Find the least positive integer  $k$  so that  $k + 25973$  is a palindrome (a number which reads the same forward and backwards).


Let  $ p > 2$  be a prime number. Find the least positive number  $ a$  which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where  $ f(X)$  and  $ g(X)$  are integer polynomials.

*Mircea Becheanu*.

Solution: \boxed{p}


A function  $f$  is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real  $x$ .  If  $x = 0$  is a root of  $f(x) = 0$ , what is the least number of roots  $f(x) = 0$  must have in the interval  $-1000 \le x \le 1000$ ?


For each positive integer  $ k$ , find the smallest number  $ n_{k}$  for which there exist real  $ n_{k}\times n_{k}$  matrices  $ A_{1}, A_{2}, \ldots, A_{k}$  such that all of the following conditions hold:

(1)  $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$ ,

(2)  $ A_{i}A_{j}= A_{j}A_{i}$  for all  $ 1 \le i, j \le k$ , and

(3)  $ A_{1}A_{2}\ldots A_{k}\ne 0$ .

Solution: \boxed{ n_k = 2^k }


There is a pile with  $15$  coins on a table. At each step, Pedro choses one of the piles in the table with  $a>1$  coins and divides it in two piles with  $b\geq1$  and  $c\geq1$  coins and writes in the board the product  $abc$ . He continues until there are  $15$  piles with  $1$  coin each. Determine all possible values that the final sum of the numbers in the board can have.

Solution: \boxed{ 1120 }


In right triangle  $ ABC$  with right angle at  $ C$ ,  $ \angle BAC < 45$  degrees and  $ AB \equal{} 4$ .  Point  $ P$  on  $ AB$  is chosen such that  $ \angle APC \equal{} 2\angle ACP$  and  $ CP \equal{} 1$ .  The ratio  $ \frac{AP}{BP}$  can be represented in the form  $ p \plus{} q\sqrt{r}$ , where  $ p,q,r$  are positive integers and  $ r$  is not divisible by the square of any prime.  Find  $ p\plus{}q\plus{}r$ .


Suppose  $a$  is a real number such that  $\sin(\pi \cdot  \cos a) = \cos(\pi \cdot \sin a)$ . Evaluate  $35 \sin^2(2a) + 84 \cos^2(4a)$ .


Given that the base- $17$  integer  $\overline{8323a02421_{17}}$  (where a is a base- $17$  digit) is divisible by  $\overline{16_{10}}$ , find  $a$ . Express your answer in base  $10$ .

*Proposed by Jonathan Liu*


Wendy randomly chooses a positive integer less than or equal to  $2020$ . The probability that the digits in Wendy's number add up to  $10$  is  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


Suppose  $ 2015= a_1 <a_2 < a_3<\cdots <a_k $  be a finite sequence of positive integers, and for all  $ m, n \in \mathbb{N} $  and  $1\le m,n \le k $ ,   $$  a_m+a_n\ge a_{m+n}+|m-n|  $$   Determine the largest possible value  $ k $  can obtain.


Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located.


A circle of radius  $4$  is inscribed in a triangle  $ABC$ . We call  $D$  the touchpoint between the circle and side BC. Let  $CD =8$ ,  $DB= 10$ . What is the length of the sides  $AB$  and  $AC$ ?


The sequence  $ (a_n)$  satisfies  $ a_1 \equal{} 1$  and  $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$  for  $ n \geq 1$ . Let  $ k$  be the least integer greater than  $ 1$  for which  $ a_k$  is an integer. Find  $ k$ .


The four faces of a tetrahedral die are labelled  $0, 1, 2,$  and  $3,$  and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled  $0,$  it disappears. If it lands on the face labelled  $1,$  nothing happens. If it lands on the face labelled  $2$  or  $3,$  there will then be  $2$  or  $3$  copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let  $p$  be the probability that they will all eventually disappear. Find  $\left\lfloor \frac{10}{p} \right\rfloor$ .


Find the natural numbers  $ n\ge 2 $  which have the property that the ring of integers modulo  $ n $  has exactly an element that is not a sum of two squares.


Find the number of pairs of integers  $x, y$  with different parities such that  $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ .


Let  $ \theta_1, \theta_2,\ldots , \theta_{2008}$  be real numbers. Find the maximum value of

 $ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$


Let  $n$  and  $k$  be integers satisfying  $\binom{2k}{2} + n = 60$ .  It is known that  $n$  days before Evan's 16th birthday, something happened.  Compute  $60-n$ .

*Proposed by Evan Chen*


Let be a natural number  $ n\ge 3. $  Find  $$  \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) ,  $$  where  $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $  and find in which circumstances this infimum is attained.


Max has a light bulb and a defective switch. The light bulb is initially off, and on the  $n$ th time the switch is flipped, the light bulb has a  $\tfrac 1{2(n+1)^2}$  chance of changing its state (i.e. on  $\to$  off or off  $\to$  on). If Max flips the switch 100 times, find the probability the light is on at the end.

*Proposed by Connor Gordon*


In the city built are  $2019$  metro stations. Some pairs of stations are connected. tunnels, and from any station through the tunnels you can reach any other. The mayor ordered to organize several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and in each station must lie at least on one line. To save money no more than  $k$  lines should be made. It turned out that the order of the mayor is not feasible. What is the largest  $k$  it could to happen?

Solution: \boxed{ 1008 }


Let  $k$  be an integer. If the equation  $(x-1)|x+1|=x+\frac{k}{2020}$  has three distinct real roots, how many different possible values of  $k$  are there?


Let  $N$  be the number of complex numbers  $z$  with the properties that  $|z|=1$  and  $z^{6!}-z^{5!}$  is a real number. Find the remainder when  $N$  is divided by  $1000$ .


Let  $ABC$  be a triangle and  $I$  its incenter. Suppose  $AI=\sqrt{2}$ ,  $BI=\sqrt{5}$ ,  $CI=\sqrt{10}$  and the inradius is  $1$ . Let  $A'$  be the reflection of  $I$  across  $BC$ ,  $B'$  the reflection across  $AC$ , and  $C'$  the reflection across  $AB$ . Compute the area of triangle  $A'B'C'$ .


The sequences  $(a_{n})$ ,  $(b_{n})$  are deﬁned by  $a_{1} = \alpha$ ,  $b_{1} = \beta$ ,  $a_{n+1} = \alpha a_{n} - \beta b_{n}$ ,   $b_{n+1} = \beta a_{n} + \alpha b_{n}$  for all  $n > 0.$  How many pairs  $(\alpha, \beta)$  of real numbers are there such that  $a_{1997} = b_{1}$  and  $b_{1997} = a_{1}$ ?

Solution: \boxed{ 1999 }


Let  $n$  be a natural number. Find the least natural number  $k$  for which there exist  $k$  sequences of  $0$  and  $1$  of length  $2n+2$  with the following property: any sequence of  $0$  and  $1$  of length  $2n+2$  coincides with some of these  $k$  sequences in at least  $n+2$  positions.

Solution: \boxed{ k = 4 }


In a right-angled triangle in which all side lengths are integers, one has a cathetus length  $1994$ . Determine the length of the hypotenuse.


Let  $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$ , where  $n$  is an integer greater than  $1$ . Find the smallest number  $k=k(n)$  such that there is a number which may be expressed as a sum of  $k$  (possibly repeated) elements in  $S_{n}$  in more than one way. (Rearrangements are considered the same.)

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


The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form  $\frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .


Compute the number of two digit positive integers that are divisible by both of their digits. For example,  $36$  is one of these two digit positive integers because it is divisible by both  $3$  and  $6$ .

*2021 CCA Math Bonanza Lightning Round #2.4*


A permutation of a finite set  $S$  is a one-to-one function from  $S$  to  $S$ .  A permutation  $P$  of the set  $\{ 1, 2, 3, 4, 5 \}$  is called a W-permutation if  $P(1) > P(2) < P(3) > P(4) < P(5)$ .  A permutation of the set  $\{1, 2, 3, 4, 5 \}$  is selected at random.  Compute the probability that it is a W-permutation.


In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a  $1$  on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins  $3$  games before Jacob wins  $3$  games can be written as  $\tfrac{p}{2^q}$ , where  $p$  and  $q$  are positive integers, and  $p$  is odd. Find the remainder when  $p+q$  is divided by  $1000$ .

*Proposed by firebolt360*


Let  $\alpha$  and  $\beta$  be positive integers such that  $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$  Find the smallest possible value of  $\beta$ .


The set  $M= \{1;2;3;\ldots ; 29;30\}$  is  divided in  $k$ subsets such that if  $a+b=n^2, (a,b \in M, a\neq b, n$  is an
integer number  $)$ , then  $a$  and  $b$  belong different subsets.
Determine the minimum value of  $k$ .


Call a  $ 3$ -digit number *geometric* if it has  $ 3$  distinct digits which, when read from left to right, form a geometric sequence.  Find the difference between the largest and smallest geometric numbers.


Suppose  $ n$  is a product of four distinct primes  $ a,b,c,d$  such that:
 $ (i)$   $ a\plus{}c\equal{}d;$
 $ (ii)$   $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
 $ (iii)$   $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine  $ n$ .


In triangle  $ABC$ , find the smallest possible value of   $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$


Alice is counting up by fives, starting with the number  $3$ . Meanwhile, Bob is counting down by fours, starting with the number  $2021$ . How many numbers between  $3$  and  $2021$ , inclusive, are counted by both Alice and Bob?


Given that nonzero reals  $a,b,c,d$  satisfy  $a^b=c^d$  and  $\frac{a}{2c}=\frac{b}{d}=2$ , compute  $\frac{1}{c}$ .

*2021 CCA Math Bonanza Lightning Round #2.2*


What is the sum of all the integers  $n$  such that  $\left|n-1\right|<\pi$ ?

*2016 CCA Math Bonanza Lightning #1.1*


Let   $x = \left( 1 + \frac{1}{n}\right)^n$  and  $y = \left( 1 + \frac{1}{n}\right)^{n+1}$  where  $n \in \mathbb{N}$ . Which one of the numbers   $x^y$ ,  $y^x$  is bigger ?

Solution: \boxed{ x^y = y^x }


$k$  is a fixed positive integer. Let  $a_n$  be the number of maps  $f$  from the subsets of  $\{1, 2, ... , n\}$  to  $\{1, 2, ... , k\}$  such that for all subsets  $A, B$  of  $\{1, 2, ... , n\}$  we have  $f(A \cap B) = \min (f(A), f(B))$ . Find  $\lim_{n \to \infty} \sqrt[n]{a_n}$ .

Solution: \boxed{k}


If   $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$ , where  $m$  and  $n$  are positive integers, what is the value of  $m+n$ ?


Find all positive reals  $x,y,z $  such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

Solution: \boxed{(2014, 2014, 2014)}


$x$  is a base- $10$  number such that when the digits of  $x$  are interpreted as a base- $20$  number, the resulting number is twice the value as when they are interpreted as a base- $13$  number. Find the sum of all possible values of  $x$ .


For a permutation  $\pi$  of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where  $\pi (i)$  denotes the  $i$ th element of the permutation. Suppose that  $M$  is the maximum possible value of  $S(\pi)$  over all permutations  $\pi$  of the integers from 1 to 10. Determine the number of permutations  $\pi$  for which  $S(\pi) = M$ .

*Ray Li*


Let  $r$  be a positive integer. Show that if a graph  $G$  has no cycles of length at most  $2r$ , then it has at most  $|V|^{2016}$  cycles of length exactly  $2016r$ , where  $|V|$  denotes the number of vertices in the graph  $G$ .


In a year that has  $365$  days, what is the maximum number of "Tuesday the  $13$ th" there can be?

Note: The months of April, June, September and November have  $30$  days each, February has  $28$  and all others have  $31$  days.


Find all functions  $f:\mathbb{N} \rightarrow \mathbb{N}$  such that for all positive integers  $n$ , there exists an unique positive integer  $k$ , satisfying  $f^k(n)\leq n+k+1$ .

Solution: \boxed{ f(n) = n + 2 }


What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?


Determine all positive integers  $n$  such that   $$ n\cdot 2^{n-1}+1 $$   is a perfect square.


Let  $f(x) = 2^x + 3^x$ . For how many integers  $1 \leq n \leq 2020$  is  $f(n)$  relatively prime to all of  $f(0), f(1), \dots, f(n-1)$ ?


The altitudes of the triangle  ${ABC}$  meet in the point  ${H}$ . You know that  ${AB = CH}$ . Determine the value of the angle  $\widehat{BCA}$ .


A collection of  $n$  squares on the plane is called tri-connected if the following criteria are satisfied:

(i) All the squares are congruent.
(ii) If two squares have a point  $P$  in common, then  $P$  is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.

How many positive integers  $n$  are there with  $2018\leq n \leq 3018$ , such that there exists a collection of  $n$  squares that is tri-connected?


Let  $T=TNFTPP$ .  Points  $A$  and  $B$  lie on a circle centered at  $O$  such that  $\angle AOB$  is right.  Points  $C$  and  $D$  lie on radii  $OA$  and  $OB$  respectively such that  $AC = T-3$ ,  $CD = 5$ , and  $BD = 6$ .  Determine the area of quadrilateral  $ACDB$ .

[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, here's the value of T$T=10$.


Determine the largest positive integer  $n$  which cannot be written as the sum of three numbers bigger than  $1$  which are pairwise coprime.

Solution: \boxed{ 17 }


For  $n$  a positive integer, denote by  $P(n)$  the product of all positive integers divisors of  $n$ . Find the smallest  $n$  for which

\[ P(P(P(n))) > 10^{12} \]


Find the largest positive integer  $n$  for which the inequality
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all  $a, b, c \in [0,1]$ . Here we make the convention  $\sqrt[1]{abc}=abc$ .


Carmen selects four different numbers from the set  $\{1, 2, 3, 4, 5, 6, 7\}$  whose sum is 11. If  $l$  is the largest of these four numbers, what is the value of  $l$ ?


Determine the largest positive integer  $n$  for which there exists a set  $S$  with exactly  $n$  numbers such that
- each member in  $S$  is a positive integer not exceeding  $2002$ ,
- if  $a,b\in S$  (not necessarily different), then  $ab\not\in S$ .

Solution: \boxed{ 1958 }


Let  $A$  and  $B$  be distinct positive integers such that each has the same number of positive divisors that 2013 has.  Compute the least possible value of  $\left| A - B \right|$ .


Let  $\triangle ABC$  be a triangle with  $AB < AC$ . Let the angle bisector of  $\angle BAC$  meet  $BC$  at  $D$  , and let  $M$  be the midpoint of  $BC$  . Let  $P$  be the foot of the perpendicular from  $B$  to  $AD$  .  $Q$  the intersection of  $BP$  and  $AM$  . Show that :  $(DQ) // (AB) $  .

Solution: \boxed{ DQ \parallel AB }


In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank   $n\times n$  grid with probability  $k$ ; he notices that the expected value of the score of the resulting grid is equal to  $k$ , too! Given that  $k > 0.9999$ , find the minimum possible value of  $n$ .

*Proposed by Andrew Wu*


Determine all positive integers  $n$  for which the equation
\[ x^n + (2+x)^n + (2-x)^n = 0 \]
has an integer as a solution.


The following sequence lists all the positive rational numbers that do not exceed  $\frac12$  by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is
\[
\frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots.
\]
Let  $m$  and  $n$  be relatively prime positive integers so that the  $2012^{\text{th}}$  fraction in the list is equal to  $\frac{m}{n}$ . Find  $m+n$ .


Find all the positive integers less than 1000 such that the cube of the sum of its digits  is equal to the square of such integer.

Solution: \boxed{1 \text{ and } 27}


A mathematical contest had  $3$  problems, each of which was given a score between  $0$  and  $7$  ( $0$  and  $7$  included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are  $7,1,2$  and  $7,1,5$ , but there might be two contestants whose ordered scores are  $7,1,2$  and  $7,2,1$ ). Find the maximum number of contestants.


Let  $A$  be a positive real number. What are the possible values of  $\displaystyle\sum_{j=0}^{\infty} x_j^2, $  given that  $x_0, x_1, \cdots$  are positive numbers for which  $\displaystyle\sum_{j=0}^{\infty} x_j = A$ ?

Solution: \boxed{ (0, A^2) }


We draw  $n$  convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.

Solution: \boxed{ 4n^2 - 4n + 2 }


There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and  $k$  meters per second, respectively, where  $k$  is some positive integer with  $7\le k \le 2013$ . Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of  $k$  are there?

*Ray Li*


More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.

Solution: \boxed{ n = 12 }


The natural numbers from  $1$  to  $50$  are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?


Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other  $2$  players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?


A natural number is called a *prime power* if that number can be expressed as  $p^n$  for some prime  $p$  and natural number  $n$ .
Determine the largest possible  $n$  such that there exists a sequence of prime powers  $a_1, a_2, \dots, a_n$  such that  $a_i = a_{i - 1} + a_{i - 2}$  for all  $3 \le i \le n$ .


Let  $S$  be the set of all positive integers from 1 through 1000 that are not perfect squares.  What is the length of the longest, non-constant, arithmetic sequence that consists of elements of  $S$ ?


In triangle  $ABC$ , angles  $A$  and  $B$  measure 60 degrees and 45 degrees, respectively.  The bisector of angle  $A$  intersects  $\overline{BC}$  at  $T$ , and  $AT=24.$   The area of triangle  $ABC$  can be written in the form  $a+b\sqrt{c},$  where  $a$ ,  $b$ , and  $c$  are positive integers, and  $c$  is not divisible by the square of any prime.  Find  $a+b+c.$


Let  $P$  be a set of  $n\ge 3$  points in the plane, no three of which are on a line. How many possibilities are there to choose a set  $T$  of  $\binom{n-1}{2}$  triangles, whose vertices are all in  $P$ , such that each triangle in  $T$  has a side that is not a side of any other triangle in  $T$ ?

Solution: \boxed{ n }


Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or  $\textit{sell}$  one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars must Misha begin with in order to build a city after three days of working?


Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is  $18$ . How many passcodes satisfy these conditions?


Find all primes  $p$  such that there exist positive integers  $q$  and  $r$  such that  $p \nmid q$ ,  $3 \nmid q$ ,  $p^3 = r^3 - q^2$ .

Solution: \boxed{ p = 7 }


The sum of the areas of all triangles whose vertices are also vertices of a  $1\times 1 \times 1$  cube is  $m+\sqrt{n}+\sqrt{p}$ , where  $m$ ,  $n$ , and  $p$  are integers.  Find  $m+n+p$ .


Calculate the following indefinite integrals.

[1]  $\int \sin x\cos ^ 3 x dx$

[2]  $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$

[3]  $\int x^2 \sqrt{x^3+1}dx$

[4]  $\int \frac{e^{2x}-3e^{x}}{e^x}dx$

[5]  $\int (1-x^2)e^x dx$

Solution: \boxed{-(x - 1)^2 e^x + C}


Let the tangent line passing through a point  $A$   outside the circle with center  $O$  touches the circle at  $B$  and  $C$ . Let  $[BD]$  be the diameter of the circle. Let the lines  $CD$  and  $AB$  meet at  $E$ . If the lines  $AD$  and  $OE$  meet at  $F$ , find  $|AF|/|FD|$ .


Determine the largest value the expression   $$  \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j}  $$  may achieve, as  $ x_1,x_2,x_3,x_4 $  run through the non-negative real numbers, and add up to  $ 1. $  Find also the specific values of this numbers that make the above sum achieve the asked maximum.

Solution: \boxed{ \frac{3}{4} }


We say that an ordered pair  $(a,b)$  of positive integers with  $a>b$  is square-ish if both  $a+b$  and  $a-b$  are perfect squares. For example,  $(17,8)$  is square-ish because  $17+8=25$  and  $17-8=9$  are both perfect squares. How many square-ish pairs  $(a,b)$  with  $a+b<100$  are there?

*Proposed by Nathan Xiong*


Let a convex polygon  $P$  be contained in a square of side one. Show that the sum of the sides of  $P$  is less than or equal to  $4$ .


A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly  $1/2$  that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?


Points  $D$  and  $E$  are chosen on the exterior of  $\vartriangle ABC$  such that  $\angle  ADC = \angle  BEC = 90^o$ . If  $\angle ACB = 40^o$ ,  $AD = 7$ ,  $CD = 24$ ,  $CE = 15$ , and  $BE = 20$ , what is the measure of  $\angle ABC $  in,degrees?


On the coordinate plane, let  $C$  be a circle centered  $P(0,\ 1)$  with radius 1. let  $a$  be a real number  $a$  satisfying  $0<a<1$ . Denote by  $Q,\ R$  intersection points of the line  $y=a(x+1) $  and  $C$ .

(1) Find the area  $S(a)$  of  $\triangle{PQR}$ .

(2) When  $a$  moves in the range of  $0<a<1$ , find the value of  $a$  for which  $S(a)$  is maximized.

*2011 Tokyo University entrance exam/Science, Problem 1*

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


Let  $n$  be a natural number divisible by  $3$ . We have a  $n \times n$  table and each square is colored either black or white. Suppose that for all  $m \times m$  sub-tables from the table ( $m > 1$ ), the number of black squares is not more than white squares. Find the maximum number of black squares.


Solve in  $ \mathbb{Z}^2 $  the following equation:  $$  (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} .  $$  *Adrian Zanoschi*

Solution: \boxed{ (0, 6) }


Define  $L(x) = x - \frac{x^2}{2}$  for every real number  $x$ .  If  $n$  is a positive integer, define  $a_n$  by
\[
  a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are  $n$  iterations of  $L$ .  For example,
\[
  a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).
\]
As  $n$  approaches infinity, what value does  $n a_n$  approach?


$ m$  and  $ n$  are positive integers. In a  $ 8 \times 8$  chessboard,  $ (m,n)$  denotes the number of grids a Horse can jump in a chessboard ( $ m$  horizontal  $ n$  vertical or  $ n$  horizontal  $ m$  vertical ). If a  $ (m,n) \textbf{Horse}$  starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a  $ \textbf{H Route}$ . For example, the  $ (1,2) \textbf{Horse}$  has its  $ \textbf{H Route}$ . Find the smallest positive integer  $ t$ , such that from any grid of the chessboard, the  $ (t,t\plus{}1) \textbf{Horse}$  does not has any  $ \textbf{H Route}$ .

Solution: \boxed{ t = 2 }


We call  $\overline{a_n\ldots a_2}$  the Fibonacci representation of a positive integer  $k$  if \[k = \sum_{i=2}^n a_i F_i,\] where  $a_i\in\{0,1\}$  for all  $i$ ,  $a_n=1$ , and  $F_i$  denotes the  $i^{\text{th}}$  Fibonacci number ( $F_0=0$ ,  $F_1=1$ , and  $F_i=F_{i-1}+F_{i-2}$  for all  $i\ge2$ ). This representation is said to be  $\textit{minimal}$  if it has fewer 1’s than any other Fibonacci representation of  $k$ . Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.


Find all positive integers  $n>1$  such that
\[\tau(n)+\phi(n)=n+1\]
Which in this case,  $\tau(n)$  represents the amount of positive divisors of  $n$ , and  $\phi(n)$  represents the amount of positive integers which are less than  $n$  and relatively prime with  $n$ .

*Raja Oktovin, Pekanbaru*

Solution: \boxed{ n = 4 }


A coin is tossed  $10$  times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.


Given two natural numbers  $ w$  and  $ n,$  the tower of  $ n$   $ w's$  is the natural number  $ T_n(w)$  defined by

\[ T_n(w) = w^{w^{\cdots^{w}}},\]

with  $ n$   $ w's$  on the right side. More precisely,  $ T_1(w) = w$  and  $ T_{n+1}(w) = w^{T_n(w)}.$  For example,  $ T_3(2) = 2^{2^2} = 16,$    $ T_4(2) = 2^{16} = 65536,$  and  $ T_2(3) = 3^3 = 27.$  Find the smallest tower of  $ 3's$  that exceeds the tower of  $ 1989$   $ 2's.$  In other words, find the smallest value of  $ n$  such that  $ T_n(3) > T_{1989}(2).$  Justify your answer.


For two sets  $A, B$ , define the operation   $$ A \otimes B = \{x \mid x=ab+a+b, a \in A, b \in B\}. $$   Set  $A=\{0, 2, 4, \cdots, 18\}$  and  $B=\{98, 99, 100\}$ . Compute the sum of all the elements in  $A \otimes B$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 7)*


On semicircle, with diameter  $|AB|=d$ , are given points  $C$  and  $D$  such that:  $|BC|=|CD|=a$  and  $|DA|=b$  where  $a, b, d$  are different positive integers. Find minimum possible value of  $d$


A math contest consists of  $9$  objective type questions and  $6$  fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly  $14$  out of  $15$  questions. Let  $O_1, O_2, \dots , O_9$  be the nine objective questions and  $F_1, F_2, \dots , F_6$  be the six fill inthe blanks questions. Let  $a_{ij}$  be the number of students who attemoted both questions  $O_i$  and  $F_j$ . If the sum of all the  $a_{ij}$  for  $i=1, 2,\dots , 9$  and  $j=1, 2,\dots , 6$  is  $972$ , then find the number of students who took the test in the school.


Find the smallest constant  $ C$  such that for all real  $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds.


Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle  $A$  inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

Solution: \boxed{ \lambda = 2 }


In triangle  $ABC$ ,  $\angle ABC$  is obtuse.  Point  $D$  lies on side  $AC$  such that  $\angle ABD$  is right, and point  $E$  lies on side  $AC$  between  $A$  and  $D$  such that  $BD$  bisects  $\angle EBC$ .  Find  $CE$  given that  $AC=35$ ,  $BC=7$ , and  $BE=5$ .


(**4**) Let  $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$  for all real numbers  $ x$ . Determine  $ f^{(2008)}(0)$  (i.e.,  $ f$  differentiated  $ 2008$  times and then evaluated at  $ x \equal{} 0$ ).


Find the largest positive integer  $n$  such that the number  $(2n)!$  ends with  $10$  more zeroes than the number  $n!$ .

*Proposed by Andy Xu*


The positive integer  $m$  is a multiple of 111, and the positive integer  $n$  is a multiple of 31. Their sum is 2017. Find  $n - m$ .


Mariana plays with an  $8\times 8$  board with all its squares blank. She says that two houses are *neighbors* if they have a common side or vertex, that is, two houses can be neighbors vertically, horizontally or diagonally. The game consists of filling the  $64$  squares on the board, one after the other, each with a number according to the following rule: she always chooses a house blank and fill it with an integer equal to the number of neighboring houses that are still in White. Once this is done, the house is no longer considered blank.
Show that the value of the sum of all  $64$  numbers written on the board at the end of the game does not depend in the order of filling. Also, calculate the value of this sum.

Note: A house is not neighbor to itself.

original wordingMariana brinca com um tabuleiro 8 x 8 com todas as suas casas em branco. Ela diz que duas
casas s˜ao vizinhas se elas possu´ırem um lado ou um v´ertice em comum, ou seja, duas casas podem ser vizinhas
verticalmente, horizontalmente ou diagonalmente. A brincadeira consiste em preencher as 64 casas do tabuleiro,
uma ap´os a outra, cada uma com um n´umero de acordo com a seguinte regra: ela escolhe sempre uma casa
em branco e a preenche com o n´umero inteiro igual `a quantidade de casas vizinhas desta que ainda estejam em
branco. Feito isso, a casa n˜ao ´e mais considerada em branco.
Demonstre que o valor da soma de todos os 64 n´umeros escritos no tabuleiro ao final da brincadeira n˜ao depende
da ordem do preenchimento. Al´em disso, calcule o valor dessa soma.
Observa¸c˜ao: Uma casa n˜ao ´e vizinha a si mesma


Find the smallest number  $k$ , such that   $ \frac{l_a+l_b}{a+b}<k$  for all triangles with sides  $a$  and  $b$  and bisectors  $l_a$  and  $l_b$  to them, respectively.

*Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov*


For every  $a \in \mathbb N$  denote by  $M(a)$  the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{  is a divisor of } ab \}.\]
Find  $\max_{a\leq 1983} M(a).$

Solution: \boxed{ 121 }


Determine the least odd number  $a > 5$  satisfying the following conditions: There are positive integers  $m_1,m_2, n_1, n_2$  such that  $a=m_1^2+n_1^2$ ,  $a^2=m_2^2+n_2^2$ , and  $m_1-n_1=m_2-n_2.$


Find all sets of integers  $n\geq 2$  and positive integers  $(a_1, a_2, \dots, a_n)$  that satisfy all of the following conditions:

- $a_1 < a_2 < \cdots < a_n$
- $a_n$  is a prime number.
- For any integer  $k$  between  $1$  and  $n$ ,  $a_k$  divides  $a_1+a_2+\cdots+a_n$ .

Solution: \boxed{ (1, 2, 3) }


Find all functions  $f:\mathbb{R}\to\mathbb{R}$  which satisfy the following equality for all  $x,y\in\mathbb{R}$  \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]*Proposed by Dániel Dobák, Budapest*

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


What is the maximal number of solutions can the equation have   $$ \max  \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 $$  where  $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$  are real numbers, all  $a_i$  not equal to  $0$ .


A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the  $b_k$  are positive integers, has the surprising property that if we multiply it out and discard all terms involving  $z$  to a power larger than  $32$ , what is left is just  $1-2z$ .  Determine, with proof,  $b_{32}$ .

Solution: \boxed{2^{27} - 2^{11}}


In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.


َA natural number  $n$  is given. Let  $f(x,y)$  be a polynomial of degree less than  $n$  such that for any positive integers  $x,y\leq n, x+y \leq n+1$  the equality  $f(x,y)=\frac{x}{y}$  holds. Find  $f(0,0)$ .


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

Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]


Find a necessary and sufficient condition on the positive integer  $n$  that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root.

Solution: \boxed{ n = 1 }


Let  $ABC$  be a right triangle with  $\angle ACB = 90^{\circ}$  and centroid  $G$ . The circumcircle  $k_1$  of triangle  $AGC$  and the circumcircle  $k_2$  of triangle  $BGC$  intersect  $AB$  at  $P$  and  $Q$ , respectively. The perpendiculars from  $P$  and  $Q$  respectively to  $AC$  and  $BC$  intersect  $k_1$  and  $k_2$  at  $X$  and  $Y$ . Determine the value of  $\frac{CX \cdot CY}{AB^2}$ .


Together, Kenneth and Ellen pick a real number  $a$ . Kenneth subtracts  $a$  from every thousandth root of unity (that is, the thousand complex numbers  $\omega$  for which  $\omega^{1000}=1$ ) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts  $a$  from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of  $a$  are there?


Let  $f: N \to  N$  satisfy  $n=\sum_{d|n} f(d), \forall n \in N$ . Then sum of all possible values of  $f(100)$  is?


Fix a sequence  $ a_1,a_2,a_3,... $  of integers satisfying the following condition:for all prime numbers  $ p $  and all positive integers  $ k $ , we have  $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of  $ a_{2013} $ .


How many distinct four letter arrangements can be formed by rearranging the letters found in the word **FLUFFY**? For example, FLYF and ULFY are two possible arrangements.


What's the largest number of elements that a set of positive integers between  $1$  and  $100$  inclusive can have if it has the property that none of them is divisible by another?

Solution: \boxed{ 50 }


Find all integer triples  $(a,b,c)$  with  $a>0>b>c$  whose sum equal  $0$  such that the number  $$ N=2017-a^3b-b^3c-c^3a $$   is a perfect square of an integer.

Solution: \boxed{(36, -12, -24)}


Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial  $f(x)$  with the following properties:

(a) for any integer  $n$ ,  $f(n)$  is an integer;
(b) the degree of  $f(x)$  is less than  $187$ .

Alice knows that  $f(x)$  satisfies (a) and (b), but she does not know  $f(x)$ . In every turn, Alice picks a number  $k$  from the set  $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of  $f(k)$ . Find the smallest positive integer  $N$  so that Alice always knows for sure the parity of  $f(0)$  within  $N$  turns.

*Proposed by YaWNeeT*


Evan has  $10$  cards numbered  $1$  through  $10$ . He chooses some of the cards and takes the product of the numbers on them. When the product is divided by  $3$ , the remainder is  $1$ . Find the maximum number of cards he could have chose.

*Proposed by Evan Chang*


Real numbers  $a$ ,  $b$ ,  $c$  which are differ from  $1$  satisfies the following conditions;
(1)  $abc =1$ (2)  $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression  $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$ .


Cynthia and Lynnelle are collaborating on a problem set. Over a  $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous  $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.

Solution: \boxed{ \frac{4}{9} }


Compute the number of ordered quadruples of complex numbers  $(a,b,c,d)$  such that
\[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \]
holds for all complex numbers  $x, y$ .

*Proposed by Evan Chen*


In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth  $1$ ,  $5$ ,  $25$ , and  $125$  quacks. How many ways can someone pay oﬀ  $125$  quacks using these coins?

*Proposed by Aaron Lin*


A and B plays the following game: they choose randomly  $k$  integers from  $\{1,2,\dots,100\}$ ; if their sum is even, A wins, else B wins. For what values of  $k$  does A and B have the same chance of winning?

Solution: \boxed{ k }


Let  $a$  be the sum of the numbers: $99 \times 0.9$  $999 \times 0.9$  $9999 \times 0.9$  $\vdots$  $999\cdots 9 \times 0.9$ where the final number in the list is  $0.9$  times a number written as a string of  $101$  digits all equal to  $9$ .
Find the sum of the digits in the number  $a$ .


4.	Find the biggest positive integer  $n$ , lesser thar  $2012$ , that has the following property:
If  $p$  is a prime divisor of  $n$ , then  $p^2 - 1$  is a divisor of  $n$ .


$ABCDEF GH$  is a regular octagon with  $10$  units side . The circle with center  $A$  and radius  $AC$  intersects  the circle  with center  $D$  and radius  $CD$  at point  $ I$ , different from  $C$ . What is the length of the segment  $IF$ ?


Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?


Given a positive integer  $n \ge 2$ , determine the largest positive integer  $N$  for which there exist  $N+1$  real numbers  $a_0, a_1, \dots, a_N$  such that $(1) \ $     $a_0+a_1 = -\frac{1}{n},$  and $(2) \ $     $(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}$  for  $1 \le k \le N-1$ .

Solution: \boxed{n}


(from The Good Soldier Svejk) Senior military doctor Bautze exposed  $abccc$  malingerers among  $aabbb$  draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?

Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.

*(1 point)*


Let  $P$  be a cubic monic polynomial with roots  $a$ ,  $b$ , and  $c$ .  If  $P(1)=91$  and  $P(-1)=-121$ , compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\]

*Proposed by David Altizio*


The sequence  $(x_n)$  is determined by the conditions: $x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$  for  $n \geq 1$ .
Find  $\sum_{n=0}^{1992} 2^nx_n$ .


A trapezium is given with parallel bases having lengths  $1$  and  $4$ . Split it into two trapeziums by a cut, parallel to the bases, of length  $3$ . We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in  $m$  and  $n$  trapeziums, respectively, so that all the  $m + n$  trapezoids obtained have the same area. Determine the minimum possible value for  $m + n$  and the lengths of the cuts to be made to achieve this minimum value.

Solution: \boxed{ 15 }


$ABCD$  is a rhombus. Take points  $E$ ,  $F$ ,  $G$ ,  $H$  on sides  $AB$ ,  $BC$ ,  $CD$ ,  $DA$  respectively so that  $EF$  and  $GH$  are tangent to the incircle of  $ABCD$ . Show that  $EH$  and  $FG$  are parallel.

Solution: \boxed{ EH \parallel FG }


Ten identical crates each of dimensions  $ 3$  ft  $ \times$   $ 4$  ft  $ \times$   $ 6$  ft.  The first crate is placed flat on the floor.  Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random.  Let  $ \frac{m}{n}$  be the probability that the stack of crates is exactly  $ 41$  ft tall, where  $ m$  and  $ n$  are relatively prime positive integers.  Find  $ m$ .


The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series.


For every integer  $n	\ge 2$  let  $B_n$  denote the set of all binary  $n$ -nuples of zeroes and ones, and split  $B_n$  into equivalence classes by letting two  $n$ -nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers  $n 	\ge 2$  for which  $B_n$  splits into an odd number of equivalence classes.

Solution: \boxed{ n = 2 }


Let  $n$  be a positive integer. Jadzia has to write all integers from  $1$  to  $2n-1$  on a board, and she writes each integer  in blue or red color. We say that pair of numbers  $i,j\in \{1,2,3,...,2n-1\}$ , where  $i\leqslant j$ , is  $\textit{good}$  if and only if number of blue numbers among  $i,i+1,...,j$  is odd. Determine, in terms of  $n$ , maximal number of good pairs.


Triangle  $ABC$  is the right angled triangle with the vertex  $C$  at the right angle. Let  $P$  be the point of reflection of  $C$  about  $AB$ . It is known that  $P$  and two midpoints of two sides of  $ABC$  lie on a line. Find the angles of the triangle.

Solution: \boxed{ 30^\circ }


Let  $ABC$  be a scalene triangle whose side lengths are positive integers. It is called *stable* if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

*Proposed by Evan Chen*


How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2 & x^2+y^2\leq 2012 \end{matrix} $ where  $ x,y $  are non-zero integers


You, your friend, and two strangers are sitting at a table. A standard  $52$ -card deck is randomly dealt into  $4$  piles of  $13$  cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces.


Let  $O$  be a circle with diameter  $AB = 2$ . Circles  $O_1$  and  $O_2$  have centers on  $\overline{AB}$  such that  $O$  is tangent to  $O_1$  at  $A$  and to  $O_2$  at  $B$ , and  $O_1$  and  $O_2$  are externally tangent to each other. The minimum possible value of the sum of the areas of  $O_1$  and  $O_2$  can be written in the form  $\frac{m\pi}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Compute  $m + n$ .


How many ordered triples of nonzero integers  $(a, b, c)$  satisfy  $2abc = a + b + c + 4$ ?


[help me] Let m and n denote the number of digits in  $2^{2007}$  and  $5^{2007}$  when expressed in base 10. What is the sum m + n?


Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability  $\tfrac23,$  and each of the other five sides has probability  $\tfrac{1}{15}.$  Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is  $\tfrac{p}{q},$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .


Define the sequence  $a_1,a_2,a_3,\ldots$  by  $a_n=\sum_{k=1}^n\sin(k)$ , where  $k$  represents radian measure. Find the index of the  $100$ th term for which  $a_n<0$ .


Compute the number of ordered triples of integers  $(a,b,c)$  between  $1$  and  $12$ , inclusive, such that, if   $$ q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}}, $$   then  $q$  is a positive rational number and, when  $q$  is written in lowest terms, the numerator is divisible by  $13$ .

*Proposed by Ankit Bisain*


Let  $ n \geq 3$  and consider a set  $ E$  of  $ 2n \minus{} 1$  distinct points on a circle. Suppose that exactly  $ k$  of these points are to be colored black.  Such a coloring is **good** if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly  $ n$  points from  $ E$ .  Find the smallest value of  $ k$  so that every such coloring of  $ k$  points of  $ E$  is good.

Solution: \boxed{n}


An arithmetic sequence consists of  $ 200$  numbers that are each at least  $ 10$  and at most  $ 100$ .  The sum of the numbers is  $ 10{,}000$ .  Let  $ L$  be the *least* possible value of the  $ 50$ th term and let  $ G$  be the *greatest* possible value of the  $ 50$ th term.  What is the value of  $ G \minus{} L$ ?


For a positive integer  $n$ , let  $s(n)$  and  $c(n)$  be the number of divisors of  $n$  that are perfect squares and perfect cubes respectively. A positive integer  $n$  is called fair if  $s(n)=c(n)>1$ . Find the number of fair integers less than  $100$ .


Three non-collinear lattice points  $A,B,C$  lie on the plane  $1+3x+5y+7z=0$ . The minimal possible area of triangle  $ABC$  can be expressed as  $\frac{\sqrt{m}}{n}$  where  $m,n$  are positive integers such that there does not exists a prime  $p$  dividing  $n$  with  $p^2$  dividing  $m$ . Compute  $100m+n$ .

*Proposed by Yannick Yao*


Let  $n$  be a positive integer. For each  $4n$ -tuple of nonnegative real numbers  $a_1,\ldots,a_{2n}$ ,  $b_1,\ldots,b_{2n}$  that satisfy  $\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n$ , define the sets
\[A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},\]
\[B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.\]
Let  $m$  be the minimum element of  $A\cup B$ . Determine the maximum value of  $m$  among those derived from all such  $4n$ -tuples  $a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}$ .

*Proposed by usjl.*


For primes  $a, b,c$  that satisfies the following, calculate  $abc$ .  $b + 8$  is a multiple of  $a$ , and  $b^2 - 1$  is a multiple of  $a$  and  $c$ . Also,  $b + c = a^2 - 1$ .


Let  $w_{1}$  and  $w_{2}$  denote the circles  $x^{2}+y^{2}+10x-24y-87=0$  and  $x^{2}+y^{2}-10x-24y+153=0$ , respectively. Let  $m$  be the smallest positive value of  $a$  for which the line  $y=ax$  contains the center of a circle that is externally tangent to  $w_{2}$  and internally tangent to  $w_{1}$ . Given that  $m^{2}=p/q$ , where  $p$  and  $q$  are relatively prime integers, find  $p+q$ .


The numbers  $1,2,\dots,10$  are written on a board. Every minute, one can select three numbers  $a$ ,  $b$ ,  $c$  on the board, erase them, and write  $\sqrt{a^2+b^2+c^2}$  in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at this point?

*Proposed by Evan Chen*


A machine-shop cutting tool has the shape of a notched circle, as shown.  The radius of the circle is  $\sqrt{50}$  cm, the length of  $AB$  is 6 cm, and that of  $BC$  is 2 cm.  The angle  $ABC$  is a right angle.  Find the square of the distance (in centimeters) from  $B$  to the center of the circle.
[asy]
size(150); defaultpen(linewidth(0.65)+fontsize(11));
real r=10;
pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C;
path P=circle(O,r);
C=intersectionpoint(B--(B.x+r,B.y),P);
draw(Arc(O, r, 45, 360-17.0312));
draw(A--B--C);dot(A); dot(B); dot(C);
label(" $A$ ",A,NE);
label(" $B$ ",B,SW);
label(" $C$ ",C,SE);
[/asy]


Let  $n>1$  be an integer and let  $f_1$ ,  $f_2$ , ...,  $f_{n!}$  be the  $n!$  permutations of  $1$ ,  $2$ , ...,  $n$ . (Each  $f_i$  is a bijective function from  $\{1,2,...,n\}$  to itself.) For each permutation  $f_i$ , let us define  $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$ . Find  $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$ .

Solution: \boxed{\frac{n^2 - 1}{3}}


Ten birds land on a  $10$ -meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any  $x$ -meter portion of the wire, there is an  $\tfrac{x}{10}$  probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?


Denote by  $d(n)$  the number of positive divisors of a positive integer  $n$ . Find the smallest constant  $c$  for which  $d(n)\le c\sqrt n$  holds for all positive integers  $n$ .


Compute the smallest positive integer  $n$  such that there do not exist integers  $x$  and  $y$  satisfying  $n=x^3+3y^3$ .

*Proposed by Luke Robitaille*


Let  $L,E,T,M,$  and  $O$  be digits that satisfy  $LEET+LMT=TOOL.$ Given that  $O$  has the value of  $0,$  digits may be repeated, and  $L\neq0,$  what is the value of the  $4$ -digit integer  $ELMO?$


A graph has  $ n$  vertices and  $ \frac {1}{2}\left(n^2 \minus{} 3n \plus{} 4\right)$  edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length  $ k$  of a circuit in such a graph.


Circle  $C$  with radius  $2$  has diameter  $\overline{AB}$ . Circle  $D$  is internally tangent to circle  $C$  at  $A$ . Circle  $E$  is internally tangent to circle  $C,$  externally tangent to circle  $D,$  and tangent to  $\overline{AB}$ . The radius of circle  $D$  is three times the radius of circle  $E$  and can be written in the form  $\sqrt{m} - n,$  where  $m$  and  $n$  are positive integers. Find  $m+n$ .


Consider the sequence:  $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$ , for  $n>1$ , where  $\text{lcm} (a,b)$  means the least common multiple of  $a$  and  $b$ . Find the greatest common divisor of  $x_{1995}$  and  $x_{1996}$ .


Triangle  $ABC$  has  $\angle BAC=90^\circ$ . A semicircle with diameter  $XY$  is inscribed inside  $\triangle ABC$  such that it is tangent to a point  $D$  on side  $BC$ , with  $X$  on  $AB$  and  $Y$  on  $AC$ . Let  $O$  be the midpoint of  $XY$ . Given that  $AB=3$ ,  $AC=4$ , and  $AX=\tfrac{9}{4}$ , compute the length of  $AO$ .


For each positive integer  $n$ , let  $r_n$  be the smallest positive root of the equation  $x^n = 7x - 4$ . There are positive real numbers  $a$ ,  $b$ , and  $c$  such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If  $100a + 10b + c = \frac{p}{7}$  for some integer  $p$ , find  $p$ .

*Proposed by Mehtaab Sawhney*


Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?


Find the least positive integer  $n$  such that  $15$  divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers  $a_1, a_2, \dots, a_{15}$ .


Let  $n_0$  be the product of the first  $25$  primes. Now, choose a random divisor  $n_1$  of  $n_0$ , where a choice  $n_1$  is taken with probability proportional to  $\phi(n_1)$ . ( $\phi(m)$  is the number of integers less than  $m$  which are relatively prime to  $m$ .) Given this  $n_1$ , we let  $n_2$  be a random divisor of  $n_1$ , again chosen with probability proportional to  $\phi(n_2)$ . Compute the probability that  $n_2\equiv0\pmod{2310}$ .


Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  and  $g$  be seven distinct positive integers not bigger than  $7$ . Find all primes which can be expressed as  $abcd+efg$


For positive integer  $n$ , let  $s(n)$  denote the sum of the digits of  $n$ . Find the smallest positive integer  $n$  satisfying  $s(n)=s(n+864)=20$ .


Point  $P$  and line  $\ell$  are such that the distance from  $P$  to  $\ell$  is  $12$ . Given that  $T$  is a point on  $\ell$  such that  $PT = 13$ , find the radius of the circle passing through  $P$  and tangent to  $\ell$  at  $T$ .


For a positive integer  $k$ , let  $d(k)$  denote the number of divisors of  $k$  and let  $s(k)$  denote the digit sum of  $k$ . A positive integer  $n$  is said to be *amusing* if there exists a positive integer  $k$  such that  $d(k)=s(k)=n$ . What is the smallest amusing odd integer greater than  $1$ ?


Let  $S$  be the sum of all positive integers that can be expressed in the form  $2^a \cdot 3^b \cdot 5^c$ , where  $a$ ,  $b$ ,  $c$  are positive integers that satisfy  $a+b+c=10$ . Find the remainder when  $S$  is divided by  $1001$ .

*Proposed by Michael Ren*


An equilateral pentagon  $AMNPQ$  is inscribed in triangle  $ABC$  such that  $M\in\overline{AB}$ ,  $Q\in\overline{AC}$ , and  $N,P\in\overline{BC}$ .

Suppose that  $ABC$  is an equilateral triangle of side length  $2$ , and that  $AMNPQ$  has a line of symmetry perpendicular to  $BC$ . Then the area of  $AMNPQ$  is  $n-p\sqrt{q}$ , where  $n, p, q$  are positive integers and  $q$  is not divisible by the square of a prime. Compute  $100n+10p+q$ .

*Proposed by Michael Ren*


Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infinity, what is the ratio between the number of triangles and the number of squares?


Given two fractions  $a/b$  and  $c/d$  we define their *pirate sum* as: $\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$  where the two initial fractions are simplified the most possible, like the result.
For example, the pirate sum of  $2/7$  and  $4/5$  is  $1/2$ .
Given an integer  $n \ge 3$ , initially on a blackboard there are the fractions: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$ .
At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction.
Determine, in function of  $n$ , the maximum and the minimum possible value for the last fraction.


Let  $ABC$  be a triangle with area  $5$  and  $BC = 10.$  Let  $E$  and  $F$  be the midpoints of sides  $AC$  and  $AB$  respectively, and let  $BE$  and  $CF$  intersect at  $G.$  Suppose that quadrilateral  $AEGF$  can be inscribed in a circle. Determine the value of  $AB^2+AC^2.$ *Proposed by Ray Li*


Let  $ABCD$  be a trapezium in which  $AB \parallel CD$  and  $AB = 3CD$ . Let  $E$  be then  midpoint of the diagonal  $BD$ . If  $[ABCD] = n \times  [CDE]$ , what is the value of  $n$ ?

(Here  $[t]$  denotes the area of the geometrical figure $ t$ .)


Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression  $\{2,5,8,11,...\}$  such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\]


Find the number of primes  $p$  between  $100$  and  $200$  for which  $x^{11}+y^{16}\equiv 2013\pmod p$  has a solution in integers  $x$  and  $y$ .


How many ways are there to write  $91$  as the sum of at least  $2$  consecutive positive integers?


Let  $n$  be an given positive integer, the set  $S=\{1,2,\cdots,n\}$ .For any nonempty set  $A$  and  $B$ , find the minimum of  $|A\Delta S|+|B\Delta S|+|C\Delta S|,$  where  $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$

Solution: \boxed{ n + 1 }


All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:
(a) There is no right triangle
(b) There is no acute triangle
having all vertices in the vertices of the 2016-gon that are still white?


Find the smallest exact square with last digit not  $0$ , such that after deleting its last two digits we shall obtain another exact square.


Let  $n$  be the smallest positive integer such that the remainder of  $3n+45$ , when divided by  $1060$ , is  $16$ . Find the remainder of  $18n+17$  upon division by  $1920$ .


Start by writing the integers  $1, 2, 4, 6$  on the blackboard. At each step, write the smallest positive integer  $n$  that satisfies both of the following properties on the board.

- $n$  is larger than any integer on the board currently.
- $n$  cannot be written as the sum of  $2$  distinct integers on the board.

Find the  $100$ -th integer that you write on the board. Recall that at the beginning, there are already  $4$  integers on the board.


Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall.  In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall.  One morning, at around  $9$  o' clock, Tony sticks the spider to the wall in the living room three feet above the floor.  Over the next few mornings, Tony moves the spider up three feet from the point where he finds it.  If the wall in the living room is  $18$  feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?


Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?


Let  $A$  denote the set of all integers  $n$  such that  $1 \le n \le 10000$ , and moreover the sum of the decimal digits of  $n$  is  $2$ . Find the sum of the squares of the elements of  $A$ .


Let  $a>1$  be a positive integer.  The sequence of natural numbers  $\{a_n\}_{n\geq 1}$  is defined such that  $a_1 = a$  and for all  $n\geq 1$ ,  $a_{n+1}$  is the largest prime factor of  $a_n^2 - 1$ .  Determine the smallest possible value of  $a$  such that the numbers  $a_1$ ,  $a_2$ , $\ldots$ ,  $a_7$  are all distinct.


In triangle  $ABC$  we have  $AB=36$ ,  $BC=48$ ,  $CA=60$ . The incircle of  $ABC$  is centered at  $I$  and touches  $AB$ ,  $AC$ ,  $BC$  at  $M$ ,  $N$ ,  $D$ , respectively.  Ray  $AI$  meets  $BC$  at  $K$ . The radical axis of the circumcircles of triangles  $MAN$  and  $KID$  intersects lines  $AB$  and  $AC$  at  $L_1$  and  $L_2$ , respectively. If  $L_1L_2 = x$ , compute  $x^2$ .

*Proposed by Evan Chen*


7. Peggy picks three positive integers between  $1$  and  $25$ , inclusive, and tells us the following information about those numbers:

- Exactly one of them is a multiple of  $2$ ;
- Exactly one of them is a multiple of  $3$ ;
- Exactly one of them is a multiple of  $5$ ;
- Exactly one of them is a multiple of  $7$ ;
- Exactly one of them is a multiple of  $11$ .

What is the maximum possible sum of the integers that Peggy picked?

8. What is the largest positive integer  $k$  such that  $2^k$  divides  $2^{4^8}+8^{2^4}+4^{8^2}$ ?

9. Find the smallest integer  $n$  such that  $n$  is the sum of  $7$  consecutive positive integers and the sum of  $12$  consecutive positive integers.


What is the difference between the median and the mean of the following data set:  $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$ ?


A positive integer  $n$  is called  $\textit{good}$  if  $2 \mid \tau(n)$  and if its divisors are    $$ 1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n,  $$   then  $d_{k+1}-d_k=2$  and  $d_{k+2}-d_{k-1}=65$ . Find the smallest  $\textit{good}$  number.


Find the value of the expression  $$ f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right) $$  assuming  $f(x) =\frac{x^2}{1 + x^2}$  .


Robert colors each square in an empty 3 by 3 grid either red or green. Find the number of colorings such that no row or column contains more than one green square.


For a positive integer  $n$ , let   $\sigma (n)$  be the sum of the divisors of  $n$  (for example   $\sigma (10) = 1 + 2 + 5 + 10 = 18$ ). For how many  $n \in \{1, 2,. .., 100\}$ ,  do we have  $\sigma (n) < n+ \sqrt{n}$ ?


A set  $S$  of positive integers is  $\textit{sum-complete}$  if there are positive integers  $m$  and  $n$  such that an integer  $a$  is the sum of the elements of some nonempty subset of  $S$  if and only if  $m \le a \le n$ .

Let  $S$  be a sum-complete set such that  $\{1, 3\} \subset S$  and  $|S| = 8$ . Find the greatest possible value of the sum of the elements of  $S$ .

*Proposed by Michael Tang*


The sides  $AD$  and  $BC$  of a convex quadrilateral  $ABCD$  are extended to meet at  $E$ .  Let  $H$  and  $G$  be the midpoints of  $BD$  and  $AC$ , respectively.  Find the ratio of the area of the triangle  $EHG$  to that of the quadrilateral  $ABCD$ .


How many polynomials of degree exactly  $5$  with real coefficients send the set  $\{1, 2, 3, 4, 5, 6\}$  to a permutation of itself?


The figure  $ABCDE$  is a convex pentagon. Find the sum  $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle  CEB$ ?


What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere? $\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$


Two vector fields  $\mathbf{F},\mathbf{G}$  are defined on a three dimensional region  $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$ .  $$ \mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0) $$  Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]


You are given  $n \ge 2$  distinct positive integers. For every pair  $a<b$  of them, Vlada writes on the board the largest power of  $2$  that divides  $b-a$ . At most how many distinct powers of  $2$  could Vlada have written?

*Proposed by Oleksiy Masalitin*

Solution: \boxed{ n-1 }


Let  $a=256$ . Find the unique real number  $x>a^2$  such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]

*Proposed by James Lin.*


Point  $A,B$  are marked on the right branch of the hyperbola   $y=\frac{1}{x},x>0$ . The straight line  $l$  passing through the origin  $O$  is perpendicular to  $AB$  and meets   $AB$  and given branch of the hyperbola at points  $D$  and  $C$  respectively. The circle through  $A,B,C$  meets  $l$  at  $F$ .
Find  $OD:CF$

Solution: \boxed{ 1:2 }


For any real numbers sequence  $\{x_n\}$  ,suppose that  $\{y_n\}$  is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$   ${(n \ge 1})$   .
Find the smallest positive number   $\lambda$  such that  for any real numbers sequence  $\{x_n\}$  and all positive integers  $m$  , have  $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Solution: \boxed{\lambda = 2}


When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck  $n$  trees and lose 1 energy, where  $n$  is her current energy, or rest (i.e. buck 0 trees) and gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed?

*Anderson Wang.*

Clarifications[list=1][*]The problem asks for the maximum *total* number of trees she can buck in 60 minutes, not the maximum number she can buck on the 61st minute.
[*]She does not have an energy cap. In particular, her energy may go above 100 if, for instance, she chooses to rest during the first minute.[/list]


For what values of the velocity  $c$  does the equation  $u_t = u -u^2 + u_{xx}$  have a solution in the form of a traveling wave  $u = \varphi(x-ct)$ ,  $\varphi(-\infty) = 1$ ,  $\varphi(\infty) = 0$ ,  $0 \le u \le 1$ ?

Solution: \boxed{ c \geq 2 }


The product of a million whole numbers is equal to million. What can be the greatest possible value of the sum of these numbers?


The numbers in the sequence 101, 104, 109, 116,  $\dots$  are of the form  $a_n = 100 + n^2$ , where  $n = 1$ , 2, 3,  $\dots$ .  For each  $n$ , let  $d_n$  be the greatest common divisor of  $a_n$  and  $a_{n + 1}$ .  Find the maximum value of  $d_n$  as  $n$  ranges through the positive integers.


Suppose that positive integers  $m,n,k$  satisfy the equations   $$ m^2+1=2n^2, 2m^2+1=11k^2. $$   Find the residue when  $n$  is divided by  $17$ .


It is given that  $x = -2272$ ,  $y = 10^3+10^2c+10b+a$ , and  $z = 1$  satisfy the equation  $ax + by + cz = 1$ , where  $a, b, c$  are positive integers with  $a < b < c$ . Find  $y.$


The  $\emph{Stooge sort}$  is a particularly inefficient recursive sorting algorithm defined as follows: given an array  $A$  of size  $n$ , we swap the first and last elements if they are out of order; we then (if  $n\ge3$ ) Stooge sort the first  $\lceil\tfrac{2n}3\rceil$  elements, then the last  $\lceil\tfrac{2n}3\rceil$ , then the first  $\lceil\tfrac{2n}3\rceil$  elements again. Given that this runs in  $O(n^\alpha)$ , where  $\alpha$  is minimal, find the value of  $(243/32)^\alpha$ .


Let the set  $A=(a_{1},a_{2},a_{3},a_{4})$  . If the sum of elements in every 3-element subset of  $A$  makes up the set  $B=(-1,5,3,8)$  , then find the set  $A$ .

Solution: \boxed{\{-3, 0, 2, 6\}}


Screws are sold in packs of  $10$  and  $12$ . Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly  $k$  screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of  $k$ ?


Two skaters, Allie and Billie, are at points  $A$  and  $B$ , respectively, on a flat, frozen lake. The distance between  $A$  and  $B$  is  $100$  meters. Allie leaves  $A$  and skates at a speed of  $8$  meters per second on a straight line that makes a  $60^\circ$  angle with  $AB$ . At the same time Allie leaves  $A$ , Billie leaves  $B$  at a speed of  $7$  meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

[asy]
defaultpen(linewidth(0.8));
draw((100,0)--origin--60*dir(60), EndArrow(5));
label(" $A$ ", origin, SW);
label(" $B$ ", (100,0), SE);
label(" $100$ ", (50,0), S);
label(" $60^\circ$ ", (15,0), N);[/asy]

Solution: \boxed{ 160 }


At what smallest  $n$  is there a convex  $n$ -gon for which the sines of all angles are equal and the lengths of all sides are different?


Given an integer  $n \geq 2$  determine the integral part of the number $ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}})	\dots ({1+\frac {k} {n})}}$  -  $\sum_{k=1}^{n-1} (1-\frac {1} {n})	\dots(1-\frac{k}{n})$

Solution: \boxed{ 0 }


Find all positive integers  $k$  for which the equation:   $$  \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n) $$   has no solution in integers positive  $(m,n)$  with  $m\neq n$ .

Solution: \boxed{ k = 2 }


Determine the number of  $ 8$ -tuples of nonnegative integers  $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$  satisfying  $ 0\le a_k\le k$ , for each  $ k \equal{} 1,2,3,4$ , and  $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$ .


There are  $ n \plus{} 1$  cells in a row labeled from  $ 0$  to  $ n$  and  $ n \plus{} 1$  cards labeled from  $ 0$  to  $ n$ . The cards are arbitrarily placed in the cells, one per cell. The objective is to get card  $ i$  into cell  $ i$  for each  $ i$ . The allowed move is to find the smallest  $ h$  such that cell  $ h$  has a card with a label  $ k > h$ , pick up that card, slide the cards in cells  $ h \plus{} 1$ ,  $ h \plus{} 2$ , ... ,  $ k$  one cell to the left and to place card  $ k$  in cell  $ k$ . Show that at most  $ 2^n \minus{} 1$  moves are required to get every card into the correct cell and that there is a unique starting position which requires  $ 2^n \minus{} 1$  moves. [For example, if  $ n \equal{} 2$  and the initial position is 210, then we get 102, then 012, a total of 2 moves.]

Solution: \boxed{ 2^n - 1 }


Restore the acute triangle  $ABC$  given the vertex  $A$ , the foot of the altitude drawn from the vertex  $B$  and the center of the circle circumscribed around triangle  $BHC$  (point  $H$  is the orthocenter of triangle  $ABC$ ).

Solution: \boxed{ ABC }


What is the maximum possible value of  $k$  for which  $2013$  can be written as a sum of  $k$  consecutive positive integers?


Five lighthouses are located, in order, at points  $A, B, C, D$ , and  $E$  along the shore of a circular lake with a diameter of  $10$  miles. Segments  $AD$  and  $BE$  are diameters of the circle. At night, when sitting at  $A$ , the lights from  $B, C, D$ , and  $E$  appear to be equally spaced along the horizon. The perimeter in miles of pentagon  $ABCDE$  can be written  $m +\sqrt{n}$ , where  $m$  and  $n$  are positive integers. Find  $m + n$ .


Find the minimum value of   $$ \big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big| $$   for real numbers  $x$  not multiple of  $\frac{\pi}{2}$ .

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


Let  $b_m$  be numbers of factors  $2$  of the number  $m!$  (that is,  $2^{b_m}|m!$  and  $2^{b_m+1}\nmid m!$ ). Find the least  $m$  such that  $m-b_m = 1990$ .

Solution: \boxed{2^{1990} - 1}


Suppose that  $x$ ,  $y$ , and  $z$  are complex numbers of equal magnitude that satisfy
\[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\]
and
\[xyz=\sqrt{3} + i\sqrt{5}.\]
If  $x=x_1+ix_2, y=y_1+iy_2,$  and  $z=z_1+iz_2$  for real  $x_1,x_2,y_1,y_2,z_1$  and  $z_2$  then
\[(x_1x_2+y_1y_2+z_1z_2)^2\]
can be written as  $\tfrac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ . Compute  $100a+b.$


Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?


The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits.  How many such numbers are there?


Concentric circles  $\Omega_1$  and  $\Omega_2$  with radii  $1$  and  $100$ , respectively, are drawn with center  $O$ . Points  $A$  and  $B$  are chosen independently at random on the circumferences of  $\Omega_1$  and  $\Omega_2$ , respectively. Denote by  $\ell$  the tangent line to  $\Omega_1$  passing through  $A$ , and denote by  $P$  the reflection of  $B$  across  $\ell$ . Compute the expected value of  $OP^2$ .

*Proposed by Lewis Chen*


The real function  $f$  is defined for  $\forall$   $x\in \mathbb{R}$  and  $f(0)=0$ . Also  $f(9+x)=f(9-x)$  and  $f(x-10)=f(-x-10)$  for  $\forall$   $x\in \mathbb{R}$ . What’s the least number of zeros  $f$  can have in the interval  $[0;2014]$ ? Does this change, if  $f$  is also continuous?


Find the sum of all possible sums  $a + b$  where  $a$  and  $b$  are nonnegative integers such that  $4^a + 2^b + 5$  is a perfect square.


The numbers  $1, 2, \dots ,N$  are arranged in a circle where  $N \geq  2$ . If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of  $N$ ? $
\textbf{a)}\ 18
\qquad\textbf{b)}\ 19
\qquad\textbf{c)}\ 28
\qquad\textbf{d)}\ 29
\qquad\textbf{e)}\ \text{None of above}
$


There are  $20$  geese numbered  $1-20$  standing in a line. The even numbered geese are standing at the front in the order  $2,4,\dots,20,$  where  $2$  is at the front of the line. Then the odd numbered geese are standing behind them in the order,  $1,3,5,\dots ,19,$  where  $19$  is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered  $1,2,\dots,20$  (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?

*Author: Ray Li*


Let  $m$  and  $n$  be positive integers with  $m\le 2000$  and  $k=3-\frac{m}{n}$ . Find the smallest positive value of  $k$ .


Consider a  $20$ -sided convex polygon  $K$ , with vertices  $A_1, A_2,...,A_{20}$  in that order. Find the number of ways in which three sides of  $K$  can be chosen so that every pair among them has at least two sides of  $K$  between them. (For example  $(A_1A_2, A_4A_5, A_{11}A_{12})$  is an admissible triple while  $(A_1A_2, A_4A_5, A_{19}A_{20})$  is not.


In the quadrilateral  $ABCD$ ,  $AB = BC = CD$  and  $\angle BMC = 90^\circ$ , where  $M$  is the midpoint of  $AD$ . Determine the acute angle between the lines  $AC$  and  $BD$ .


In triangle  $ABC$ ,  $AB = 28$ ,  $AC = 36$ , and  $BC = 32$ . Let  $D$  be the point on segment  $BC$  satisfying  $\angle BAD = \angle DAC$ , and let  $E$  be the unique point such that  $DE \parallel AB$  and line  $AE$  is tangent to the circumcircle of  $ABC$ . Find the length of segment  $AE$ .

*Ray Li*


An 8-by-8 square is divided into 64 unit squares in the usual way.  Each unit square is colored black or white.  The number of black unit squares is even.  We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black.  We call this operation a *step*.  If  $C$  is the original coloring, let  $S(C)$  be the least number of steps required to make all the unit squares black.  Find with proof the greatest possible value of  $S(C)$ .


Let  $ n$  be a given positive integer.  Find the smallest positive integer  $ u_n$  such that for any positive integer  $ d$ , in any  $ u_n$  consecutive odd positive integers, the number of them that can be divided by  $ d$  is not smaller than the number of odd integers among  $ 1, 3, 5, \ldots, 2n \minus{} 1$  that can be divided by  $ d$ .

Solution: \boxed{ u_n = 2n-1 }


Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?


Let  $ X$  be the set of all positive integers greater than or equal to  $ 8$  and let  $ f: X\rightarrow X$  be a function such that  $ f(x\plus{}y)\equal{}f(xy)$  for all  $ x\ge 4, y\ge 4 .$  if  $ f(8)\equal{}9$ , determine  $ f(9) .$


Let  $a, b, c, p, q, r$  be positive integers with  $p, q, r \ge 2$ . Denote
\[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \]
Initially, some pieces are put on the each point in  $Q$ , with a total of  $M$  pieces. Then, one can perform the following three types of operations repeatedly:
(1) Remove  $p$  pieces on  $(x, y, z)$  and place a piece on  $(x-1, y, z)$  ;
(2) Remove  $q$  pieces on  $(x, y, z)$  and place a piece on  $(x, y-1, z)$  ;
(3) Remove  $r$  pieces on  $(x, y, z)$  and place a piece on  $(x, y, z-1)$ .

Find the smallest positive integer  $M$  such that one can always perform a sequence of operations, making a piece placed on  $(0,0,0)$ , no matter how the pieces are distributed initially.

Solution: \boxed{p^a q^b r^c}


Let  $a$ ,  $b$ , and  $c$  be real numbers such that  $0\le a,b,c\le 5$  and  $2a + b + c = 10$ . Over all possible values of  $a$ ,  $b$ , and  $c$ , determine the maximum possible value of  $a + 2b + 3c$ .

*Proposed by Andrew Wen*


$ 2^n $  coins are given to a couple of kids. Interchange of the coins occurs when some of the kids has at least half of all the coins. Then from the coins of one of those kids to the all other kids are given that much coins as the kid already had. In case when all the coins are at one kid there is no possibility for interchange. What is the greatest possible number of consecutive interchanges? ( $ n $  is natural number)

Solution: \boxed{ n }


We call a path Valid if
i. It only comprises of the following kind of steps:
A.  $(x, y) \rightarrow (x + 1, y + 1)$ B.  $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis.
Let  $M(n)$   = set of all valid paths from  $(0,0) $ , to  $(2n,0)$ , where  $n$  is a natural number.
Consider a Valid path  $T \in M(n)$ .
Denote  $\phi(T) = \prod_{i=1}^{2n} \mu_i$ ,
where  $\mu_i$ =
a)  $1$ , if the  $i^{th}$  step is  $(x, y) \rightarrow (x + 1, y + 1)$ b)  $y$ ,  if the  $i^{th} $  step is  $(x, y) \rightarrow (x + 1, y - 1)$
Now Let  $f(n) =\sum _{T \in M(n)} \phi(T)$ . Evaluate the number of zeroes at the end in the decimal expansion of  $f(2021)$


Let n be a non-negative integer. Define the *decimal digit product* \(D(n)\) inductively as follows:

- If \(n\) has a single decimal digit, then let \(D(n) = n\).

- Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\).

Let \(P_k(1)\) be the probability that \(D(i) = 1\) where \(i\) is chosen uniformly randomly from the set of integers between 1 and \(k\) (inclusive) whose decimal digit products are not 0.

Compute \(\displaystyle\lim_{k\to\infty} P_k(1)\).

*proposed by the ICMC Problem Committee*


*Version 1*. Let  $n$  be a positive integer, and set  $N=2^{n}$ . Determine the smallest real number  $a_{n}$  such that, for all real  $x$ ,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
*Version 2*. For every positive integer  $N$ , determine the smallest real number  $b_{N}$  such that, for all real  $x$ ,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]

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


Find the sum of all integers  $n$  for which  $n - 3$  and  $n^2 + 4$  are both perfect cubes.


Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?


Let  $z_1,z_2,z_3,\dots,z_{12}$  be the 12 zeroes of the polynomial  $z^{12}-2^{36}$ . For each  $j$ , let  $w_j$  be one of  $z_j$  or  $i z_j$ . Then the maximum possible value of the real part of  $\displaystyle\sum_{j=1}^{12} w_j$  can be written as  $m+\sqrt{n}$  where  $m$  and  $n$  are positive integers. Find  $m+n$ .


Determine the polynomial  $P\in \mathbb{R}[x]$  for which there exists  $n\in \mathbb{Z}_{>0}$  such that for all  $x\in \mathbb{Q}$  we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]

*Dumitru Bușneag*

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


The real-valued function  $f$  is defined for positive integers, and the positive integer  $a$  satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$  for all positive integers  $n$ .
(i) Show that  $f(n+ 4a) = f(n)$  for all positive integers  $n$ .
(ii) Determine the smallest possible  $a$ .


The function  $f: \mathbb{N}\to\mathbb{N}_{0}$  satisfies for all  $m,n\in\mathbb{N}$ : \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine  $f(1982)$ .


Find  $ [\sqrt{19992000}]$  where  $ [x]$  is the greatest integer less than or equal to  $ x$ .


The inhabitants of a planet speak a language only using the letters  $A$  and  $O$ . To avoid mistakes, any two words of equal length differ at least on three positions. Show that there are not more than  $\frac{2^n}{n+1}$  words with  $n$  letters.


Find all triplets of real numbers  $(x,y,z)$  such that the following equations are satisfied simultaneously:
\begin{align*}
 x^3+y=z^2
y^3+z=x^2
z^3+x =y^2
\end{align*}

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


Find all real numbers  $a$  for which there exists a function  $f$  defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality  $$ f(f(x))=x^2f(x)+ax^2 $$  for all real  $x$ .

Solution: \boxed{a = 0}


Let  $P(x)$  be the polynomial of degree at most  $6$  which satisfies  $P(k)=k!$  for  $k=0,1,2,3,4,5,6$ . Compute the value of  $P(7)$ .


A right triangle with perpendicular sides  $a$  and  $b$  and hypotenuse  $c$  has the following properties: $a = p^m$  and  $b = q^n$  with  $p$  and  $q$  prime numbers and  $m$  and  $n$  positive integers,  $c = 2k +1$  with  $k$  a positive integer.
Determine all possible values of  $c$  and the associated values of  $a$  and  $b$ .


For  $n \in \mathbb{N}$ , let  $P(n)$  denote the product of the digits in  $n$  and  $S(n)$  denote the sum of the digits in  $n$ . Consider the set $A=\{n \in \mathbb{N}: P(n)$  is non-zero, square free and  $S(n)$  is a proper divisor of  $P(n)\}$ .
Find the maximum possible number of digits of the numbers in  $A$ .


Denote by  $S(n)$  the sum of the digits of the positive integer  $n$ . Find all the solutions of the equation  $n(S(n)-1)=2010.$


Given is a non-isosceles triangle  $ABC$  with  $\angle ABC=60^{\circ}$ , and in its interior, a point  $T$  is selected such that  $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$ . Let  $M$   the intersection point of the medians in  $ABC$ . Let  $TM$  intersect  $(ATC)$  at  $K$ . Find  $TM/MK$ .

Solution: \boxed{\frac{TM}{MK} = 2}


Let  $A$  be the set  $\{1,2,\ldots,n\}$ ,  $n\geq 2$ . Find the least number  $n$  for which there exist permutations  $\alpha$ ,  $\beta$ ,  $\gamma$ ,  $\delta$  of the set  $A$  with the property: \[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \]
*Marcel Chirita*

Solution: \boxed{ n = 28 }


Given triangle  $ABC$ , let  $D$ ,  $E$ ,  $F$  be the midpoints of  $BC$ ,  $AC$ ,  $AB$  respectively and let  $G$  be the centroid of the triangle. For each value of  $\angle BAC$ , how many non-similar triangles are there in which  $AEGF$  is a cyclic quadrilateral?


Each positive integer number  $n \ ge 1$  is assigned the number  $p_n$  which is the product of all its non-zero digits. For example,  $p_6 = 6$ ,  $p_ {32} = 6$ ,  $p_ {203} = 6$ . Let  $S = p_1 + p_2 + p_3 + \dots + p_ {999}$ . Find the largest prime that divides  $S $ .


Let  $ABC$  be a triangle with  $\angle C=90^\circ$ , and  $A_0$ ,  $B_0$ ,  $C_0$  be the mid-points of sides  $BC$ ,  $CA$ ,  $AB$  respectively. Two regular triangles  $AB_0C_1$  and  $BA_0C_2$  are constructed outside  $ABC$ . Find the angle  $C_0C_1C_2$ .


Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a  $50\%$  chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team  $A$  beats team  $B$ . The probability that team  $A$  finishes with more points than team  $B$  is  $m/n$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .


A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?


Some language has only three letters -  $A, B$  and  $C$ . A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all  $A$ . What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?


A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?


Square  $S_1$  is inscribed inside circle  $C_1$ , which is inscribed inside square  $S_2$ , which is inscribed inside circle  $C_2$ , which is inscribed inside square  $S_3$ , which is inscribed inside circle  $C_3$ , which is inscribed inside square  $S_4$ .

[center]<see attached>[/center]

Let  $a$  be the side length of  $S_4$ , and let  $b$  be the side length of  $S_1$ . What is  $\tfrac{a}{b}$ ?


A four-digit number  $n$  is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by  $5$ , the result is  $1434$ . For example,  $1984$  is *literally 1434* because  $1$  mod  $5$  is  $1$ ,  $9$  mod  $5$  is  $4$ ,  $8$  mod  $5$  is  $3$ , and  $4$  mod  $5$  is  $4$ . Find the sum of all four-digit positive integers that are *literally 1434*.

*Proposed by Evin Liang*

Solution*Solution.*  $\boxed{67384}$ The possible numbers are  $\overline{abcd}$  where  $a$  is  $1$  or  $6$ ,  $b$  is  $4$  or  $9$ ,  $c$  is  $3$  or  $8$ , and  $d$  is  $4$  or  $9$ . There are  $16$  such numbers and the average is  $\dfrac{8423}{2}$ , so the total in this case is  $\boxed{67384}$ .


Given an integer  $n\ge\ 3$ , find the least positive integer  $k$ , such that there exists a set  $A$  with  $k$  elements, and  $n$  distinct reals  $x_{1},x_{2},\ldots,x_{n}$  such that  $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$  all belong to  $A$ .

Solution: \boxed{ k = 3 }


A pen costs  $11$  € and a notebook costs  $13$  €. Find the number of ways in which a person can spend exactly  $1000$  € to buy pens and notebooks.


A *Beaver-number* is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of *Beaver-numbers* differing by exactly  $1$  a *Beaver-pair*. The smaller number in a *Beaver-pair* is called an *MIT Beaver*, while the larger number is called a *CIT Beaver*. Find the positive difference between the largest and smallest *CIT Beavers* (over all *Beaver-pairs*).


Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence  $(a_1, a_2, \ldots , a_n)$ , where all  $a_k$  equal  $0$  or  $1$ . After that Tanya writes down a sequence  $(b_1, b_2, \ldots , b_n)$ , where all  $b_k$  also equal  $0$  or  $1$ . Then Masha either does nothing or says “Mutabor” and replaces both sequences: her own sequence by  $(a_n, a_{n-1}, \ldots , a_1)$ , and Tanya’s sequence by  $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$ . Masha’s sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya’s sequence and tell the sequence covered by the napkin. For what  $n$  Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word “Mutabor” has been pronounced.

Solution: \boxed{ n }


Victor has  $3$  piles of  $3$  cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the  $3$  piles.) In how many orders can Victor draw the cards?


Let  $T_k = \frac{k(k+1)}{2}$  be the  $k$ -th triangular number. The infinite series  $$ \sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)} $$  has the value  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


Find the least positive integer  $n$  ( $n\geq 3$ ), such that among any  $n$  points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.

Solution: \boxed{ n = 7 }


Let  $n$  be positive integer and  $S$ = { $0,1,…,n$ }, Define set of point in the plane.   $$ A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \}  $$  , We want to place a electricity post on a point in  $A$  such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in  $A$  is in shine area

Solution: \boxed{ n+1 }


**H**orizontal parallel segments  $AB=10$  and  $CD=15$  are the bases of trapezoid  $ABCD$ . Circle  $\gamma$  of radius  $6$  has center within the trapezoid and is tangent to sides  $AB$ ,  $BC$ , and  $DA$ . If side  $CD$  cuts out an arc of  $\gamma$  measuring  $120^{\circ}$ , find the area of  $ABCD$ .


Two congruent right circular cones each with base radius  $3$  and height  $8$  have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance  $3$  from the base of each cone. A sphere with radius  $r$  lies inside both cones. The maximum possible value for  $r^2$  is  $\frac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .


A convex equilateral pentagon with side length  $2$  has two right angles. The greatest possible area of the pentagon is  $m+\sqrt{n}$ , where  $m$  and  $n$  are positive integers. Find  $100m+n$ .

*Proposed by Yannick Yao*


Find the integer represented by  $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $ . Here  $[x]$  denotes the greatest integer less than or equal to  $x.$


Compute the number of ways to erase 24 letters from the string ``OMOMO $\cdots$ OMO'' (with length 27), such that the three remaining letters are O, M and O in that order. Note that the order in which they are erased does not matter.

[i]Proposed by Yannick Yao


There are seven cards in a hat, and on the card  $k$  there is a number  $2^{k-1}$ ,  $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds  $124$ . What is the most probable sum he can get?


 $A$  and  $B$  plays a game, with  $A$  choosing a positive integer  $n \in \{1, 2, \dots, 1001\} = S$ .  $B$  must guess the value of  $n$  by choosing several subsets of  $S$ , then  $A$  will tell  $B$  how many subsets  $n$  is in.  $B$  will do this three times selecting  $k_1, k_2$  then  $k_3$  subsets of  $S$  each.

What is the least value of  $k_1 + k_2 + k_3$  such that  $B$  has a strategy to correctly guess the value of  $n$  no matter what  $A$  chooses?


Let  $ABCD$  be a cyclic quadrilateral with  $AB = 5$ ,  $BC = 10$ ,  $CD = 11$ , and  $DA = 14$ . The value of  $AC + BD$  can be written as  $\tfrac{n}{\sqrt{pq}}$ , where  $n$  is a positive integer and  $p$  and  $q$  are distinct primes. Find  $n + p + q$ .


Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer  $n$  for which  $n+9,16n+9,27n+9$  are also perfect squares.


You are given  $n$  not necessarily distinct real numbers  $a_1, a_2, \ldots, a_n$ . Let's consider all  $2^n-1$  ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to  $1$ ?

For example, if  $a = [-1, 2, 2]$ , then we got  $3$  once,  $4$  once,  $2$  twice,  $-1$  once,  $1$  twice, so the total number of ones here is  $2$ .

*(Proposed by Anton Trygub)*


Each row of a  $24 \times 8$  table contains some permutation of the numbers  $1, 2, \cdots , 8.$  In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)*

Solution: \boxed{8 \cdot (8!)^3}


Find the number of pairs  $(a,b)$  of natural nunbers such that  $b$  is a 3-digit number,  $a+1$  divides  $b-1$  and  $b$  divides  $a^{2} + a + 2$ .


Let  $P$  be the set of all  $2012$  tuples  $(x_1, x_2, \dots, x_{2012})$ , where  $x_i \in \{1,2,\dots 20\}$  for each  $1\leq i \leq 2012$ . The set  $A \subset P$  is said to be decreasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in A$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \leq x_i (1\leq i \leq 2012)$  also belongs to  $A$ . The set  $B \subset P$  is said to be increasing if for each  $(x_1,x_2,\dots ,x_{2012} ) \in B$  any  $(y_1,y_2,\dots, y_{2012})$  satisfying  $y_i \geq x_i (1\leq i \leq 2012)$  also belongs to  $B$ . Find the maximum possible value of  $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where  $A$  and  $B$  are nonempty decreasing and increasing sets ( $\mid \cdot \mid$  denotes the number of elements of the set).


The Lucas numbers  $L_n$  are defined recursively as follows:  $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}$  for  $n\geq2$ . Let  $r=0.21347\dots$ , whose digits form the pattern of the Lucas numbers. When the numbers have multiple digits, they will "overlap," so  $r=0.2134830\dots$ , **not**  $0.213471118\dots$ . Express  $r$  as a rational number  $\frac{p}{q}$ , where  $p$  and  $q$  are relatively prime.


Find the largest possible number of rooks that can be placed on a  $3n \times 3n$  chessboard so that each rook is attacked by at most one rook.

Solution: \boxed{ 4n }


Let  $V_n=\sqrt{F_n^2+F_{n+2}^2}$ , where  $F_n$  is the Fibonacci sequence
( $F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$ )

Show that  $V_n,V_{n+1},V_{n+2}$  are the sides of a triangle with area  $1/2$


For real  $\theta_i$ ,  $i = 1, 2, \dots, 2011$ , where  $\theta_1 = \theta_{2012}$ , find the maximum value of the expression

\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}.
\]

*Proposed by Lewis Chen*


Find the smallest value of  $n$  for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds  $3^{2007}$ .


Find all positive integers  $x$  such that the product of all digits of  $x$  is given by  $x^2 - 10 \cdot x - 22.$


In a group of  $n > 20$  people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of  $n$  and  $k,$  this group has a peculiar property: If any  $20$  people are removed from the group, the number of pairs of people that know each other is
at most  $\frac{n-k}{n}$  times that of the original group of people.
(a) If  $k = 41,$  for what positive integers  $n$  could such a group exist?
(b) If  $k = 39,$  for what positive integers  $n$  could such a group exist?

Solution: \boxed{n \ge 381}


Find all natural  $ x $  for which  $ 3x+1 $  and  $ 6x-2 $  are perfect squares, and the number  $ 6x^2-1 $  is prime.


Define  $ P(\tau ) = (\tau + 1)^3$  . If  $x + y = 0$ , what is the minimum possible value of  $P(x) + P(y)$ ?


The eleven members of a cricket team are numbered  $1,2,...,11$ . In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?


Find  $a$  if  $a$  and  $b$  are integers such that  $x^2 - x - 1$  is a factor of  $ax^{17} + bx^{16} + 1$ .


A number  $p$  is  $perfect$  if the sum of its divisors, except  $p$  is  $p$ . Let  $f$  be a function such that:

 $f(n)=0$ , if n is perfect
 $f(n)=0$ , if the last digit of n is 4
 $f(a.b)=f(a)+f(b)$

Find  $f(1998)$


For which integers  $n>1$  does there exist a rectangle that can be subdivided into  $n$  pairwise noncongruent rectangles similar to the original rectangle?

Solution: \boxed{ n \ge 3 }


Suppose  $x$  and  $y$  are nonzero real numbers simultaneously satisfying the equations $x + \frac{2018}{y}= 1000$  and  $ \frac{9}{x}+ y = 1$ .
Find the maximum possible value of  $x + 1000y$ .


Let  $K$  be a point on the angle bisector, such that  $\angle BKL=\angle KBL=30^\circ$ . The lines  $AB$  and  $CK$  intersect in point  $M$  and lines  $AC$  and  $BK$  intersect in point  $N$ . Determine   $\angle AMN$ .


Three circles, each of radius 3, are drawn with centers at  $(14,92)$ ,  $(17,76)$ , and  $(19,84)$ .  A line passing through  $(17,76)$  is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it.  What is the absolute value of the slope of this line?


Let  $p$  be a prime number.  $T(x)$  is a polynomial with integer coefficients and degree from the set  $\{0,1,...,p-1\}$  and such that  $T(n) \equiv T(m)  (mod p)$  for some integers m and n implies that  $ m \equiv n (mod p)$ . Determine the maximum possible value of degree of  $T(x)$

Solution: \boxed{ p-2 }


Line  $\ell$  passes through  $A$  and into the interior of the equilateral triangle  $ABC$ .   $D$  and  $E$  are the orthogonal projections of  $B$  and  $C$  onto  $\ell$  respectively.  If  $DE=1$  and  $2BD=CE$ , then the area of  $ABC$  can be expressed as  $m\sqrt n$ , where  $m$  and  $n$  are positive integers and  $n$  is not divisible by the square of any prime.  Determine  $m+n$ .

[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label(" $A$ ",A,S);
label(" $B$ ",B,S);
label(" $C$ ",C,N);
label(" $D$ ",D,dir(B--D));
label(" $E$ ",E,dir(C--E));
[/asy]


With the digits  $1, 2, 3,. . . . . . , 9$  three-digit numbers are written such that the sum of the three digits is  $17$ . How many numbers can be written?


An integer  $n>0$  is written in decimal system as  $\overline{a_ma_{m-1}\ldots a_1}$ . Find all  $n$  such that
\[n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)\]