can do  $k$  conveniently thought steps in such a way that with the received information, he can determine the area of each one of the  $n^2$  smaller rectangles.

Solution: \boxed{2n - 1}


Consider a standard ( $8$ -by- $8$ ) chessboard. Bishops are only allowed to attack pieces that are along the same diagonal as them (but cannot attack along a row or column). If a piece can attack another piece, we say that the pieces threaten each other. How many bishops can you place a chessboard without any of them threatening each other?


For all positive integers  $n > 1$ , let  $f(n)$  denote the largest odd proper divisor of  $n$  (a proper divisor of  $n$  is a positive divisor of  $n$  except for  $n$  itself). Given that  $N=20^{23}\cdot23^{20}$ , compute
\[\frac{f(N)}{f(f(f(N)))}.\]


In triangle  $ABC$  we have  $\angle C = 90^o$  and  $AC = BC$ . Furthermore  $M$  is an interior pont in the triangle so that  $MC = 1 , MA = 2$  and  $MB =\sqrt2$ . Determine  $AB$


Given an integer  $n\geq 2$ , determine the maximum value the sum  $x_1+\cdots+x_n$  may achieve, as the  $x_i$  run through the positive integers, subject to  $x_1\leq x_2\leq \cdots \leq x_n$  and  $x_1+\cdots+x_n=x_1 x_2\cdots x_n$ .

Solution: \boxed{ S = 2n }


On the planet Mars there are  $100$  states that are in dispute. To achieve a peace situation, blocs must be formed that meet the following two conditions:
(1) Each block must have at most  $50$  states.
(2) Every pair of states must be together in at least one block.
Find the minimum number of blocks that must be formed.


Let  $m, n, a, k$  be positive integers and  $k>1$  such that the equality   $$ 5^m+63n+49=a^k $$  holds. Find the minimum value of  $k$ .


$8$  singers take part in a festival. The organiser wants to plan  $m$  concerts. For every concert there are  $4$  singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises  $m$ .

Solution: \boxed{ m = 14 }


The diagram below shows a circle with center  $F$ . The angles are related with  $\angle BFC = 2\angle AFB$ ,  $\angle CFD = 3\angle AFB$ ,  $\angle DFE = 4\angle AFB$ , and  $\angle EFA = 5\angle AFB$ . Find the degree measure of  $\angle BFC$ .
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
dotfactor=4;
draw(unitcircle);
pair A,B,C,D,E,F;
A=dir(90);
B=dir(66);
C=dir(18);
D=dir(282);
E=dir(210);
F=origin;
dot(" $F$ ",F,NW);
dot(" $A$ ",A,dir(90));
dot(" $B$ ",B,dir(66));
dot(" $C$ ",C,dir(18));
dot(" $D$ ",D,dir(306));
dot(" $E$ ",E,dir(210));
draw(F--E^^F--D^^F--C^^F--B^^F--A);
[/asy]


Find all prime numbers  $p$ , for which the number  $p + 1$  is equal to the product of all the prime numbers which are smaller than  $p$ .

Solution: \boxed{ p = 5 }


As in the following diagram, square  $ABCD$  and square  $CEFG$  are placed side by side (i.e.  $C$  is between  $B$  and  $E$  and  $G$  is between  $C$  and  $D$ ). If  $CE = 14$ ,  $AB > 14$ , compute the minimal area of  $\triangle AEG$ .
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(real x, real y) {
 pair P = (x,y);
 dot(P,linewidth(3)); return P;
}
int big = 30, small = 14;
filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));
draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);
label(" $A$ ",D2(0,big),NW);
label(" $B$ ",D2(0,0),SW);
label(" $C$ ",D2(big,0),SW);
label(" $D$ ",D2(big,big),N);
label(" $E$ ",D2(big+small,0),SE);
label(" $F$ ",D2(big+small,small),NE);
label(" $G$ ",D2(big,small),NE);
[/asy]


How many integers  $n$  are there subject to the constraint that  $1 \leq n \leq 2020$  and  $n^n$  is a perfect square?


A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled  $A$ . The three vertices adjacent to vertex  $A$  are at heights 10, 11, and 12 above the plane. The distance from vertex  $A$  to the plane can be expressed as  $\tfrac{r-\sqrt{s}}{t}$ , where  $r$ ,  $s$ , and  $t$  are positive integers, and  $r+s+t<1000$ . Find  $r+s+t$ .


Let  $n$  be a fixed positive integer. How many ways are there to write  $n$  as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with  $k$  an arbitrary positive integer and  $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$ ? For example, with  $n = 4$ , there are four ways:  $4$ ,  $2 + 2$ ,  $1 + 1 + 2$ ,  $1 + 1 + 1 + 1$ .

Solution: \boxed{n}


At the meeting, each person is familiar with 22 people. If two persons  $A$  and  $B$  know each with one another, among the remaining people they do not have a common friend. For each pair individuals  $A$  and  $B$  are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?


Let  $ \mathcal P$  be a parabola, and let  $ V_1$  and  $ F_1$  be its vertex and focus, respectively. Let  $ A$  and  $ B$  be points on  $ \mathcal P$  so that  $ \angle AV_1 B \equal{} 90^\circ$ . Let  $ \mathcal Q$  be the locus of the midpoint of  $ AB$ . It turns out that  $ \mathcal Q$  is also a parabola, and let  $ V_2$  and  $ F_2$  denote its vertex and focus, respectively. Determine the ratio  $ F_1F_2/V_1V_2$ .

Solution: \boxed{ \frac{7}{8} }


Compute  $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)

*Proposed by Tristan Shin*


In how many ways can you write  $12$  as an ordered sum of integers where the smallest of those integers is equal to  $2$ ? For example,  $2+10$ ,  $10+2$ , and  $3+2+2+5$  are three such ways.


A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now  $264$ , how old is the tortoise?


Consider a sequence  $\{a_n\}$  of integers, satisfying  $a_1=1, a_2=2$  and  $a_{n+1}$  is the largest prime divisor of  $a_1+a_2+\ldots+a_n$ . Find  $a_{100}$ .


The *subnumbers* of an integer  $n$  are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime.

*Proposed by Lewis Chen*


Jay notices that there are  $n$  primes that form an arithmetic sequence with common difference  $12$ . What is the maximum possible value for  $n$ ?

*Proposed by James Lin*


$A_1, A_2, ..., A_n$  are the subsets of  $|S|=2019$  such that union of any three of them gives  $S$  but if we combine two of subsets it doesn't give us  $S$ . Find the maximum value of  $n$ .


How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set  $\{ 1,4,7,10 \}$ , where each number in the set is chosen at least once? (For example, one possible sum is  $1+1+1+4+7+7+10+10=41$ .)


Given  $n \in\mathbb{N}$ , ﬁnd all continuous functions  $f : \mathbb{R}\to \mathbb{R}$  such that for all  $x\in\mathbb{R},$
\[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]

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


Nine weights are placed in a scale with the respective values  $1kg,2kg,...,9kg$ . In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one?


Let  $\ell$  be a line with negative slope passing through the point  $(20,16)$ . What is the minimum possible area of a triangle that is bounded by the  $x$ -axis,  $y$ -axis, and  $\ell$ ?

*Proposed by James Lin*


A square of side  $n$  is formed from  $n^2$  unit squares, each colored in red, yellow or green. Find minimal  $n$ , such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).

Solution: \boxed{ n = 7 }


Find all polynomials  $W$  with real coefficients possessing the following property: if  $x+y$  is a rational number, then  $W(x)+W(y)$  is rational.

Solution: \boxed{ W(x) = ax + b }


For each prime  $p$ , let  $\mathbb S_p = \{1, 2, \dots, p-1\}$ . Find all primes  $p$  for which there exists a function  $f\colon \mathbb S_p \to \mathbb S_p$  such that
\[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \]
for all  $n \in \mathbb S_p$ .

*Andrew Wen*


A *palindromic table* is a  $3 \times 3$  array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
  O & M & O
  N & M & N
  O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters  $O$  and  $M$ ?  (The table may contain only a single letter.)

*Proposed by Evan Chen*


Jackson begins at  $1$  on the number line. At each step, he remains in place with probability  $85\%$  and increases his position on the number line by  $1$  with probability  $15\%$ . Let  $d_n$  be his position on the number line after  $n$  steps, and let  $E_n$  be the expected value of  $\tfrac{1}{d_n}$ . Find the least  $n$  such that  $\tfrac{1}{E_n}
> 2017$ .


The expression  $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$  is equal to  $p\sqrt{5}/2^{50}$ , where  $p$  is an integer. Find  $p$ .


The function  $f(n)$  is defined on the positive integers and takes non-negative integer values.  $f(2)=0,f(3)>0,f(9999)=3333$  and for all  $m,n:$  \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine  $f(1982)$ .


Determine all positive integers  $a,b,c$  such that  $ab + ac + bc$  is a prime number and  $$ \frac{a+b}{a+c}=\frac{b+c}{b+a}. $$

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


Let  $\Omega_1$  and  $\Omega_2$  be two circles in the plane.  Suppose the common external tangent to  $\Omega_1$  and  $\Omega_2$  has length  $2017$  while their common internal tangent has length  $2009$ .  Find the product of the radii of  $\Omega_1$  and  $\Omega_2$ .

*Proposed by David Altizio*


Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=?


Find all natural numbers  $a>1$ , with the property that every prime divisor of  $a^6-1$  divides also at least one of the numbers  $a^3-1$ ,  $a^2-1$ .

*K. Dochev*

Solution: \boxed{ a = 2 }


Triangle  $ABC$  has side lengths  $AB=13, BC=14,$  and  $CA=15$ . Let  $\Gamma$  denote the circumcircle of  $\triangle ABC$ . Let  $H$  be the orthocenter of  $\triangle ABC$ . Let  $AH$  intersect  $\Gamma$  at a point  $D$  other than  $A$ . Let  $BH$  intersect  $AC$  at  $F$  and  $\Gamma$  at point  $G$  other than  $B$ . Suppose  $DG$  intersects  $AC$  at  $X$ . Compute the greatest integer less than or equal to the area of quadrilateral  $HDXF$ .

*Proposed by Kenan Hasanaliyev (claserken)*


Each face of a tetrahedron is a triangle with sides  $a, b,$ c and the tetrahedon has circumradius 1. Find  $a^2 + b^2 + c^2$ .


Let  $n$  be a natural number. Find all real numbers  $x$  satisfying the equation   $$ \sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4. $$

Solution: \boxed{x = 1}


Find the sum of all positive integers whose largest proper divisor is  $55$ . (A proper divisor of  $n$  is a divisor that is strictly less than  $n$ .)


In a class with  $23$  students, each pair of students have watched a movie together. Let the set of movies watched by a student be his *movie collection*. If every student has watched every movie at most once, at least how many different movie collections can these students have?


Let  $x_1,x_2,y_1,y_2$  be real numbers satisfying the equations  $x^2_1+5x^2_2=10$ ,  $x_2y_1-x_1y_2=5$ , and  $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of  $y_1^2+5y_2^2$


$(MON 1)$  Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.


Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)


A right rectangular prism has integer side lengths  $a$ ,  $b$ , and  $c$ . If  $\text{lcm}(a,b)=72$ ,  $\text{lcm}(a,c)=24$ , and  $\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?

*Proposed by Deyuan Li and Andrew Milas*


Let  $ m,n\in \mathbb{N}^*$ . Find the least  $ n$  for which exists  $ m$ , such that rectangle  $ (3m \plus{} 2)\times(4m \plus{} 3)$  can be covered with  $ \dfrac{n(n \plus{} 1)}{2}$  squares, among which exist  $ n$  squares of length  $ 1$ ,  $ n \minus{} 1$  of length  $ 2$ ,  $ ...$ ,  $ 1$  square of length  $ n$ . For the found value of  $ n$  give the example of covering.

Solution: \boxed{ n = 8 }


Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in  $3$ , Anne and Carl in  $5$ .  How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?


Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?


Let  $1=d_1<d_2<d_3<\dots<d_k=n$  be the divisors of  $n$ . Find all values of  $n$  such that  $n=d_2^2+d_3^3$ .


$4.$  Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend  $3$  galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend  $24$  galleons in total. Everyone has a whole number of galleons, and the number of galleons between the three of them is  a multiple of  $7$ . What are all the possible number of galleons that Harry can have?


Let  $ABC$  be a triangle such that  $AB = 7$ ,  $BC = 8$ , and  $CA = 9$ . There exists a unique point  $X$  such that  $XB = XC$  and  $XA$  is tangent to the circumcircle of  $ABC$ . If  $XA = \tfrac ab$ , where  $a$  and  $b$  are coprime positive integers, find  $a + b$ .

*Proposed by Alexander Wang*


$PS$  is a line segment of length  $4$  and  $O$  is the midpoint of  $PS$ . A semicircular arc is drawn with  $PS$  as diameter. Let  $X$  be the midpoint of this arc.  $Q$  and  $R$  are points on the arc  $PXS$  such that  $QR$  is parallel to  $PS$  and the semicircular arc drawn with  $QR$  as diameter is tangent to  $PS$ . What is the area of the region  $QXROQ$  bounded by the two semicircular arcs?

Solution: \boxed{2\pi - 2}


Find all pairs  $(x, y)$  of real numbers satisfying the system : $\begin{cases} x + y = 3
x^4 - y^4 = 8x - y \end{cases}$

Solution: \boxed{(2, 1)}


A new website registered  $2000$  people. Each of them invited  $1000$  other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?

*(5 points)*


In the coordinate plane, the line passing through points  $(2023,0)$  and  $(-2021,2024)$  also passes through  $(1,c)$  for a constant  $c$ . Find  $c$ .

*Proposed by Andy Xu*


Let  $a, b, c, d$  be an increasing arithmetic sequence of positive real numbers with common difference  $\sqrt2$ . Given that the product  $abcd = 2021$ ,  $d$  can be written as  $\frac{m+\sqrt{n}}{\sqrt{p}}$  , where  $m, n,$  and  $p$  are positive integers not divisible by the square of any prime. Find  $m + n + p$ .


$2021$  people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up  $1000$  positions to the left of their original position.


Consider the following one-person game: A player starts with score  $0$  and writes the number  $20$  on an
empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers
(call them  $b$  and  $c$ ) such that  $b + c = a$ . The player then adds  $b\times c$  to her score. She repeats the step
several times until she ends up with all  $1$ 's on the whiteboard. Then the game is over, and the final score is
calculated. Let  $M, m$  be the maximum and minimum final score that can be possibly obtained respectively.
Find  $M-m$ .


Given a positive integer  $n$ , determine the maximum number of lattice points in the plane a square of side length  $n +\frac{1}{2n+1}$  may cover.


For a positive integer  $n$ , let  $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest  $n$  such that  $\left|f_n^{\prime \prime}(0)\right|>2023$ .


Peter and Basil play the following game on a horizontal table  $1\times{2019}$ . Initially Peter chooses  $n$  positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by  $s$  cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by  $s$  cells neither to the left, nor to the right, the coin stays in the current cell. Find the least  $n$  such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.

Solution: \boxed{ n = 2 }


For any natural number, let  $S(n)$  denote sum of digits of  $n$ . Find the number of  $3$  digit numbers for which  $S(S(n)) = 2$ .


Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains?


In trapezoid  $ ABCD$  with  $ \overline{BC}\parallel\overline{AD}$ , let  $ BC\equal{}1000$  and  $ AD\equal{}2008$ . Let  $ \angle A\equal{}37^\circ$ ,  $ \angle D\equal{}53^\circ$ , and  $ m$  and  $ n$  be the midpoints of  $ \overline{BC}$  and  $ \overline{AD}$ , respectively. Find the length  $ MN$ .


In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]

pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);

pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw(" $O$ ", B, dir(180));
draw(" $B$ ", A, dir(45));
draw(" $A$ ", C, dir(45));
draw(" $Q$ ", E, dir(45));
draw(" $P$ ", D, dir(45));
draw(" $R$ ", Z, dir(45));
draw(" $S$ ", F, dir(45));
draw(" $a$ ", Y, dir(210));
draw(" $b$ ", J, dir(100));
[/asy]

Solution: \boxed{2 : 1}


Let  $a$  be real number. A circle  $C$  touches the line  $y=-x$  at the point  $(a, -a)$  and passes through the point  $(0,\ 1).$ Denote by  $P$  the center of  $C$ . When  $a$  moves, find the area of the figure enclosed by the locus of  $P$  and the line  $y=1$ .


Let  $\underline{xyz}$  represent the three-digit number with hundreds digit  $x$ , tens digit  $y$ , and units digit  $z$ , and similarly let  $\underline{yz}$  represent the two-digit number with tens digit  $y$  and units digit  $z$ .  How many three-digit numbers  $\underline{abc}$ , none of whose digits are 0, are there such that  $\underline{ab}>\underline{bc}>\underline{ca}$ ?


The polynomial  $x^3-kx^2+20x-15$  has  $3$  roots, one of which is known to be  $3$ . Compute the greatest possible sum of the other two roots.

*2015 CCA Math Bonanza Lightning Round #2.4*


If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example,  $3333$ ), two values (for example,  $2666$ ), three values (for example,  $5215$ ), or four values (for example,  $4236$ ). The mean number of values that show is  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


Find the smallest constant M, so that for any real numbers  $a_1, a_2, \dots a_{2023} \in [4, 6]$  and  $b_1, b_2, \dots b_{2023} \in [9, 12] $  following inequality holds:  $$  \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot  \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots  + a_{2023} b_{2023} \right)  $$  *Proposed by Zaza Meliqidze, Georgia*


Let  $x_1, x_2, . . . , x_{2022}$  be nonzero real numbers. Suppose that  $x_k + \frac{1}{x_{k+1}} < 0$  for each  $1 \leq k \leq 2022$ , where  $x_{2023}=x_1$ . Compute the maximum possible number of integers  $1 \leq n \leq 2022$  such that  $x_n > 0$ .


Given any triangle  $ABC$  and any positive integer  $n$ , we say that  $n$  is a *decomposable* number for triangle  $ABC$  if there exists a decomposition of the triangle  $ABC$  into  $n$  subtriangles with each subtriangle similar to   $\triangle ABC$ . Determine the positive integers that are decomposable numbers for every triangle.

Solution: \boxed{ n = 1 }


$2019$  circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?

Solution: \boxed{ 2 }


For a real number  $a$  and an integer  $n(\geq 2)$ , define  $$ S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} $$  Find every value of  $a$  s.t. sequence  $\{S_n(a)\}_{n\geq 2}$  converges to a positive real.


What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)

Alexandru Mihalcu


There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number  $k$  such that no matter how I select and color  $k$  points, you can always color the remaining  $100-k$  points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.

Solution: \boxed{ k = 50 }


Two beads, each of mass  $m$ , are free to slide on a rigid, vertical hoop of mass  $m_h$ . The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio  $m/m_h$  such that the hoop always remains in contact with the ground? Neglect friction.
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,0)--(3,0));
draw(circle((1.5,1),1));
filldraw(circle((1.4,1.99499),0.1), gray(.3));
filldraw(circle((1.6,1.99499),0.1), gray(.3));
[/asy]


The diagonals  $AC$  and  $BD$  of a convex quadrilateral  $ABCD$  with  $S_{ABC} = S_{ADC}$  intersect at  $E$ . The lines through  $E$  parallel to  $AD$ ,  $DC$ ,  $CB$ ,  $BA$ meet  $AB$ ,  $BC$ ,  $CD$ ,  $DA$  at  $K$ ,  $L$ ,  $M$ ,  $N$ , respectively. Compute the ratio  $\frac{S_{KLMN}}{S_{ABC}}$


Al told Bob that he was thinking of  $2011$  distinct positive integers. He also told Bob the sum of those  $2011$  distinct positive integers. From this information, Bob was able to determine all  $2011$  integers. How many possible sums could Al have told Bob?

*Author: Ray Li*


Let  $x$  be the largest root of  $x^4 - 2009x + 1$ . Find the nearest integer to  $\frac{1}{x^3-2009}$  .


Find the greatest positive integer  $m$  with the following property:
For every permutation  $a_1, a_2, \cdots, a_n,\cdots$  of the set of positive integers, there exists positive integers  $i_1<i_2<\cdots <i_m$  such that  $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$  is an arithmetic progression with an odd common difference.

Solution: \boxed{ 3 }


There are  $15$  people, including Petruk, Gareng, and Bagong, which will be partitioned into  $6$  groups, randomly, that consists of  $3, 3, 3, 2, 2$ , and  $2$  people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.


Determine the integers  $x$  such that  $2^x + x^2 + 25$  is the cube of a prime number

Solution: \boxed{ x = 6 }


 Let  $f(x) = (x - 5)(x - 12)$  and  $g(x) = (x - 6)(x - 10)$ .
Find the sum of all integers  $n$  such that  $\frac{f(g(n))}{f(n)^2}$  is defined and an integer.


Consider the  $4\times4$  array of  $16$  dots, shown below.

[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]

Counting the number of squares whose vertices are among the  $16$  dots and whose sides are parallel to the sides of the grid, we find that there are nine  $1\times1$  squares, four  $2\times2$  squares, and one  $3\times3$  square, for a total of  $14$  squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the  $14$  squares is missing at least one vertex?


In the coordinate plane, suppose that the parabola  $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$  touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of  $y\geq 0$  of  $C$  and the  $x$ -axis.

Solution: \boxed{ \sqrt{3} }


Let  $n\geq 3$  be an integer. Find the largest real number  $M$  such that for any positive real numbers  $x_1,x_2,\cdots,x_n$ , there exists an arrangement  $y_1,y_2,\cdots,y_n$  of real numbers satisfying
\[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\]
where  $y_{n+1}=y_1,y_{n+2}=y_2$ .

Solution: \boxed{ n-1 }


Find all functions  $f:\mathbb{R}\rightarrow \mathbb{R}$  such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$  where  $y\in \mathbb{R}$ .

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


Given any two positive real numbers  $x$  and  $y$ , then  $x\Diamond y$  is a positive real number defined in terms of  $x$  and  $y$  by some fixed rule.  Suppose the operation  $x\Diamond y$  satisfies the equations  $(x\cdot y)\Diamond y=x(y\Diamond y)$  and  $(x\Diamond 1)\Diamond x=x\Diamond 1$  for all  $x,y>0$ .  Given that  $1\Diamond 1=1$ , find  $19\Diamond 98$ .


Let  $A$  and  $B$  be sets such that there are exactly  $144$  sets which are subsets of either  $A$  or  $B$ . Determine the number of elements  $A \cup B$  has.


Let  $ f(n,k)$  be the number of ways of distributing  $ k$  candies to  $ n$  children so that each child receives at most  $ 2$  candies. For example  $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$ . Determine the value of  $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$ .


Among all fractions (whose numerator and denominator are positive integers) strictly between  $\tfrac{6}{17}$  and  $\tfrac{9}{25}$ , which one has the smallest denominator?


An ordered pair  $(n,p)$  is *juicy* if  $n^{2} \equiv 1 \pmod{p^{2}}$  and  $n \equiv -1 \pmod{p}$  for positive integer  $n$  and odd prime  $p$ . How many juicy pairs exist such that  $n,p \leq 200$ ?

Proposed by Harry Chen (Extile)


Let  $G$  be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs  $(n,k)$ ,  $n\ge 3$ , such that:

a) For all graph  $G$  with  $n$  vertex and  $k$  edges, it is always possible to perform a series of switching process so that all edges are eventually blue.

b) There exist a graph  $G$  with  $n$  vertex and  $k$  edges and it is possible to perform a series of switching process so that all edges are eventually blue.

Solution: \boxed{ (n, k) }


How many 3-term geometric sequences  $a$ ,  $b$ ,  $c$  are there where  $a$ ,  $b$ , and  $c$  are positive integers with  $a < b < c$  and  $c = 8000$ ?


For positive integers  $a>b>1$ , define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least  $d$  such that for any  $a,b$ , the sequence  $x_n$  does not contain  $d$  consecutive prime numbers.

*V. Senderov*


For an integer  $n>3$  denote by  $n?$  the product of all primes less than  $n$ . Solve the equation  $n?=2n+16$ .

*V. Senderov*

Solution: \boxed{ n = 7 }


Consider all  $ m \times n$  matrices whose all entries are  $ 0$  or  $ 1$ . Find the number of such matrices for which the number of  $ 1$ -s in each row and in each column is even.


In the  $xy$ -coordinate plane, the  $x$ -axis and the line  $y=x$  are mirrors.  If you shoot a laser beam from the point  $(126, 21)$  toward a point on the positive  $x$ -axis, there are  $3$  places you can aim at where the beam will bounce off the mirrors and eventually return to  $(126, 21)$ .  They are  $(126, 0)$ ,  $(105, 0)$ , and a third point  $(d, 0)$ .  What is  $d$ ?  (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.)


Let  $A$  be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from  $A$  differs at most in 1 position from one of the chosen?

Solution: \boxed{ 4 }


Let  $a_n$  be the closest to  $\sqrt n$  integer.
Find the sum  $1/a_1 + 1/a_2 + ... + 1/a_{1980}$ .


A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.


We have  $x_i >i$  for all  $1 \le i \le n$ .
Find the minimum value of  $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$


In square  $ABCD,$  points  $E$  and  $F$  are chosen in the interior of sides  $BC$  and  $CD$ , respectively. The line drawn from  $F$  perpendicular to  $AE$  passes through the intersection point  $G$  of  $AE$  and diagonal  $BD$ . A point  $K$  is chosen on  $FG$  such that  $|AK|= |EF|$ . Find  $\angle EKF.$


In the sequence  $00$ ,  $01$ ,  $02$ ,  $03$ ,  $\cdots$ ,  $99$  the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by  $1$  (for example,  $29$  can be followed by  $19$ ,  $39$ , or  $28$ , but not by  $30$  or  $20$ ). What is the maximal number of terms that could remain on their places?


Let  $A$  be a set of natural numbers, for which for  $\forall n\in \mathbb{N}$  exactly one of the numbers  $n$ ,  $2n$ , and  $3n$  is an element of  $A$ . If  $2\in A$ , show whether  $13824\in A$ .

Solution: \boxed{ 13824 \notin A }


How many positive integer solutions does the equation have   $$ \left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1? $$  ( $\lfloor x \rfloor$  denotes the integer part of  $x$ ,  for example  $\lfloor 2\rfloor = 2$ ,  $\lfloor \pi\rfloor = 3$ ,  $\lfloor \sqrt2 \rfloor =1$ )


Let  $n$  be a positive integer such that  $1 \leq n \leq 1000$ . Let  $M_n$  be the number of integers in the set  $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let  $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$  Find  $a-b$ .


There is a rectangular field that measures  $20\text{m}$  by  $15\text{m}$ . Xiaoyu the butterfly is sitting at the perimeter of the field on one of the  $20\text{m}$  sides such that he is  $6\text{m}$  from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?


Let  $n$  be an integer greater than or equal to  $1$ . Find, as a function of  $n$ , the smallest integer  $k\ge 2$  such that, among any  $k$  real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than  $1 / n$ , either  strictly greater than  $n$ .

Solution: \boxed{ n^2 + 2 }


$101$  numbers are written on a blackboard:  $1^2, 2^2, 3^2, \cdots, 101^2$ . Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.


In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag.  Tony then pours all the jelly beans out of the five bags and begins making patterns with them.  One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:

\[\begin{array}{ccccccccc}&&&&*&&&&&&&*&*&*&&&&&*&*&*&*&*&&&*&*&*&*&*&*&*& *&*&*&*&*&*&*&*&*&&&&\vdots&&&&\end{array}\]

Continuing in this way, Tony finishes a row with none left over.  For instance, if Tony had exactly  $25$  jelly beans, he could finish the fifth row above with no jelly beans left over.  However, when Tony finishes, there are between  $10$  and  $20$  rows.  Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number.  How many jelly beans are in each bag?  (Assume that no marble gets put inside more than one bag.)


Find all pairs  $(p,q)$  of prime numbers which  $p>q$  and  $$ \frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1} $$  is an integer.

Solution: \boxed{ (3, 2) }


How many distinct sets are there such that each set contains only non-negative powers of  $2$  or  $3$  and sum of its elements is  $2014$ ? $
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 54
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ \text{None of the preceding}
$


Let  $ABCD$  be a parallelogram that is not rhombus. We draw the symmetrical half-line of  $(DC$  with respect to line  $BD$ . Similarly we draw the symmetrical half- line of  $(AB$  with respect to  $AC$ . These half- lines intersect each other in  $P$ . If  $\frac{AP}{DP}= q$  find the value of  $\frac{AC}{BD}$  in function of  $q$ .

Solution: \boxed{\sqrt{q}}


Five numbers are chosen from  $\{1, 2, \ldots, n\}$ . Determine the largest  $n$  such that we can always pick some of the 5 chosen numbers so that they can be made into two groups whose numbers have the same sum (a group may contain only one number).

Solution: \boxed{ n = 15 }


Find all integers  $n$  satisfying  $n \geq 2$  and  $\dfrac{\sigma(n)}{p(n)-1} = n$ , in which  $\sigma(n)$  denotes the sum of all positive divisors of  $n$ , and  $p(n)$  denotes the largest prime divisor of  $n$ .


For  $ n \equal{} 1,\ 2,\ 3,\ \cdots$ , let  $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$  be the point of intersection of  $ y \equal{} \ln (nx)$  and  $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$ .

(1) Show that  $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$  to find  $ \lim_{n\to\infty} q_n$ .

(2) Find  $ \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx$ .


Let  $A$ ,  $B$ ,  $C$ , and  $P$  be points in the plane such that no three of them are collinear. Suppose that the areas of triangles  $BPC$ ,  $CPA$ , and  $APB$  are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle  $ABC$ .

*Proposed by Ankan Bhattacharya*


A rectangle  $ABCD$  has side lengths  $AB=6 \text{ miles}$  and  $BC=9\text{ miles}.$  A pigeon hovers at point  $P$ , which is 5 miles above some randomly chosen point inside  $ABCD$ . Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as  $\tfrac{a}{b}$ , what is  $ab$ ?

*2022 CCA Math Bonanza Lightning Round 2.2*


The area of a circle centered at the origin, which is inscribed in the parabola  $y=x^2-25$ , can be expressed as  $\tfrac ab\pi$ , where  $a$  and  $b$  are coprime positive integers.  What is the value of  $a+b$ ?


Cirlce  $\Omega$  is inscribed in triangle  $ABC$  with  $\angle BAC=40$ . Point  $D$  is inside the angle  $BAC$  and is the intersection of exterior bisectors of angles  $B$  and  $C$  with the common side  $BC$ . Tangent form  $D$  touches  $\Omega$  in  $E$ . FInd  $\angle BEC$ .


Let n be the natural number ( $n\ge 2$ ). All natural numbers from  $1$   up to  $n$  ,inclusive, are written on the board   in some order:  $a_1$ ,  $a_2$  ,  $...$  ,  $a_n$ . Determine all natural numbers  $n$  ( $n\ge 2$ ), for which the product  $$ P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n) $$   is an even number, whatever the arrangement of the numbers written on the board.

Solution: \boxed{ n }


Find all quadruplets of positive integers  $(m,n,k,l)$  such that  $3^m = 2^k + 7^n$  and  $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$ .

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


A sequence of positive integers  $a_1,a_2,\ldots $  is such that for each  $m$  and  $n$  the following holds: if  $m$  is a divisor of  $n$  and  $m<n$ , then  $a_m$  is a divisor of  $a_n$  and  $a_m<a_n$ . Find the least possible value of  $a_{2000}$ .


Find all pairs  $ (x, y)$  of integers such that  $ x^3 \minus{} y^3 \equal{} 2xy \plus{} 8$ .

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


Today, Ivan the Confessor prefers continuous functions  $f:[0,1]\to\mathbb{R}$  satisfying  $f(x)+f(y)\geq |x-y|$  for all pairs  $x,y\in [0,1]$ . Find the minimum of  $\int_0^1 f$  over all preferred functions.

(Proposed by Fedor Petrov, St. Petersburg State University)


Let  $a_{10} = 10$ , and for each integer  $n >10$  let  $a_n = 100a_{n - 1} + n$ . Find the least  $n > 10$  such that  $a_n$  is a multiple of  $99$ .


Let  $g:[0,1]\rightarrow \mathbb{R}$  be a continuous function and let  $f_{n}:[0,1]\rightarrow \mathbb{R}$  be a
sequence of functions defined by  $f_{0}(x)=g(x)$  and  $$ f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt. $$  Determine  $\lim_{n\to \infty}f_{n}(x)$  for every  $x\in (0,1]$ .

Solution: \boxed{ g(0) }


Let  $ ABCD$  be an isosceles trapezoid with  $ \overline{AD}\parallel{}\overline{BC}$  whose angle at the longer base  $ \overline{AD}$  is  $ \dfrac{\pi}{3}$ . The diagonals have length  $ 10\sqrt {21}$ , and point  $ E$  is at distances  $ 10\sqrt {7}$  and  $ 30\sqrt {7}$  from vertices  $ A$  and  $ D$ , respectively. Let  $ F$  be the foot of the altitude from  $ C$  to  $ \overline{AD}$ . The distance  $ EF$  can be expressed in the form  $ m\sqrt {n}$ , where  $ m$  and  $ n$  are positive integers and  $ n$  is not divisible by the square of any prime. Find  $ m \plus{} n$ .


We call a rectangle of size  $2 \times 3$  (or  $3 \times 2$ ) without one cell in corner a  $P$ -rectangle. We call a rectangle of size  $2 \times 3$  (or  $3 \times 2$ ) without two cells in opposite (under center of rectangle) corners a  $S$ -rectangle. Using some squares of size  $2 \times 2$ , some  $P$ -rectangles and some  $S$ -rectangles, one form one rectangle of size  $1993 \times 2000$  (figures don’t overlap each other). Let  $s$  denote the sum of numbers of squares and  $S$ -rectangles used in such tiling. Find the maximal value of  $s$ .


Find the number of squares in the sequence given by  $ a_0\equal{}91$  and  $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$  for  $ n \ge 0.$


$201$  positive integers are written on a line, such that both the first one and the last one are equal to  $19999$ . Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.


Holding a rectangular sheet of paper  $ABCD$ , Prair folds triangle  $ABD$  over diagonal  $BD$ , so that the new location of point  $A$  is  $A'$ . She notices that  $A'C =\frac13 BD$ . If the area of  $ABCD$  is  $27\sqrt2$ , find  $BD$ .


Let  $a$  and  $b$  be real numbers bigger than  $1$ . Find maximal value of  $c \in \mathbb{R}$  such that   $$ \frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c $$

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


suppose that polynomial  $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$  does not have a real root. what is the maximum number of coefficients to be  $-1$ ?(14 points)


How many positive five-digit integers are there that have the product of their five digits equal to  $900$ ?

(Karl Czakler)


Find  $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$

where  $_iC_j$  is a binominal coefficient which means  $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$ .


Find all the pairs of positive numbers such that the last
digit of their sum is 3, their difference is a primer number and
their product is a perfect square.

Solution: \boxed{ (9, 4) }


The numbers 1 through 10 can be arranged along the vertices and sides of a pentagon so that the sum of the three numbers along each side is the same. The diagram below shows an arrangement with sum 16. Find, with proof, the smallest possible value for a sum and give an example of an arrangement with that sum.

[asy]
int i;
pair[] A={dir(18+72*0), dir(18+72*1), dir(18+72*2),dir(18+72*3), dir(18+72*4), dir(18+72*0)};
pair O=origin;
int[] v = {7,1,10,4,3};
int[] s = {8, 5, 2, 9, 6};
for(i=0; i<5; i=i+1) {
label(string(v[i]), A[i], dir(O--A[i]));
label(string(s[i]), A[i]--A[i+1], dir(-90)*dir(A[i]--A[i+1]));
}
draw(rotate(0)*polygon(5));[/asy]


You are given  $n \ge 2$  distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of  $2$ . For every  $n$  find the largest possible number of elegant pairs.

*Proposed by Oleksiy Masalitin*

Solution: \boxed{ n-1 }


Calvin makes a number. He starts with  $1$ , and on each move, he multiplies his current number by  $3$ , then adds  $5$ . After  $10$  moves, find the sum of the digits (in base  $10$ ) when Calvin's resulting number is expressed in base  $9$ .

*Proposed by Calvin Wang*


Determine all natural numbers $ n> 1$  with the property:
For each divisor  $d> 1$   of number  $n$ , then  $d - 1$  is a divisor of  $n - 1$ .

Solution: \boxed{ n }


Let  $XYZ$  be a  right triangle of  area  $1$  m $^2$  . Consider the triangle  $X'Y'Z'$  such that  $X'$  is the symmetric of X wrt side  $YZ$ ,  $Y'$  is the symmetric of  $Y$  wrt side  $XZ$  and  $Z' $  is the symmetric  of  $Z$   wrt side  $XY$ . Calculate the area of the triangle  $X'Y'Z'$ .


Let  $T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2}$  where  $x,y,z$  are real numbers such that  $1 \leq x,y,z \leq 4$  and  $x-y+z=4$ .
Find the smallest value of  $10 \times T$ .


In  $\vartriangle ABC$  points  $D, E$ , and  $F$  lie on side  $\overline{BC}$  such that  $\overline{AD}$  is an angle bisector of  $\angle BAC$ ,  $\overline{AE}$  is a median, and  $\overline{AF}$  is an altitude. Given that  $AB = 154$  and  $AC = 128$ , and  $9 \times DE = EF,$  find the side length  $BC$ .


Find all injective functions  $ f : \mathbb{N} \to \mathbb{N}$  which satisfy
\[ f(f(n)) \le\frac{n \plus{} f(n)}{2}\]
for each  $ n \in \mathbb{N}$ .

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


Two real numbers  $x$  and  $y$  are such that  $8y^4+4x^2y^2+4xy^2+2x^3+2y^2+2x=x^2+1$ .
Find all possible values of  $x+2y^2$


Find all functions  $f:\mathbb{R}\to\mathbb{R}$  such that
                          $f(f(x)f(1-x))=f(x)$     and      $f(f(x))=1-f(x)$ ,
for all real  $x$ .

Solution: \boxed{f(x) = \frac{1}{2}}


Let  $ a\in (1,\infty) $  and a countinuous function  $ f:[0,\infty)\longrightarrow\mathbb{R} $  having the property:  $$  \lim_{x\to \infty} xf(x)\in\mathbb{R} .  $$  **a)** Show that the integral  $ \int_1^{\infty} \frac{f(x)}{x}dx $  and the limit  $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $  both exist, are finite and equal.**b)** Calculate  $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

Solution: \boxed{ \ln 2 }


Quadrilateral  $ABCD$  has both an inscribed and a circumscribed circle and sidelengths  $BC = 4, CD = 5, DA = 6$ . Find the area of  $ABCD$ .


On the round necklace there are  $n> 3$  beads, each painted in red or blue. If a bead has adjacent beads painted the same color, it can be repainted (from red to blue or from blue to red). For what  $n$  for any initial coloring of beads it is possible to make a necklace in which all beads are painted equally?

Solution: \boxed{ n }


Unit square  $ABCD$  is divided into four rectangles by  $EF$  and  $GH$ , with  $BF = \frac14$ .  $EF$  is parallel to  $AB$  and  $GH$  parallel to  $BC$ .  $EF$  and  $GH$  meet at point  $P$ . Suppose  $BF + DH = FH$ , calculate the nearest integer to the degree of  $\angle FAH$ .
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55.
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $C$ ",D2(C),SE);
label(" $D$ ",D2(D),NE);
label(" $E$ ",D2(E),plain.N);
label(" $F$ ",D2(F),S);
label(" $G$ ",D2(G),W);
label(" $H$ ",D2(H),plain.E);
label(" $P$ ",D2(P),SE);
[/asy]


Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers  $x$ .

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


Let  $ABC$  be a triangle with  $\angle BAC=117^\circ$ .  The angle bisector of  $\angle ABC$  intersects side  $AC$  at  $D$ .  Suppose  $\triangle ABD\sim\triangle ACB$ .  Compute the measure of  $\angle ABC$ , in degrees.


The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.


A rock travelled through an n x n board, stepping at each turn to the cell neighbouring the previous one by a side, so that each cell was visited once. Bob has put the integer numbers from 1 to n^2 into the cells, corresponding to the order in which the rook has passed them. Let M be the greatest difference of the numbers in neighbouring by side cells. What is the minimal possible value of M?

Solution: \boxed{ 2n-1 }


Consider a regular pentagon  $ABCDE$ , and let the intersection of diagonals  $\overline{CA}$  and  $\overline{EB}$  be  $F$ . Find  $\angle AFB$ .

*Proposed by Justin Chen*


Matrices  $A$ ,  $B$  are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0  1 & 2 & 0  0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0  2 & 4 & 0  0 & 0 & 12\end{pmatrix}\]
Find volume of  $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ .


Find the smallest  $\lambda \in \mathbb{R}$  such that for all  $n \in \mathbb{N}_+$ , there exists  $x_1, x_2, \ldots, x_n$  satisfying  $n = x_1 x_2 \ldots x_{2023}$ , where  $x_i$  is either a prime or a positive integer not exceeding  $n^\lambda$  for all  $i \in \left\{ 1,2, \ldots, 2023 \right\}$ .


Let  $ABCD$  be a rectangle with sides  $AB,BC,CD$  and  $DA$ . Let  $K,L$  be the midpoints of the sides  $BC,DA$  respectivily. The perpendicular from  $B$  to  $AK$  hits  $CL$  at  $M$ . Find  $$ \frac{[ABKM]}{[ABCL]} $$


Find maximum value of number  $a$  such that for any arrangement of numbers  $1,2,\ldots ,10$  on a circle, we can find three consecutive numbers such their sum bigger or equal than  $a$ .


The NIMO problem writers have invented a new chess piece called the *Oriented Knight*.  This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right.  How many ways can the knight move from the bottom-left square to the top-right square of a  $16\times 16$  chess board?

*Proposed by Tony Kim and David Altizio*


In the game of Guess the Card, two players each have a  $\frac{1}{2}$  chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered  $1,2,\dots,16$  clockwise. They participate in an  $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players  $1$  and  $9$  face each other in the last round is  $\frac{m}{n}$  where  $m,n$  are positive integers, find  $100m+n$ .

*Proposed by Evan Chen*


Given that  $O$  is a regular octahedron, that  $C$  is the cube whose vertices are the centers of the faces of  $O$ , and that the ratio of the volume of  $O$  to that of  $C$  is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime integers, find  $m+n$ .


We have a  $6 \times 6$  square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?


Let  $n$  be a positive integer. Determine, in terms of  $n,$  the largest integer  $m$  with the following property: There exist real numbers  $x_1,\ldots, x_{2n}$  with  $-1<x_1<x_2<\ldots<x_{2n}<1$  such that the sum of the lengths of the  $n$  intervals   $$ [x_1^{2k-1},x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \ldots, [x_{2n-1}^{2k-1},x_{2n}^{2k-1}] $$   is equal to 1 for all integers  $k$  with  $1\leq k \leq m.$

Solution: \boxed{n}


At first, on a board, the number  $1$  is written  $100$  times. Every minute, we pick a number  $a$  from the board, erase it, and write  $a/3$  thrice instead. We say that a positive integer  $n$  is *persistent* if after any amount of time, regardless of the numbers we pick, we can find at least  $n$  equal numbers on the board. Find the greatest persistent number.

Solution: \boxed{ 67 }


Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December  $3$  and January  $31$  were  $5^\circ \text C$  and  $2^\circ \text C$  respectively, find the low temperature in December  $25$ .


Let  $\mathcal S$  be a set of  $16$  points in the plane, no three collinear. Let  $\chi(S)$  denote the number of ways to draw  $8$  lines with endpoints in  $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of  $\chi(\mathcal S)$  across all such  $\mathcal S$ .

*Ankan Bhattacharya*


Problem 3 (MAR CP 1992) :
From the digits  $1,2,...,9$ , we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows :  $123456789$ ,  $123456798$ , ...,  $987654321$ . What is the  $100000th$  number ?


Let  $m\ge 3$  and  $n$  be positive integers such that  $n>m(m-2)$ . Find the largest positive integer  $d$  such that  $d\mid n!$  and  $k\nmid d$  for all  $k\in\{m,m+1,\ldots,n\}$ .

Solution: \boxed{ m-1 }


Suppose  $n \ge 0$  is an integer and all the roots of  $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$  are integers. Find all possible values of  $\alpha$ .

Solution: \boxed{\alpha = -3}


A certain state issues license plates consisting of six digits (from 0 to 9).  The state requires that any two license plates differ in at least two places.  (For instance, the numbers 027592 and 020592 cannot both be used.)  Determine, with proof, the maximum number of distinct license plates that the state can use.


Let the set of all bijective functions taking positive integers to positive integers be  $\mathcal B.$  Find all functions  $\mathbf F:\mathcal B\to \mathbb R$  such that   $$ (\mathbf F(p)+\mathbf F(q))^2=\mathbf F(p \circ p)+\mathbf F(p\circ q)+\mathbf F(q\circ p)+\mathbf F(q\circ q) $$   for all  $p,q \in \mathcal B.$

*Proposed by ckliao914*

Solution: \boxed{ \mathbf{F}(p) = 1 }


For each integer  $n\geqslant2$ , determine the largest real constant  $C_n$  such that for all positive real numbers  $a_1, \ldots, a_n$  we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]

*(4th Middle European Mathematical Olympiad, Team Competition, Problem 2)*


A point  $P$  is chosen uniformly at random in the interior of triangle  $ABC$  with side  lengths  $AB = 5$ ,  $BC = 12$ ,  $CA = 13$ . The probability that a circle with radius  $\frac13$  centered at  $P$  does not intersect the perimeter of  $ABC$  can be written as  $\frac{m}{n}$  where  $m, n$  are relatively prime positive integers. Find  $m + n$ .


Determine all functions  $f : \mathbb{R} \to \mathbb{R}$  satisfying
    \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all  $a,b \in \mathbb{R}$ .

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


Determine the smallest positive real  $K$  such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers  $0 \le a,b,c \le 1$ .

*Proposed by Fajar Yuliawan, Indonesia*


For each positive integer  $m$ , we define  $L_m$  as the figure that is obtained by overlapping two  $1 \times m$  and  $m \times 1$  rectangles in such a way that they coincide at the  $1 \times 1$  square at their ends, as shown in the figure.

[asy]
pair h = (1, 0), v = (0, 1), o = (0, 0);
for(int i = 1; i < 5; ++i)
{
  o = (i*i/2 + i, 0);
  draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle);
  string s = " $L_" + (string)(i) + "$ ";
  label(s, o + ((i / 2), -1));
  for(int j = 1; j < i; ++j)
  {
    draw(o + j*v -- o + j*v + h);
    draw(o + j*h -- o + j*h + v);
  }
}
label("...", (18, 0.5));
[/asy]

Using some figures  $L_{m_1}, L_{m_2}, \dots, L_{m_k}$ , we cover an  $n \times n$  board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of  $m_1 + m_2 + \dots + m_k$ .

Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.


$ABC$  is a triangle with points  $D$ ,  $E$  on  $BC$  with  $D$  nearer  $B$ ;  $F$ ,  $G$  on  $AC$ , with  $F$  nearer  $C$ ;  $H$ ,  $K$  on  $AB$ , with  $H$  nearer  $A$ . Suppose that  $AH=AG=1$ ,  $BK=BD=2$ ,  $CE=CF=4$ ,  $\angle B=60^\circ$  and that  $D$ ,  $E$ ,  $F$ ,  $G$ ,  $H$  and  $K$  all lie on a circle. Find the radius of the incircle of triangle  $ABC$ .


Determine all polynomials  $P(x)$  with real coefficients and which satisfy the following properties:
i)  $P(0) = 1$ ii) for any real numbers  $x$  and  $y,$ \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\]

Solution: \boxed{ P(x) = x^2 + 1 }


Let  $\mathcal{P}$  be a set of monic polynomials with integer coefficients of the least degree, with root  $k \cdot \cos\left(\frac{4\pi}{7}\right)$ , as  $k$  spans over the positive integers. Let  $P(x) \in \mathcal{P}$  be the polynomial so that  $|P(1)|$  is minimized. Find the remainder when  $P(2017)$  is divided by  $1000$ .

*Proposed by **eisirrational***


Let  $A,B,C,D$ , be four different points on a line  $\ell$ , so that  $AB=BC=CD$ . In one of the semiplanes determined by the line  $\ell$ , the points  $P$  and  $Q$  are chosen in such a way that the triangle  $CPQ$  is equilateral with its vertices named clockwise. Let  $M$  and  $N$  be two points of the plane be such that the triangles  $MAP$  and  $NQD$  are equilateral (the vertices are also named clockwise). Find the angle  $\angle MBN$ .


In  $\triangle ABC$ , points  $E$  and  $F$  lie on  $\overline{AC}, \overline{AB}$ , respectively. Denote by  $P$  the intersection of  $\overline{BE}$  and  $\overline{CF}$ . Compute the maximum possible area of  $\triangle ABC$  if  $PB = 14$ ,  $PC = 4$ ,  $PE = 7$ ,  $PF = 2$ .

*Proposed by Eugene Chen*


Let  $a,b,c$  be distinct positive integers such that  $b+c-a$ ,  $c+a-b$  and  $a+b-c$  are all perfect squares. What is the largest possible value of  $a+b+c$  smaller than  $100$  ?


Let  $\otimes$  be a binary operation that takes two positive real numbers and returns a positive real number.  Suppose further that  $\otimes$  is continuous, commutative  $(a\otimes b=b\otimes a)$ , distributive across multiplication  $(a\otimes(bc)=(a\otimes b)(a\otimes c))$ , and that  $2\otimes 2=4$ .  Solve the equation  $x\otimes y=x$  for  $y$  in terms of  $x$  for  $x>1$ .


Find all functions  $f:\mathbb{Z}^{+}\to\mathbb{Z}^{+}$  such that   $$ m^{2}+f(n)^{2}+(m-f(n))^{2}\geq f(m)^{2}+n^{2} $$   for all pairs of positive integers  $(m,n)$ .

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


Don Miguel places a token in one of the  $(n+1)^2$  vertices determined by an  $n \times n$  board. A *move* consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most  $\sqrt2$  away, as long as it stays on the board. A *path* is a sequence of moves such that the token was in each one of the  $(n+1)^2$  vertices exactly once. What is the maximum number of diagonal moves (those of length  $\sqrt2$ ) that a path can have in total?

Solution: \boxed{ n^2 }


Find all positive integers  $k>1$ , such that there exist positive integer  $n$ , such that the number  $A=17^{18n}+4.17^{2n}+7.19^{5n}$  is product of  $k$  consecutive positive integers.

Solution: \boxed{ k = 2 }


Let  $ABCD$  be a nondegenerate isosceles trapezoid with integer side lengths such that  $BC \parallel AD$  and  $AB=BC=CD$ . Given that the distance between the incenters of triangles  $ABD$  and  $ACD$  is  $8!$ , determine the number of possible lengths of segment  $AD$ .

*Ray Li*


Define  $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$ . Show that each polynomial has just three distinct real roots. Let  $A$  be the largest root of  $p(x)$  and  $B$  the largest root of  $q(x)$ . Show that  $A^{2}+3 B^{2}= 4$ .

Solution: \boxed{ A^2 + 3B^2 = 4 }


Let  $ABC$  be a triangle with  $BC = 5$ ,  $CA = 3$ , and  $AB = 4$ . Variable points  $P, Q$  are on segments  $AB$ ,  $AC$ , respectively such that the area of  $APQ$  is half of the area of  $ABC$ . Let  $x$  and  $y$  be the lengths of perpendiculars drawn from the midpoint of  $PQ$  to sides  $AB$  and  $AC$ , respectively. Find the range of values of  $2y + 3x$ .

Solution: \boxed{[6, 6.5]}


Find, with proof, the smallest real number  $C$  with the following property:
For every infinite sequence  $\{x_i\}$  of positive real numbers such that  $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$  for  $n = 1, 2, 3, \cdots$ , we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n}  \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]

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


Solve in positive integers the equation
\[ n \equal{} \varphi(n) \plus{} 402 ,
\]
where  $ \varphi(n)$  is the number of positive integers less than  $ n$  having no common prime factors with  $ n$ .

Solution: \boxed{ n = 802 }


Given are  $n$  points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with  leading coefficient equal to  $1$ . A parabola is called  $good$  if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it).  What is the greatest number of  $good$  parabolas?


For what positive integers  $n$  and  $k$  do there exits integers  $a_1,a_2,\dots,a_n$  and  $b_1,b_2,\dots,b_k$  such that the products  $a_ib_j$  ( $1\le i\le n,1\le j\le k$ ) give pairwise different residues modulo  $nk$ ?

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


Let  $n$  be a positive integer and let  $G$  be the set of points  $(x, y)$  in the plane such that  $x$  and  $y$  are integers with  $1 \leq x, y \leq n$ . A subset of  $G$  is called *parallelogram-free* if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of  $G$  can have?

Solution: \boxed{ 2n-1 }


Two thieves stole an open chain with  $2k$  white beads and  $2m$  black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets  $k$  white beads and  $m$  black beads. What is the minimal number of cuts that is always sufficient?


Determine the largest positive integer  $n$  such that the following statement is true:

There exists  $n$  real polynomials,  $P_1(x),\ldots,P_n(x)$  such that the sum of any two of them have no real roots but the sum of any three does.

Solution: \boxed{ n = 3 }


Let  $n$  be a positive integer. There are  $n$  purple and  $n$  white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?

Solution: \boxed{ n }


What is the largest possible rational root of the equation  $ax^2 + bx + c = 0{}$  where  $a, b$  and  $c{}$  are positive integers that do not exceed  $100{}$ ?


Let  $f_1(x)=x^2-1$ , and for each positive integer  $n \geq 2$  define  $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial  $f_{2004}$  have?


A set of  $k$  integers is said to be a *complete residue system modulo*  $k$  if no two of its elements are congruent modulo  $k$ . Find all positive integers  $m$  so that there are infinitely many positive integers  $n$  wherein  $\{ 1^n,2^n, \dots , m^n \}$  is a complete residue system modulo  $m$ .

Solution: \boxed{ m }


Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:

- The sum of the fractions is equal to  $2$ .
- The sum of the numerators of the fractions is equal to  $1000$ .

In how many ways can Pedro do this?


A man disposes of sufficiently many metal bars of length  $2$  and wants to construct a grill of the shape of an  $n \times n$  unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

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


Let  $P_1, P_2, \ldots, P_6$  be points in the complex plane, which are also roots of the equation  $x^6+6x^3-216=0$ . Given that  $P_1P_2P_3P_4P_5P_6$  is a convex hexagon, determine the area of this hexagon.


Find all positive integers  $m$  for which there exist three positive integers  $a,b,c$  such that  $abcm=1+a^2+b^2+c^2$ .

Solution: \boxed{ m = 4 }


Let  $T$  be a right triangle with sides having lengths  $3$ ,  $4$ , and  $5$ . A point  $P$  is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of  $T$ . What is the area of the set of awesome points?


given a positive integer  $n$ .
the set  $\{ 1,2,..,2n \}$  is partitioned into  $a_1<a_2<...<a_n $  and  $b_1>b_2>...>b_n$ .
find the value of :  $ \sum_{i=1}^{n}|a_i - b_i| $


Let  $k<<n$  denote that  $k<n$  and  $k\mid n$ . Let  $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$  be such that, if  $n\leq 2013$  and  $k<<n$ , then  $f(k)<<f(n)$ . What’s the least possible value of  $M$ ?


Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?

*Proposed by Michael Tang*


Find the largest  $k$  for which there exists a permutation  $(a_1, a_2, \ldots, a_{2022})$  of integers from  $1$  to  $2022$  such that for at least  $k$  distinct  $i$  with  $1 \le i \le 2022$  the number  $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$  is an integer larger than  $1$ .

*(Proposed by Oleksii Masalitin)*


Jane tells you that she is thinking of a three-digit number that is greater than  $500$  that has exactly  $20$  positive divisors. If Jane tells you the sum of the positive divisors of her number, you would not be able to figure out her number. If, instead, Jane had told you the sum of the \textit{prime} divisors of her number, then you also would not have been able to figure out her number. What is Jane's number? (Note: the sum of the prime divisors of  $12$  is  $2 + 3 = 5$ , not  $2 + 2 + 3 = 7$ .)


Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers  $1$  to  $16$ ?

Solution: \boxed{ 11 }


Derek fills a square  $10$  by  $10$  grid with  $50$   $1$ s and  $50$   $2$ s. He takes the product of the numbers in each of the  $10$  rows. He takes the product of the numbers in each of the  $10$  columns. He then sums these  $20$  products up to get an integer  $N.$  Find the minimum possible value of  $N.$


The teacher gave each of her  $37$  students  $36$  pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed.


2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero.  At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right.  Suppose the new numbers on the ducklings sum to 1000.  What is the maximum possible sum of the original numbers on all 2018 slips?


In a rectangular array of points, with 5 rows and  $N$  columns, the points are numbered consecutively from left to right beginning with the top row.  Thus the top row is numbered 1 through  $N,$  the second row is numbered  $N+1$  through  $2N,$  and so forth.  Five points,  $P_1, P_2, P_3, P_4,$  and  $P_5,$  are selected so that each  $P_i$  is in row  $i.$   Let  $x_i$  be the number associated with  $P_i.$   Now renumber the array consecutively from top to bottom, beginning with the first column.  Let  $y_i$  be the number associated with  $P_i$  after the renumbering.  It is found that  $x_1=y_2,$   $x_2=y_1,$   $x_3=y_4,$   $x_4=y_5,$  and  $x_5=y_3.$   Find the smallest possible value of  $N.$


Find the smallest positive integer  $n$  that has at least  $7$  positive divisors  $1 = d_1 < d_2 < \ldots < d_k = n$ ,  $k \geq 7$ , and for which the following equalities hold:  $$ d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1 $$  *Proposed by Mykyta Kharin*


Let  $S$  be the set of positive integers  $n$  such that
\[3\cdot
 \varphi (n)=n,\]
where  $\varphi (n)$  is the number of positive integers  $k\leq n$  such that  $\gcd (k, n)=1$ . Find
\[\sum_{n\in S} \, \frac{1}{n}.\]

[i]Proposed by Nathan Ramesh


A square pyramid with base  $ABCD$  and vertex  $E$  has eight edges of length 4.  A plane passes through the midpoints of  $\overline{AE}$ ,  $\overline{BC}$ , and  $\overline{CD}$ .  The plane's intersection with the pyramid has an area that can be expressed as  $\sqrt{p}$ .  Find  $p$ .

Solution: \boxed{ p = 80 }


Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).


Find all positive integers  $n$  such that there exists a sequence of positive integers  $a_1$ ,  $a_2$ , $\ldots$ ,  $a_n$  satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every  $k$  with  $2\leq k\leq n-1$ .

*Proposed by North Korea*

Solution: \boxed{ n = 3 }


We denote by  gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd $(4, 6, 8)=2$  and gcd  $(12, 15)=3$ .) Suppose that positive integers  $a, b, c$  satisfy the following four conditions: $\bullet$  gcd  $(a, b, c)=1$ , $\bullet$  gcd  $(a, b + c)>1$ , $\bullet$  gcd  $(b, c + a)>1$ , $\bullet$  gcd  $(c, a + b)>1$ .
a) Is it possible that  $a + b + c = 2015$ ?
b) Determine the minimum possible value that the sum  $a+ b+ c$  can take.

Solution: \boxed{ 30 }


Let  $p$  be a prime number and  $F=\left \{0,1,2,...,p-1 \right \}$ . Let  $A$  be a proper subset of  $F$  that satisfies the following property:  if  $a,b \in A$ , then  $ab+1$  (mod  $p$ )  $ \in A$ .
How many elements can  $A$  have? (Justify your answer.)

Solution: \boxed{ |A| = 1 }


Define  $m(n)$  to be the greatest proper natural divisor of  $n\in \mathbb{N}$ . Find all  $n \in \mathbb{N} $  such that  $n+m(n) $  is a power of  $10$ .

*N. Agakhanov*


Let  $n > 1$  be an integer and  $X = \{1, 2, \cdots , n^2 \}$ . If there exist  $x, y$  such that  $x^2\mid y$  in all subsets of  $X$  with  $k$  elements, find the least possible value of  $k$ .

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


At a recent math contest, Evan was asked to find  $2^{2016} \pmod{p}$  for a given prime number  $p$  with  $100 < p < 500$ . Evan has forgotten what the prime  $p$  was, but still remembers how he solved it:

- Evan first tried taking  $2016$  modulo  $p - 1$ , but got a value  $e$  larger than  $100$ .
- However, Evan noted that  $e - \frac{1}{2}(p - 1) = 21$ , and then realized the answer was  $-2^{21} \pmod{p}$ .

What was the prime  $p$ ?


Let  $a,b,c$  be positive integers such that  $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$  are  $7$  distinct primes. The sum of two of  $a,b,c$  is  $800$ . If  $d$  be the difference of the largest prime and the least prime among those  $7$  primes, find the maximum value of  $d$ .


Find all values of the parameter  $m$  such that the equations  $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$  have only one root.

Solution: \boxed{ m = 0 }


Given a figure  $F: x^2+\frac{y^2}{3}=1$  on the coordinate plane. Denote by  $S_n$  the area of the common part of the  $n+1' s$  figures formed by rotating  $F$  of  $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$  radians counterclockwise about the origin. Find  $\lim_{n\to\infty} S_n$ .

Solution: \boxed{\pi}


Let  $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$  such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of  $ a, b, c, d$  is  $ n^2.$  Determine, with proof, the values  of  $m$  and  $ n.$

Solution: \boxed{ m = 9 }


In triangle  $ABC$ ,  $AB>AC$ . The bisectors of  $\angle{B},\angle{C}$  intersect the sides  $AC,AB$  at  $P,Q$ , respectively. Let  $I$  be the incenter of  $\Delta ABC$ . Suppose that  $IP=IQ$ . How much isthe value of  $\angle A$ ?


For any permutation  $p$  of set  $\{1, 2, \ldots, n\}$ , define  $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$ . Denoted by  $i(p)$  the number of integer pairs  $(i, j)$  in permutation  $p$  such that  $1 \leqq < j \leq n$  and  $p(i) > p(j)$ . Find all the real numbers  $c$ , such that the inequality  $i(p) \leq c \cdot d(p)$  holds for any positive integer  $n$  and any permutation  $p.$

Solution: \boxed{ 1 }


Three circles  $C_i$  are given in the plane:  $C_1$  has diameter  $AB$  of length  $1$ ;  $C_2$  is concentric and has diameter  $k$  ( $1 < k < 3$ );  $C_3$  has center  $A$  and diameter  $2k$ .  We regard  $k$  as fixed.  Now consider all straight line segments  $XY$  which have one endpoint  $X$  on  $C_2$ , one endpoint  $Y$  on  $C_3$ , and contain the point  $B$ .  For what ratio  $XB/BY$  will the segment  $XY$  have minimal length?

Solution: \boxed{ \frac{XB}{BY} = 1 }


$ABC$  is triangle.  $l_1$ - line passes through  $A$  and parallel to  $BC$ ,  $l_2$  -  line passes through  $C$  and parallel to  $AB$ . Bisector of  $\angle B$  intersect  $l_1$  and  $l_2$  at  $X,Y$ .  $XY=AC$ . What  value can take  $\angle A- \angle C$  ?


Let  $ABC$  be an equilateral triangle . Let  point  $D$  lie on side  $AB,E$   lie on side  $AC, D_1$  and  $E_1$  lie on side BC such that  $AB=DB+BD_1$  and  $AC=CE+CE_1$ . Calculate the smallest angle between the lines  $DE_1$  and  $ED_1$ .


An integer   $n>1$  is given .  Find the smallest positive number  $m$  satisfying the following conditions： for any set  $\{a,b\}$   $\subset \{1,2,\cdots,2n-1\}$  ,there are non-negative integers    $ x, y$   ( not all zero)  such  that    $2n|ax+by$  and  $x+y\leq m.$

Solution: \boxed{ n }


Let  $p$  be an odd prime. An integer  $x$  is called a *quadratic non-residue* if  $p$  does not divide  $x-t^2$  for any integer  $t$ .

Denote by  $A$  the set of all integers  $a$  such that  $1\le a<p$ , and both  $a$  and  $4-a$  are quadratic non-residues. Calculate the remainder when the product of the elements of  $A$  is divided by  $p$ .

*Proposed by Richard Stong and Toni Bluher*


Find all functions  $f: \mathbb R \to \mathbb R$  such that for any  $x,y \in \mathbb R$ , the multiset  $\{(f(xf(y)+1),f(yf(x)-1)\}$  is identical to the multiset  $\{xf(f(y))+1,yf(f(x))-1\}$ .

*Note:* The multiset  $\{a,b\}$  is identical to the multiset  $\{c,d\}$  if and only if  $a=c,b=d$  or  $a=d,b=c$ .

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


What is the least positive integer  $n$  such that  $25^n +16^n$  leaves a remainder of  $1$  when divided by  $121$ ?


Let  $n$  be an integer greater than  $1$ , and let  $p$  be a prime divisor of  $n$ . A confederation consists of  $p$  states, each of which has exactly  $n$  airports. There are  $p$  air companies operating interstate flights only such that every two airports in different states are joined by a direct (two-way) flight operated by one of these companies. Determine the maximal integer  $N$  satisfying the following condition: In every such confederation it is possible to choose one of the  $p$  air companies and  $N$  of the  $np$  airports such that one may travel (not necessarily directly) from any one of the  $N$  chosen airports to any other such only by flights operated by the chosen air company.

Solution: \boxed{ n }


Find the smallest positive integer  $n,$  such that    $3^k+n^k+ (3n)^k+ 2014^k$  is a perfect square for all natural numbers  $k,$  but not a perfect cube, for all natural numbers  $k.$


Let  $n\geq 2$  be a positive integer. Find the minimum value of positive integer  $m$  for which there exist positive integers  $a_1,\ a_2,\ \cdots, a_n$  such that : $\bullet\ a_1<a_2<\cdots <a_n=m$  $\bullet \ \frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2}$  are all square numbers.

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


Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin  $O$  that cuts them both, then these polygons are called "properly placed". Find the least  $m \in \mathbb{N}$ , such that for any group of properly placed polygons,  $m$  lines can drawn through  $O$  and every polygon is cut by at least one of these  $m$  lines.

Solution: \boxed{ m = 2 }


A subset of a student group is called an *ideal company* if
1) in this subset, all girls are liked by all young men,
2) no one can be added to this subset without violating condition  $1$ .
In a certain group,  $9$  female students and  $15$  students study. Warden of the group made a list of all kinds of ideal companies in this group. What is the largest number of companies on this list?


Let  $\tau (n)$  denote the number of positive integer divisors of  $n$ . Find the sum of the six least positive integers  $n$  that are solutions to  $\tau (n) + \tau (n+1) = 7$ .


Let  $P$  be a polynomial with integer coefficients such that  $P(0)=0$  and
\[\gcd(P(0), P(1), P(2), \ldots ) = 1.\]
Show there are infinitely many  $n$  such that
\[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

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


A deck of  $2n$  cards numbered from  $1$  to  $2n$  is shuffled and n cards are dealt to  $A$  and  $B$ .  $A$  and  $B$  alternately discard a card face up, starting with  $A$ . The game when the sum of the discards is first divisible by  $2n + 1$ , and the last person to discard wins. What is the probability that  $A$  wins if neither player makes a mistake?


Let  $\sigma(n)$  be the number of positive divisors of  $n$ , and let  $\operatorname{rad} n$  be the product of the distinct prime divisors of  $n$ . By convention,  $\operatorname{rad} 1 = 1$ .  Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural*


Positive integers  $a_0<a_1<\dots<a_n$ , are to be chosen so that  $a_j-a_i$  is not a prime for any  $i,j$  with  $0 \le i <j \le n$ . For each  $n \ge 1$ , determine the smallest possible value of  $a_n$ .

Solution: \boxed{ 4n + 1 }


Find all values of the positive integer  $ m$  such that there exists polynomials  $ P(x),Q(x),R(x,y)$  with real coefficient satisfying the condition: For every real numbers  $ a,b$  which satisfying  $ a^m-b^2=0$ , we always have that  $ P(R(a,b))=a$  and  $ Q(R(a,b))=b$ .

Solution: \boxed{ m = 1 }


In scalene  $\triangle ABC$ ,  $I$  is the incenter,  $I_a$  is the  $A$ -excenter,  $D$  is the midpoint of arc  $BC$  of the circumcircle of  $ABC$  not containing  $A$ , and  $M$  is the midpoint of side  $BC$ . Extend ray  $IM$  past  $M$  to point  $P$  such that  $IM = MP$ . Let  $Q$  be the intersection of  $DP$  and  $MI_a$ , and  $R$  be the point on the line  $MI_a$  such that  $AR\parallel DP$ . Given that  $\frac{AI_a}{AI}=9$ , the ratio  $\frac{QM} {RI_a}$  can be expressed in the form  $\frac{m}{n}$  for two relatively prime positive integers  $m,n$ . Compute  $m+n$ .

*Ray Li.*

Clarifications[list=1][*]"Arc  $BC$  of the circumcircle" means "the arc with endpoints  $B$  and  $C$  not containing  $A$ ".[/list]


For each of the  $9$  positive integers  $n,2n,3n,\dots , 9n$  Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected  $n$  so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to?

Solution: \boxed{ 4 }


Let  $N \geqslant 3$  be an integer. In the country of Sibyl, there are  $N^2$  towns arranged as the vertices of an  $N \times N$  grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.

*Proposed by Sutanay Bhattacharya and Anant Mudgal*

Solution: \boxed{ N }


Let  $S$  be a set of  $2017$  points in the plane. Let  $R$  be the radius of the smallest circle containing all points in  $S$  on either the interior or boundary. Also, let  $D$  be the longest distance between two of the points in  $S$ . Let  $a$ ,  $b$  be real numbers such that  $a\le \frac{D}{R}\le b$  for all possible sets  $S$ , where  $a$  is as large as possible and  $b$  is as small as possible. Find the pair  $(a, b)$ .

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


$n$  coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?

Solution: \boxed{ 2 }


A point  $ M$  is taken on the perpendicular bisector of the side  $ AC$  of an acute-angled triangle  $ ABC$  so that  $ M$  and  $ B$  are on the same side of  $ AC$ . If  $ \angle BAC\equal{}\angle MCB$  and  $ \angle ABC\plus{}\angle MBC\equal{}180^{\circ}$ , find  $ \angle BAC.$


Find all functions  $f:\mathbb{N}_0\to\mathbb{N}_0$  for which  $f(0)=0$  and
\[f(x^2-y^2)=f(x)f(y) \]
for all  $x,y\in\mathbb{N}_0$  with  $x>y$ .

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


Determine all rational numbers  $a$  for which the matrix  $$ \begin{pmatrix}
    a       & -a & -1 & 0
    a       & -a & 0 & -1
    1  & 0 & a & -a
    0      & 1 & a &  -a
\end{pmatrix} $$  is the square of a matrix with all rational entries.

*Proposed by Daniël Kroes, University of California, San Diego*

Solution: \boxed{ a = 0 }


Find the smallest natural number  $ k$  for which there exists natural numbers  $ m$  and  $ n$  such that  $ 1324 \plus{} 279m \plus{} 5^n$  is  $ k$ -th power of some natural number.


Two planes,  $P$  and  $Q$ , intersect along the line  $p$ . The point  $A$  is given in the plane  $P$ , and the point  $C$  in the plane  $Q$ ; neither of these points lies on the straight line  $p$ . Construct an isosceles trapezoid  $ABCD$  (with  $AB \parallel CD$ ) in which a circle can be inscribed, and with vertices  $B$  and  $D$  lying in planes  $P$  and  $Q$  respectively.

Solution: \boxed{ ABCD }


The equation  $$ (x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016) $$  is written on the board, with  $2016$  linear factors on each side. What is the least possible value of  $k$  for which it is possible to erase exactly  $k$  of these  $4032$  linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Solution: \boxed{ k = 2016 }


A square is divided into  $25$  unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?


Suppose that a sequence  $(a_n)_{n=1}^{\infty}$  of integers has the following property: For all  $n$  large enough (i.e.  $n \ge N$  for some  $N$  ),  $a_n$  equals the number of indices  $i$ ,  $1 \le i < n$ , such that  $a_i + i \ge n$ . Find the maximum possible number of integers which occur infinitely many times in the sequence.

Solution: \boxed{ 2 }


Let  $a$  and  $b$  be the roots of the quadratic  $x^2-7x+c$ . Given that  $a^2+b^2=17$ , compute  $c$ .


Moor has  $2016$  white rabbit candies. He and his  $n$  friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that  $n$  is a positive integer (Moor has at least  $1$  friend), how many possible values of  $n$  exist?


Given is a rectangle with perimeter  $x$  cm and side lengths in a  $1:2$  ratio. Suppose that the area of the rectangle is also  $x$   $\text{cm}^2$ . Determine all possible values of  $x$ .


Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 10 mph.  If he completes the first three laps at a constant speed of only 9 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?

Solution: \boxed{15 \text{ mph}}


Amanda has the list of even numbers  $2, 4, 6, \dots 100$  and Billy has the list of odd numbers  $1, 3, 5, \dots 99$ . Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list?

*2016 CCA Math Bonanza Individual #3*


There are 1000 doors  $D_1, D_2, . . . , D_{1000}$  and 1000 persons  $P_1, P_2, . . . , P_{1000}$ .
Initially all the doors were closed. Person  $P_1$  goes and opens all the doors.
Then person  $P_2$  closes door  $D_2, D_4, . . . , D_{1000}$   and leaves the odd numbered doors open. Next  $P_3$  changes the state of every third door, that
is,  $D_3, D_6, . . . , D_{999}$  . (For instance,  $P_3$  closes the open door  $D_3$  and opens
the closed door D6, and so on). Similarly,  $P_m$  changes the state of the
the doors  $D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .$  while leaving the other doors untouched. Finally,  $P_{1000}$  opens  $D_{1000}$  if it was closed or closes it if it were
open. At the end, how many doors will remain open?


The nine horizontal and nine vertical lines on an  $8\times8$  checkerboard form  $r$  rectangles, of which  $s$  are squares.  The number  $s/r$  can be written in the form  $m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$


What is the largest possible distance between two points, one on the sphere of radius 19 with center  $(-2, -10, 5)$  and the other on the sphere of radius 87 with center  $(12, 8, -16)$ ?


Suppose there exist constants  $A$ ,  $B$ ,  $C$ , and  $D$  such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers  $n\geq 4$ .  What is  $A+B+C+D$ ?

*Proposed by David Altizio*


Given a quadratic function  $y$  with respect to  $x$ .   $y$  takes the values of  $-1, -2$  and  $1$  for the values of  $x=-1,\ 0,\ 1$  respectively.
Find the length of the line segment of the graph of the curve cut by the  $x$ -axis.

1956 Tokyo Institute of Technology entrance exam


The sum of the two perfect cubes that are closest to  $500$  is  $343+512=855$ .  Find the sum of the two perfect cubes that are closest to  $2008$ .


A student at Harvard named Kevin
Was counting his stones by  $11$ He messed up  $n$  times
And instead counted  $9$ s
And wound up at  $2007$ .

How many values of  $n$  could make this limerick true?


The vertex  $ A$  of the acute triangle  $ ABC$  is equidistant from the circumcenter  $ O$  and the orthocenter  $ H.$  Determine all possible values for the measure of angle  $ A.$


A rogue spaceship escapes.  $54$  minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels  $12\%$  faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?


Let  $f$  be a monic cubic polynomial such that the sum of the coefficients of  $f$  is  $5$  and such that the sum of the roots of  $f$  is  $1$ . Find the absolute value of the sum of the cubes of the roots of  $f$ .


Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be  $a$ ,  $b$ ,  $c$ . Define  $Q(x)$  to be the monic cubic polynomial with roots  $ab$ ,  $bc$ ,  $ca$ . Compute  $|Q(1)|$ .

*Proposed by Nathan Xiong*


How many positive integers divide the number  $10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$  ?


Let  $1,7,19,\ldots$  be the sequence of numbers such that for all integers  $n\ge 1$ , the average of the first  $n$  terms is equal to the  $n$ th perfect square. Compute the last three digits of the  $2021$ st term in the sequence.

*Proposed by Nathan Xiong*


Find the sum of all positive integers less than  $2006$  which are both multiples of six and one more than a multiple of seven.


David, when submitting a problem for CMIMC, wrote his answer as  $100\tfrac xy$ , where  $x$  and  $y$  are two positive integers with  $x<y$ . Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's!  What is the smallest possible value of  $x+y$ ?


Let  $\phi(n)$  denote  $\textit{Euler's phi function}$ , the number of integers  $1\leq i\leq n$  that are relatively prime to  $n$ .  (For example,  $\phi(6)=2$  and  $\phi(10)=4$ .)  Let \[S=\sum_{d|2008}\phi(d),\] in which  $d$  ranges through all positive divisors of  $2008$ , including  $1$  and  $2008$ .  Find the remainder when  $S$  is divided by  $1000$ .


Define  $f(x)$  as  $\frac{x^2-x-2}{x^2+x-6}$ .   $f(f(f(f(1))))$  can be expressed as  $\frac{p}{q}$  for relatively prime positive integers  $p,q$ . Find  $10p+q$ .


The sides of triangle are  $x$ ,  $2x+1$  and  $x+2$  for some positive rational  $x$ . Angle of triangle is  $60$  degree. Find perimeter


Let  $n$  be a positive integer.  In  $n$ -dimensional space, consider the  $2^n$  points whose coordinates are all  $\pm 1$ .  Imagine placing an  $n$ -dimensional ball of radius 1 centered at each of these  $2^n$  points.  Let  $B_n$  be the largest  $n$ -dimensional ball centered at the origin that does not intersect the interior of any of the original  $2^n$  balls.  What is the smallest value of  $n$  such that  $B_n$  contains a point with a coordinate greater than 2?


If the probability that the sum of three distinct integers between  $16$  and  $30$  (inclusive) is even can be written as  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers, find  $m + n$ .


When flipped, coin A shows heads  $\textstyle\frac{1}{3}$  of the time, coin B shows heads  $\textstyle\frac{1}{2}$  of the time, and coin C shows heads  $\textstyle\frac{2}{3}$  of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as  $\textstyle\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Compute  $p + q$ .

*Proposed by Eugene Chen*


It takes a steamer  $5$  days to go from Gorky to Astrakhan downstream the Volga river and  $7$  days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?


It is given polygon with  $2013$  sides  $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any  $9$  consecutive vertices is constant and its value is  $300$ . If we know that  $A_{13}$  is marked with  $13$  and  $A_{20}$  is marked with  $20$ , determine with which number is marked  $A_{2013}$


Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem  $k$  takes each  $k$  minutes to solve. If for any given problem there is a  $\frac13$  chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?


Let  $ABCD$  be a square with side length  $4$ . Consider points  $P$  and  $Q$  on segments  $AB$  and  $BC$ , respectively, with  $BP=3$  and  $BQ=1$ . Let  $R$  be the intersection of  $AQ$  and  $DP$ . If  $BR^2$  can be expressed in the form  $\frac{m}{n}$  for coprime positive integers  $m,n$ , compute  $m+n$ .

*Proposed by Brandon Wang*


A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?


A positive number  $\dfrac{m}{n}$  has the property that it is equal to the ratio of  $7$  plus the number’s reciprocal and  $65$  minus the number’s reciprocal.  Given that  $m$  and  $n$  are relatively prime positive integers, find  $2m + n$ .


\[\left(1 + \frac{1}{1+2^1}\right)\left(1+\frac{1}{1+2^2}\right)\left(1 + \frac{1}{1+2^3}\right)\cdots\left(1 + \frac{1}{1+2^{10}}\right)= \frac{m}{n},\] where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


A point  $(a,b)$  in the plane is called *sparkling* if it also lies on the line  $ax+by=1$ .  Find the maximum possible distance between two sparkling points.

*Proposed by Evan Chen*


**O**n a blackboard a stranger writes the values of  $s_7(n)^2$  for  $n=0,1,...,7^{20}-1$ , where  $s_7(n)$  denotes the sum of digits of  $n$  in base  $7$ . Compute the average value of all the numbers on the board.


Start with a three-digit positive integer  $A$ . Obtain  $B$  by interchanging the two leftmost digits of  $A$ . Obtain  $C$  by doubling  $B$ . Obtain  $D$  by subtracting  $500$  from  $C$ . Given that  $A + B + C + D = 2014$ , find  $A$ .


Given a regular polygon with  $p$  sides, where  $p$  is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for  $p$ ?


Let  $m$  be the smallest positive integer such that  $m^2+(m+1)^2+\cdots+(m+10)^2$  is the square of a positive integer  $n$ . Find  $m+n$